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SLOW LIGHT IN FIBER BRAGG GRATINGS FOR SENSING
THERMAL PHASE NOISE, ATTOSTRAINS
AND OTHER APPLICATIONS
A DISSERTATION
SUBMITTED TO THE DEPARTMENT OF
ELECTRICAL ENGINEERING
AND THE COMMITTEE ON GRADUATE STUDIES
OF STANFORD UNIVERSITY
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
Georgios Skolianos
March 2016
http://creativecommons.org/licenses/by-nc/3.0/us/
This dissertation is online at: http://purl.stanford.edu/gd710fc6670
© 2016 by Georgios Skolianos. All Rights Reserved.
Re-distributed by Stanford University under license with the author.
This work is licensed under a Creative Commons Attribution-Noncommercial 3.0 United States License.
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I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.
Michel Digonnet, Primary Adviser
I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.
Olav Solgaard
I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.
Jelena Vuckovic
Approved for the Stanford University Committee on Graduate Studies.
Patricia J. Gumport, Vice Provost for Graduate Education
This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file inUniversity Archives.
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Abstract
Strain sensors have many applications in structural health monitoring, civil engineering,
geoscience, and gravitational wave detection. To address the needs of high-end applications,
and to push the resolution of fiber strain sensors well beyond their current picostrain/√Hz
level, we have explored the potential of utilizing the narrow slow-light resonances that exist in
strong fiber Bragg gratings (FBGs) and the noise limitations of this kind of sensor.
Prior to this work the strain resolution of slow-light FBG strain sensors were limited by the
frequency noise of the laser used to interrogate them. The main goals of this thesis were first
to reduce this source of noise by utilizing a probe laser with a much narrower linewidth, which
necessitated the design of new gratings matched to the new laser; and second, to study
theoretically and experimentally the next noise source below the laser frequency noise, which
is thermodynamic phase noise.
This work was broadly divided into four major tasks. First, we studied theoretically, from
basic thermodynamic principles, the magnitude and frequency dependence of the thermal
phase-noise in Fabry-Perot-like resonances similar to the slow-light resonances available in
strong FBGs. This study showed that the thermal phase noise is proportional to the group
index of the resonance, and that when expressed in units of strain it is proportional to 1/√L,
where L is the length of the sensor. Thus the shorter the FBG the higher the thermal phase
noise in units of strain. Second, we improved the design and the fabrication of our FBGs in
order to achieve very high group delays, and hence very high sensitivities in short fibers. This
step was crucial because to measure thermal phase noise in units of strain (normalized output
power noise to input power times the sensitivity), a short FBG with high sensitivity is needed
for the sensor to be limited by thermal phase noise, otherwise it would be limited from other
noise source, i.e. laser intensity noise. We achieved this result by using strongly apodized
FBGs written with a femtosecond laser in deuterium-loaded fiber, and thermally annealing the
FBG optimum sensitivity was achieved. Using this technique, we were able to achieve a 42-ns
group delay, an eight-fold improvement compared to what was reported previously in similar
FBGs. Third, we had to modify our experimental setup to improve its stability and reduce the
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dominant laser frequency noise of the sensor. To this end we probed our sensors with a new
low-noise laser from OrbitsLightwave with a 200-Hz linewidth, we placed our sensor in an
anechoic enclosure, and we used a low-noise photo-detector. The last effort was to use these
various developments and improvements to design, fabricate, and test two FBGs, one to
measure the thermal phase noise in an FBG for first time, and the other to observe the smallest
strain ever measured in an FBG-based sensor (an minimum detectable strain (MDS) of
110 fε/√Hz at 2 kHz and 30 fε/√Hz at 30 kHz). This sensor was so stable that it exhibited no
drift in its Allan variance after a four-day measurement. By integrating a 4-day output trace
with an 8-hour integration time, we were able to measure an absolute MDS of 250 attostrains,
the lowest value ever measured in an FBG.
While we were aiming to measure thermal phase noise and reduce the MDS, some other
applications presented themselves. Because these devices confine light not only temporally
but also spatially, they can be used in applications that benefit from extremely high intensities
and confinement, in particular in quantum electrodynamic experiments and nonlinear optic
applications such as optical signal processing. As a proof of concept, I report in the end of this
thesis the performance of two FBGs optimized for maximum field enhancement, maximum
Purcell factor, maximum group delay, and minimum group velocity (a record of 300 km/s).
The measured values for these parameters are the highest reported in an all-fiber device. These
properties enable robust novel devices that are simple to fabricate and in which light can be
coupled easily and efficiently.
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Acknowledgement
First of all I would like to thank Professor Michel Digonnet for his help and support
throughout all the years I spent at Stanford University and for letting me join his research
group from the moment when I arrived at Stanford University. From this first moment, until
the very end of writing this thesis he was always available for me when I needed his help and
guidance. In our interactions in our meetings, which were often several per week, he patiently
showed me how to thoroughly think about a problem from every angle, how to solve it, how
to pay attention to the details and then step back and explain my work to someone else without
taking for granted that this person would know what I already knew. I am really thankful for
all his efforts, without his support in every aspect, my Ph.D. journey would not have been
possible.
I would also like to thank Professor Jelena Vuckovic and Professor Olav Solgaard for being
on my reading committee. They were the instructors in my first courses at Stanford, and both
my interactions with them and the knowledge I gained through their courses helped me with
my research and enabled me to realize other possible applications of my research. Also I
would like to thank Professor Nick Bambos for being the chair of my oral defense committee
and Amir Safavi-Naeini for being on my oral committee.
I would like to thank Professor Martin Bernier at University Laval for the close interaction we
had, and for fabricating all the fiber Bragg gratings I studied in this thesis. Without his help
we wouldn’t be able to realize our gratings and make this research a reality.
I would also like to thank my professors at the Aristotle University of Thessaloniki in Greece,
where I did my undergraduate studies, and especially my undergraduate advisor Professor
Emmanouil Kriezis who introduced me to optical fibers and provided me with the right skills
and knowledge to attend Stanford University.
I would like to specially thank He Wen, who started this work of slow-light in fiber Bragg
gratings and guided me in my first steps in this work. I would also to thank Arushi Arora for
her help in the last years of my Ph.D. journey, and especially for taking over some time-
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consuming but very important aspects of this work. Her help sped up my progress a lot. Also I
would like to thank the rest of our research group (Josh, Kiarash, Olive, Jacob, Therice).
I would like to thank Northrop Grumman for their financial support for my research and the
interactions we had in our biannual meetings. I also thank Ingrid Tarien for her administrative
help.
I would like to thank all my friends and the Hellenic community at Stanford that made the
years here a very enjoyable experience.
Last but not least I would like to specially thank my parents and my brother for all their
support, not only during my time at Stanford University but throughout all my studies and my
life.
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Table of contents
Abstract iv
Acknowledgement vi
Table of contents viii
List of Figures x
Chapter 1: Introduction 1 1.1 Background and motivation 1 1.2 Fiber Bragg gratings as a strain sensor 1 1.3 Evolution of FBG strain sensing and their limitations 2 1.4 Summary of this thesis work 9 References 11
Chapter 2: Basic theory of strain sensing using a slow-light FBG 13 2.1 Fiber Bragg Grating 13 2.2 Strain sensing using FBGs 17
2.2.1 General principle of strain sensing using an FBG 17 2.2.2 Using a sharp resonance to increase the strain sensitivity 20 2.2.3 Noise sources in an ultra-high sensitive strain sensor 21
2.3 Temperature sensitivity of slow-light FBGs 23 2.4 Slow-light resonances in an FBG 24
2.4.1 Slow-light resonances in a Fabry-Perot interferometer 25 2.4.2 Forming an FP and highly sensitive slow-light resonances 27
References 30
Chapter 3: Modeling the thermal phase noise in a passive Fabry-Perot resonator 32 3.1 Modeling phase noise in a Fabry-Perot interferometer 33
3.1.1 Background 33 3.1.2 Intuitive picture 34 3.1.3 Phase noise in the transmitted signal 35 3.1.4 Phase noise on resonance 41 3.1.5 Phase noise off resonance 45 3.1.6 Phase noise in the reflected signal 50
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3.1.7 Applicability to FBGs and FBG-based FPs 52 3.1.8 Phase noise in experimental fiber FP sensors 53
3.2 Conclusions 54 References 56
Chapter 4: Improving the performance of slow-light FBGs 59 4.1 Apodized FBGs 59 4.2 Realization of improved FBGs 64
4.2.1 Fabrication of slow-light FBGs with femtosecond lasers in deuterium-loaded fibers
65 4.2.2 FBG annealing 69
4.3 Modeling the index profile of FBGs written with a femtosecond laser 71 4.4 Conclusions 76 References 77
Chapter 5: Measuring the intrinsic thermal phase noise and 250 attostrains using slow-
light FBGs 79 5.1 Experimental Setup 80 5.2 Measuring the strain sensitivity and MDS of FBG sensors 83 5.3 Measuring thermal phase-noise in a 5-mm FBG 85 5.4 Measuring an absolute strain of 250 attostrains 88 5.5 Conclusions 92 References 93
Chapter 6: Other applications of slow-light FBGs 94 6.1 Optimizing slow-light FBGs for high Purcell factor and intensity enhancement 96 6.2 Fabrication and characterization of two FBGs optimized for high Purcell factor and
intensity enhancement 98 6.3 Conclusions 102 References 102
Chapter 7: Conclusions and future work. 105
Appendix: Power spectral density on resonance 109
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List of Figures
Figure 1.1. a) Generic transmission spectrum of a uniform FBG. b) Shift of the transmission
spectrum of an FBG due to an applied strain. In solid line is the initial transmission and
in dashed line the spectrum after a strain has been applied to the FBG. Probing the edge
of the FBG bandgap with a laser results in a large transmission change when strain is
applied on the FBG. ........................................................................................................... 2
Figure 1.2. Evolution of the MDS of selected strain sensors using FBGs reported over the
years. .................................................................................................................................. 4
Figure 1.3. Conversion of frequency noise into power noise through a device with a steep
transmission slope. The resulting power noise is proportional to the laser frequency noise
and to the sensitivity (i.e., the slope). ................................................................................. 5
Figure 1.4. Noise dependence at the output on detected power at 23 kHz [8]. .......................... 6
Figure 2.1. Periodically varying refractive index in a uniform FBG. ..................................... 14
Figure 2.2. Simulated power (a) reflection and (b) transmission spectrum for a uniform FBG
of length L = 2 mm, index modulation Δnac = Δndc = 5 x 10-4, period Λ= 534.3 nm, and
loss γ = 0.1 m-1. ................................................................................................................. 16
Figure 2.3. Shift of the Bragg wavelength and the whole spectrum due to an applied strain.
The initial transmission is indicated with the solid line while the spectrum after a strain
has been applied to the FBG is shown with the dashed line. ........................................... 18
Figure 2.4. Using a sharp peak for strain sensing. ................................................................... 20
Figure 2.5. Schematic of a generic Fabry-Perot interferometer ............................................... 25
Figure 2.6. Calculated reflection spectrum of a representative π -shifted grating (the grating’s
index profile is shown schematically in the inset). .......................................................... 28
Figure 2.7. a) For wavelengths near the band edge of a strong FBG, multiple reflections from
both ends of the FBG cause recirculation of light inside the FBG. The situation is
analogous to a Fabry-Perot (FP) interferometer. b) Similar to a FP, sharp resonances are
formed in the transmission spectrum. .............................................................................. 29
Figure 3.1. Schematic of the computation of the phase noise in the signal transmitted by a
Fabry-Perot interferometer ............................................................................................... 36
Figure 3.2. Phase-noise PSD of the signal on resonance transmitted by a 1-cm fiber Fabry-
Perot interferometer (see text for details). ....................................................................... 43
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Figure 3.3. Phase-noise PSD spectrum at the transmission port of a silica-fiber FP with
reflectivities r1 = r2 = √0.99, normalized to the single-pass PSD spectrum. Solid curves
are the numerical solutions for different detuning δΦ0 from resonance. ......................... 47
Figure 3.4. PSD spectrum on resonance (dashed curve) and off resonance for δΦ0 = 0.05
(solid curve) illustrating the splitting that occurs in the phase-noise resonances when the
optical signal is detuned from a Fabry-Perot resonance. ................................................. 49
Figure 3.5. PSD spectra in reflection (black curves) for the same FP as in Fig. 3.3, calculated
for two values of the detuning from resonance. The PSD spectra in transmission for the
same two detunings (solid red curves, reproduced from Fig. 3.3) are also shown for
comparison. ...................................................................................................................... 51
Figure 3.6. Theoretical predictions of the phase noise dependence on group index for three
highly sensitive strain sensors utilizing FBGs, and experimental noise measured for each
of them. ............................................................................................................................. 54
Figure 4.1. Creating strong slow-light resonances in an apodized FBG. a) ac and dc index-
modulation profiles of an apodized fiber Bragg grating. b) Dependence of the Bragg
wavelength on position along the grating. c) Effective mirrors in the apodized FBG
forming equivalent FPs. d) The transmission slow-light resonances formed as a result of
these multiple equivalent FPs. .......................................................................................... 60
Figure 4.2. a) Index profile of a uniform FBG (period Λ not to scale) and b) simulated
transmission and group index spectra of this FBG. ......................................................... 63
Figure 4.3. a) Index profile of a Gaussian-apodized FBG (period Λ not to scale), and b)
simulated transmission and group index spectra of this FBG. ......................................... 64
Figure 4.4. Power loss coefficients versus ac index modulation for different writing
techniques. ........................................................................................................................ 65
Figure 4.5. Setup and exposure conditions used to write the deuterium-loaded fiber Bragg
gratings using a femtosecond laser. (Courtesy Martin Bernier). ...................................... 67
Figure 4.6. Measured evolution with annealing temperature of (a) the transmission, (b) the
group delay, and (c) their product, for the slow-light resonances of a particular FBG. ... 71
Figure 4.7. Index modulation profile example of a fiber Bragg grating written with a
femtosecond laser; (a) convoluted profile, half-peak-to-peak ac index profile, and dc
index profile; (b) upper and lower envelopes of the index modulation. .......................... 72
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Figure 4.8. a) Measured and simulated transmission spectra of the FBG. b) Magnified portion
of slow-light peak region of the transmission spectra shown in a). c) Measured and
simulated group-delay spectra in the same wavelength range as in b). ........................... 73
Figure 4.9. Experimental setup used to measure the group delay and transmission of the FBG
(see text for details). ......................................................................................................... 74
Figure 5.1. Experimental setup used to characterize the noise spectra and the sensitivity of
FBG strain sensors. The PZT plate excited by the function generator induces a known
strain on the FBG to calibrate its sensitivity. The lock-in amplifier measures the sensor’s
response ............................................................................................................................ 80
Figure 5.2. Noise contributions in a 5-mm FBG slow-light sensor. ........................................ 85
Figure 5.3. Transmission spectrum of the FBG used for measuring thermal phase noise,
measured after fabrication (before any annealing took place). ........................................ 86
Figure 5.4. Evolution of Τ0τg/c versus annealing temperature for the FBG that was used to
measure phase noise. ........................................................................................................ 87
Figure 5.5. Measured and calculated noise spectrum contributions in a 20-mm slow-light FBG
strain sensor in units of strain. .......................................................................................... 89
Figure 5.6. Time trace of the 20-mm sensor’s response at 30 kHz. ......................................... 90
Figure 5.7. Generic Allan deviation curve. .............................................................................. 91
Figure 5.8. Allan deviation in units of strain calculated from the data of Fig. 5.6. ................. 91
Figure 6.1. Simulated dependence on the laser beam width of (a) the Purcell factor and
intensity enhancement, and (b) the transmission of the best slow-light resonance of a
saturated FBG, evaluated at the peak of the slowest resonance. Solid curves simulate an
FBG with the parameters of [11], and dashed curves the experimental FBGs. ............... 98
Figure 6.2. Measured and fitted (a) transmission and (b) group-delay spectra of the first FBG.
Inset shows its inferred index-modulation profiles. ......................................................... 99
Figure 6.3. Calculated distribution of the intensity distribution along the FBG for the three
lowest modes. ................................................................................................................. 100
Figure 7.1 Evolution of the MDS of selected strain sensors using FBGs reported over the
years. .............................................................................................................................. 106
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Chapter 1: Introduction
1.1 Background and motivation
Sensing strain (ε = ΔL/L), namely a strain-induced change in length ΔL relative to some initial
length L, is useful in several applications, including structural health monitoring in aerospace
[1] and civil engineering [2], monitoring volcanoes and sensing earthquakes in the geosciences
[3], and gravitational wave sensing [4]. The requirement for the minimum detectable strain
(MDS), or equivalently the strain resolution, is different for different applications. In civil
engineering, an MDS of a few µε (10-6) is required, while in aerospace and in geosciences an
MDS of a few pε (10-12) up to a few nε (10-9) is necessary. For higher end applications such as
gravitational-waves detection, the resolution requirement is considerably lower, around 10-22 ε.
Optical fibers, especially fiber Bragg gratings (FBGs), have been used for decades as strain
sensors because they can be very sensitive, small, and they can be used in harsh environments,
they are immune to electromagnetic interference, and they operate equally well at low and
high temperatures, among other advantages common to most fiber sensors. In order to achieve
very low MDS, in the sub-femtostrain/√Hz range, very long interferometric optical sensors, on
the order of a few kilometers, are used [4], which is obviously exceedingly costly and of no
practical use for most applications. Thus it is important to investigate methods to reach this
range of strain resolution in a small and simple sensor. Such practical ultra-sensitive sensors
can enable new applications and fundamental observations that were not attainable before.
1.2 Fiber Bragg gratings as a strain sensor
A conventional FBG is a short length of single-mode fiber with a spatially periodic
modulation of the refractive index in the core region. At each of these index modulations
multiple Fresnel reflections take place. Over a limited wavelength range, centered around
what is known as the Bragg wavelength, most of these partial reflections are in phase, and
they add essentially constructively to create a strong reflected signal. Thus an FBG acts as a
reflector, and little to no light is transmitted (see Fig. 1.1a). This wavelength range is called
the bandgap. Outside the bandgap, the partial reflections are out of phase, leading to
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essentially no reflection: the incident light is fully transmitted. But at some specific
wavelengths close to the bandgap some of the partial reflections happen to be in phase,
leading to a partial reflection and a dip in the transmission. If the wavelength is slightly
detuned from these weaker resonances, the partial reflections are abruptly out of phase, and all
the light is transmitted. This phenomenon leads to the ripples shown in Fig. 1.1a. Large index
modulations and long FBGs, which cause stronger and more Fresnel reflections, respectively,
lead to higher reflections in this wavelength range. Also, a longer FBG exhibits a narrower
bandgap because the reflections can only be in phase over a smaller range of wavelengths.
When a strain is applied to an FBG, the whole spectrum shifts (see Fig. 1.1b). By measuring
this wavelength shift, the applied strain can be inferred. An easy way to measure this shift is
with an optical spectrum analyzer (OSA). But because of the limited resolution of even high-
end OSAs (~0.01 nm), this approach produces strain resolutions only down to a few
microstrains/√Hz. To reduce the MDS, significantly smaller wavelength shifts must be
detected. This can become possible by increasing the sensitivity of the sensor and by reducing
the noise in the detected signal, which was one of the objectives of this thesis work.
Figure 1.1. a) Generic transmission spectrum of a uniform FBG. b) Shift of the transmission spectrum
of an FBG due to an applied strain. In solid line is the initial transmission and in dashed line the
spectrum after a strain has been applied to the FBG. Probing the edge of the FBG bandgap with a laser
results in a large transmission change when strain is applied on the FBG.
1.3 Evolution of FBG strain sensing and their limitations
In the early 1990s Kersey et al. used an imbalanced Mach-Zehnder interferometer (MZI) to
sense the strain-induced shift in the bandgap if an FBG [5] with a far greater precision that can
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be done even today with a commercial OSA. The FBG was interrogated with a broadband
light source, and only the portion of the spectrum of the light that overlaps with the FBG’s
narrow bandgap was reflected. The reflected light was sent through an MZI with an arm
imbalance. When a strain is applied to the FBG, the bandgap shifts and so does the center
wavelength of the reflected light spectrum. The MZI converts this wavelength change into a
power change at its output port, which is measured with a power meter. This power change is
proportional to the wavelength shift and to the length mismatch between the two arms of the
MZI. Thus a smaller wavelength shift can be detected by using an MZI with a large length
mismatch between its two arms. But by increasing the length mismatch, two deleterious
effects take place. First, the thermal stability of the MZI deteriorates. Second, the linewidth of
the reflected signal entering the MZI is approximately equal to the bandwidth of the FBG
bandgap. If the length mismatch is larger than the coherence length of the reflected signal
(which is inversely proportion to this linewidth), the light signals at the ends of the two arms
of the MZI are no longer mutually coherent and no longer interfere at the output of the MZI,
and the latter no longer functions as a wavemeter. This limitation can be pushed back by
reducing the linewidth of the FBG (i.e., increasing the coherence length of the reflected
signal). However, this goes hand in hand with extremely long FBGs that have limited use as
sensors, or with a reduction in reflected power: this improvement can only be taken so far
before the reflected signal is so weak that the signal-to-noise ratio (SNR) at the detector
increases, resulting in a reduced MDS. By using an MZI with a 10-mm length mismatch and
an FBG with a ~0.4-nm reflection linewidth, Kersey et al. achieved an MDS of ~600 pε/√Hz
[5], which was a record at that time. This data point is shown in Fig. 1.1, which illustrates the
evolution of the resolution of selected high-end FBG strain sensors reported over the past 25
years.
In 1998 Lissak et al. used the edge of the bandgap to make a significantly more sensitive FBG
sensor [6]. At the edge of the bandgap, the spectrum has a steep slope, i.e., the transmission
goes from ~100% to ~0% over a small change of wavelength (see Fig. 1.1b). Thus, if the
interrogating laser wavelength is selected to fall at the point with the steepest slope, as the
bandgap shifts relative to the (fixed) laser wavelength in response to a small applied strain, the
output power changes by an amount proportional to the strain (see Fig. 1.1b). With a steep
slope, even a slight strain-induced shift will produce a very large change in the output power.
Using this technique, the steeper the slope the larger the output change for a given wavelength
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shift (and equivalently a given strain). The ultimate limitation of this general scheme is noise
sources that affect the relative position of the spectrum and the frequency of the interrogating
laser, i.e., laser frequency noise, thermal phase noise, and environmental noise: these noise
sources are proportional to the slope of the transmission, and for a steep slope they become
dominant. In this case MDS is limited by the noise in the relative position of the spectrum and
the frequency of the interrogating laser which is independent of the slope and the spectrum.
Thus increase of the slope does not affect the MDS. This technique led to a higher sensitivity
and decreased the MDS down to ~50 pε/√Hz [6], one order of magnitude smaller than
achieved by Kersey et al. in [5].
Figure 1.2. Evolution of the MDS of selected strain sensors using FBGs reported over the years.
In 2008 Gatti [7] used the sharp transmission resonance that exists in the reflection spectrum
of a π-shifted FBG to further increase the slope of the spectrum and hence the strain
sensitivity. The sensing principle is exactly the same as in [6]: the steep slope of this
resonance converts a small strain-induced wavelength shift into a large power change. The
narrower this resonance is, the steeper its slope is, or equivalently the lower the group velocity
of light traveling at the frequency of this resonance. A higher sensitivity then requires a
resonance with a narrow linewidth, or equivalently a low group velocity vg, or equivalently
still a high group index, which is by definition ng = c/vg, where c is the velocity of light in
vacuum. A higher sensitivity also requires a high resonance transmission (since obviously a
resonance with near zero transmission, even if extremely narrow, would transmit nearly no
light, leading to no detected signal and hence zero sensitivity). In [7] the MDS was limited by
the frequency noise of the laser. The latter scales like the laser linewidth. This linewidth
introduces an uncertainty in the laser wavelength. When the laser is transmitted through a
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steep slope, this uncertainty is converted into an uncertainty in the output power, namely a
power noise. Thus the power noise due to laser frequency noise is proportional to the
sensitivity (see Fig. 1.3). The laser used in [7] had a linewidth of 2 MHz. The fact that Gatti et
al. used a sharp resonance to increase the sensitivity led to a reduction of the MDS down to
5 pε/√Hz [7], another order of magnitude compared to Lissak’s measured MDS in [6]. So in a
period of about 15 years, the MDS of FBG strain sensors was reduced by roughly two orders
of magnitude, to the few pε/√Hz level.
In 2011 He Wen, a former graduate student of our research group, was first to introduce a new
principle to greatly improve on these results [8]. When the index modulation of an FBG is
high (high reflectivity) and the propagation loss is low, the FBG does not behave like a simple
reflector but like a Fabry-Perot (FP) interferometer, and it exhibits sharp resonances either just
outside or inside the bandgap, as described in relation to Fig. 1.1a. Although the existence of
these resonances was known [9,10], He Wen was first to design FBGs with very sharp
resonances (very steep slopes) and to exploit them to demonstrate very low strain resolutions.
This method produced resonances much sharper than any reported until then in FBGs, and
therefore FBGs with unprecedented group indices (hence the term of slow-light resonances)
and strain sensors with record sensitivities. By interrogating such an FBG with a laser with a
much lower frequency noise than used by Gatti et al. (~8-kHz linewidth [8] compared to
2 MHz in [7]) He Wen was able to measure a record MDS of 280 fε/√Hz at 23 kHz [8], more
than one order of magnitude lower than the previous record in a single FBG [7].
Figure 1.3. Conversion of frequency noise into power noise through a device with a steep transmission
slope. The resulting power noise is proportional to the laser frequency noise and to the sensitivity (i.e.,
the slope).
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Figure 1.4. Noise dependence at the output on detected power at 23 kHz [8].
In Fig. 1.4 the noise dependence on the output power of Wen’s FBG sensor at 23 kHz is
shown [8]. The filled blue circles indicate the experimental points and the curves indicate the
predicted or measured main noise sources present in the detected signal. The solid black curve
is the total predicted noise, calculated by adding geometrically all the noise sources. At low
output power the noise is limited by the photodetector thermal noise. This noise source
depends solely on the photodetector and is completely independent of the incident light power
and of the sensor. At high power the noise is limited by the laser frequency noise. As already
discussed, the laser frequency noise depends on (increases with) the laser linewidth and it is
proportional to the sensor’s sensitivity. All the other noise sources shown in Fig. 1.4, namely
laser intensity noise, electrical shot noise, and optical shot noise, are negligible, because the
high sensitivity reduces their impact relative to the sources of noise that depend on sensitivity.
The laser intensity noise depends only on the laser type and on its operating power, whereas
the electrical shot noise and optical shot noise are due to the quantum nature of the light and
(for a given detector) depend only on detected power. If the sensor were shot-noise limited, by
increasing the operating power the MDS could be reduced, since the noise would increase
with increasing power (as the square root of power) slower than the sensitivity (proportional to
power, as plotted in green in Fig. 1.4). Unfortunately, this limit is hard to reach because other
noise sources are usually higher. But even if a sensor is shot-noise limited in a specific power
range, we could not indefinitely decrease the MDS by increasing the detected power because
at high enough power (above ~80 µW in the specific example of Fig. 1.4) other noise sources
7
proportional to power, such as laser intensity noise and frequency noise, would become
dominant.
In 2010, around the same time as Wen’s experiment, Gagliardi et al. [11] reported a fiber
strain sensor utilizing one of the slow-light resonances of a 10-cm FP made of two FBGs
fabricated on the same fiber. Using the same general principle as described in Fig. 1.1a, the
authors were able to measure an MDS of ~220 fε/√Hz at 1.5 kHz and ~700 fε/√Hz at 2 Hz,
which are the lowest MDS reported in a passive, relatively short fiber sensor prior this work.
These figures are remarkably low, especially considering the low frequencies of these
measurements. One downside of this sensor was that it was 5 times longer than Wen’s sensor,
which reduced its ability to detect a strain confined to a small area. Furthermore, to reduce the
laser frequency noise a relative complicated approach was used that involved locking the
interrogating laser to an ultrastable frequency comb. That publication [11] also claimed that
the sensor noise was limited by the fiber thermal phase noise, namely noise arising from
thermodynamic temperature fluctuations of the fiber. Specifically, when light propagates
through a medium, thermodynamic temperature fluctuations induce random fluctuations of
both the refractive index and length of the medium, which in turn impart noise to the phase of
the light traveling through it. In 1992 Wanser published a simple formula for the power
spectral density (PSD) of the thermal phase noise of a signal that has traveled once through an
optical fiber [12]. In this case the thermal phase noise is proportional to the square root of the
length of the fiber, making it extremely difficult to measure it and to verify this expression in
short fibers. Nevertheless, this expression can be conveniently used to predict the phase-noise-
limited minimum detectable phase shift in the large number of fiber sensors in which light
travels through a fiber only once, as in a Mach-Zehnder interferometer [13]. Wanser
subsequently extended this study to the phase noise of a signal that has traveled twice through
the same fiber [13], as occurs for example in a Michelson interferometer. It was, however, not
straightforward to extend it to the case of a multi-pass interferometer like an FP.
Gagliardi’s claim of a sensor limited by phase noise limited was subsequently challenged and
proven to be incorrect by Cranch and Foster [14], who showed via theoretical considerations
that in [11] the phase noise was calculated incorrectly and was in reality negligible. These
results raised a controversy in the community regarding how the phase noise scales with group
index in a resonant cavity, a calculation that had not been published at the time. Gagliardi et
8
al. assumed that the phase noise is proportional to the square root of the group index.
Therefore they calculated the phase noise of their sensor by replacing the length of the fiber in
Wanser’s formula with an effective length equal to the length of their FP times the group
index. This calculation inherently assumed that the phase noise accumulated by the light
during different passes through the FP were uncorrelated, but this is not true. As we showed
subsequently in a detailed study of the phase noise in FP interferometers [15], these
contributions are correlated, and as a result the total phase noise at the output of an FP is
proportional to the group index. On the other hand, Cranch and Foster [14] assumed that the
phase noise doesn’t scale with group index at all, because they used a different definition of
phase noise without explicitly stating it, as explained in [16, 17]. When the work reported in
this thesis started, the dependence of the phase noise at the output of a resonant cavity was
therefore an open question that needed to be answered carefully in order to first understand the
magnitude of the contribution of phase noise in Wen’s slow-light sensors, and second to
design a short passive strain sensor limited by phase noise, which had not been done before.
This was another major objective of this thesis work.
Another elegant and highly sensitive method for detecting a very small strain with an FBG is
to write the FBG in an active fiber to create a fiber laser [18,19]. The lasing wavelength
depends on the Bragg wavelength. Thus this sensing technique is similar to what has been
described for passive slow-light FBGs. A strain applied to the fiber laser induces a shift in the
laser wavelength, and an interrogation system is set up to measure this shift. The MZI
approach described earlier [5] has been used extensively for interrogating active FBG strain
sensors [18,19]. As discussed earlier, in passive FBG strain sensors utilizing an MZI as a
wavelength readout the main limitation is that the bandgap of the FBG can only be made so
narrow before its reflectivity becomes too small for useful lengths. But this is not the case for
active fiber laser sensors. Fiber lasers can have very narrow linewidths (less than 200 Hz)
[20], thus this interrogation technique is not limited by the coherence length of the light input
into the MZI. As a result, an MZI with a large length imbalance can be used, which leads to
much larger sensitivities and therefore much smaller resolvable wavelength shifts and strains.
Using an active FBG-based laser as a strain sensor, a measured MDS of ~60 fε/√Hz at 7 kHz
was reported in [18], and ~120 fε/√Hz at 2 kHz in [19] (see Fig. 1.2). At the start of this work
these were the lowest reported strain resolutions in a fiber sensor.
9
1.4 Summary of this thesis work
At the end of Wen’s work in 2011, it was clear that in order to reduce the MDS of slow-light
FBG sensors and reach or even exceed the resolution level of active FBG-based sensors, two
objectives needed to be met. We needed to reduce the frequency noise of the laser, and to gain
a sound understanding of the next noise source that would potentially limit our MDS, which
was phase noise. Thus the main objective of this thesis was first to study theoretically, then to
measure, the thermal phase noise in a passive fiber resonator, with a view to reduce the
measured MDS in a slow-light FBG strain sensor. This was a four-pronged effort. First, a
theoretical model was develop to predict the thermal phase noise of an FP cavity and its
relationship to the FP length and group index. Second, we showed how to increase the group
index in very short FBGs in order to achieve high group delays and high strain sensitivities.
This study was necessary to observe phase noise in an FBG and to explore other possible
applications of these slow-light FBGs, for example as high-Q resonators for nonlinear optics
and as optical delay lines. The third prong was to improve the experimental sensor developed
by He Wen to (1) reduce the laser frequency noise; (2) reduce the environmental noise; and
(3) improve the long-term stability of the sensor. The fourth and final prong was to use these
various developments and improvements to design, fabricate, and test two FBGs, one to
measure the thermal phase noise in an FBG for first time, and the other to observe the smallest
strain ever measured in an FBG-based sensor.
The first part of this effort, the development of a theoretical model of phase noise in a Fabry-
Perot interferometer, is discussed in Chapter 3. We show that at least at low frequencies, the
phase noise in proportional to the group index, and that when expressed in units of strain it is
inversely proportional to the square root of the length of the FP cavity. Thus, longer cavities
have a lower phase noise expressed in units of strain. These results enabled us to ultimately
design and test FBG sensors with a noise limited by phase noise and record-high strain
sensitivities.
The second part of this work, the design of FBGs with very strong resonances, is discussed in
Chapter 4. Since a short FBG was required to measure phase noise and verify our theory, we
needed to maintain the same sensitivity as, or achieve a higher sensitivity than, in previous
work but in a much shorter FBG. As discussed in Chapter 2, where the basic theory of strain
10
sensing using FBG is presented, to achieve this result the group index in an FBG had to be
increased to increase the sensitivity and/or amplify the phase noise. We were able to achieve
high group delays and sensitivities in short FBGs by using strongly apodized FBGs that
exhibit a high index modulation while maintaining approximately the same low loss as in
earlier FBGs. These FBGs were realized in collaboration with Prof. Martin Bernier at
University Laval in Québec, who fabricated all the FBGs tested in this work according to our
specifications. The FBGs were written using an infrared femtosecond laser in a deuterium-
loaded fiber, a technique that has been shown, in part as an outcome of this work, to increase
the maximum achievable index modulation and reduce the loss for the same index modulation.
These FBGs were thermally annealed to further reduce their internal loss and optimize them
for our applications. These improvements led to a record high group delay of 42 ns [21] and
revealed other interesting properties of slow-light FBGs, in particular high Purcell factor and
high intensity enhancement, that are discussed in Chapter 6. These properties can enable new
applications of slow-light FBGs, such as low-threshold lasers, enhanced nonlinear effects, etc.,
which can be useful in telecommunications.
The last two efforts are reported in Chapter 5. Specifically, first we discuss the improvements
made in the experimental sensor, including the acquisition of a new laser with very low
frequency noise, isolating the sensor in an anechoic enclosure to reduce the environmental
noise and improve the overall stability and noise performance of the sensor, and reducing the
detector noise. Next, we combined all the conclusions from the previous investigations to
design, fabricate, and test two FBGs compatible with our improved setup, one with a high
enough sensitivity and a large enough phase noise to measure the thermal phase noise in an
FBG for first time, and the other with a low enough overall noise to observe the smallest strain
ever measured in an FBG-based sensor. The first, 5 mm in length, enabled us to confirm the
magnitude of the phase noise, and its frequency dependence in the few-kHz range, predicted
by the model presented in Chapter 3. The second one, 20 mm in length, exhibited the smallest
MDS ever reported in a passive FBG, namely 110 fε/√Hz at 2 kHz and 30 fε/√Hz at 30 kHz,
which is about two [11] and about ten [8] times smaller than the previous records,
respectively. The output of this sensor was so stable (no sign of drift in four days) that a
record-low absolute MDS of 250 attostrains at 30 kHz was observed by integrating a 4-day
output trace with an 8-hour integration time. This sensor was so sensitive that it enabled us to
determine that the noise level in our laboratory was higher during the night than during the
11
day, an observation that could not be made with the previous generation of sensors. This result
shows that this new generation of ultra-sensitive sensors can be used to study phenomena that
were not detectable before.
Finally, Chapter 7 summarizes the salient results of this work and proposes possible future
research directions in slow-light FBGs.
References [1] R. Di Sante, “Fibre Optic sensors for structural health monitoring of aircraft cmposite
structures: recent advances and applications,” Sensors, 15, 18666 (2015).
[2] C. I. Merzbacher, A. D. Kersey, and E. J. Friebele, “Fiber optic sensors in concrete
structures: a review,” Smart Materials and Structure. 5, 196 (1996).
[3] N. Beverini, Calamai, D. Carbone, G. Carelli, N. Fotino, F. Francesconi, S. Gambino,
R. Grassi, E. Maccioni, A. Messina, M. Morganti, and F. Sorrentino, “Strain sensors
based on Fiber Bragg Gratings for volcano monitoring,” in Fotonica AEIT Italian
Conference on Photonics Technologies, 2015, 1 (2015).
[4] B. Abbott et al., “LIGO: the laser interferometer gravitational-wave observatory,”
Reports on Progress in Physics, 72, 076901 (2009).
[5] A. D. Kersey, T. A. Berkoff, and W. W. Morey, "High resolution fibre-grating based
strain sensor with interferometric wavelength-shift detection," Electronic Letters, 28,
136 (1992).
[6] B. Lissak, A. Arie, and M. Tur, " Highly sensitive dynamic strain measurement by
locking lasers to fiber Bragg gratings," Optics Letters, 23, 1930 (1998).
[7] D. Gatti, G. Galzerano, D. Janner, S. Longhi, and P. Laporta, "Fiber strain sensor
based on a π-phase shifted Bragg grating and the Pound-Drever-Hall technique,"
Optics Express 16, 1945 (2008).
[8] H. Wen, G. Skolianos, S. Fan, M. Bernier, R. Vallée, and M. J. F. Digonnet, “Slow-
light fiber-Bragg-grating strain sensor with a 280-femtostrain/√ Hz resolution,”
Journal of Lightwave Technology, 31, 1804 (2013).
[9] T. Erdogan, “Fiber grating spectra,” Journal of Lightwave Technology, 15, 1277
(1997).
[10] J. E. Sipe, L. Poladian, and C. M. De Sterke “Propagation through nonuniform grating
structures,” Journal of Optical Society of America A 11, 1307 (1994).
12
[11] G. Gagliardi, M. Salza, S. Avino, P. Ferraro, and P. De Natale, “Probing the ultimate
limit of fiber-optic strain sensing,” Science, 330, 1081 (2010).
[12] K. H. Wanser, “Fundamental phase noise limit in optical fibres due to temperature
fluctuations,” Electronic Letters, 28, 53 (1992).
[13] K. H. Wanser, “Theory of thermal phase noise in Michelson and Sagnac fiber
interferometers,” in Proceedings of Tenth Fibre Sensors Conference, 584 (1994).
[14] G. A. Cranch, and S. Foster, “Comment on “Probing the ultimate limit of fiber-optic
strain sensing,” Science, 335, 6066 (2012).
[15] G. Skolianos, H. Wen, and M. J. F. Digonnet, “Thermal phase noise in Fabry-Pérot
resonators and fiber Bragg gratings,” Physical Review A, 89, 033818, (2014).
[16] S. Foster, and G. A. Cranch “Comment on “Thermal phase noise in Fabry-Pérot
resonators and fiber Bragg gratings”,” Physical Review A, 92, 017801 (2015).
[17] G. Skolianos, and M. J. F. Digonnet, “Reply to “Comment on ‘Thermal phase noise in
Fabry-Pérot resonators and fiber Bragg gratings,” Physical Review A, 92, 017802
(2015).
[18] K. P. Koo, and A. D. Kersey, “Bragg grating-based laser sensors systems with
interferometric interrogation and wavelength division multiplexing,” Journal of
Lightwave Technology, 13, 1243 (1995).
[19] G.A . Cranch, G. M. H. Flockhart, and C. K. Kirkendall, “Distributed feedback fiber
laser strain sensors,” IEEE Sensors Journal, 8, 1161 (2008).
[20] H. Jiang, P. Lemonde, G. Santarelli, and F. Kefelian, "Ultra low frequency noise
laser stabilized on optical fiber spool," Frequency Control Symposium, 2009 Joint
with the 22nd European frequency and time forum. IEEE International, 815, Besançon
(2009).
[21] G. Skolianos, A. Arora, M. Bernier, and M. J. Digonnet, “Slowing down light to 300
km/s in a deuterium-loaded fiber Bragg grating,” Optics Letters, 40, 1524 (2015).
13
Chapter 2: Basic theory of strain sensing
using a slow-light FBG
In this chapter we discuss how a slow-light FBG can be used for strain sensing. Specifically,
the fundamentals of FBG theory are presented: what is an FBG, and how its transmission and
reflection spectra are simulated from its postulated index-modulation and loss profiles. We
then discuss the basic principles of conventional FBG strain sensors and of the more recent
FBG strain sensors that utilize slow-light resonances involved in this work. For the latter, we
establish the dependence of their strain sensitivity on the group index and transmission of the
resonance, and discuss their main sources of noise. The final section reviews the various
methods that have been developed to induce slow-light resonances in an FBG prior to this
work, including the introduction of a π phase shift in the approximate middle of the grating
and strong uniform FBGs. The use of strong apodized FBGs to induce slow-light resonances,
the approach that is central to this thesis work, will be discussed in detail in Chapter 4. For
these schemes we also present the scaling laws of the group index dependence on the grating
parameters, which will be useful later to maximize the strain sensitivity.
2.1 Fiber Bragg Grating
A fiber Bragg grating (FBG) is a fiber in which the refractive index of the core region is
periodically modulated over a short length, usually a few mm up to 10 cm, with a period Λ.
This modulation is illustrated in Fig. 2.1 for the particular case of a grating with a uniform
index modulation. The effective refractive index of the unperturbed fiber (before the grating
was written) is n0, and Δnac is the half peak-to-peak amplitude of the index modulation. Because
this modulation does not generally have zero average value, the modulation in the mean
effective refractive index of the fiber also has some finite value Δndc. When these profiles are
constant, the grating is said to be uniform. In all other cases it is generally referred to as
apodized.
For a uniform grating with a sinusoidal modulation the general function that describes the
effective refractive index n(z) as a function of the position z along the grating, is [1]:
14
n(z) = n0 +Δndc +Δnac cos2π zΛ
#
$%
&
'( (2.1)
Figure 2.1. Periodically varying refractive index in a uniform FBG.
As light passes through the FBG, the refractive index is constantly changing. As a
consequence, light inside the FBG gets partially reflected due to Fresnel reflection wherever
and index change takes place, and the reflected light is coupled to the same fundamental mode
but traveling in the opposite direction. The reflections are strongest where the change in the
refractive index is steepest. Thus two strong reflections take place per period with opposite
phases, one on each side of the middle of each modulation period. The phase difference
between two reflections that occur at two different points one period apart is 2π<n>2Λ/λ,
where <n> is the mean effective refractive index, equal to n0+Δndc, and λ is the wavelength of
the incident light in vacuum. This is true because the light that is reflected at the second point
in the FBG has traveled an extra round-trip distance equal to twice the period. Thus when the
wavelength is equal to 2(neff + Δndc)Λ all the reflections from different periods add in phase.
This wavelength is called the Bragg wavelength, and it is given by [1]:
λB (z) = 2(n0 +Δndc )Λ (2.2)
Using the coupled-mode theory for the two counter-propagating modes, a transfer matrix that
connects the forward and the backward electric fields at the output ( Eout+ and Eout
− , respectively)
and the input ( Ein+ and Ein
− , respectively) of an FBG with length L can be calculated. This
transfer matrix is [1]:
15
Ein+
Ein−
"
#$$
%
&''=
cosh κ 2 −σ 2 ⋅L( )-i σ
κ 2 −σ 2sinh κ 2 −σ 2 ⋅L( ) -i κ
κ 2 −σ 2sinh κ 2 −σ 2L( )
i κ
κ 2 −σ 2sinh κ 2 −σ 2L( ) cosh κ 2 −σ 2 ⋅L( )+ i σ
κ 2 −σ 2sinh κ 2 −σ 2 ⋅L( )
"
#
$$$$$
%
&
'''''
Eout+
Eout−
"
#$$
%
&''
(2.3)
where κ and σ are the ac and dc coupling coefficients, respectively, given by:
σ = 2π n0 +Δndc( ) 1λ−1λB
#
$%
&
'(− i
γ2
(2.4a)
κ = πΔnacλ
(2.4b)
where γ is the power loss coefficient that characterizes propagation in the grating. From Eq.
2.3 the field reflection coefficient r and the field transmission coefficient t can be calculated
as:
r =−κsinh κ 2 −σ 2L( )
σsinh κ 2 −σ 2L( )+ i κ 2 −σ 2 cosh κ 2 −σ 2L( ) (2.5a)
t = i κ 2 −σ 2
σsinh κ 2 −σ 2L( )+ i κ 2 −σ 2 cosh κ 2 −σ 2L( ) (2.5b)
In the case of a lossless propagation through the FBG, it can be easily calculated that the
power that is transmitted plus the power that is reflected equals the total power, namely
r 2 + t 2 =1 (energy conservation).
Using Eqs. 2.5a and 2.5b, we plotted in Fig. 2.2 the power reflection and transmission spectra
of a uniform FBG with a period Λ = 534.3 nm, a length L = 2 mm, Δnac = Δndc = 0.5x10-3, and
a uniform loss of γ = 0.1 m-1, as an example. As expected, there is a maximum reflectivity at
the Bragg wavelength defined in Eq. 2.2. In the immediate vicinity of this wavelength, a
region known as the bandgap, most of the partial reflections are in phase, and only a small
fraction of the light is transmitted. Far away from the Bragg wavelength all the light is
transmitted and the grating does not affect the propagating light. Close to the bandgap but
outside the bandgap some ripples occur; the reason for their existence is discussed in detail in
16
section 2.3.2b. If a stress is applied to the grating, the two spectra retain their respective shape
but they shift to a new Bragg wavelength by a usually relatively small amount (a few nm or
less). This property has been exploited extensively in FBG sensors, as discussed later.
The bandwidth of the bandgap, defined as the bandwidth between the first zero reflectivities
on either side of the Bragg wavelength, can be calculated using [1]:
Δλ = λBΔnacn0
1+ 2n0ΛΔnacL#
$%
&
'( (2.6)
In the limit of strong and/or long gratings (ΔnacL>>λB) of most interest in this work, the
bandwidth Δλ can be reduced from Eq. 2.6 to:
Δλ ≈ λBΔnacn0
(2.7)
Thus, in this limit, as Δnac increases the bandwidth of the bandgap also increases.
Figure 2.2. Simulated power (a) reflection and (b) transmission spectrum for a uniform FBG of length
L = 2 mm, index modulation Δnac = Δndc = 5 x 10-4, period Λ= 534.3 nm, and loss γ = 0.1 m-1.
Δnac also determines the maximum reflectivity of the FBG. From Eqs. 2.4a and 2.5a it can be
seen that in the lossless case, at λ = λΒ, we have σ = 0 and r = itanh(κL). Thus the reflectivity
17
increases as κ increases. Consequently, a high κ, which is proportional to Δnac, leads to a high
reflectivity. This is an important property that we use extensively later to design ultra-sensitive
sensors. Also, this last equation states that the reflected light is phase shifted by π/2 relative to
the incident light. When light is slightly detuned from the Bragg wavelength, the total phase
shift of the reflected light is different from π/2, because each reflection has a slightly different
phase. This is why in equation 2.5 the reflection and the transmission coefficients are complex
numbers and their phase is wavelength dependent. Using this phase, the group delay τg and the
group index ng in transmission can be calculated from the following equations
τ g = −dφtdω
=ngLc (2.8a)
ng =cvg (2.8b)
where φt is the phase of the transmitted electric field relative to the input field, ω is the angular
frequency of the light, c the speed of light in free space, and vg the group velocity. Using an
equation similar to Eq. 2.8a, the group delay in reflection can also be calculated [1].
2.2 Strain sensing using FBGs
2.2.1 General principle of strain sensing using an FBG
If a longitudinal strain ε is applied to an FBG, three effects take place [2]:
1) The period of the FBG is changed by a quantity ΔΛ which, assuming that the applied
strain is uniform, is given by:
ΔΛΛ
=ΔLL= ε (2.9)
2) The refractive index n of the material is changed due to the applied strain through the
elasto-optic effect
3) The diameter of the fiber changes because of the change in fiber length, causing the
propagating mode to slightly change and hence the effective refractive index to
change. This effect is negligible.
18
The second and third effects can be lumped together as a change in the effective refractive
index of the propagating mode n0 expressed as:
Δn0n0
=1n0dn0dε
ε (2.10)
By taking the partial derivative of λB (Eq. 2.2), it is clear that these three effects result in a
shift in the Bragg wavelength given by the sum of Eq. 2.9 and 2.10:
ΔλBλB
=ΔΛΛ
+Δn0n0
⇒ΔλB = λB 1+1n0dn0dε
$
%&
'
()ε (2.11)
The wavelength shift, which applies to the Bragg wavelength and the entire spectrum, is
proportional to the applied strain ε, the Bragg wavelength λΒ, and a material constant,
(1+1/n0•dn0/dε). For silica fibers, which are most commonly used in telecommunications and
fiber sensors, this material constant has been measured to be 0.78 [2]: the change in grating
period accounts for a positive contribution equal to 1, and the strain-induced change for a
negative contribution equal to -0.22. Thus, for a silica fiber, the reflection-spectrum shift due
to an applied strain is given by:
ΔλB = 0.78λBε (2.12)
Figure 2.3. Shift of the Bragg wavelength and the whole spectrum due to an applied strain. The initial
transmission is indicated with the solid line while the spectrum after a strain has been applied to the
FBG is shown with the dashed line.
19
The strain applied to an FBG is then almost always sensed by measuring the strain-induced
shift in the reflection or transmission spectrum (see Fig. 2.3). An obvious way to do this is by
launching broadband light into the FBG and using an optical spectrum analyzer (OSA) to
measure the spectral shift induced by a strain. However, because of the relatively low
resolution of OSAs, the strain resolution is not very good. A typical OSA has a resolution of
~10 pm. For an FBG with a Bragg wavelength at 1.55 µm, Eq. 2.12 gives ΔλB=1.2 pm/µε,
leading to a minimum detectable strain (MDS) of only 8 µε.
To overcome the limited resolution of OSAs, the FBG can interrogated with a laser tuned to
the steepest slope of the FBG reflection or transmission spectrum, as discussed in the
introduction. In this scheme the change in the output power is measured, and the applied strain
can be recovered from this measurement and the knowledge of the spectrum’s slope at the
probe wavelength, which is also obtained easily with a measurement of the spectrum.
For this general type of sensing scheme, the normalized strain sensitivity SN is defined as:
SN (λprobe ) =1Pin
dPoutdε
!
"#
$
%&λprobe
(2.13)
where dPout is the change in output power at the probe wavelength λprobe in response to a small
applied strain dε, and Pin is the power input into the FBG. Since the transmission is by
definition T = Pout/Pin, this expression can be rewritten as:
SN (λprobe ) =1Pin
dPoutdε
!
"#
$
%&λprobe
=dTdε
!
"#
$
%&λprobe
=dTdλ
dλdε
!
"#
$
%&λprobe
=dλBdε
dTdλ!
"#
$
%&λprobe
(2.14)
From Eq. 2.14 it can be seen that the sensitivity is proportional to the slope of the transmission
spectrum dT/dλ evaluated at the probe wavelength. Essentially, this slope converts the
wavelength shift into a change in transmission, and therefore a change in output power. In this
case the MDS is limited by how well a small change in the output power can be resolved,
namely by the output power noise Pnoise, since this is the smallest change in the output power
that can be measured. Thus the MDS, or equivalently the strain resolution of the sensor, is
defined as:
20
MDS= PnoisePin
1SN (2.15)
Stated differently, the MDS is the noise of the sensor expressed in units of strain. To achieve a
low MDS, the output noise power (relative to the input power) must be low and/or the
sensitivity must be high. Lissak et al. were first to report the use of the steep slope in the
FBG’s band edge to achieve a very low MDS (~50 pε/√Hz) [3], as discussed in the
introduction.
2.2.2 Using a sharp resonance to increase the strain sensitivity
The slope and the sensitivity can be significantly increased by using a sharp resonance.
Assume that we can fabricate an FBG with a very narrow transmission peak, as illustrated in
Fig. 2.4. As discussed previously, if the peak is interrogated on the steepest slope a
wavelength shift is converted to a transmission change. This change increases as the slope of
the peak is increased. From Fig. 2.4 one can easily see that if the peak linewidth is decreased,
and/or if the peak transmission is increased, the slope is increased, the sensitivity increases, as
stated by Eq. 2.14.
Figure 2.4. Using a sharp peak for strain sensing.
If the peak is narrow, by definition the quality factor is high too, and so are the group delay
and the group index of the resonance. So we expect intuitively that the sensitivity scales like
the group index, or the group delay. The relationship between sensitivity and group delay can
21
be readily expressed in a closed form in the particular case of a resonance with a Lorentzian
lineshape, which is applicable to many structural slow-light resonances (when the Q factor of
the cavity is spoiled, the energy decays exponentially, hence the lineshape is Lorentzian). In
this case it is easy to show, by taking the derivative relative to the wavelength of the general
expression of a Lorentzian transmission function and looking for the wavelength where this
derivative is maximum, that the maximum strain sensitivity is [4]:
SNmax =
3.22cT0τ g,maxλ
(2.16)
where T0 is the maximum transmission of the resonance (i.e., at the linecenter of the
resonance) and τg,max is the maximum group delay of the resonance (i.e., at the linecenter of the
resonance). Equation 2.16 shows that the strain sensitivity is proportional to the peak
transmission and the peak group delay of the resonance, as was expected based on the
previous discussion. It is worth mentioning that even if the peak is not exactly a Lorentzian,
the scaling laws of Eq. 2.16 are still valid; only the numerical factor (3.22) differs a little. Thus
for a highly sensitive sensor a slow-light peak with a large peak transmission times group
delay product is needed. From now on, we will concentrate solely on strain sensors that utilize
a sharp resonance to achieve a high sensitivity.
2.2.3 Noise sources in an ultra-high sensitive strain sensor
According to Eq. 2.15, to reduce the MDS we not only have to increase the sensitivity, but
also to reduce the noise. Noise sources can be grouped in two general categories: (1) noise
sources that are independent of the sensor’s strain sensitivity and (2) noise sources that depend
on the sensitivity.
The first category, sensitivity-independent noise sources, includes the intensity noise of the
laser, the photodetector noise, and the shot noise (electrical and optical). The intensity noise of
a laser depends solely on the laser type and its output power. The photodetector noise is a
characteristic of the photodetector alone. Finally, the shot noise depends on the power of the
detected optical signal and exists because of the quantum nature of the light. Because these
22
noise sources are independent of the sensitivity, their individual contribution to the MDS
decreases as the sensitivity is increased (see Eq. 2.15).
The second category, sensitivity-dependent noise sources, includes the intrinsic thermal phase
noise of the fiber, the laser frequency noise, and environmental noise. The thermal phase noise
is caused by random thermodynamic fluctuations of the fiber temperature, which induce
variations in the optical path length experienced by light that propagates through the fiber.
This noise can be interpreted as a random strain applied to the FBG, since the strain has the
same effect—a change in optical path length— and also causes the slow-light resonance
frequency to fluctuate randomly around a mean value. The laser frequency noise, as discussed
in the introduction, is random fluctuations in the laser mean wavelength. Thus the position of
the laser wavelength relative to the resonant wavelength of the resonance fluctuates randomly.
This has the same effect as if the resonance used for sensing was randomly fluctuating around
a mean value. Finally, the environmental noise, such as vibrations, acoustic noise, etc., is an
actual signal that is not useful but constitutes a real strain signal that is also detected and
constitutes a source of noise. Thus, these three noise sources introduce an uncertainty in the
relative spectral position of the slow-light resonance and the laser signal, as a true strain signal
would do. The FBG being interrogated on the slope of this resonance, it converts each of these
three sources of noise into a power noise at the detector. Consequently, each of these three
noise sources produces power noise that is proportional to the slope of the transmission at the
probe wavelength, and therefore to the sensitivity.
According to this analysis the output power noise, and consequently the MDS, can be
decomposed into two main components, one due to the sensitivity-independent noise sources
Pi, and one due to the sensitivity-dependent noise sources Pd=ASN, where A is a factor that
doesn’t depend on the sensitivity. The MDS can therefore be written formally as:
MDS = PnoiseSNPin
=Pi2 +Pd
2
SNPin=1Pin
Pi2
SN2 + A
2 (2.17)
From Eq . 2.17 it can be clearly seen that if the sensitivity is high enough (SΝ >> Pi/A), the
sensitivity-independent noise sources become negligible. On the other hand at low sensitivities
the noise sources that are sensitivity dependent become dominant. In this limit, the MDS is
23
given by A/Pin and is independent of the sensitivity. Thus, increasing the sensitivity further
does not improve (reduce) the MDS.
2.3 Temperature sensitivity of slow-light FBGs
The analysis in the previous section can be applied also for other quantities that change the
Bragg wavelength of the FBG, and hence shift the spectrum, like humidity, acoustic waves,
temperature, etc. As long as a change in the environment shifts the Bragg wavelength, this
change is converted to a change in the output power of the FBG—which is proportional to the
slope of the operating point and therefore the group delay—via the shift in the FBG’s
spectrum. In this section we focus on thermal sensitivity and temperature sensing because it is
important for the thermal stability of our strain sensor.
As in the previous case of an applied strain, when the temperature of an FBG changes three
effects take place [5]:
1) The FBG expands longitudinally, which changes its period;
2) The refractive index of the fiber material changes, and hence so does the effective
refractive index of the propagating mode;
3) The diameter of the fiber changes, which changes the mode’s effective index too.
All three effects on the Bragg wavelength can be modeled as [5]:
ΔλB = λB ξ +1n0dn0dK
"
#$
%
&'ΔK (2.18)
where ξ is the thermal expansion coefficient of the fiber and ΔK is the temperature change.
For a silica fiber, ξ=5x10-7 K-1and dn0/dK=1.1x10-5 K-1. Thus, at a wavelength of 1.55 µm
ΔλB = 12.5 pm/˚K.
The normalized temperature sensitivity ST of the FBG is defined similarly to its normalized
strain sensitivity (Eq. 2.14):
24
ST (λprobe ) =1Pin
dPoutdK
!
"#
$
%&λprobe
=dTdK!
"#
$
%&λprobe
=dTdλ
dλdK
!
"#
$
%&λprobe
=dλBdK
dTdλ!
"#
$
%&λprobe
(2.19)
Hence, the temperature sensitivity is also proportional to the slope dT/dλ of the resonance at
the operating wavelength, as expected. Equation 2.19 and 2.14 differ only by the factor in
front of the slope. Therefore the analysis of sections 2.2.2 and 2.2.3 is directly applicable to
the temperature sensitivity and temperature sensing. In fact it can be used to model sensing of
any parameter, as long as the factor for this specific parameter is used to scale the slope. This
shows that as the strain sensitivity is increased by increasing the slope of the resonance, the
temperature sensitivity is increased too, and the thermal stability of the sensor is degraded. On
the other hand, this high thermal sensitivity allowed us to measure random temperature
fluctuations in the atomic level of the fiber (phase noise), as we discuss in Chapter 5.
In practice other changes in the environment of the fiber besides temperature drift have the
same effect: they change the output power of the FBG when the latter is probed on the edge of
a resonance. In general it is hard to distinguish the different sources that cause the power
change. It can be done, however, by using multiple FBG sensors co-located on the same or
nearby fibers and comparing their outputs [6-8]. The thermal drift can also be greatly reduced
by mounting the FBG on a support that compensates for the direct effect of the temperature
change on the FBG (athermal FBGs) [9]. In our sensor, because we were interested in
measuring a dynamic strain, we were able to easily isolate the slowly-varying thermal drift
from the fast strain using a lock-in amplifier, which isolates the specific frequency of the
strain and effectively filters out the signals at other frequencies, in particular the slow thermal
drift. We were also able to cancel out the effect of the drift on the sensitivity by using a
feedback loop with a proportional-integral-derivative (PID) controller, as discussed in detail in
Chapter 5.
2.4 Slow-light resonances in an FBG
Until now we have explained how a highly sensitive sensor can be achieved in principle by
making use of a slow-light resonance with a high transmission. A well-known optical device
that exhibits such resonances is a Fabry-Perot (FP) interferometer. In this section the group
delay and group index of the resonances of an FP are derived algebraically. This derivation
25
provides general scaling laws for the group delay and consequently the strain sensitivity of an
FP. We then discuss how an FP can be formed in a single FBG, and how the scaling laws
derived for an FP apply to the FP-like resonances of slow-light FBGs.
2.4.1 Slow-light resonances in a Fabry-Perot interferometer
We consider a conventional Fabry-Perot interferometer with two parallel reflectors M1 and M2
with field reflection coefficients r1 and r2, and transmission coefficients t1 and t2, respectively.
The mirrors M1 and M2 can be two FBGs, or any number of conventional optical reflectors. In
the particular case of two FBGs of interest here, the field reflection and transmission
coefficients are the quantities defined in Eqs. 2.5a and 2.5b, respectively. The distance
between the two mirrors is L and the medium between the mirrors has a uniform refractive
index n (Fig. 2.5).
Figure 2.5. Schematic of a generic Fabry-Perot interferometer
These four coefficients (r1, r2, t1 and t2,) implicitly incorporate the medium’s single-pass
transmission exp(-αlossL), where αloss is the medium’s loss coefficient. In such an FP the
transmission after the second mirror is given by [10]:
Et
E0
=t1t2
1− r1r2ej2Φ1
ej Φ1−π−
ωnLc
#
$%
&
'(
Φ1 = 2πnL / λ +ϕr1 +ϕr2
2
(2.20)
where φr1 and φr2 are the phase shift associated with reflection on the first and the second
mirror, respectively. The factor π in the phase term arises from the phase associated with
26
transmission through each of the two mirrors, φt1 = φr1 - π/2 and φt2 = φr2 - π/2. The group delay
of the transmitted signal is by definition (Eq. 2.8a) proportional to the derivative of the
argument of the right-hand side of Eq. 2.20, or
ng =ddω
2Φ1 −π −ωnLc
− tan−1 r 1r2 sin2Φ1
r 1r2 cos2Φ1 −1#
$%
&
'(
#
$%%
&
'((cL (2.21)
After some straightforward algebraic manipulations, it is easy to prove from Eq. 2.21 that
ng = n 2 1− r1r2 cos(2Φ1 )1+ r1r2( )
2− 2r1r2 cos(2Φ1 )
−1#
$%%
&
'(( (2.22)
On a resonance, Φ1 is a multiple of π, and this expression reduces to:
ng = n2
1− r1r2−1
"#$
%&' = n
2Fπ r1r2
−1"
#$
%
&' (2.23)
where F is the finesse of the FP.
In the limit of high finesse:
ng ≈ n2Fπ
(2.24)
A similar derivation applied to the signal reflected by the FP gives the relationship for the
group index in reflection ng ' :
ng' =
2r2t12
1− r1r2( ) r2 r12 + t1
2( ) − r1( ) (2.25)
In the limit of high finesse, Eq. 2.25 reduces to Eq. 2.24.
27
Equation 2.23 states that in order to achieve a large group delay, the round-trip loss of the
cavity (1-r1r2) must be small, namely the mirrors must have high reflectivity, and the
absorption loss in the mirrors and in the medium between the mirrors must be low. Also, from
Eq. 2.20 it can be seen that for the transmission to be high the absorption loss must be low,
namely the round-trip loss of the cavity must be as close as possible to the light that is
transmitted outside the cavity from the mirrors. Thus, if the two mirrors were FBGs, a high
sensitivity requires that the two FBGs have a high Δnac and a low loss.
2.4.2 Forming an FP and highly sensitive slow-light resonances
In this section we discuss how FP-like resonances can be created in a single FBG. To create
slow-light resonances an effective FP needs to be formed. Thus the analysis of section 2.3.1 is
directly applicable to FBGs as well. An effective FP can be formed in a single FBG using one
of the following approaches:
1) A π-shifted FBG
2) A strong uniform FBG
3) A strong apodized FBG
The first two approaches, which have been investigated before to demonstrate ultra-sensitive
strain sensors [4, 11], are described in this chapter. The third approach, using an apodized
FBG, is described in Chapter 4.
2.4.2a Slow-light in π-shifted FBGs
A π-shifted FBG is an FBG with a π phase shift in the middle of the grating’s refractive index
profile. This type of grating can be conceptualized as two nominally identical FBGs separated
by a gap of length λ/4 (a phase shift of π/2 rad), which means that light traveling in the gap
accumulates a phase of π per round trip. Each grating acts as a reflector, and the two gratings
together form an FP separated in the phase space by π. This FP is so short that it supports a
single transmission resonance, located around the middle of the grating’s bandgap (Fig. 2.6).
This resonance can have a high sensitivity to strain because this type of FBG can exhibit a
high Δnac.
28
Figure 2.6. Calculated reflection spectrum of a representative π -shifted grating (the grating’s index
profile is shown schematically in the inset).
Very often in a π-shifted FBG, because of fabrication imperfections, for example if the phase
shift is not exactly π or not exactly in the middle of the FBG, the slow-light resonance is not
exactly in the middle of the bandgap. This can lower the peak transmission of the resonance
because the FP is not symmetric (the two mirrors have different reflectivities), thus the
maximum transmission is lower than in the symmetric case. The merit of this application was
demonstrated with the report of an MDS of 5 pε/√Hz by Gatti et al. in 2008 [11].
2.4.2b Slow-light in uniform FBGs
As we have already discussed earlier in this chapter, in a uniform grating in the middle of the
bandgap the FBG is essentially fully reflective, while far away from the bandgap it is
essentially fully transmissive. Between these two extremes, over a short span of wavelengths
on the sharp edges of the bandgap, the FBG is partially transmissive. Thus the FBG acts as a
distributed partial mirror. The light component that is not transmitted at each point of the FBG
is reflected, travels through the FBG back towards the input, where the portion of it that is not
transmitted is partially reflected again, and so on and so forth. Light therefore makes multiple
round-trip passes through the FBG, as illustrated in Fig. 2.7.
29
Figure 2.7. a) For wavelengths near the band edge of a strong FBG, multiple reflections from both ends
of the FBG cause recirculation of light inside the FBG. The situation is analogous to a Fabry-Perot (FP)
interferometer. b) Similar to a FP, sharp resonances are formed in the transmission spectrum.
At certain specific wavelengths, the multiple small output fields that are transmitted at each
pass are in phase and add coherently to each other, which leads to a strong output. The FBG
behaves like a distributed FP [12]. The two mirrors are distributed and spatially overlap. The
optical distance between them is defined through the phase of the light, as in the π-shifted
FBGs, where the physical space between the two FBGs is essentially zero. Like in an FP, the
phase-matching condition for constructive interference occurs only at discrete wavelengths.
Additionally, like in an FP, to produce substantial group delays and therefore high
sensitivities, the FBG must reflect sufficiently over a fairly short distance, which requires in
practice a strong ac index modulation (~10-3 or greater) and a low loss.
A uniform FBG is not the most effective way to generate slow light and high sensitivities.
When the index modulation is uniform, at all slow-light wavelengths the grating reflects
strongly throughout its length, and light experiences a large reflective “loss” instead of being
transmitted by the nominally lossless medium between the two reflectors as it is in typical
free-space or fiber FPs. Although a uniform FBG is not the optimal way to increase the
sensitivity, He Wen used an almost uniform FBG to achieve a record low MDS of 280 fε/√Hz
[4]. In Chapter 4, we will discuss, among other things, how apodizing the FBG’s index-
30
modulation profile helps to solve this problem and increases the group delay and consequently
the sensitivity even further.
References [1] T. Erdogan, “Fiber grating spectra,” Journal of Lightwave Technology, 15, 8, 1277
(1997).
[2] A. Kersey, M. A. Davis, H. J. Patrick, M. Leblanc, K. P. Koo, C. G. Askins, M. A.
Putnam, and E. J. Friebele, “Fiber grating sensors,” Journal of Lightwave Technology
15, 1442 (1997).
[3] B. Lissak, A. Arie, and M. Tur, “Highly sensitive dynamic strain measurement by
locking lasers to fiber Bragg gratings,” Optics Letters, 23, 1930 (1998).
[4] H. Wen, G. Skolianos, S. Fan, M. Bernier, R. Vallée, and M. J. Digonnet, “Slow-light
fiber-Bragg-grating strain sensor with a 280-femtostrain/√ Hz resolution,” Journal of
Lightwave Technology, 31, 1804 (2013).
[5] E. J. Friebele, M. A. Putnam, H. J. Patrick, A. D. Kersey, A. S. Greenblatt, G. P.
Ruthven, M. H. Krim, and K. S. Gottschalck, “Ultrahigh-sensitivity fiber-optic strain
and temperature sensor,” Optics Letters, 23, 222 (1998).
[6] J. D. C. Jones, “Review of fibre sensor techniques for temperature-strain
discrimination,” Optical Society of America Technical Digest Series in 12th
International Conference on Optical Fiber Sensors, 16, 36 (1997).
[7] M. Song, S. B. Lee, S. S. Choi, and B. Lee, “Simultaneous measurement of
temperature and strain using two fiber Bragg gratings embedded in a glass tube,”
Optical Fiber Technology, 3, 194 (1997).
[8] W. C. Du, X. M. Tao, and H. Y. Tam, “Fiber Bragg grating cavity sensor for
simultaneous measurement of strain and temperature,” IEEE Photonics Technology
Letters, 11, 105 (1999).
[9] Y. L. Lo, and C. P. Kuo, “Packaging a fiber Bragg grating with metal coating for an
athermal design,” Journal of Lightwave Technology, 21, 1377 (2003).
[10] G. Skolianos, H. Wen, and M. J. F. Digonnet, “Thermal phase noise in Fabry-Pérot
resonators and fiber Bragg gratings,” Physical Review A, 89, 033818, (2014).
[11] D. Gatti, G. Galzerano, D. Janner, S. Longhi, and P. Laporta, “Fiber strain sensor
based on a π-phase shifted Bragg grating and the Pound-Drever-Hall technique,”
Optics Express, 16, 1945-1950 (2008).
31
[12] J. E. Sipe, L. Poladian, and C. M. De Sterke “Propagation through nonuniform grating
structures,” Journal of Optical Society of America A ,11, 1307 (1994)
32
Chapter 3: Modeling the thermal phase noise
in a passive Fabry-Perot resonator
As discussed in the two previous chapters, the intrinsic thermal phase noise of the fiber is
expected to be the next lowest source of noise after laser frequency noise in the slow-light
fiber Bragg gratings we have been developing for ultra-sensitive strain sensing. The phase
noise being generally exceedingly small, in most cases smaller than 50 fε/√Hz in terms of
strain, and given the wide disagreement in the predictions of the magnitude of this phase noise
reported in the literature, it was important to perform our own theoretical predictions of its
magnitude and spectral dependence.
This chapter presents an original derivation of the thermal phase noise in a fiber Fabry-Perot
resonator operated in transmission or in reflection. It reveals a number of interesting properties
that remained unpublished until these results were reported in Physics Review A in 2014 [1].
It shows that at low frequencies the power spectral density (PSD) of the phase noise of an FP
interferometer is proportional to the group index of the resonance [1]. As mentioned in the
introduction, this was anticipated. When light propagates through a resonance, it bounces back
and forth in the resonator several times and accumulates phase noise at every pass. Thus the
phase noise in the output light signal is expected to be the phase noise picked up in a single
pass, which is proportional to the square root of the length, times the number of passes, which
is proportional to the group index. Thus, in a resonator the phase noise is expected to be
“amplified” by the group index. The MDS of a sensor is defined as the noise over the
sensitivity (see Chapter 2). Therefore if the MDS is limited by phase noise, it will be
proportional to the square root of the length of the resonator, since the sensitivity is
proportional to the group index times the length of the resonator. These results were later
verified in [2,3] and enabled us to design the sensor presented in Chapter 5, in which the
dominant noise source is the phase noise of the fiber.
Specifically in this chapter, it is shown that although in the most general case the PSD of the
phase noise in an FP can only be expressed as an infinite series, this series can be written in a
simple and convenient closed-form expression under two broad general regimes, namely at
low noise frequencies for any optical frequency (on or off an FP resonance), and at any
33
frequency for a signal on resonance. This distinction is important, because for maximum
sensitivity most FP sensors operate slightly off resonance, as discussed in Chapter 2, so this
first closed-form expression is critical to this work. Numerical evaluations of this infinite
series in the case where it cannot be expressed in a closed form (off resonance at high
frequencies) demonstrates that the PSD has a sinc dependence on frequency [1]. This
dependence is valid for an FP of arbitrary finesse. The PSD of the phase noise picked up by a
light signal that has traveled through a fiber twice therefore does not have a cosine dependence
on frequency, unlike reported in [4], but a sinc dependence [1]. Although the two dependences
are asymptotically identical in the low-frequency limit, this correction is significant when
dealing with high-frequency sensors; we predict, unlike previous models, that the phase noise
does not have zeros at certain high frequencies. While all previous publications have been
concerned with FPs operated in reflection, in this chapter a derivation of the phase-noise PSD
for an FP operated in transmission is provided as well. This is of direct relevance in particular
to the ultra-sensitive slow-light FBG sensors reported in this work, which operate in
transmission. This analysis is also instrumental in that it shows how to design FBGs in order
to (1) achieve the lowest possible MDS using our sensing principle, (2) measure the thermal
phase noise in an FBG, and (3) use this measurement to verify the magnitude of the phase
noise predicted by our theory.
3.1 Modeling phase noise in a Fabry-Perot interferometer 3.1.1 Background
As we briefly discussed in the introduction, when light propagates through a fiber,
thermodynamic temperature fluctuations change randomly both the refractive index and length
of the fiber, which in turn translates into noise in the phase of light propagating through the
fiber. Wanser studied this phenomenon and published in 1992 a simple formula for the PSD of
the thermal phase noise of a signal that has traveled once through an optical fiber [5], which
we refer to as the single-pass phase noise. He showed that the single-pass phase noise is
proportional to the square root of the length of the fiber. This dependence is based on the
assumption that the phase noise accumulated in two different points of the fiber is completely
uncorrelated to each other, which is correct at all except possibly exceedingly low (sub-mHz)
noise frequencies. This expression was used to predict the phase-noise-limited minimum
detectable phase shift in a number of fiber sensors in which light travels through a fiber only
34
once, as in a Mach-Zehnder interferometer [4]. Wanser later extended this study to the phase
noise of a signal that has traveled twice through the same fiber [4], as occurs, for example, in a
Michelson interferometer. Because slow-light FBG sensors utilize FP-like resonances, it is of
interest to further extend this work to these multi-pass interferometers.
Expressions for the thermal phase noise in multi-pass interferometers used in reflection have
been published [6–9]. However, careful comparison shows that these expressions do not agree
with each other, and often disagree markedly. A recent manifestation of this lack of consensus
is the report by Gagliardi et al. of an FP sensor that was allegedly limited by phase noise [6].
This claim was challenged in [7], which provided mathematical arguments that the phase
noise in the FP resonator of [6] was significantly lower than claimed. The error in [6] came
from the fact that the authors misused Wanser’s formula. They replaced the length of the fiber
in Wanser’s formula with an effective length equal to the group index times the length of the
fiber. This led the authors to the wrongful conclusion that the phase noise is proportional to
the square root of the group index, since they inadvertently made the assumption embedded in
Wanser’s formula that the phase noise accumulated between different passes is completely
uncorrelated. This assumption is correct for two adjacent points seen by light propagating in
one direction, the case studied by Wanser, but as we shall see below it is not for light that sees
the same point when traveling several times through the same fiber, as in an FP. The authors
of [7] also presented an expression for the phase noise in an FP, but they used a different
definition of phase noise without explicitly mentioning it, which led to further confusion.
They essentially calculated the noise in the resonant frequency of the cavity, and called it
phase noise, which resulted in a “phase noise” that is off by a factor of √2 and independent of
group index.
3.1.2 Intuitive picture
In a fiber Fabry-Perot interferometer probed near resonance, photons travel multiple times
back and forth between the two reflectors. During each pass through the medium contained in
the cavity, they pick up a phase noise that adds to the phase noise already accumulated during
previous passes. Two types of correlation play a role in this system, namely (1) correlation of
the temperature at any two different points along the fiber at a specific time (spatial
correlation), and (2) correlation of the temperature at two different times at a specific point
35
(temporal correlation). Regarding the spatial correlation, the temperature variations at two
different points are generally considered to be fully uncorrelated [10]. A consequence of this
property is that for light propagating once through a fiber the phase noise PSDs at each point
add up, and the total PSD is proportional to the fiber length L. Regarding the temporal
correlation, for low perturbation (phase noise) frequencies f, the round-trip transit time through
the cavity is short compared to the perturbation period 1/f. During the time it takes light to
travel through the very short cavity (typically a few cm or less), there is not enough time for
the temperature to change, and therefore the noise components picked up in successive passes
are fully correlated. A corollary of this property is that the total phase noise is expected to be
the phase noise of a fiber of length L multiplied by the number of passes N through the cavity,
and therefore for a high-finesse cavity it is expected to be significantly larger than the single-
pass phase noise. The PSD should therefore be proportional to the square of the number of
passes through the cavity. The number of passes through an FP cavity is approximately equal
ng/n, where ng is the group index of the light in the cavity and n the refractive index of the
medium between the mirrors. The PSD at low frequencies should therefore be proportional to
(ng/n)2. Since ng depends on the frequency of the light, and is maximum on a resonance, so
should the phase noise. The PSD at low frequencies, proportional to ng2, should be a very
strong function of optical frequency near a resonance. Detailed algebraic manipulations of the
expressions of the phase-noise power and/or PSD at low frequencies published by several
other groups prior to this work show that they do not support these physical predictions [6–8].
However, the predictions published in [1] have been confirmed by others [2,3].
3.1.3 Phase noise in the transmitted signal
We consider a generic fiber Fabry-Perot interferometer consisting of a medium of length L
and uniform refractive index n placed between two parallel reflectors M1 and M2 with field
reflection coefficients r1 and r2, and transmission coefficients t1 and t2, respectively (Fig. 3.1).
The medium’s single-pass transmission exp(-alossL), where aloss is the medium’s loss
coefficient, is implicitly incorporated in these four coefficients. Without loss of generality, we
assume that the medium is an optical fiber; the results are, however, applicable to any linear
intra-cavity medium. The phase noise in the reflectors is assumed to be negligible, as
supported by [9], compared to the phase noise of any reasonable length of fiber. The general
process used to compute the phase noise of this FP is essentially the same as the text-book
method to calculate the electric field at the two outputs of an FP, namely by adding the fields
36
produced after each successive round trip through the cavity [11]. The difference is that the
phase of the fields is now not just the propagation term, but this term plus the phase noise
arising from thermodynamic fluctuations inside the medium, the latter being given by the
aforementioned analytical expression of the single-pass phase noise derived by Wanser [5].
Figure 3.1. Schematic of the computation of the phase noise in the signal transmitted by a Fabry-Perot
interferometer
The phase variation per unit length at a point z along the fiber is φ(z, t) . If this variation is
caused by temperature fluctuations ΔT(z,t), then it is given by [4]:
φ(z, t) = 2πλ
dndT
+ nα!
"#
$
%&ΔT z, t − L − z
v!
"#
$
%& (3.1)
where dn/dT is the dependence on temperature T of the fiber mode effective index n, α is the
thermal expansion coefficient of the fiber, and v = c/n is the phase velocity of light in the fiber.
Integrating this incremental phase noise along the fiber length yields the phase fluctuations per
pass
Φ(t) = φ(z, t)dz0
L
∫ =2πλ
dndT
+ nα#$%
&'( ΔT (z, t − L − z
v)dz
0
L
∫
(3.2)
The PSD of the phase noise of the signal that has traveled through the fiber only once
(“single-pass PSD” for short) is the Fourier transform (denoted by F) of the autocorrelation of
37
Φ(t) [5]
Sf (ω ) =F < Φ(t)Φ(t + τ ) >( ) (3.3)
where τ is the variable of integration in the autocorrelation, and ω is the thermodynamic
fluctuations frequency. Equation 3.3 can be rewritten in terms of the autocorrelation of dφ(z, t)
<Φ(t)Φ(t +τ )>= < φ(z, t − L − zv)dzφ(z ', t − L − z '
v+τ )dz ' >
0
L
∫0
L
∫ (3.4)
As discussed in section 3.1.2, thermodynamic fluctuations along a fiber are typically assumed
to be uncorrelated in space [4], which means that the autocorrelation of φ(z, t) is a delta
function of z - z’ and only a function of the time delay τ of light through the fiber
< φ(z, t)φ(z ', t + τ ) >= Rφ(τ )δ(z − z ') (3.5)
In [5], Wanser described the noise statistics with a particular function Rφ(τ), or Rw(τ), which he
omitted to specify but which looks roughly like an exponential symmetric about τ = 0.
Entering this function in Eq. 3.5, then entering this autocorrelation in Eqs. 3.4 and 3.3, yields
the single-pass PSD presented in [5]
Sϕϕ (ω) =F Rw (τ )δ(z− z ')dzdz '0
L
∫0
L
∫#
$%
&
'(=F Rw (τ )( )L = 4πL
λ 2kBT
2
k(dndt+ nα)2 log
kmax2 + (ω
v)2
#
$%
&
'(2
+ωD#
$%
&
'(2
kmin2 + (ω
v)2
#
$%
&
'(2
+ωD#
$%
&
'(2
#
$
%%%%
&
'
((((
(3.6)
where kB is the Boltzmann constant, k is the thermal conductivity, and D the thermal
diffusivity of the fiber material, kmax = 2/W0, where W0 is the 1/e mode field radius, and
kmin = 2.405/af, where af is the fiber cladding radius. The fiber jacket is assumed to have
negligible effect on the thermal phase noise, since little to no energy of the mode propagates in
the jacket.
This expression has been widely used in the literature. It has been shown, however, to be
38
incorrect at very low frequencies (typically under ~1 kHz) because it ignores 1/f noise,
dominant at these frequencies [12]. The physical origin of the 1/f noise and consequently its
expression are still a subject of debate [12-16]. However, the physical mechanism that best
matches experimental observations of phase noise in optical fibers at low frequencies is
thermomechanical noise, which is spontaneous length fluctuations caused by mechanical
dissipation, or equivalently random extensions and contractions of the fiber due to internal
friction [12-15]. The PSD of this contribution to phase noise is described analytically by
[12,15]:
STM (ω) =nλ 22kΒTLφ03E0Af
2πω
(3.7)
where E0 is the bulk modulus of the material, Af the cross-sectional area of the fiber including
the polymeric jacket, and φ0 the loss angle that characterizes mechanical dissipation [12,15].
Unlike the main component of phase noise, the 1/f thermomechanical noise does depend on
the jacket thickness because in a first order approximation the fiber is modeled as a glass rod,
and the larger the area of the rod the smaller the mechanical energy dissipation (less internal
friction), which causes the length change and hence the phase change. The total phase noise
PSD of a fiber, Sf(ω), is then the sum of the thermodynamic fluctuations contribution (Eq. 3.6)
and this 1/f thermomechanical contribution (Eq. 3.7):
Sf (ω) = Sϕϕ (ω)+ STM (ω) (3.8)
This sum has been shown to predict correctly the phase noise PSD measured in an optical
fiber down to 20 Hz, the resolution of the instruments [12]. Above a few kHz, and often above
1 kHz [12], the thermomechanical contribution is negligible and the total single-pass PSD
measured experimentally is very well represented by Wanser’s formula (thermodynamic
fluctuations) alone [12]. An example of the spectral shape of this total single-pass PSD will be
given in Section 3.1.4. In any case, the results presented in this chapter do not depend on the
specific shape of the total single-pass PSD.
We are interested in calculating the phase noise in both the signals reflected and transmitted
by the FP. For the transmitted signal, we must calculate the phase noise in the circulating field
39
Ec2 at the output reflector M2 by summing all the fields traveling in the cavity (shown in Fig.
3.1), then transmit this signal through this reflector to obtain the transmitted signal and its
phase noise. The transmitted field being simply Et = t2Ec2 (see Fig. 3.1), the transmitted and
intra-cavity fields have exactly the same phase (except for a constant, deterministic phase shift
on transmission). Consequently, the transmitted phase noise PSD SFPt is equal to the intra-
cavity phase noise PSD SFP. The derivation of the phase noise PSD of the circulating field at
reflector M2, and hence of the transmitted field, is presented in the rest of this section and in
sections 3.1.4, and 3.1.5. The more complex case of the reflected field is dealt with in section
3.1.6.
In the Fabry-Perot interferometer of Fig. 3.1, the intra-cavity field Ec2(t) just before mirror M2
can be calculated classically by adding all the fields that have traveled back and forth an odd
number of times between the mirrors. For this derivation, we invert the coordinates in Fig. 3.1,
so that z = 0 is at mirror M2 and z = L is at mirror M1, which produces simpler equations. If E0
is the field incident on the input mirror M1, the first term in this series is
t1 exp −iΦ0( )exp −i φ(z, t − zv)dz
0
L
∫$%&
'() , where Φ0 = 2πnL / λ +ϕ t1 is the single-pass phase through
the fiber plus the phase shift ϕ t1 associated with the transmission through the first mirror, and
the integrated phase term is the total phase noise picked up by the signal as it traveled between
z = 0 and z = L. The time variable t - z/v accounts for the fact that the field picked up the phase
noise at different points and therefore at different times, since the field propagates.
Iterating this process over an infinite number of passes n through the FP yields the intra-cavity
field
Ec2
E0
= t1e−iΦ0 exp[−i φ(z, t − z
v0
L
∫ )dz]{1+ (r1r2e−i2Φ1 )n
n=1
∞
∑ ⋅ exp[−i φ(z, t − 2mL − zv
)+φ(z, t − 2mL + zv
)'()
*+,dz
0
L
∫m=1
n
∑ ]}
Φ1 = 2πnL / λ +ϕr1 +ϕr 2
2
(3.9)
where φr1 and φr2 are the phase shift associated with reflection on the first and the second
mirror, respectively.
40
Since the phase noise terms in the last exponential terms are small, the latter can be expanded
to first order in φ(z, t) to get
Ec2
E0
= A{1− i φ(z, t − zv)dz
0
L
∫ − i (r1r2e−i2Φ1 )m
m=1
∞
∑ ⋅ φ(z, t − 2mL − zv
)+φ(z, t − 2mL + zv
)'()
*+,dz
0
L
∫ } (3.10)
where A is the usual field enhancement factor provided by an FP
A = t1e−iΦ0
1− r1r2e−2iΦ1
(3.11)
The phase variations due to random temperature fluctuations are described by the argument
θ(t) on the right side of Eq. 3.10 without the inclusion of A (whose phase term does not
include any phase noise). The PSD of the phase noise of the intra-cavity field is the Fourier
transform of the autocorrelation of θ(t). The general form of the phase noise PSD is therefore
SFPt =F [< θ (t)θ (t + τ ) >]=F [< Arg{1− i φ(z, t − zv)dz
0
L
∫ − i (r1r2e−i2Φ1 )m
m=1
∞
∑ .
φ(z, t − 2mL − zv
)+φ(z, t − 2mL + zv
)&'(
)*+dz}
0
L
∫ Arg{1− i φ(z ', t − z 'v+ τ )dz '
0
L
∫ − i (r1r2e−i2Φ1 )m
m=1
∞
∑ .
φ(z ', t − 2mL − z 'v
+ τ )0
L
∫ +φ(z ', t − 2mL + z 'v
+ τ )dz '} >]
(3.12)
As a check, since this expression is valid for any values of the reflectivities, in the limit of
r1 = r2 = 0 it should give the single-pass PSD. In this limit, the two infinite series vanish and
Eq. 3.12 reduces to
SFPt =F [Arg{1− i φ(z, t − zv)dz
0
L
∫ } ⋅ Arg{1− i φ(z ', t − z 'v+ τ )dz '
0
L
∫ }] (3.13)
Since the phase noise ε is small, in Eq. 3.13 the argument of the two terms of the form (1 – iε)
is -ε, and Eq. 3.13 is simply
41
SFPt =F [ φ(z, t − zv)dz
0
L
∫ φ(z ', t − z 'v+ τ )dz '
0
L
∫ ] (3.14)
which is by definition the single-pass PSD (see Eqs. 3.3 and 3.4), as expected.
Note that since Eq. 3.12 is the phase noise inside the cavity at the output reflector, it cannot be
used to calculate the PSD of the phase noise of a signal that has traveled twice through the
fiber (as one might be tempted to do by setting r1 = 0 and r2 = 1). To do so, one must calculate
the PSD of the intra-cavity at the input reflector, which we do in Appendix.
Under the most general conditions, Eq. 3.12 does not have a simple closed form, and it must
be evaluated numerically. As described in Chapter 2, we are interested in FBGs strain sensors
interrogated at one of the two optical frequencies on either side of a resonance where the slope
of the transmission (or reflection) spectrum is steepest. Calculating the phase noise at these
interrogation frequencies requires evaluating Eq. 3.12 (or equivalently 3.10) away from
resonance. In other devices, operated on a dark fringe for example, the phase noise must be
evaluated on resonance. These two cases are investigated separately in the following sub-
sections.
3.1.4 Phase noise on resonance
On a resonance, Φ1 = pπ, where p is an integer, and Eq. 3.10 can be greatly simplified. It can
be shown, after some lengthy but straightforward calculation (see Appendix), that in this case
the intra-cavity phase noise PSD can be written as
SFPt (ω)= Sf (ω)[1+ cos 2ωLv
+sinc 2ωLv
!
"#
$
%&. 2r1r21+ (r1r2 )
2 −2r1r2 cos2ωLv
] (3.15)
As expected, on resonance the PSD of an FP is equal to the single-pass PSD Sf(ω) times a
scale factor that depends on both the frequency and the mirror reflectivities, and hence on the
finesse. This expression is valid at all frequencies, but again only on resonance.
42
To get a physical sense for this scale factor, it is instructive to look at Eq. 3.15 in the limit of
low frequency, when 2ωL/v << π (a condition roughly equivalent to a frequency f = ω/(2π)
much smaller than the reciprocal of the single-pass transit time through the FP). For a fiber FP
sensor made of a silica fiber (n ≈ 1.45), the FP length is typically less than 1 m, and this
condition is met for all frequencies much smaller than ~50 MHz, which covers the frequency
range of the overwhelming majority of sensors. In this frequency range, the three
trigonometric terms in Eq. 3.15 are close to unity, and the PSD becomes independent of
frequency. The reason for this independence is that at low frequency the temperature
essentially does not fluctuate during the very short transit time of the light through the fiber,
and all frequencies produce the same phase noise. Equation 3.15 then becomes
SFPt (ω ) = Sf (ω ) 1− 21− r1r2
"#$
%&'
2
when ω <<πv2L
(3.16)
The factor in parentheses is the ratio of the group velocity ng (at resonance) to the mode
effective index n, which is the slowing-down factor η of the FP on resonance (see Eq. 2.23).
Hence
SFPt (ω ) = Sf (ω )ngn
!
"#
$
%&
2
= Sf (ω )η2 when ω <<πv2L
(3.17)
As predicted earlier from basic physical principles (see Section 3.1.2), the phase noise of an
FP, which is proportional to the square root of the PSD, is proportional to the group index,
hence to the finesse (in an FP the group index on resonance is related to the finesse F by
ng(ω0) = 2nF/π for a large finesse, see Εq. 2.24).
To illustrate the properties of phase noise, we simulated an FP made with a silica fiber
(n = 1.45 at 1.55 µm) with a length L = 1 cm, a cladding diameter 2af = 125 µm, reflectivities
|r1|2 = |r2|2 = 0.99 with φr1 = φr2 = 0, a 1/e mode field radius W0 = 5.2 µm, k = 1.37 W/m/K, and
D = 0.86x10-6 m2/s for silica. The thermodynamic component was calculated using Eq. 3.15.
To calculate the 1/f noise component of Sf(ω) (Eq. 3.7), we used a fiber cross-sectional
diameter of 250 µm (Af = 4.91 10-4 cm2) to account for the presence of the 62.5-µm-thick
jacket, and an effective bulk modulus E0 =1.9x1010 Pa for the composite silica-jacket fiber, as
43
done in [12,15]. The fiber temperature was taken to be T = 300K. In this wavelength region
the fiber loss is negligible compared to the reflectors’ transmissions, hence the group index is
imposed by the latter. From the values of r1 and r2, the slowing-down factor is η= ng/n = 199
(see Eq. 1.19), corresponding to a group index ng = 289. Figure 3.2 plots the PSD of this fiber
FP used in transmission as a function of the fluctuation frequency f in Hz. The lower red curve
is the total single-pass PSD Sf(ω) given by Eq. 3.8. It decays as 1/f up to ~80 Hz, and above
this frequency exhibits the same dependence predicted by Wanser (dashed red curve) [5],
namely it is constant up to ~10 kHz and it decays as 1/f2 above this value. Thus, even though
Wanser’s formula ignores the thermomechanical contribution, in this particular example it
predicts the phase noise well for any frequencies above ~80 Hz, which covers most sensor
applications. The dashed blue curve is the low-frequency limit of the FP PSD given by Eq.
3.17. It is the same as the single-pass PSD (red lower curve) but multiplied (translated in the
logarithmic space) by η2 ≈ 104.
Figure 3.2. Phase-noise PSD of the signal on resonance transmitted by a 1-cm fiber Fabry-Perot
interferometer (see text for details).
The exact PSD (solid black middle curve), calculated with Eq. 3.15, exhibits three distinct
regions. Up to ~10 MHz, it overlaps perfectly with the low-frequency PSD. Above ~10 MHz,
it decreases rapidly toward the single-pass PSD. At frequencies above ~10 GHz it exhibits
sharp resonances confined in amplitude roughly between 3-dB below the low-frequency limit
and 3-dB below the single-pass PSD.
44
The physical reason for the rapid decrease in the second region, followed by the oscillations in
the third region, is successively destructive and constructive interference. The total phase
noise is the sum Zf of the noise accumulated by all the forward waves, which are correlated to
each other, plus the sum Zb of the noise accumulated by the backward propagating signals,
which are also correlated to each other. Because the noise at high frequency accumulated in
the forward and in the backward directions are uncorrelated, the PSD is the sum of the square
of these two terms
SFPt (ω p ') = Sf (ω p ') Z f2 + Zb
2( ) (3.18)
The magnitudes of Zf and Zb depend strongly on how the noisy waves interfere in the FP,
which in turn depends strongly on the noise frequency ω. On resonance and at dc noise
(ω = 0), and assuming the common condition φr1 = φr2 = 0, the optical signal resonates in the
FP at frequencies given by the usual condition 2ω0L/v = ±2mπ, where m is an integer. At noise
frequency ω, the resonance condition becomes 2(ω0+ω)L/v = 2mπ. For certain high enough
noise frequencies ωp, the noisy signal resonates on a higher order of interference m + p, where
p is an integer, i.e., 2(ω0+ωp)L/v = 2(m+p)π. The phase noise is therefore expected to exhibit
resonant frequencies ωp = pπv/L. At these frequencies the correlated noise components
(forward together, and backward together) add up in phase at each consecutive round trip,
leading to a large total noise. These resonant frequencies agree well with the sharp peaks
above ~1010 Hz in Fig. 3.2. For example, for the FP modeled in Fig. 3.2 (L = 1 cm, n = 1.45)
the first resonance (p = 1) is expected at ω1 = πv/L, or ω1/(2π) = 10.3 GHz, in agreement with
Fig. 3.2. Similarly, for certain noise frequencies ωp’ the noisy signal is anti-resonant, i.e.,
when 2(ω0+ωp’)L/v = ±2(m+p)π + π. These anti-resonant frequencies are then given by
ωp’ = (p+1/2)πv/L. At these frequencies the correlated noise components picked up in
successive round trips add up out of phase, leading to a weak noise. This sequence of anti-
resonant frequencies also agrees well with the sequence of minima in Fig. 3.2. For example,
the first anti-resonant frequency (p = 0) is expected at ω1’ = πv/(2L), or ω1’/(2π)=5.17 GHz.
This value is in broad agreement with the first minimum in Fig. 3.2 (6.85 GHz). The
discrepancy arises from the sinc dependence of SFPt and the Sf term, which were ignored in this
simplified argument.
45
Each of Zf and Zb is an infinite series of terms diminishing exponentially in amplitude. At an
anti-resonant noise frequency, these terms have opposite phase, i.e.,
Z f = Zb = (1− t + t2 − t3 + ...) = 1
1+ t (3.19)
where t is the round-trip transmission of the FP cavity (i.e., t = 1 - δ where δ is the round-trip
loss). Since the loss δ assumed in the simulations of Fig. 3.2 is small, the right hand side of
Eq. 3.19 equals 1/2. Inserting this value in Eq. 3.18 yields SFPt (ω p ') = Sf (ω p ') / 2 , which
explains why the minima in the PSD are Sf/2 (3 dB below the red curve Sf(f), see Fig. 3.2). It is
worth mentioning that although the phase noise is clearly very low at these anti-resonant
frequencies, the sensitivity of a sensor should also be minimum, for the same physical reason.
So we expect no net benefit in terms of minimum detectable signal (or signal-to-noise ratio)
when operating at one of these frequencies. A similar argument can be made to explain why
the noise maxima at the resonant frequencies occur at half the low-frequency limit.
In summary, the PSD is never larger than the low-frequency limit. As a consequence, in an FP
with a known finesse F (related to the slowing-down factor by η = 2F/π , see Eq. 2.24), an
upper bound value of the PSD can be quickly evaluated by multiplying the single-pass PSD by
(2F/π)2. At certain (anti-resonant) frequencies the PSD drops below the single-pass PSD. The
phase noise can therefore be drastically reduced compared to its low-frequency limit by
operating a device at one of these frequencies.
3.1.5 Phase noise off resonance
In the more general case where the FP is probed off resonance, the terms containing Φ0 and
Φ1, in the general expression of the PSD (Eq. 3.12), are no longer equal to unity and Eq. 3.12
cannot be simplified. However, in the limit of low frequencies (2ωL/v << π), it can be shown
that the intra-cavity field (Eq. 3.9) takes the simple form
Ec2
E0
=t1 exp −iΦ0
'( )1− r1r2 exp −i2Φ1
'( ) (3.20)
46
where Φ'0 = Φ0 +Φ(L, t) , Φ
'1 = Φ1 +Φ(L, t) and Φ(L, t) is the single-pass phase noise integrated
along the length of the fiber, defined in Eq. 3.2. This equation, valid off and on resonance but
at low frequencies only, states that the expression of the noisy intra-cavity field has exactly the
same form as the expression of the intra-cavity field in the absence of phase noise, provided
the usual phase term Φ0 is replaced by Φ0 plus the integrated phase noise, which makes
physical sense.
If we make the further assumption that the single-pass phase noise is small (i.e., Φ(L, t) << π),
which is very well satisfied for many practical situations, we can approximate the argument of
Eq. 3.20 by taking its Taylor-series expansion around Φ(L, t) at low frequency. Taking the
Fourier transform of the autocorrelation of the noise component of this argument gives the
PSD
SFPt (ω ) = Sf (ω )ng (λ)n
!
"#
$
%&
2
when ω <<πv2L
(3.21)
This result generalizes the result of Eq. 3.17 (which was derived only on resonance using a
different method) in that it applies at any arbitrary wavelength (on or off resonance), at low
frequency, and when the phase noise is small (it makes no other assumption). As long as we
are only concerned with low frequencies, which again covers almost all practical applications
of FP, the phase noise PSD is simply proportional to ng2(λ), and the phase noise to ng(λ), i.e.,
to the number of passes through the Fabry-Perot. Since this derivation makes no assumption
about the phase noise itself, it is true for each of the two contributions of the single-pass phase
noise individually, namely the thermodynamic noise (Eq. 3.6) and the 1/f noise (Eq. 3.7).
The salient and most important finding is that in a resonator such as an FP cavity, the phase
noise PSD is enhanced by a factor of (ng/n)2. The thermal phase noise, which is proportional to
Φ(L, t) , is enhanced by a factor of ng/n in the presence of slow light (in contrast to √ng claimed
in [17]).
In summary, simple closed-form expressions have been derived for the phase-noise PSD of
the transmitted signal in two different regimes, namely on resonance at any frequency, and off
47
resonance at low frequencies. Off resonance, the PSD is very simply the single-pass PSD Sf
multiplied by η2, the square of the slowing-down factor η, which is proportional to the group
index at the optical frequency. The thermal phase noise is highest on resonance, and it
decreases away from resonance in proportion to the group index. On resonance, the PSD is
also proportional to Sf. The proportionality factor is also η2 but only at low frequencies; at high
frequencies it depends on η in a more complex fashion. In this last case there is no simple
closed-form expression for the PSD, and one must resort to numerical simulations.
Figure 3.3. Phase-noise PSD spectrum at the transmission port of a silica-fiber FP with reflectivities
r1 = r2 = √0.99, normalized to the single-pass PSD spectrum. Solid curves are the numerical solutions
for different detuning δΦ0 from resonance.
Figure 3.3 plots the phase-noise PSD spectrum for the same symmetric FP as in Fig. 3.2 but
now calculated for non-zero detuning from a resonance up to high frequencies, in which case
no closed-form solution exists. These plots were calculated numerically using Eq. 3.9 and the
following procedure. The FP cavity length L was divided into p = 101 segments of equal
length. For each segment, a temporal sequence of random phase fluctuations δφ(ti) was
generated at q = 100001 equidistant times ti ranging from t = 0 to t = 100001T, where T is the
reciprocal of the sampling frequency. The sampling frequency was chosen to be a multiple of
v/(2L). The random fluctuations at a given point had the same PSD spectrum as shown in Fig.
3.2 (single-pass curve). These p distributions were independent, since thermal fluctuations are
uncorrelated in space. In order to generate these random sequences of noisy phase with a
specific PSD spectrum, we followed the process described in [18]. The ratio Ec2/E0 was
48
calculated for 1001 passes. This is equal to ~5 times the number of passes through the FP,
which is more than enough to reach convergence (increasing the number of passes changed
the outcome by less than 1%). We then calculated the autocorrelation of the phase and its
Fourier transform to obtain the PSD spectrum. This process was repeated many times (~1000),
each time using a different set of p temporal distributions. These PSD spectra were finally
averaged to obtain the final PSD spectrum.
This process was carried out for two different fiber PSDs, namely first assuming that the
single-pass PSD is given by Eq. 3.6, i.e., that it does not include the thermomechanical
contribution (an assumption certainly valid in most cases at frequencies above ~1 kHz, as
illustrated in the example of Figure 3.2 where an ~80-Hz cutoff was predicted), then assuming
that it does include the thermomechanical noise (and is given by Eq. 3.8). As suggested by Eq.
3.15, and as verified with numerical simulations, in both cases when normalized to the single-
pass PSD the transmitted PSD depends on frequency as ωL/v, which explains the
normalizations of each of the two axes in Fig. 3.3. All curves are the numerical solutions
plotted for increasing detuning from resonance δω0, characterized by the round-trip phase
detuning δΦ0 = 2δω0L/v. The top black solid curve is plotted on resonance (δΦ0 = 0); it is the
same as the oscillatory solid curve in Fig. 3.2, except normalized to Sφφ(ω) and plotted versus
ωL/v instead of ω. At low frequency (ωL/v ≤ ~10-3 rad) the numerical simulations confirm the
prediction of Eq. 3.17 that the PSD is proportional to the ratio (ng(λ)/n)2, i.e, it decreases
rapidly with increasing detuning. Above this frequency, in the off-resonance case an
additional resonance occurs at low frequency (e.g., at ωN ≈ 1.65x10-2π rad for δΦ0 = 0.05 rad),
and the high-frequency resonances each split into two.
The origin of this splitting can be better understood by re-plotting Fig. 3.3 with a linear
coordinate (Fig. 3.4). This figure clearly shows that off resonance all the noise resonances
split, by the same amount, and that this amount increases with increasing detuning, up to the
maximum possible detuning of δΦ0 = π. The extra resonance that appears at low frequency
simply arises from the splitting of the resonance at dc. The reason for this splitting is again
interferometric. As stated earlier, at dc noise the optical signal resonates at frequencies given
by 2ω0L/v = ±2mπ. When the signal is detuned from resonance by δω0, it is the sum of the
signal and the noise that resonates, at frequencies given by 2(ω + ω0+ δω0)L/v = ±2mπ.
Combining these last two equations shows that the noise is expected to resonate at frequencies
49
ω = ±δω0, i.e., the splitting in the noise peaks is constant and equal to 2δω0. This prediction is
consistent with the splitting in Fig. 3.4. For example, for δΦ0 = 0.05 rad the splitting should be
2δω0 = δΦ0v/L =1.03x109 rad/s, which is precisely the peak splitting predicted numerically in
Fig. 3.4. Note that the split peaks are decreased by a factor of 4 relative to the original peaks
(energy conservation dictates a factor of two reduction, which is squared in a PSD). The split
peaks are not quite symmetric only because of numerical errors.
Figure 3.4. PSD spectrum on resonance (dashed curve) and off resonance for δΦ0 = 0.05 (solid curve)
illustrating the splitting that occurs in the phase-noise resonances when the optical signal is detuned
from a Fabry-Perot resonance.
The normalized FP PSD spectra plotted in Figs. 3.3-3.4 depend on η, but they are independent
of the fiber length as long as the fiber loss has a negligible contribution to the total round-trip
cavity loss (so that η remains the same). In this case, when the length is increased from the
value used to generate the figures, first, the single-pass PSD increases in proportion to L, and
so does the FP PSD, but their ratio remains the same. Second, the resonant frequencies also
increase proportionally to L, so the resonances occur at different frequencies but the
normalized frequencies also remain unchanged. If the length becomes so long that the fiber
loss increases the total round-trip cavity loss noticeably, then one other contribution comes in:
the group index decreases, and as a result the normalized FP PSD spectrum also decreases in
proportion to the ratio η2 (see Fig. 3.2). The normalized FP PSD spectra in Figs. 3.3-3.4 then
need to be recalculated for the new η.
50
3.1.6 Phase noise in the reflected signal
The case of the reflected field (Er) is more complex, because this field is the sum of two terms,
the fraction t1 of the intra-cavity field Ec1 at the input mirror that is transmitted by the input
mirror, and the fraction r1 of the incident field reflected by the input mirror. Compared to the
transmission case, calculating Er therefore requires passing the signal one more time through
the fiber, from M2 to M1. When the FP has a high finesse, this additional pass is expected to
have negligible effect on Ec1, and therefore on the t1Ec1 contribution. When the finesse is low,
it is expected to have a significant impact. On the other hand, the reflected contribution is
noiseless (since it has not traveled through the cavity), so it is expected to reduce the relative
phase noise (relative to the total reflected power) compared to the phase noise in the
transmitted signal.
The derivation of Ec1 mirrors the derivation of the intra-cavity field Ec2 (see Appendix). The
end result is that the phase-noise PSD of the signal reflected by an FP at low frequency and at
any wavelength, on or off resonance, is again given by Eq. 3.21, i.e., it is equal to the single-
pass PSD times the square of the slowing-down factor at the wavelength of interest. For an
arbitrary frequency (including low frequency) and for an asymmetric or lossy FP, on
resonance the FP field reflection RFP is non-zero, and it can be shown, similarly to the
derivation in Appendix for the phase noise in transmission, that the PSD on resonance and at
any frequency is
SFPr (ω ) = 2Sf (ω )r2t1
2
1− r1r2( )RFP
"
#$
%
&'
2 1+ sinc 2ωLv
1+ r1r2( )2− 2r1r2 cos
2ωLv
(3.22)
The right-hand side of Eq. 3.22 can be recast in terms of the group index of the reflected
signal ng ' (see Eq. 2.24):
51
SFPr (ω ) = Sf (ω )ng 'n
!
"#
$
%&2 1− r1r2( )
21+ sinc 2ωL
v!
"#
$
%&
2 1+ r1r2( )2− 2r1r2 cos
2ωLv
!
"#
$
%&
(3.23)
Note that for a lossless and symmetric FP these last two expressions do not apply on
resonance because the reflected power is zero and therefore the phase noise cannot be defined.
Off resonance the expression at low frequencies is the same as Eq. 3.21 but using the group
index in reflection ( ng ' ) instead of in transmission (ng).
Setting r1 = 0, t1 = 1, and r2 = 1 in Eq. 3.23 gives the PSD of a signal that has traveled twice
through a fiber. Equation 3.23 shows that this PSD depends on frequency as sinc(2ωL/v). This
result contradicts [4], which predicts a cos(2ωL/v) dependence (in our opinion arising from an
error in the Fourier transform calculation in [4]). It is consistent, however, with the sinc
dependence of the PSD of a Sagnac interferometer [10] (also a two-pass device).
Figure 3.5. PSD spectra in reflection (black curves) for the same FP as in Fig. 3.3, calculated for two
values of the detuning from resonance. The PSD spectra in transmission for the same two detunings
(solid red curves, reproduced from Fig. 3.3) are also shown for comparison.
The PSD spectrum in reflection, calculated numerically using the same technique as in the
previous section, is shown in Fig. 3.5 for the same symmetric and lossless FP as in the
previous figures. The phase noise is only plotted off resonance, where there is a non-zero
52
reflected signal from the FP. Because of the first reflection off the input mirror M1 is noiseless,
at low frequencies the PSD in reflection is exactly the same as in transmission, since it is
proportional to the group index (and the latter is the same in reflection and transmission for
the present case of a lossless symmetric FP). At high frequencies, however, it is lower than the
phase noise in transmission. We hypothesize that this reduction is a result of the partial
correlation of the various contributions to the phase noise at high frequencies. Halfway
between resonances (i.e., for δΦ0 = π/2, a case not shown in Fig. 3.5), the phase noise is zero,
since the intracavity field is zero and there is therefore no reflection from the cavity.
3.1.7 Applicability to FBGs and FBG-based FPs
The PSD expressions derived so far have assumed that the two mirrors are so thin that their
contributions to phase noise are negligible, as is the case for dielectric coatings for example.
When one or both of the reflectors are FBGs, there are two differences in the derivation of the
FP phase noise. First, the phases on reflection φr1 and φr2 may be different from what they are
for dielectric (or metallic) coatings. This may shift the resonance frequencies at high
frequencies, but it will not affect the phase noise at a particular optical frequency with respect
to a resonance. The second difference is that each mirror now has a finite length and therefore
adds its own phase noise to the signal. This effect is discussed in the next paragraph.
Calculating the phase noise of an FP made of two FBGs separated by a length of fiber
therefore requires calculating first the phase noise picked up by the signal reflected by an FBG.
This calculation can be done by summing the incremental fields reflected by each of the
periods in the FBG, in much the same way as the transmitted fields were summed in Section
3.1.3. The same remarks apply to the phase noise picked up by a signal with a wavelength in
the vicinity of the slow-light resonances of an FBG [19].
In this last device, the light is concentrated in the middle of the FBG, as discussed later on in
this thesis. At each pass through the FBG, the light therefore travels an effective length that is
shorter than the physical length LFBG of the FBG. As a consequence, an upper bound value of
the phase noise of this device (in reflection or in transmission) can be obtained by equating the
FBG to an FP made with thin reflectors spaced by a length LFBG. For the same reasons, an
upper bound value of the phase noise of the signal reflected or transmitted by an FP made of
two FBGs spaced by a length of fiber can be obtained by equating the device to an FP made
53
with thin reflectors spaced by a length equal to the total length of the device, including the
length of the two FBGs.
3.1.8 Phase noise in experimental fiber FP sensors
To compare the predictions of this study to experimental noise data, we plot in Fig. 3.6 the
predicted dependence on group index of the phase noise in transmission for three highly
sensitive fiber strain sensors reported in three references [6,20,21], and compare these
predictions to the measured noise reported in the same references. In order to calculate the
predicted phase noise we used Wanser’s Formula (Eq. 3.6), with and without the
thermomechanical noise (Eq. 3.7). The difference between the two results was negligible,
showing in particular that the 1/f noise was negligible in these sensors. It is worth mentioning
that when expressed in units of strain, the phase noise scales as the reciprocal of the square
root of the FBG length, since the sensitivity is proportional to thelength, and the phase noise is
proportional to the square root of the length.
The first sensor (the work of our research group) consists of a 2-cm FBG with a large index
modulation, which created sharp slow-light peaks on the edges of its bandgap as explained in
section 2.3.2b [20]. It was probed at the wavelength of steepest slope of one of these
resonances (to maximize its strain sensitivity), and tested for strain sensitivity at 23 kHz. The
measured group index at the peak of this resonance was 58.2. The predicted dependence of the
phase noise on group index for this sensor is shown in Fig. 3.6 as the solid black curve.
Because this curve was generated assuming that the effective length of the FBG was equal to
its physical length, as discussed in the previous section, this curve (slightly) overestimates the
phase noise. As expected from the previous sections, the phase noise increases linearly with
increasing group index. The measured total noise of this sensor (solid circle in Fig. 3.6) is 13.5
times larger from the predicted phase noise (solid curve). This is consistent with the noise
measurements reported in [20], which concluded that the noise was dominated by laser
frequency noise.
The second sensor is an FP-based sensor made with two FBGs spaced by 13 cm, with a
finesse of 110 [6], or a group index of 101.5. This sensor was probed at a wavelength tuned to
the steepest slope of a resonance. The open circle in Fig. 3.6 is the measured noise of this
sensor at 1.5 kHz, calculated by converting the measured minimum detectable strain reported
54
in [6] into a phase error. The predicted phase noise for this sensor (dashed red curve) is much
higher than for the previous sensor because its length is longer and the frequency lower. The
measured total noise of this sensor is 9 times higher than the predicted phase noise. Unlike
claimed in [6], the noise of this sensor was not limited by phase noise, as demonstrated in [7].
Figure 3.6. Theoretical predictions of the phase noise dependence on group index for three highly
sensitive strain sensors utilizing FBGs, and experimental noise measured for each of them.
The third highly sensitive experimental sensor is the fiber Fabry-Perot sensor reported in [21].
This sensor is similar to the one reported in [6], with a length of 2 cm and a finesse of 48, and
it was operated at 216 Hz. Once again the experimental noise is larger than the predicted phase
noise, here by a factor of 4.5.
This study shows that in the three most sensitive passive fiber strain sensors reported prior to
this work, the measured noise is a factor of 4.5–13.5 higher than the predicted phase noise.
Thus this gives significant room for further decreasing the minimum detectable signal of these
three sensors before the phase noise limit is reached.
3.2 Conclusions
In this chapter a rigorous derivation of the PSD spectrum of the phase noise in the signals
55
transmitted and reflected by a Fabry-Perot interferometer was reported. For practical reasons,
we endeavored to express this PSD in terms of the group index ng of the light and of the
detuning of the optical signal from a Fabry-Perot resonance. We found, as supported by
physical arguments, that in transmission the PSD at low frequencies (much smaller than the
reciprocal of the single-pass transit time through the FP, or typically under 10 MHz for a
typical cm-long FBG) is equal to the PSD of the phase noise acquired by a signal that has
traveled once through the FP (single-pass PSD Sf(ω)) times the square of the FP’s slowing
down factor, (ng/n)2, where n is the phase index of the light inside the FP. Equivalently, the
phase noise (expressed in radian), which is the square root of the PSD, is equal to the single-
pass phase noise times ng/n. The higher the finesse F (or equivalently the group index) of the
FP, the larger the number of passes light makes through the FP, and the larger the phase noise
it acquires, in proportion to this number of passes, i.e., in proportion to ng or F. At low
frequency, when the signal is on an optical resonance, ng is maximum and so is the PSD.
When the signal is detuned from a resonance, the PSD is described by the same expression,
i.e., it is still proportional to (ng/n)2, except that ng is lower because of the detuning.
Application of this model to three highly sensitive FP strain sensors reported in the literature
shows that in these sensors, the total noise was 4.5–13.5 times higher than their respective
predicted phase noise. The conclusion is that the phase-noise limit had not been reached by
anyone in a passive FBG-based sensor, unlike claimed in the literature [6], before the work
reported in this thesis.
At high frequency (above ~10 MHz for a typical 1-cm FBG), we showed that the PSD
spectrum can also be expressed in a simple closed-form expression in transmission when the
signal is on resonance, and that off resonance the PSD must be calculated numerically.
Numerical simulations showed that in this frequency range, the PSD oscillates between 3-dB
below the low-frequency PSD and 3-dB below the single-pass PSD. This behavior is
attributed to constructive and destructive interference between the signals co- and counter-
propagating in the FP. Equivalent expressions are derived for the phase noise in the signal
reflected by a Fabry-Perot, which is found to have almost the same magnitude at low
frequencies, but to be lower at high frequencies, because of the presence of the noise-free
reflection off the input FP mirror. Other than this detailed analysis of the phase noise spectrum
dependence on detuning, the main value of this study is that it provides simple expressions to
calculate the PSD spectrum of the phase noise in any linear Fabry-Perot interferometer
56
(including slow-light fiber Bragg gratings) at low frequencies (which turns out to be the range
of frequency of interest to the overwhelming majority of FP applications), in reflection or in
transmission, from the knowledge of the group index or finesse alone.
This study was the first prong of a four-pronged effort to observe phase noise in a short
passive FBG and measure the smallest MDS in an FBG. Specifically, it enabled us to properly
design the FBGs reported in Chapter 5 to (1) measure thermal phase noise in an FBG and
verify this theory, and (2) achieve the lowest possible MDS using our slow-light FBG sensors.
The second prong was to increase the group index. Since we need to use a short FBG to
measure phase noise, we needed to maintain the same or achieve a higher sensitivity in a very
short fiber. Thus there was an immediate need to increase the group index in an FBG to
increase the sensitivity and/or to “amplify” the phase noise relative to other sources of noise.
The way we achieved this is discussed in the next Chapter 4. The third prong was to improve
our experimental sensor to (1) reduce the laser frequency noise, (2) reduce the environmental
noise, and (3) improve the stability of the experimental setup. The fourth and final prong was
to combine these results and design two FBGs that are compatible with our improved
experimental setup, with high enough sensitivities, and with a high and low phase noise,
respectively, and measure the thermal phase noise in an FBG, as well as the smallest strain
measured ever in an FBG sensor. These last two efforts are discussed in Chapter 5.
References
[1] G. Skolianos, H. Wen, and M. J. F. Digonnet, “Thermal phase noise in Fabry-‐Pérot
resonators and fiber Bragg gratings,” Physical Review A, 89, 033818 (2014).
[2] S. Foster, and G. A. Cranch “Comment on “Thermal phase noise in Fabry-‐Pérot
resonators and fiber Bragg gratings”,” Physical Review A, 92, 017801 (2015).
[3] G. Skolianos, and M. J. F. Digonnet, “Reply to “Comment on ‘Thermal phase noise in
Fabry-‐Pérot resonators and fiber Bragg gratings,” Physical Review A, 92, 017802
(2015).
[4] K. H. Wanser, “Theory of thermal phase noise in Michelson and Sagnac fiber
interferometers,” Proc. SPIE 2360, Tenth International Conference on Optical Fibre
Sensors, 584 (1994).
[5] K. H. Wanser, “Fundamental phase noise limit in optical fibres due to temperature
57
fluctuations,” Electronic Letters, 28, 53 (1992).
[6] G. Gagliardi, M. Salza, S. Avino, P. Ferraro, and P. De Natale, “Probing the ultimate
limit of fiber-‐optic strain sensing,” Science, 330, 1081 (2010).
[7] G. A. Cranch, and S. Foster, “Comment on “Probing the ultimate limit of fiber-‐optic
strain sensing,” Science, 335, 286 (2012).
[8] V. Annovazzi-‐Lodi, S. Donati, and S. Merlo, “Thermodynamic phase noise in fiber
interferometers,” Optical and Quantum Electronics, 28, 43 (1996).
[9] N. Nakagawa, E. K. Gustafson, P. T. Beyersdorf, and M. M. Fejer, “Thermal noise in
half-‐infinite mirrors with nonuniform loss: A slab of excess loss in a half-‐infinite
mirror,” Physical Review D, 65, 102001 (2002).
[10] K. Kråkenes, and K. Blotekjaer, “Comparison of fiber-‐optic Sagnac and Mach-‐
Zehnder interferometers with respect to thermal processes in the fiber,” Journal of
Lightwave Technology, 13, 682 (1995).
[11] A. Siegman, “Lasers”, Chapter 11.3, University Science Books Sausalito, CA (1986).
[12] R. E. Bartolo, A. B. Tveten, and A. Dandridge, “Thermal phase noise measurements
in optical fiber interferometers,” IEEE Journal of Quantum Electronics, 48, 720
(2012).
[13] S. Foster, “Low-‐frequency thermal noise in optical fiber cavities,” Physical Review A
86, 043801 (2012).
[14] L. Duan, “General treatment of the thermal noises in optical fibers,” Physical
Review A, 86, 023817 (2012).
[15] L. Duan, “Intrinsic thermal noise of optical fibres due to mechanical dissipation,”
Electronic Letters, 46, 1515 (2010).
[16] S. Foster, A. Tikhomirov, and M. Milnes, “Fundamental thermal noise in distributed
feedback fiber lasers,” IEEE Journal of Quantum Electronics, 43, 378 (2007).
[17] G. Gagliardi, M. Salza, S. Avino, P. Ferraro, and P. De Natale, “Response to Comment
on “Probing the ultimate limit of fiber-‐optic strain sensing,” Science, 335, 286
(2012).
[18] W. H. Tranter, K. S. Shanmugan, T. S. Rappaport, and K. L. Kosbar, “Principles of
communication systems simulation with wireless applications,” Prentice Hall,
Upper Saddle River, NJ, , Chap. 7, 278-‐282 (2003).
[19] H. Wen, G. Skolianos, S. Fan, and M. J. F. Digonnet, “Slow light in fiber Bragg
58
gratings,” Proc. SPIE 7949, Advances in Slow and Fast Light IV, 79490E (2011).
[20] H. Wen, G. Skolianos, S. Fan, M. Bernier, R. Vallée, and M. J. F. Digonnet, “Slow-‐light
fiber-‐Bragg-‐grating strain sensor with a 280-‐femtostrain/√ Hz resolution,” Journal
of Lightwave Technology, 31, 1804 (2013).
[21] J. H. Chow, D. E. McClelland, and M. B. Gray, “Demonstration of a passive
subpicostrain fiber strain sensor,” Optics Letters, 30,1923 (2005).
59
Chapter 4: Improving the performance of slow-light FBGs
In this chapter we discuss the practical improvements that were made in the slow-light FBGs
used as sensors in order to achieve even lower MDSs than our group’s previous world record,
and observe and measure quantitatively the thermal phase noise in an FBG sensor for the first
time. To this end, we needed FBGs with stronger resonances and achieve higher sensitivities
and lower noise, mainly lower laser frequency noise and detector noise. As we discussed in
Chapter 2, to obtain strong resonances the gratings needed to have (1) a weak internal loss (as
in all resonators), (2) as large a Δnαc as possible, and (3) a suitably apodized profile. In this
chapter we discuss how proper apodization leads to an increase in group delay. Next, we show
that to satisfy these three conditions it is preferable to use femtosecond gratings, which
produce high index modulations and a much lower loss (for a given index modulation) than
traditional UV-written gratings [1]. It is also beneficial to write the gratings in a deuterium-
loaded fiber [2], a technique that yields much higher photosensitivity and index modulations
[3]. Furthermore, the grating should be annealed to reduce the loss introduced by the writing
process. Finally, we describe in detail the process used to fabricate the gratings tested in this
thesis and how we modeled the index-modulation and loss profiles of the FBGs written with
this technique.
4.1 Apodized FBGs
As discussed in Chapter 2, high group delays can be achieved in a uniform grating. The main
two disadvantages of a uniform grating are first that the slow-light peaks are outside the
bandgap, which leads to a lower reflectivity, and second that the whole FBG is reflective at
the same wavelengths, resulting in a higher cavity loss.
A solution to both of these problems is to shift the Bragg wavelength in the middle of the
FBG, namely to use an FBG with an apodized index profile (Fig. 4.1a). In such a grating the
average index modulation varies along the grating. The Bragg condition is then z dependent:
λB (z) = 2(n0 +Δndc (z))Λ (4.1)
60
where n0 is the effective index of the fiber mode before the grating was written, Λ is again the
grating period, z is the position along the grating, and Δndc(z) is the average index modulation
at z. Since Δndc depends on position, so does the Bragg wavelength, according to the solid
green curve in Fig. 4.1b, which is linearly related to the Δndc(z) curve in Fig. 4.1a. For a given
Bragg wavelength, for example λB1, there are two positions, z1 and z’1, around which the FBG
reflects at λB1. The FBG therefore supports an infinite number of FPs, with mirror spacings
varying from zero to some maximum value equal to the physical length of the FBG, and
centered on a Bragg wavelength that ranges from the value at the top of the green curve to the
value at the bottom of the curve in Fig. 4.1b.
Figure 4.1. Creating strong slow-light resonances in an apodized FBG. a) ac and dc index-modulation
profiles of an apodized fiber Bragg grating. b) Dependence of the Bragg wavelength on position along
the grating. c) Effective mirrors in the apodized FBG forming equivalent FPs. d) The transmission
slow-light resonances formed as a result of these multiple equivalent FPs.
In this continuum of Bragg wavelength pairs, only for some pairs (shown as i = 1 to 4 in Fig.
4.1b) is the optical path length at λBi between zi and z’i and back to zi a multiple of 2π. For
these pairs, the transmitted fields add in phase to produce a resonance. The FBG then behaves
like the superposition of these few resonant FPs (four are represented in the example of Fig.
61
4.1c). The mirror pair closest to the center of the FBG (z1 and z1’ in Fig. 4.1c) has the largest
Δnac of all mirrors, and therefore the highest reflectivity of all phase-matched pairs, and hence
this FP has the resonance with the narrowest linewidth (see Fig. 4.1d). This mirror pair also
has the shortest spacing; therefore, it supports the fundamental mode (the electric field
distribution along the FBG exhibits a single lobe). The other phase-matched pairs have a
reflectivity that decreases with increasing distance from the center of the FBG, since Δnac is
decreasing, and they support a broader resonance (see Fig. 4.1d). Since they have a larger
mirror spacing, they support a higher order mode (i.e., their electric field distributions have
multiple lobes). A strong apodized FBG therefore exhibits a series of resonances narrower and
generally stronger toward the center of the grating. One advantage compared to uniform FBGs
is that between zi and z’i the Bragg wavelength is not equal to λBi (see green curve in Fig. 4.1b)
and the grating is essentially transparent at λBi, which leads to much stronger slow-light
resonances than if the index-modulation profile were uniform. For this to be true, the rate at
which the Bragg wavelength changes with position should be larger than the rate at which the
size of the bandgap increases with position. Thus, the rate of change of Δndc—proportional to
Bragg wavelength—with position must be larger than the rate of change of Δnac—proportional
to the bandwidth of the bandgap—with position. If this condition is not satisfied, the fiber
between zi and z’i will still act as a reflector, and therefore an effective FP cannot be formed
and the FBG does not slow down light as effectively as it can.
For this kind of FBGs, and in general for an FBG with arbitrary index and loss profiles, there
is no analytical solution to calculate the field reflection and transmission coefficients, unlike in
the uniform case discussed in Chapter 2. In order to model these FBGs and to predict their
transmission and group index/delay spectra we developed a Matlab program. This application
numerically calculates these spectra based on the piecewise method described in [4]. As an
input it uses the Δnac, the Δndc and the loss, all of which are functions of the position in the
fiber. We derived these functions for the particular FBG that was modeled based on their
fabrication technique, described below in section 4.2. This derivation is explained thoroughly
in section 4.3. After the three desired profiles were obtained, we divided the FBG along its
length into a small number N of uniform FBGs of equal length δL, in our case approximately
100. Each uniform FBG has the Δnac, Δndc and loss of the original FBG in its position. The
number of the uniform FBGs must be chosen so as to fulfill two requirements: (1) each
uniform FBG must include several periods of the grating (L>>Λ) in order for the analytical
62
solution for the uniform FBG (Eq. 2.3) to be valid, and (2) the Δnac, Δndc and loss of the
original FBG must not change significantly between the beginning and the end of any of the N
uniform FBGs, otherwise this piecewise approximation breaks down. We can then calculate
the transfer matrix of each uniform FBG (Eq. 2.3), which connects the forward (denoted with
“+” superscript) and backward (denoted with “-” superscript) electric fields at its output and
input at one particular wavelength λ. The output of the ith uniform FBG is the input of the
(i+1)th uniform FBG. Using the matrix in Eq. 2.3 we can then calculate the input electric field
of the ith uniform FBG from the input of the (i+1)th uniform FBG:
Ei+
Ei−
"
#$$
%
&''=
cosh κ i2 −σ i
2 ⋅δL( )-i σ i2
κ i2 −σ i
2sinh κ i
2 −σ i2 ⋅δL( ) -i κ i
2
κ i2 −σ i
2sinh κ i
2 −σ i2δL( )
i κ i2
κ i2 −σ i
2sinh κ i
2 −σ i2δL( ) cosh κ i
2 −σ i2 ⋅δL( )+ i σ i
2
κ i2 −σ i
2sinh κ i
2 −σ i2 ⋅δL( )
"
#
$$$$$$
%
&
''''''
Ei+1+
Ei+1−
"
#$$
%
&''
= AiEi+1+
Ei+1−
"
#$$
%
&''
(4.2)
where κι and σι are the ac and dc coupling coefficients, respectively, for the ith uniform FBG
given by:
σ i = 2π n0 +Δndci( ) 1
λ−1λB
#
$%
&
'(− i
γ i
2 (4.3a)
κ i = πΔnac
i
λ (4.3b)
where γ is the power loss coefficient that characterizes propagation in the grating.
Afterwards, by multiplying the transfer matrices for all the N segments, a single matrix C is
obtained that relates the input and output electric fields of the whole FBG:
Ein+
Ein−
"
#$$
%
&''=E1+
E1−
"
#$$
%
&''= A1A2!AN
Eout+
Eout−
"
#$$
%
&''=C
Eout+
Eout−
"
#$$
%
&'' (4.4)
From Eq. 4.4 assuming no backward propagating ( Eout− =0) field is injected into the FBG (as is
the case in our sensing and other applications) the field transmission and reflection coefficient
63
can be easily calculated by evaluating C numerically. Then it is trivial to calculate the group
delay and index spectra, as described in Chapter 2 (see Eqs. 2.8).
As an example of this process, and to illustrate the benefits of apodization by the same token,
we show in Figs. 4.2 and 4.3 the simulated group-index and transmission spectra of two
FBGs, one with a uniform index-modulation profile and the other with an apodized profile.
The gratings have the same length L = 2 cm, a half peak-to-peak ac index modulation
Δnac ≈ 10-3, a period Λ = 532.98 nm, and a uniform loss with a power loss coefficient
γ = 0.4 m-1, close to the loss measured in FBGs of this strength [5].
Figure 4.2a shows the index profile of the uniform FBG, and Fig. 4.2b its simulated spectra.
The slow-light resonances occur just outside the bandgap, as described in Chapter 2. The
strongest resonance is the one closest to the band edge (see insets in Fig. 4.2b). Its maximum
group index is 146.9. In a uniform FBG, Δnac = Δndc (see Fig. 4.2a), and as discussed in
Chapter 2, the effective FP has a relatively low finesse, which explains this relatively low
group index.
Figure 4.2. a) Index profile of a uniform FBG (period Λ not to scale) and b) simulated transmission and
group index spectra of this FBG.
Figure 4.3a shows the profile of the second FBG, a Gaussian-apodized grating with
Δnac = Δndc = 10-3 and a Gaussian’s full width at half maximum (FWHM) equal to twice the
64
FBG length. The loss profile was assumed to be uniform, with the same power loss coefficient
of 0.4 m-1. Both the transmission and the group-index spectra (Fig. 4.3b) exhibit sharp
resonances on the short-wavelength side of the bandgap. The resonances in the group index
and in the transmission spectra occur at the same wavelengths, as expected (this is true for any
index-modulation profile). The strongest slow-light resonance is the one furthest inside the
bandgap; its peak group index is as large as 361.5, i.e., ~2.5 times slower than in the uniform
grating of Fig. 4.2. When the power loss coefficient is reduced from 0.4 m-1 to 0.12 m-1, the
peak group index increases to 1204. These simulations illustrate the importance of apodization
to obtain slow-light resonances with large group indices, and the critical importance of
reducing the loss (a requirement that applies to all gratings, apodized or not).
Figure 4.3. a) Index profile of a Gaussian-apodized FBG (period Λ not to scale), and b) simulated
transmission and group index spectra of this FBG.
4.2 Realization of improved FBGs
It has been discussed and explained that producing higher group indices and larger
transmission leading to high sensitivity in an FBG generally requires (1) high index
modulation, (2) strong apodization (e.g., Gaussian rather than uniform), and (3) reducing the
internal loss. In this section we discuss how we met these goals, namely by using femtosecond
FBGs, fabricating them in deuterium-loaded fibers, and annealing the gratings.
65
4.2.1 Fabrication of slow-light FBGs with femtosecond lasers in deuterium-loaded fibers
The FBGs used in this work were fabricated at Université Laval using a femtosecond laser and
a conventional phase-mask technique. He Wen, a former graduate student in our research
group, started the study of slow light in FBGs. After characterizing several FBGs written by
different methods and studying the literature she concluded that femtosecond-written FBGs
have a much lower loss for the same index of modulation than gratings written with other
techniques, in particular using UV lasers [6]. This property is very useful to achieve strong
resonances. In Fig. 4.4 the summary of this study is plotted in the form of the loss versus Δnac.
The data combine values found in the literature as well as He Wen’s measured values (filled
red shapes) and a new representative data point for a femtosecond FBG written in a
deuterium-loaded fiber that we characterized in this thesis. It can be seen first that FBGs
written with a femtosecond laser clearly produce lower loss for the same Δnac, by an
approximate factor of 2–5. However, a larger sample is needed to establish a more precise
quantitative comparison, and further studies to understand the origin of this lower loss.
Second, there is a trend, especially with the conventional writing methods, that as Δnac
increases the loss increases too. This is expected because the same defects that induce the
increase in the fiber refractive index also introduce loss. These defects are introduced when
light with a high intensity is concentrated on a specific point of the fiber.
Figure 4.4. Power loss coefficients versus ac index modulation for different writing techniques.
66
In general it is believed that during fabrication of an FBG using light two main
photosensitivity mechanisms are responsible for the change in refractive index of the fiber [7],
namely color centers and glass densification. In the first mechanism light breaks specific
bonds (also known as precursors) in some of the glass-matrix molecules, which creates color
centers, namely species that exhibit one or more strong absorption peaks, usually in the ultra-
violet region [8]. This induced absorption forces a change in the refractive index of the
material around 1.55 µm through the Kramers-Kronig relationship [9]. This mechanism, which
is the only one present in conventional non-femtosecond FBGs, is directly connected with the
internal loss of the FBG, thus the higher the index change the higher the loss, which is in
agreement with the trend in Fig. 4.4. Furthermore, this mechanism needs a dopant—it is not
present in pure silica fibers—to create bonds that can be broken. In conventional
telecommunication fibers the core is usually doped with germanium to increase its refractive
index relative to the cladding. This dopant provides the necessary bonds (for example, Ge-Si
bonds) [9] that change the refractive index of the fiber when they are broken. Because of this
need for dopants, this mechanism can be greatly enhanced by loading the fiber with hydrogen
or deuterium [10–12]. The downside of loading the fiber with hydrogen is that hydrogen
bonds with oxygen in the glass matrix to form hydroxyl radicals (OH-), which absorb strongly
around 1.4 µm and introduce undesirable loss at 1.55 µm where the FBGs operate [13]. In
contrast, when using deuterium instead of hydrogen, OD- radicals are formed, but they absorb
at a longer wavelength (~1.9 µm [12]) and have little effect on the FBG loss around 1.5 µm.
Figure 4.4 illustrates, as has been known for a long time, that doping the fiber with H2 or D2
produces much larger index modulations. It also shows the newer result that for the same
index modulation, deuterium-loaded FBGs have a significantly lower internal loss when
fabricated with a femtosecond laser than with a UV laser, by a factor of 4 to 5, which as we
shall see has a major benefit for slow-light generation.
The second mechanism responsible for the formation of a grating, glass densification, is
present only when the FBGs are written with high-intensity light, namely with a femtosecond
laser. In this mechanism the high incident intensity causes compaction to the glass, and hence
changes its density. This density change causes the refractive index to change. In femtosecond
FBGs, the index modulation is believed to arise in general from a combination of these two
mechanisms, although further studies are needed to determine their exact contributions and the
dependence of these contributions on other fabrication parameters such as the pulse width,
67
pulse energy, exposure time, and glass composition. At this time the physical reasons why
compaction introduces a lower loss than color-center formation have not been fully elucidated.
Figure 4.5. Setup and exposure conditions used to write the deuterium-loaded fiber Bragg gratings
using a femtosecond laser. (Courtesy Martin Bernier).
The FBGs reported in this thesis were realized thanks to Professor Martin Bernier at
Université Laval in Québec using the method as follows. They were written through the jacket
in a single-mode silica fiber from OFS (SMF-28 compatible germanosilicate fiber BF04446)
with a polyimide jacket. The fiber has a core/cladding/jacket diameter of 8.4/125/155 microns,
and a numerical aperture of 0.11. Polyimide was used because during subsequent annealing of
the grating, it can withstand higher temperatures than conventional jacket materials, up to
~400°C for short periods of time, and the jacket did not have to be removed for post-
fabrication annealing. By preserving the jacket during the entire grating fabrication process,
the mechanical strength of the FBG was not compromised [14].
To enhance the germanium-doped core photosensitivity, as discussed above, the fiber was first
loaded with deuterium in a pressurized chamber at 2000 psi and room temperature for 14 days.
The FBGs were written using the method described in [14] and illustrated in Fig. 4.5. It uses a
femtosecond near-IR laser for efficiently writing through the jacket, combined with the
scanning phase-mask technique [15] to extend and control the FBG length. The laser was an
806-nm Ti:sapphire laser (Coherent Legend-HE) producing 34-fs pulses at a repetition rate of
1 kHz. The laser beam was focused through an acylindrical lens with an 8-mm focal length,
and sent through a phase mask placed about 150 µm in front of the fiber. The pulse energy
measured before the lens was adjusted to a typical value of 75 µJ. This value was selected to
be (1) sufficiently higher than the type-I writing threshold, estimated to ~40 µJ, to benefit
68
from a significant writing speed; and (2) sufficiently lower than the Type-II damage threshold,
estimated to ~150 µJ, to ensure that the FBG is free of void-like defects, which would have
significantly increased photo-induced losses [16]. Under such exposure conditions in
deuterium-loaded fibers, the FBG formation is a result of a combination of color-centers and
glass densification, as discussed above [17]. The laser induces a narrow channel of refractive
index change about 1-µm wide that propagates along the focusing axis. Since this channel is
significantly narrower than the ~10-µm field diameter of the fiber core mode, in order to
maximize the effective index modulation the beam was scanned over the core’s cross section
during the writing process. This was achieved by mounting the lens on a piezoelectric actuator
(not shown in Fig. 4.5) to sweep the focused beam around the fiber core over ±10 µm at a
frequency of 1 Hz. The change in refractive index remains non-uniform across the cross-
section of the fiber, and therefore birefringence is induced [5,18,19]. The phase mask was
fabricated in-house at Université Laval using holographic lithography to obtain a uniform
pitch of 1070 nm and produce a first-order Bragg resonance around 1550 nm. The Gaussian
beam incident on the lens had an FWHM of 5 mm, which defines the exposure length along
the non-focusing axis of the lens. To produce gratings of various lengths, both the incident
beam and the lens were mounted on an air-bearing linear motion stage (not shown in Fig. 4.5)
and scanned along the fiber’s main axis while both the mask and the fiber were kept stationary
(to avoid vibrations, which would have reduced the fringe visibility). The scanning length was
adjusted up to 20 mm. The scanning speed was adjusted to control the total exposure fluence
and consequently the index modulation, typically from 4 mm/min down to a minimum of
1 mm/min to reach saturation and maximize the index modulation to ~3.5x10-3. The FBG
samples were left at room temperature for about two weeks following their fabrication to
make sure that most of the deuterium had out-gassed and the material properties stabilized
before measurements and annealing.
It has been observed that in femtosecond FBGs, for both mechanisms the index change
evolves with laser fluence, namely the incident power times the exposure time, in the same
fashion. Specifically, below a certain threshold fluence the glass suffers no change in
refractive index [17]. Above this threshold the change in the refractive index grows linearly
with incident fluence. Finally for high fluences the index change saturates, namely any further
increase in the fluence has no effect on the index (presumably because all the color-centers
69
precursors have been converted into color centers and because the density of the fiber cannot
change anymore). In section 4.3 we explain how we used a sigmoid function to model this
behavior in an abstract way and lump both mechanisms together. A more detailed model,
based on basic physical principles, is being developed currently in our research laboratory by
Arushi Arora.
4.2.2 FBG annealing
To further reduce the loss and improve the sensitivity of our FBGs they were annealed by
heating them up to 400°C. Annealing is believed to mitigate the loss via at least two
mechanisms, namely by reducing the concentration of shallow defects and the local stresses
introduced by exposure to intense light during fabrication [20]. Annealing reduces the internal
loss of the grating, but it also reduces its index modulation. Our experience shows that the rate
at which the index modulation is reduced is lower than the rate of loss reduction, so that
annealing generally produces stronger and more transmissive slow-light resonances, if it is
done optimally, meaning up to the right temperature for the right amount of time.
The annealing process gave us a post-fabrication knob to empirically optimize our FBGs for
each particular application (i.e., sensitivity, group delay, etc.), by controlling the loss and the
reflectivity of our effective FPs. We were able to find an optimal point, where the pair of loss
and index modulation maximizes a particular figure of merit, by annealing the FBGs in small
temperature steps (down to 10 °C) for 30 minutes each. At the point where our figure of merit
reached its maximum, as determined by measurements, we stopped the annealing process.
This method enabled us to optimize our FBGs with a controllable method, something that was
not possible to do during the fabrication since it is not accurately modeled yet. Thus we cannot
predict the exact impact of the writing conditions on the loss and the index change, i.e.,
because of alignment issues in the fabrication setup, etc. In the future with the new model that
is being developed from Arushi Arora we expect to overcome this problem and have better
control over the loss and the index change during the fabrication process itself.
Figure 4.6 shows an example of the evolution with annealing temperature of the peak
transmission T0 and peak group delay τg,max of the first seven slow-light resonances of a
particular FBG, and their product. The grating was 12.5 mm long and was written with the
70
method mentioned in section 4.2.1. It was annealed at a particular temperature in a fiber oven
for 30 minutes. After the oven was switched off the FBG was taken out of it quickly (~1 s)
and rapidly cooled down to room temperature, at which point the transmission and group
delay spectra were re-measured. As the FBG was annealed to higher and higher temperatures,
the transmission (see Fig 4.6a) and the T0τg,max (see Fig 4.6c) product increased monotonically,
whereas the group delay reached a maximum then decreased (see Fig 4.6b). These trends are
representative of the behavior of a large number of femtosecond FBGs that were annealed in
the course of this work. The transmission increases monotonically because both effects of the
annealing process (reducing both the loss and the index modulation) lead to an increase in the
transmission. In contrast, these two effects have opposite impacts on the group delay. When
the loss is reduced the group delay increases, but when the index modulation (and hence the
reflectivity of the effective mirrors) is reduced, the group delay decreases. At one specific pair
of loss and index modulation, the group delay reaches a maximum similar to a normal FP in
which the group delay reaches its maximum when it is matched, namely the transmission of
the mirrors is equal to the round-trip loss of the FP [21]. It is worth mentioning that the closer
the peak is to the middle of the bandgap the higher the annealing temperature where the group
delay reaches its maximum. Also, as we annealed this FBG new resonances arose inside the
bandgap that we were not able to observe before because of their very low transmitted power.
Annealing reduced the loss sufficiently that these peaks acquired a sufficiently high
transmission to be detected. Figure 4.6c is informative when designing a strain sensor, whose
sensitivity is proportional to the T0τg,max product. It shows in particular up to what maximum
temperature the grating must be annealed in order for the best resonance (the one with the
highest product) will reach its highest sensitivity. In this particular grating it was resonance #7
after annealing it up to (~230 °C).
To illustrate the significance of these three improvements, in an apodized femtosecond-written
FBG 2 cm in length we were able to achieve a record group delay of 20 ns [18], 4 times larger
than our previous record of 5.1 ns reported in [5]. Then by introducing deuterium in the
fabrication process and annealing the FBGs with a goal to achieve the highest possible group
delay, we were able to double the group delay up to 42 ns in a 12.5-cm long FBG [22]. This
shows that with these combined improvements we were able to have an eight-fold
improvement in group delay. This value is the current world record for a group delay in an
optical fiber.
71
Figure 4.6. Measured evolution with annealing temperature of (a) the transmission, (b) the group
delay, and (c) their product, for the slow-light resonances of a particular FBG.
4.3 Modeling the index profile of FBGs written with a femtosecond laser
This section discusses how we modeled the index modulation of the FBGs written with the
technique described in the previous section.
As discussed in section 4.2 the index change is related to the laser fluence. If the index change
induced by the femtosecond laser was proportional to the laser fluence, the ac index
modulation of the FBG would have the same profile as the convolution f(z) of the focused
laser intensity profile (a Gaussian with a FWHM W ) and a rectangular function (to account
for the scanning of the laser beam over a length L ):
f (z) = exp(−4 ln(2) z2
W 2 )∗Rect(z / L) (4.5)
where z is the longitudinal axis of the fiber, and Rect(z/L) = 1 for 0 ≤ z ≤ L and 0 elsewhere.
This convolution f(z) is plotted (not to scale) as the dashed curve in Fig. 4.7a for L = 10 mm
and W = 5 mm.
However, as discussed in section 4.2.1 the index change is not proportional to the fluence, and
two effects reshape this profile. First, there is a threshold intensity below which the fiber
refractive index is virtually unmodified [17]; this effect shortens the profile and gives it
sharper edges. Second, the index change saturates at high fluence, which flattens the index-
modulation profile in its central region. Since the exact dependence of Δnac on the fluence is
not known, these two physical effects were incorporated in the model of the index-modulation
profile Δnac(z) by processing the convolution f(z) through a sigmoid function [23], which
yielded the following trial index-modulation profile:
72
Δnac (z) = Δn01
1+ exp(−af (z)+ b)−
11+ exp(b)
#
$%
&
'( (4.6)
Figure 4.7. Index modulation profile example of a fiber Bragg grating written with a femtosecond laser;
(a) convoluted profile, half-peak-to-peak ac index profile, and dc index profile; (b) upper and lower
envelopes of the index modulation.
The parameters a and b control the threshold and the saturation applied to the original profile.
The second term in the bracket ensures that the ac index modulation is zero outside of the
FBG. The factor Δn0 is a constant scaling parameter. The dc index profile was assumed to
have the form:
Δndc (z) =αΔnac (z) (4.7)
where α is a constant.
By definition of the ac and dc index-modulation profiles, the index modulation of the grating
is confined between an upper envelope Δn+(z) = Δndc(z) + Δnac(z) and a lower envelope
Δn–(z) = Δndc(z) – Δnac(z), as illustrated in Fig. 4.7b. Note that, as expected from the symmetry
of the laser scanning during fabrication, these profiles are symmetric about the middle of the
grating (Δnac(z) = Δnac(L - z)). Also, the lower envelope Δn–(z) is negative along some of the
grating, as expected for a grating induced by a nonlinear process [24]. The index modulation
in our gratings is at least partly induced by a multi-photon process [25], and as such it is a
73
nonlinear function of the laser intensity. The index modulation therefore has frequency
components not only at the fundamental frequency, but also at harmonics of this frequency.
The fitted ac index profile contains only the fundamental frequency (since harmonics do not
affect the FBG properties around 1550 nm), which explains the negative lower envelope. In all
simulations the FBG loss was assumed to have the same profile shape as Δnac(z), on the basis
that the loss increases with index modulation, as discussed in relation to Fig. 4.4.
Figure 4.8. a) Measured and simulated transmission spectra of the FBG. b) Magnified portion of slow-
light peak region of the transmission spectra shown in a). c) Measured and simulated group-delay
spectra in the same wavelength range as in b).
The general profiles of Eqs. 4.6 and 4.7 were used as an input to our code (described in detail
in section 4.1) to predict the grating’s transmission and group-index spectra. Specifically, to
infer the index-modulation profile of a particular fabricated FBG, its group delay and
transmission spectra were first measured. The profiles calculated using the index-modulation
profiles of Eqs. 4.6 and 4.7 were then visually best fitted to these measured spectra by
74
adjusting the values of the five free parameters in the model, namely Δn0, a, and b in Eq. 4.6,
α in Eq. 4.7, and the maximum loss coefficient γmax (at the center of the grating) in Eq. 4.3a.
As an example of the performance of our modeling, in Fig. 4.8 we plot the measured spectra
(solid line) and the predicted spectra (dashed line) using our model for the FBG that gave us
the record-high group delay of 42 ns. The measured spectra were measured using the
experimental shown in Fig. 4.9. Light from a fiber-coupled tunable 1550-nm laser was
amplitude modulated at a frequency fm of 1 MHz with the help of a function generator. The
modulated light was passed through a fiber polarization controller, then coupled into the FBG.
The output signal from the FBG was sent through a 3-dB fiber splitter, which was connected
at one output end to an optical power meter to measure the output power, and at the other
output end to a detector followed by a lock-in amplifier to measure the phase of the output
signal. The laser was first tuned to a wavelength far away from the FBG bandgap (1555 nm),
which provided a point of reference where the output transmission (unaffected from the FBG)
was high and wavelength independent, and set to unity, and the output phase (unaffected by
the FBG dispersion) was set to zero. Both outputs were acquired using a data acquisition
system. The laser wavelength was then scanned and the measurements repeated at individual
wavelengths across the slow-light region. The measured spectrum of the phase delay φ
provided the spectrum of the group delay τg, related to the phase delay by τg = φ/(2πfm).
Figure 4.9. Experimental setup used to measure the group delay and transmission of the FBG (see text
for details).
The transmission spectrum exhibits a steep-walled reflection bandgap, with several slow-light
resonances on its short-wavelength edge, and none on the long-wavelength side (solid green
curve in Fig. 4.8a). Figure 4.8b zooms in on the slow-light transmission resonances on the
75
short-wavelength edge. Toward the center of bandgap (longer wavelengths), the sharpness of
the resonances increases (solid green curve). Their peak transmissions of the resonances
decrease toward the Bragg wavelength (longer wavelengths). The slow-light resonances occur
in pairs. Each peak in a pair corresponds to a resonance of one of the FBG’s two
eigenpolarizations. This resonance splitting occurs as a result of birefringence that develops in
the FBG during writing, as described in section 4.2.1. The polarization controller (see Fig. 4.9)
was adjusted prior to each spectrum measurement to launch as much of the light as possible in
one of the eigenpolarizations of the FBGs, so as to minimize these spurious peaks, but the
latter could not all be eliminated. Figure 4.8c shows the measured group delay spectrum of the
same grating. The group delay generally increases for resonances with longer wavelengths,
then decreases. The slowest resonance exhibited the aforementioned record group delay of
42 ns.
The index profiles Δnac(z) and Δndc(z) used for the fitting are shown in Fig. 4.7. They gave the
best fits to the measured spectra. These profiles were calculated using the following values:
Δn0 = 2.4x10-3, a = 0.135, b = 1.6, and α = 0.667 in Eq. 4.6, and a very low peak power loss
coefficient of 0.19 m-1 (a single-pass loss as low as 0.010 dB) in Eq. 4.3a. The Gaussian width
and scan length in Eq. 4.2 were kept at their measured values of W = 5 mm and L = 10 mm,
respectively. Because of the high peak intensity (~5.2x1012 W/cm2) of the laser beam and
fairly slow scan rate (~1 mm/min) used during fabrication for this particular FBG, the profiles
exhibit a strong saturation over a sizeable fraction of the length of the grating (see Fig. 4.7).
The FWHM length of the apodized profiles is 12.5 mm, which is the FWHM of the index
profile that yielded the best fit (Fig. 4.8). The maximum half-peak-to-peak ac index
modulation is 1.98x10-3. The grating period, known from the fabrication conditions
(Λ ≈ 538 nm) was also finely adjusted in the simulations (to 537.99 nm) to fine-tune the Bragg
wavelength (this parameter produces only a shift in the spectra). The noise floor of the
measured transmission spectra (imposed by the instrument) was simulated by adding a
constant offset to the simulated spectrum (-32.2 dB in Fig. 4.8a and -34.8 dB in Fig. 4.8 b)
(the noise floors differ because of different settings in the measurement apparatus).
There is a reasonable agreement between the two sets of measured and predicted spectra,
especially the FBG’s bandgap (Fig. 4.8a) and the locations and amplitudes of both the
transmission peaks (Fig. 4.8b) and the group-delay peaks (Fig. 4.8c). The model predicts that
76
this grating should have a few more slow-light peaks (deeper in the bandgap). However, these
peaks have too low a transmission to be observed. A similar agreement for the spurious peaks
due to the second eigenpolarization can be obtained by suitably adjusting the Δndc profile for
this second polarization, as described in [18]. It is important to point out that the predicted
spectra are fairly sensitive to the fitted values and to the shape of the index profile. Other
profile shapes close to the one we used gave fits of noticeably different quality. Thus our
model can predict experimental spectra very accurately.
4.4 Conclusions
In this chapter we discussed the improvements we made in the design of the FBGs and their
fabrication process. Specifically, we showed that apodized FBGs can significantly increase the
group delay and consequently the sensitivity of FBGs compared to uniform FBGs. We also
showed that deuterium-loaded FBGs written with a femtosecond laser have some of the largest
Δnac reported in an FBG and at the same time very low internal losses, two properties that are
very important to achieve high group delays and high sensitivities to strain. Finally, we
described in detail how the FBGs used in this thesis were fabricated and how we modeled
their index-modulation profiles, their loss profiles, and their transmission and group-delay
spectra.
At this point we have all the needed parts in place. In Chapter 3 we studied the thermal phase
noise in an FP-like resonance. In this chapter we showed how we can get very strong
resonances by fulfilling the requirements discussed in Chapter 2. In the next chapter we
combine the results from these chapters to demonstrate how we designed and fabricated an
FBG that verifies our phase-noise theory, and observed with it the thermal phase-noise of a
grating for the first time. We also show how, based on the results of Chapter 3, we reduced
this thermal phase noise in a second FBG to achieve a record low MDS of 30 fε/√Hz and an
absolute measurement of strain of 250 attostrains, both at 30 kHz.
77
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79
Chapter 5: Measuring the intrinsic thermal phase noise and
250 attostrains using slow-light FBGs
In the previous chapters we have discussed the properties that an FBG needs in order to
achieve a high strain sensitivity (Chapter 2), and how we can realize such FBGs in practice
(Chapter 4). Also, in Chapter 3, we studied theoretically the thermal phase noise in a Fabry-
Perot cavity. This study showed that the phase noise in a FP-like resonance is proportional to
the group index of the resonance and to the square root of the length of the resonator L. As we
have shown in Chapter 2, the sensitivity in a slow-light resonance is proportional to the group
delay, which is proportional to the group index and the length of the resonator, and the
minimum detectable strain (MDS) is by definition the noise over the sensitivity. Thus in a
strain sensor with an MDS limited by phase noise, the MDS increases as the sensor becomes
shorter as 1/√L. Consequently, to make the thermal-phase noise the dominant noise in our
sensor and therefore be able to measure it, we had to fulfill two requirements: (1) our sensor
must have high sensitivity to be limited by sensitivity-dependent noise sources, which include
thermal phase noise; (2) our sensor must be short enough so that the thermal phase noise is
dominant. To fulfill both requirements simultaneously was challenging, since as the length of
the sensor becomes shorter and shorter, the sensitivity becomes smaller. We were able to
achieve both—high sensitivity in a short sensor— by increasing the group index using the
techniques discussed in Chapter 4. In a second device, to measure the lowest possible MDS
we decreased the thermal phase by increasing the length of the sensor, keeping all the other
noise sources at low levels. This second sensor had to be compatible with the rest of the
experimental setup, namely the probe laser, which raised another challenge because this laser
had a narrow tuning range, hence the FBGs’ slow-light resonances and the laser wavelength
had to be carefully matched.
In this chapter we report the observation and quantitative measurement of the thermal phase
noise in a 5-mm slow-light FBG over a range of frequencies between 1 and 10 kHz. The
measured phase-noise spectrum is in excellent agreement with the theoretical model presented
in Chapter 3, which provides experimental evidence of its validity. To further verify this
result, we replaced the 5-mm FBG with a longer FBG (2 cm) in which the thermal phase noise
in terms of MDS was predicted to be significantly lower, and the total noise also lower. As a
80
result of this lower total noise, we measured in this FBG sensor a record low MDS of
110 fε/√Hz at 2 kHz, and 30 fε/√Hz at 30 kHz. When integrating the output of this sensor at
30 kHz for about 8 hours, a record low absolute strain of 250 attostrains was measured. These
last two figures at 30 kHz are the lowest ever reported in a fiber strain sensor.
Figure 5.1. Experimental setup used to characterize the noise spectra and the sensitivity of FBG strain
sensors. The PZT plate excited by the function generator induces a known strain on the FBG to
calibrate its sensitivity. The lock-in amplifier measures the sensor’s response
5.1 Experimental Setup
Figure 5.1 shows the experimental setup that we used to characterize our sensors. Light from
the narrow-linewidth, low-noise laser first traveled through a variable optical attenuator. This
allowed us first to operate the laser at the maximum output power, which gave the lowest
noise performance, and second to control the input power to our sensor for reasons explained
below. Then the light passed through an isolator, to protect the laser from back-reflections,
and a polarization controller (PC), which was used to tune the light polarization to excite only
one of the two eigenpolarizations of the FBG, as discussed in Chapter 4. At this point the light
was in its proper state (proper power and polarization) and was launched into the FBG sensor.
All components were fiber-pigtailed and connectorized and/or spiced together. The sensor was
mounted on a linear PZT plate, needed to induce a known strain to the FBG, and was placed
inside an anechoic enclosure to isolate it from the environment. A function generator applied a
voltage to the PZT. Finally, the optical output from the sensor was converted into an electrical
signal using two photodetectors. The first photodetector is an optical power meter (OPM),
which monitored the operating point of our sensor. For long-term measurements (several days)
the output of the OPM was used as an input to the PID controller in order to stabilize the
sensor. The second photodetector was connected to the lock-in amplifier, which measured the
response of the sensor at one specific frequency of interest. By switching the frequency we
81
were able to reconstruct the spectrum of the noise or of the MDS. The outputs from the lock-in
amplifier and the OPM were captured using a data acquisition system (DAQ).
The probe laser was a new laser from Orbits Lightwave with a linewidth below 200 Hz. The
previous laser used in earlier work had a linewidth of 8 kHz [1]. This 40-fold reduction in the
linewidth was expected to reduce the laser frequency noise, and therefore the MDS of the
FBG strain sensor, by at least 6.5 times, since the frequency noise is approximately
proportional to the square root of the linewidth (this approximation is valid for a
semiconductor laser [2]). This laser had a limited tunability of ±0.2 nm. Its actual laser
frequency noise spectrum, provided by Orbits Lightwave and converted in units of strain, is
showed as the red curves in Figs. 5.2 and 5.3.
To match the operating wavelength of the laser and of the FBG, we actually order a custom-
made laser to match the wavelength of our FBGs. The main issue is that the slow-light
resonances of a typical strong FBG span a fraction of a nm, and the control over the
fabrication parameters is not accurate enough to predict from these parameters alone the exact
wavelength of the steepest slope of the best resonance of a grating. We therefore could not
predict that this wavelength would fall within the 400-pm tuning range of the laser. The
approach we took to solve this problem is to fabricate a large number of strong apodized
FBGs in D2-loaded fibers, and annealed them one by one to maximize the T0τg,max product.
We then purchased thr custom-made laser at the wavelength that maximizes the number of
FBGs whose best resonance (in terms of the highest T0τg,max product) fall within the tuning
range of the laser.
As discussed briefly in Chapter 4, the method used to inscribe the FBGs induces strong
birefringence in the fiber, causing the slow-light peaks to split [3,4]. If both eigenpolarizations
were excited, only a portion of the input light would be transmitted and would be useful for
the sensor. The rest of it would be reflected. Thus in this case the sensitivity would be greatly
reduced. In the extreme, unrealistic case that the probing wavelength is at the steepest slope of
one eigenpolarization but the input light is polarized orthogonal to this eigenpolarization the
sensitivity would be zero, since no light is transmitted. Thus, in order to achieve the
maximum possible output power and hence the maximum sensitivity, we used the PC to excite
only one eigenpolarization.
82
The second reason for the use of an optical attenuator is as follows. If the input power of the
light prior to the FBG is too high, because of the expected intensity enhancement in a
resonator with a high Q factor, as explained in Chapter 6 the intensity inside the FBG is very
high. This intensity is converted into heat via absorption loss, causing the fiber to heat up. The
temperature change shifts the Bragg wavelength, causing the output power to vary. Then
because the laser is off resonance, the intensity inside the FBG decreases to a very low value,
the FBG cools down, and the Bragg wavelength shifts back to its initial, normal-temperature
value. This intensity-induced power variation results in an instability in the FBG output, as
observed experimentally for example in [5]. Thus in order to reduce this unwanted source of
noise, we had to reduce the input power and hence reduce the power that is converted into
heat. Consequently, the presence of the attenuator was crucial. It is worth mentioning that the
attenuator was needed mainly for the 5-mm FBG, since, as described in Chapter 6, the
intensity enhancement of the FBGs written with our method reaches its maximum around this
length.
In order to stabilize our sensor against laboratory temperature fluctuations, we used a PID
feedback loop. We did not use the derivative component in the PID controller because it
drastically increased the noise in the kHz range. The PID controller monitored the output
power of the FBG using the OPM (see Fig. 5.1). We first measured the output power of the
FBG at the point where the sensitivity was maximum, and we set this value as the target value
in the PID controller. If the relative position between the resonance and the laser drifted, either
because the laser wavelength or the slow-light resonance drifted, the output power of the FBG
changed and the PID controller applied a correction voltage to the dedicated input of the laser
in order to counter-act the power change. This applied voltage finely tuned the laser
wavelength by changing the length of the laser cavity. Thus the PID controller forces the laser
wavelength to be always the same as the wavelength where the resonance has its steepest
slope, and hence cancel out any drift in the sensitivity of the FBG sensor. This feedback loop
was very effective: as discussed further on, it enabled us operate the sensor continuously up to
four days.
83
5.2 Measuring the strain sensitivity and MDS of FBG sensors
As described in Chapter 2 the sensitivity is defined as:
SN (λprobe ) =1Pin
dPoutdε
!
"#
$
%&λprobe
=dλBdε
dTdλ!
"#
$
%&λprobe
(5.1)
Thus, in order to measure the sensitivity of a very sharp resonance, we need to measure the
change in the input power when a known strain is applied to the FBG. To apply a strain to the
FBG, the fiber was attached to a linear PZT plate. In our earlier work [1], a circular PZT was
used instead, which induced a lateral strain on the FBG that shifted the transmission spectrum.
This constant shift was easily corrected by adjusting the wavelength of the tunable laser.
However, in this work the limited tunability of the new laser made it impossible to correct for
this bend-induced shift. This problem was corrected by using a linear PZT instead of a PZT
ring. A specific peak-to-peak voltage of 0.01 V at 27 kHz was applied to the PZT, using the
function generator shown in Fig. 5.1. This frequency is close to the resonant frequency of the
PZT, thus a relatively high strain was applied (~5 nε), as explained later in detail. The
calibration of the strain applied to the sensor was achieved by using both of the following two
methods:
1. We used a thermally stable fiber Mach-Zehnder interferometer (MZI). The FBG
mounted on the linear PZT was placed in the first arm of the MZI. The length of the
second arm was carefully matched to the arm that included the FBG to minimize the
impact of differential thermal fluctuations between the two arms and have a relatively
thermally stable MZI. The MZI was further stabilized by controlling the length of the
second arm using a PZT and a feedback loop to stabilize the length mismatch between
the two arms. This measurement was only a few minutes long so we were not
interested in achieving long-term stability. Then we used a tunable laser tuned far
away from the FBG’s bandgap, where the FBG acts like a normal fiber, to interrogate
the MZI. Afterwards, we applied a known voltage to the PZT, which applied an
unknown strain (the strain we needed to calibrate) to the FBG. This unknown strain
changed the path mismatch between the two arms, causing the output power of the
84
MZI to oscillate at the same frequency as the applied strain. From the measured
amplitude of the modulated output power we calculated the phase modulation, and
hence the length modulation of the first arm. Finally, by dividing this length
modulation by the total length of the fiber that was attached to the PZT (which
includes the FBG), we calculated the applied strain to the FBG.
2. In this second method, we tuned a tunable laser to the maximum sensitivity of a
relatively broad slow-light resonance (low strain-sensitivity) of the FBG of interest,
which was attached to the PZT. We had previously measured the transmission
spectrum of this resonance, as described in section 4.3, which is possible to do
accurately because this resonance was by choice broad. Then, we were able to
calculate the sensitivity numerically with high accuracy. This was done by using the
rightmost equation in Eq. 5.1, namely by taking the derivative of this measured
transmission spectrum and multiplying it by the constant term dλB/dε (equal to 1.2
pm/µε, see Chapter 2). Then, as in the previous method we applied a known voltage to
the PZT, which induced an unknown strain to the FBG. This applied strain caused a
power modulation at the FBG output, as described in Chapter 2. The applied strain
was then easily inferred by dividing the measured output modulation by the measured
sensitivity and solving the middle equation Eq. 5.1 for the only unknown, the applied
strain dε=dPout/(SNPin).
Both methods gave similar results, namely an rms applied strain of ~5 nε for a peak-to-peak
voltage of 0.01 V applied to the PZT, or a response of 500 nε/V.
Knowing the strain that the PZT applied to our FBG for a given applied voltage, we could
proceed to measure the highest sensitivity of the FBG. To do that we tuned our laser at the
point where the response of our FBGs was maximum when we applied an rms strain of
Δε = 5 nε. At this point, we measured the response of our sensor ΔPout. Afterwards, using the
middle equation of Eq. 5.1, we inferred the sensitivity SN=ΔPout/(PinΔε). The noise spectrum
was measured by repeating this measurement after turning off the voltage applied to the PZT,
and tuning the operating frequency of the lock-in amplifier. The MDS spectrum was obtained
by dividing the measured noise spectrum by the measured sensitivity.
85
5.3 Measuring thermal phase-noise in a 5-mm FBG
As discussed in Chapter 3, the phase noise in an FP-like resonance in terms of strain is
inversely proportional to the square root of the resonator’s length. Thus for the thermal phase
noise to be the dominant noise source, we had to increase it above the frequency noise of our
laser. Simulations have shown that FBGs shorter than 2-3 mm do not support slow light—with
the current technology they are too short to exhibit strong enough reflections. Thus we chose
to study a 5-mm long FBG since it was relative easy to fabricate, as discussed in Chapter 4.
Figure 5.2 shows the different noise components in units of strain per square root of Hz versus
noise frequency. The green dashed curve represents the predicted phase noise in a FBG 5-mm
long, as described in Chapter 3, whereas the red solid curve represents the frequency noise.
The frequency noise of our laser was provided to us by the manufacturer, so it was
independently measured. It can be seen that between 1 kHz and 10 kHz the FBG’s thermal
phase noise is higher than the laser frequency noise. Thus, as long as the sensor is limited by
the sensitivity-dependent noise sources—which occurs when the sensor has high sensitivity—
and as long as the environmental noise is low, the dominant noise source would be the
intrinsic thermal phase noise of the FBG. The predicted total noise for such a grating is
represented by the black dashed curve in Fig. 5.2.
Figure 5.2. Noise contributions in a 5-mm FBG slow-light sensor.
Based on the foregoing, we fabricated a saturated 5-mm FBG using a femtosecond laser in a
deuterium-loaded fiber, as described in section 4.2.1. Our goal was to have an FBG with the
86
highest possible index modulation. The length of the FBG (5 mm) was achieved by writing the
FBG with a stationary laser beam with a 5-mm FWHM. Figure 5.3 shows the measured
transmission spectrum of this FBG. This spectrum was measured as described in section 4.3,
before annealing the FBG. At short wavelengths the transmission is much lower than the
expected 0 dB (100%) transmission. This is due to the fact light is coupling to cladding modes
of the fiber, which are lossy, and it does not couple back to the propagating mode. The
bandwidth of the bandgap was 3.9 nm. From this number and using Eq. 2.7 the half-peak-to-
peak index modulation prior to annealing was inferred to be 3.7x10-3, close to the highest ever
reported (~5x10-3) in [6], which met our goal of a very high index modulation.
Figure 5.3. Transmission spectrum of the FBG used for measuring thermal phase noise, measured after
fabrication (before any annealing took place).
The FBG was then annealed at 100°C, 200°C, 300°C, and 400°C for 30 minutes at each
temperature. As annealing proceeded the product Τ0τg,max was observed to constantly increase,
and therefore the sensitivity did too (see Fig 5.4). We stopped the annealing process at 400°C
since further annealing would have deteriorated the fiber and the sensitivity was already high
enough. The resonance used for sensing had a measured group delay of ~8 ns and a measured
transmission of ~0.65. The group delay was measured using our previous laser as described in
Chapter 4, therefore it may have been underestimated because of the limited resolution of that
laser (0.1 pm). Using Eq. 2.16 the sensitivity can be calculated from these two values to be
~3.25x106. Using the method described in section 5.2 the measured sensitivity was slightly
higher, namely ~4x106. The small discrepancy is mainly due to the fact that Eq. 2.16 is an
approximation, and also that the group delay is underestimated.
87
With the experimental setup described earlier the noise in this FBG (black open circles line in
Fig. 5.2) was measured. The blue solid curve represents the measured noise due to all
sensitivity-independent sources, namely the intensity noise of our laser, the photodetector
noise, and the shot noise (both electrical and optical). These noise sources were measured
using exactly the same setup, shown in Fig. 5.1, with the same output power but by replacing
the FBG with a normal fiber with zero sensitivity, and therefore in which the sensitivity-
dependent noise sources were negligible. The combined sensitivity-independent noise sources
are clearly negligible. The total noise (dashed black curve) was calculated by geometrically
summing up all noise components, namely the measured laser frequency noise (solid red
curve), the predicted FBG phase noise (dashed green curve), and the sensitivity-independent
noise sources. There is only a small discrepancy around 8 kHz due to a resonance observed in
the measured data, which is due to an environmental noise in our laboratory. This resonance
was due to environmental noise in our laboratory: we have observed the same resonance in
other FBG sensors (see Fig 5.5 for example). Since the noise is dominated by the FBG phase
noise over most of the measurement frequency range, the conclusion is that this sensor was
dominated over this range by phase noise.
Figure 5.4. Evolution of Τ0τg/c versus annealing temperature for the FBG that was used to measure
phase noise.
This observation constitutes the first measurement of the thermal phase noise in a short
passive FBG sensor (it has been observed in active FBG sensors [7]). The single-pass phase
88
noise in this 5-mm fiber in this frequency range is of the order of 1.6 nrad/√Hz, which is
extremely small and hence was difficult to measure; the use of the “amplification” by a factor
of ng (about 480 in this FBG) is what made the measurement possible. The good agreement
between the measured and predicted noise spectra gives credence to the validity of the model
on which this noise spectrum was predicted, namely the theory presented in Chapter 3. In the
next section, we provide further validation in a grating with a longer length in which the phase
noise is indeed found to be negligible, as predicted by the model.
5.4 Measuring an absolute strain of 250 attostrains
In the previous section we discussed a sensor having thermal phase noise as a dominant noise
source in the frequency range of 1-10 kHz. According to the analysis in Chapter 3, if we
increase the length L of our FBG by a factor of N while maintaining the same high sensitivity,
the thermal phase noise should be reduced by 1/√N. Thus in a ~20 mm sensor (4 times longer
than the previous sensor) the thermal phase noise should be reduced by a factor of ~2. In Fig.
5.5 we can see the noise components in this case. The color code is the same as in Fig. 5.2,
with the green dashed curve representing the thermal phase noise, the frequency noise
represented with the red solid curve, and black dashed curve representing the geometrical sum
of all noise sources. In this figure the sensitivity-independent noise sources have been omitted
for clarity purposes since they are negligible. With this longer FBG the dominant noise source
is clearly the frequency noise of the laser for the whole range of frequencies between 1 kHz
and 30 kHz.
To verify these predictions experimentally, we fabricated a new, longer FBG sensor in
deuterium-loaded fiber by scanning the femtosecond laser for 2 cm, as described in section
4.2.1. Because this FBG was longer than the previous FBG we did not aim for the highest
possible index modulation, since the index modulations was not so important to achieve a high
enough sensitivity. The FBG round-trip loss increases with length, and so does the FBG
reflectivity. Thus by reducing the index modulation in a longer FBG we can reduce the loss
while maintaining the same reflectivity as in a shorter FBG. As in the previous FBG, from the
measured bandwidth of the bandgap prior to any annealing, we inferred the half peak-to-peak
index modulation to be ~1.7 x10-3, approximately half that of the previous FBG. We then
annealed the grating at 50°C, 80°C, 120°C, and 140°C for 30 minutes at each temperature. We
89
stopped at this lower temperature because the sensitivity was already high enough for the
noise to be limited by sensitivity-dependent noise sources. Further annealing would have had
two possible outcomes: (1) an increase in sensitivity, leading to a narrower peak and a smaller
dynamic range of the measured strain with no improvement in the MDS); (2) a decrease in
sensitivity because the group delay deteriorates, possibly leading to a smaller MDS. The
resonance that was used for strain sensing had a peak transmission of ~0.55 and a peak group
delay of 11.2 ns. These values lead to a predicted sensitivity of ~3.8x106. The measured
sensitivity was about the same as in the previous FBG (~4x106). The solid black curve with
the circles in Fig. 5.5 represents the noise in units of strain measured in this sensor. It can be
seen that there is a very good agreement between the measured and the predicted total noise.
The spurious resonance at ~8 kHz is observable in this case too, as well as one additional
resonance at ~ 25 kHz, again due to environmental noise. Figure 5.5 shows a measured MDS
of 110 fε/√Hz at 2 kHz, which is a factor of 2 lower than what has been reported in a Fabry-
Perot made with two FBGs [8], and 30 fε/√Hz at 30 kHz, about one order of magnitude lower
than what has been reported in our previous work [1]. This measurement also shows for the
first time that in a passive FBG sensor the thermal phase noise expressed in units of strain is
reduced as the length increases, a counter-intuitive result of the analysis in Chapter 3.
Figure 5.5. Measured and calculated noise spectrum contributions in a 20-mm slow-light FBG strain
sensor in units of strain.
Using the stabilization setup described in section 5.1 we monitored the output of our sensor
for four days. Figure 5.6 shows the time trace of this measurement. An interesting observation
90
of Fig. 5.6 is that the noise level during the night was higher than during the day. This
behavior was quite periodic and repeatable, showing that it was not just a glitch in our setup
but an actual repeatable signal from the environment. This noise variation may have been
caused by machinery that was operating during the night; we have no explanation for its
origin. This is an observation that we were not able to make before, because our previous
sensor had a poorer resolution (280 fε/√Hz at 23 kHz), which is much higher than the noise
level during both night and day. This observation shows the significance of developing
simple, low-cost ultra-sensitive strain sensors: they enable us to make observations that were
impossible before. The sharp peaks recorded by our sensor during the day (see Fig. 5.6) could
have been caused by a construction project that was taking place in an adjoining laboratory. It
is worth mentioning that in spite of these spikes the sensor continued to operate at its optimal
point, further demonstrating the efficacy of the feedback control system.
Figure 5.6. Time trace of the 20-mm sensor’s response at 30 kHz.
Using the time trace in Fig. 5.6 the Allan deviation can be calculated. The Allan deviation is a
plot of the noise in a system as a function of integration time [9]. It is obtained by integrating
the output signal of an instrument recorded over a long time with and increasingly longer
integration time, and plotting the resulting integrated noise as a function of integration time.
Using the Allan deviation we can derive some interesting conclusions about the limiting
factors in our sensor, for example the optimal integration time to achieve the lowest minimum
detectable absolute strain. In Fig. 5.7 we plot a generic Allan deviation curve to illustrate its
main characteristics. If the noise in a system is white, as the integration time increases the
91
Allan deviation decreases with a slope of -1/2: as expected for white noise, the integrated
Figure 5.7. Generic Allan deviation curve.
noise decreases as the square root of the measurement bandwidth. If there is a drift in the
instrument under test (because of some temperature change for example) around the time
constant characteristic of this drift the Allan deviation starts to increase with a slope that
depends on the physical origin of the drift, but which is typically 0, 1/2 (as in the hypothetical
example of Fig. 5.7), or 1. The absolute minimum value that can be measured occurs for the
integration time where the Allan deviation is minimum: for larger integration times the noise
decreases because of the reduced bandwidth, but integration also incorporates the drift into the
integrated noise, leading to an increased integrated noise.
Figure 5.8. Allan deviation in units of strain calculated from the data of Fig. 5.6.
Using the data from Fig 5.6 the Allan deviation in units of strain was calculated for our FBG
sensor (see Fig. 5.8). Figure 5.8 shows that even after four days there was no drift in our
92
sensor output whatsoever, in spite of the extraordinary temperature sensitivity of our sensor
(which also increases proportional to the slope of the resonance, see section 2.3). When we
integrate its output signal for ~8 hours, the absolute strain that we were able to resolve is
250 attostrains (2.5x10-16). This value is only slightly larger than the absolute strain calculated
by dividing the measured MDS of 30 fε/√Hz by the square root of the integration time
(8 hours) expressed in second (176 attostrains). This discrepancy is due to the fact that during
the night the noise was significantly increased (almost doubled), whereas the MDS of
30 fε/√Hz was measured during the day. Nevertheless, to best of our knowledge, this value of
250 attostrains is the lowest experimental absolute strain ever reported in an FBG sensor.
5.5 Conclusions
In this chapter we experimentally characterized two FBG strain-sensors. One was specifically
designed to be limited by thermal phase noise, and the second one to reduce the thermal phase
noise and achieve a record-breaking strain resolution. These measurements enabled us to
verify our theory of phase noise developed in Chapter 3, and to show in particular that longer
sensors have a lower phase noise in units of strain. Also, we experimentally verified that the
next noise source—below the laser frequency noise—that limits our ultra-sensitive FBG strain
sensors is the thermal phase noise. In an FBG with a longer length, in which by design the
phase noise was suppressed, we measured a record low MDS of 110 fε/√Hz at 2 kHz and
30 fε/√Hz at 30 kHz, two times and one order of magnitude lower than the previous records at
those frequencies, respectively [1,8]. Finally we monitored the output of our sensor for four
days at 30 kHz, and observed no sign of drift, thus demonstrating the efficacy of our electronic
control system at correcting for temperature-induced drift in the FBG spectrum. This
measurement led us to a measurement of a record absolute strain resolution of 250 attostrains.
This measurement also allowed us to make an interesting and novel observation: the noise
level during the night was much higher than during the daytime.
This chapter completes the report of the main goal of this thesis, which was to observe thermal
phase noise in an FBG sensor and to reduce the MDS significantly below the previous records.
All the previous chapters and the topics we discussed were building up to the results reported
in this chapter. In the next chapter we discuss other very interesting facts and potential
applications of these slow-light FBGs.
93
References
[1] H. Wen, G. Skolianos, S. Fan, M. Bernier, R. Vallée, and M. Digonnet, “Slow-light
fiber-Bragg-grating strain sensor with a 280-femtostrain/√ Hz resolution,” Journal of
Lightwave Technology, 31, 1804 (2013).
[2] G. Skolianos, M. Bernier, R. Vallée, and M. J. F. Digonnet, “Observation of ~20-ns
group delay in a low-loss apodized fiber Bragg grating,” Optics Letters, 39, 3978
(2014).
[3] H. Wen, G. Skolianos, M. J. F. Digonnet, and S. Fan, “Slow Light in Fiber Bragg
Gratings,” Procedings SPIE 7949, 79490E (2011).
[4] G. Skolianos, M. Bernier, R. Vallée, and M. J. F. Digonnet, “Observation of ~20-ns
group delay in a low-loss apodized fiber Bragg grating,” Optics Letters, 39, 3978
(2014).
[5] J. Upham, I. D. Leon, D. Grobnic, E. Ma, M. N. Dicaire, S. A. Schulz, S. Murugkar,
and R. W. Boyd, “Enhancing optical field intensities in Gaussian-profile fiber Bragg
gratings,” Optics Letters, 39, 849 (2014).
[6] P. J Lemaire, A. M Vengsarkar, W. A. Reed, and V. Mizrahi, “Refractive index
changes in optical fibers sensitized with molecular hydrogen,” Conference on Optical
Fiber Communication of 1994, Optical Society of America Technical Digest Series, 4,
47 (1994).
[7] G. A. Miller, G. A. Cranch, and C. K. Kirkendall, “High-Performance Sensing Using
Fiber Lasers,” Optics & Photonics News, 23, 30 (2012).
[8] G. Gagliardi, M. Salza, S. Avino, P. Ferraro and P. De Natale, “Probing the ultimate
limit of fiber-optic strain sensing,” Science 330, 1081 (2010).
[9] D. W. Allan, “Statistics of atomic frequency standards,” Proceedings of the IEEE, 54,
221 (1966).
94
Chapter 6: Other applications of slow-light FBGs
So far we have discussed how we can achieve very high group delays in an apodized FBG and
how we can use this kind of FBGs as strain sensors with unprecedented strain resolutions that
can resolve the fiber’s extremely small thermodynamic phase noise. In this chapter we discuss
other applications for this kind of slow-light FBGs [1]. Since structural slow light is often
associated with an increase in optical energy density, this research can also be directed
towards devices offering strong three-dimensional confinement. This is particularly important
to couple light efficiently to quantum emitters, of interest for low-threshold lasers [2], cavity
quantum electrodynamics [3], and enhanced nonlinear interactions [4]. These applications
generally benefit from a high Purcell factor (or cooperativity), defined as [3,5]:
f = 3λ3
4π 2n3QVm
(6.1)
where n is the medium’s refractive index, Q is the resonance quality factor, Vm is the mode
volume, and λ is again the wavelength in vacuum. Achieving a high Purcell factor therefore
requires a high Q factor, or equivalently a large group delay, in a small mode volume, which
previous chapters have shown is now readily achievable in the strong FBG that have emerged
from this work.
Purcell factor have been measured in several types of nanophotonic structures. A low value
(1.2) was observed in a photonic-crystal microcavity embedded with InAs/InP quantum wires
[6]. In a GaP microdisk a Purcell factor of 13 was reported [7]. Although these devices are
valuable because they involve materials with interesting optical properties, they tend to
produce lower Purcell factor because of higher loss. Larger Purcell factors are possible in
fiber-based devices because of the very low propagation loss of fibers around 1.5 µm. These
devices are also advantageous because light can be coupled into them easily and efficiently. In
a very short free-space Fabry Perot (FP) resonator consisting of two high reflectors deposited
on the curved tip of two single-mode fibers, a Purcell factor of 206 was achieved [8]
(calculated from Eq. 6.1 and experimental values supplied in [8]). This device, however,
requires extreme care in mirror surface preparation, and it is very sensitive to misalignment.
95
All-fiber devices are preferable for practical applications because of their immunity to
misalignment. A Purcell factor of 29.8 was measured in an all-fiber FP made of a tapered fiber
placed between FBGs [9]. This result stemmed mostly from the very strong confinement
afforded by the taper’s 500-nm waist, the downside being a fragile and less practical device
for some applications. A more appealing approach is strong FBGs, because when designed
properly they exhibit large Q factors [10,11]. FBGs are also much more robust than fiber
tapers and free-space FPs, and simpler to fabricate. An intensity enhancement of 45 was
reported in a π-shifted FBG [12], and 800 in an apodized FBG designed to exhibit slow-light
resonances [13].
In this chapter, we investigate theoretically and demonstrate experimentally that much
stronger confinements and higher Purcell factors can be obtained in apodized femtosecond
FBGs fabricated in deuterium-loaded fibers, as described in Chapter 2. Because of their large
index modulations (>3x10-3), low internal loss (as low as 0.02 m-1), and strong apodization,
these FBGs act as FP resonators and support slow-light resonances with very high group
indices, as demonstrated in Chapter 2. They should therefore also exhibit large Purcell factors
[1]. They are particularly attractive because they can be written in any fiber material, e.g.,
chalcogenide [14] or bismuth oxide [15], which would lead to interesting low-power
nonlinear-optic applications in mm-scale devices. They can also be fabricated in fibers doped
with PbS quantum dots [16] for low-threshold laser and amplifier applications. These gratings
can be written through the jacket, hence they retain the fiber’s mechanical strength [17], of
interest to many applications. This chapter begins with a numerical study showing that for
their current loss and index modulation, if their length is optimized in the range of 5–6 mm
these FBGs can support intensity enhancements in excess of 2000 and Purcell factors as large
as ~40 [1]. These predictions are verified experimentally with two FBGs with near optimum
length (~4–5 mm). In both gratings we report observing for the first time the fundamental
slow-light mode. This mode has a measured Purcell factor of 38.7 in the first grating, and an
intensity enhancement in the second grating of 1525, to our knowledge both new records for
an all-fiber device.
96
6.1 Optimizing slow-light FBGs for high Purcell factor and intensity
enhancement
The FBGs were written through the jacket in a single-mode SMF-28 compatible germano-
silicate fiber loaded with deuterium, as detailed in Chapter 4. In brief, the beam of an 806-nm
mode-locked Ti:sapphire laser was focused with an acylindrical lens through a phase mask
onto the fiber. The measured pulse energy incident on the lens was 75 µJ, high enough to
produce type-I FBGs at a fairly high writing speed while remaining below the threshold of
type-II FBGs (~150 µJ) [18]. The phase mask had a pitch of 1070 nm (Bragg wavelength of
~1550 nm).
In a femtosecond FBG the Q factor and group delay of the resonances have a complex
dependence on the shape, width, and peak magnitude of the ac and dc index-modulation
profiles, and on the loss profile as described in Chapters 2 and 4. As we discussed in Chapter 4
the index-modulation profiles themselves depend on the laser beam width and fluence, and on
the nonlinear processes relating this fluence to the photo-induced index change. To achieve
the lowest possible mode volumes, we investigated only short FBGs, i.e., fabricated with a
stationary beam. As shown in Chapter 4 the measured transmission and group-delay spectra of
these FBGs are well predicted by an ac index-modulation profile of the form
Δnac (z) = Δn01
1+ exp(−af (z)+ b)−
11+ exp(b)
#
$%
&
'( (6.2)
where the constants a and b model the nonlinear response of the index change to the laser
intensity, and Δn0 is a constant scaling factor. The function f(z) is the laser intensity profile,
which is a Gaussian with a FWHM W
f (z) = exp(−4 ln(2) z2
W 2 ) (6.3)
97
where z is the location along the FBG measured from the center. The dc index-modulation
profile is Δndc(z) = aΔnac(z), where a is a constant. Finally, the power loss coefficient is
assumed to have the same z dependence, with a maximum value (at the center) γmax.
To make headway through this large parameter space, we first modeled an FBG with the
parameter values of [11], which is representative of the best strong FBGs we currently
produce (Δn0 = 2.4×10-3, a = 0.135, b = 1.6, a = 0.667, γmax = 0.19 m-1). In these simulations,
the beam width W was the only free parameter, and it controlled the grating length. We then
asked the question of what beam widths produce the largest Purcell factor and the largest
enhancement. The width was varied from 1 to 10 mm. For each value the transfer-matrix
model described in Chapter 4 was used to calculate numerically the group delay for the
fundamental mode (the resonance closest to the bandgap), along with its peak transmission.
These simulations required dividing the FBG in segments containing many index-modulation
periods each, and therefore multiplying a modest number of matrices (~100). Calculating the
mode volume and intensity enhancement required computing the electric-field distribution
along the FBG, as done for a stack of thin films [19]. Each period was then divided into 50
segments. These simulations were far more computer intensive, requiring of the order of
100,000 matrices per mm of grating.
The predicted dependence of the Purcell factor on the laser beam width is plotted in Fig. 6.1a
(solid black curve). As the beam width (and hence the FBG width) is increased, the Purcell
factor first increases, because the reflectivity of the FBG increases and hence the Q factor
increases rapidly (for the same reason that the finesse of an FP increases when the mirrors’
reflectivities are increased), while the mode volume increases more slowly. When the width is
increased too much, the grating is longer and lossier, the Q factor decreases, and so does the
Purcell factor. There is therefore a beam width that maximizes the Purcell factor, as shown in
Fig. 6.1a. This maximum Purcell factor occurs for W ≈ 6.4 mm and is equal to ~41.4. The
intensity enhancement shows a similar dependence (solid red curve in Fig. 6.1a), for the same
physical reasons: there is a beam width (4.5 mm) that maximizes the intensity enhancement, to
a value of ~2180, nearly three times larger than previously reported [13]. As the beam width is
increased from zero the transmission at the center of the resonance (solid curve in Fig. 6.1b) is
nearly constant at first, then it decreases monotonically for widths above ~4 mm. For most
applications, a low transmission is undesirable because it leads to deterioration of the noise in
98
the output signal. As a compromise between a larger Purcell factor/field enhancement and a
lower transmission, we opted to fabricate FBGs with a beam width of 5 mm.
Figure 6.1. Simulated dependence on the laser beam width of (a) the Purcell factor and intensity
enhancement, and (b) the transmission of the best slow-light resonance of a saturated FBG, evaluated at
the peak of the slowest resonance. Solid curves simulate an FBG with the parameters of [11], and
dashed curves the experimental FBGs.
6.2 Fabrication and characterization of two FBGs optimized for high
Purcell factor and intensity enhancement
Two FBGs were fabricated. The laser delivered ~34-fs pulses at 1 kHz with a peak intensity of
~5x1012 W/cm2 and a beam width W = 5 mm. The writing time was 300 s. Both FBGs were
saturated (an increase in fluence would have led to an insignificant increase in peak index
modulation). For each FBG the transmission and group-index spectra were measured as
described in section 4.3, then the grating was annealed at 210°C, 220°C, 230°C, 240°C, and
250°C for 30 minutes each, and the two spectra re-measured after each step. The spectra
measured after annealing at 250°C are shown in Fig. 6.2. The pattern of resonances is similar
to earlier observations in strong FBGs [10,11,20,21,22]. After each annealing step all
transmission and group-delay resonances were found to have increased. Between 240°C and
250°C the group delay of the slowest resonance (1550.15 nm, see Fig. 6.2b) stopped
increasing, while the transmission resonance slightly increased. Based on previous experience,
this behavior indicated that further annealing would have degraded the group delay of this and
other resonances, and hence the Purcell factor. Annealing was thus discontinued at 250°C. All
results reported for this FBG pertain to this temperature.
99
The measured peak group delay of the slowest resonance is 22 ns (see Fig. 6.2b). Its Q factor,
obtained from its measured FWHM linewidth (0.11 pm), is 1.33×107. Experimental evidence
that this resonance is the fundamental mode is that although it has a reasonable transmission (–
15 dB, see Fig. 6.1b), no other resonance appeared on its long-wavelength side as annealing
proceeded. In contrast, in longer FBGs additional resonances continue to appear as the
annealing temperature is gradually increased (and the loss gradually decreases). Fitting these
spectra to a model, presented in the next paragraph, also confirms that this resonance is the
fundamental mode.
Figure 6.2. Measured and fitted (a) transmission and (b) group-delay spectra of the first FBG. Inset
shows its inferred index-modulation profiles.
The dashed red curves in Figs. 6.2a and 6.2b are the simulated spectra obtained by best fitting
Eqs. 6.2 to the measured spectra, as described in Chapters 2 and 3. The beam width W was
taken to have the experimentally measured value (5 mm). The ac and dc index-modulation
profiles that produced these fits are plotted in the inset of Fig. 6.2b. The parameter values of
these profiles, fitted to obtain the best visual match to the measured spectra, are
Δn0 = 3.3×10-3, a = 3.75, b = 2.289, and a = 0.689. The fitted loss is γmax = 0.44 m-1. The
higher loss compared to the FBG of [11] (0.19 m-1) is consistent with the higher index
modulation (2.38×10-3 vs. 1.98×10-3). The measured and predicted spectra agree reasonably
well, especially the amplitude, linewidth, and group delay of the peaks, which gives credence
to the fitted profiles. The FWHM of the inferred index profiles (4.5 mm) is a little narrower
than the beam width (5 mm) because of the nonlinear response of the fiber index to the
fluence, which is taken into account in the sigmoid of Eq. 6.2 but likely with not quite the
same intensity dependence as in the actual FBG.
100
The mode volume is Vm = Aeff Leff, where Aeff is the area of the fiber transverse mode and Leff
the effective length of the FBG’s longitudinal mode. The mode area is the area integral of the
intensity across the fiber [23]
Aeff = 2π exp −2 2r /D( )2( )r dr = πD2
80
∞
∫ (6.4)
where r is the radial direction in the fiber’s cylindrical coordinates and D is the 1/e field
diameter of the transverse fiber mode. For the SMF-28 fiber, D = 10.5 µm, which gives
Aeff = 43.3 µm2. The mode effective length was obtained by calculating the intensity
distribution of the fundamental mode along the z axis of the FBG, as described in relation to
Fig. 6.1 and using the inferred index-modulation profile (inset of Fig. 6.2b). Figure 6.3 plots
the envelope of the amplitude of the intensity distribution, normalized to the input intensity,
for this mode (red curve). (The actual intensity profile has a high-frequency sinusoidal
variation that follows the FBG’s index modulation.) The mode is concentrated symmetrically
near the center of the FBG. Its effective length is obtained by integrating numerically this
intensity profile along z. This gave a value of Leff = 750 µm, much shorter than the FBG
length, as expected. The mode volume is then Vm = Aeff Leff = 3.28×104 µm3. Inserting this
value and the measured Q factor (1.33x107) in Eq. 6.1 gives a Purcell factor of 38.7. This
compares favorably to the Purcell factor of ~30 observed in a bitapered fiber FP [9], since the
device reported here is simpler, smaller, significantly more robust, and much easier to
fabricate.
Figure 6.3. Calculated distribution of the intensity distribution along the FBG for the three lowest
modes.
101
For comparison, we also plotted in Fig. 6.3 the intensity distribution of the next two
longitudinal modes. The maximum intensity in the fundamental mode is ~1000 times higher
than the input intensity. The second mode exhibits two maxima, and its maximum intensity is
slightly higher (~1100), as observed and explained in [13]. Specifically, the second mode has
two lobes because as we explained in Chapter 4 it corresponds to a longer effective FP, and
the second mode in an FP has two maxima too. The maximum intensity is higher because, as
explained in Chapter 4, this peak corresponds to an effective FP with lower reflectivities since
the Δnac is smaller. The third mode has three maxima, all about a third as strong as the
fundamental mode. Higher order modes have a longer effective length, and a lower Purcell
factor.
Since the solid curves in Fig. 6.1 were generated for the parameter values of a different FBG,
these curves were re-simulated using the values of Δn0, a, b, a, and γmax inferred from our
measurements. The new simulations (dashed curves in Fig. 6.1) show similar trends, with only
slightly shorter optimum FBG widths and slightly lower maxima, for both the Purcell factor
and the intensity enhancement. The top vertical axis is the FWHM of the FBG (which is
always a little shorter than the beam width). The value of the FBG’s Purcell factor inferred
from measurements (38.7, filled circle in Fig. 6.1a) is in good agreement with the predicted
value (31.8, open circle) predicted for the actual FBG length of 4.5 mm (see inset of Fig.
6.2b). These two values differ because the fitted resonance predicts a group delay (and
therefore a Q factor) ~20% lower than the measured value (see Fig. 6.2b).
The second FBG was fabricated under nominally identical conditions. However, it was
expected to differ slightly because of small differences in the alignment of the beam with
respect to the fiber, and in the amount of time the D2-loaded fibers were left at room
temperature before the FBGs were written. The second FBG was annealed up to 300°C in
steps of 100°C for 30 minutes each. Compared to the first FBG, its inferred index-modulation
profile has a similar shape, its maximum ac index modulation is slightly higher (2.43x10-3)
and its peak loss is the same (0.44 m-1). It is also ~20% shorter (3.9 mm), likely due to the
aforementioned differences. Its measured Q is 7.3x106 (lower by ~2). The reason is that it is
shorter, hence its reflectivity is lower, and its fundamental mode experiences a higher effective
round-trip loss. Its mode volume, calculated from its inferred index profile, is 3.2×104 µm3,
almost the same as the first FBG. Its measured Purcell factor is 21 (solid square in Fig. 6.1a),
102
which is lower by ~2, because its Q factor is half as large. Because this FBG has sensibly the
same loss and index modulation as the first FBG, its Purcell factor and intensity enhancement
are also predicted by the dashed curves in Fig. 6.1a. The predicted Purcell factor (from Fig.
6.1a with an FBG length of 3.9 mm) is 24.2 (open square). It is slightly higher than the
measured value (by 15%), again because the fitted group delay is 15% higher than the
measured group delay. Its intensity enhancement is 1525. It is higher than for the first FBG
because the fundamental mode is closer to a matched FP mode.
6.3 Conclusions
In conclusion, numerical simulations of strong FBGs using index profiles and loss inferred
from measurements show that the slow-light resonances of these devices can exhibit large
intensity enhancements and Purcell factors when their length is optimized (5–6 mm). These
predictions are verified experimentally in two short type-I FBGs fabricated in D2-loaded fiber
and annealed to reduce their loss. For each grating the measured transmission and group-delay
spectra are fitted to a model to infer the index-modulation profiles and loss, from which the
Purcell factor and intensity enhancement are calculated. The fundamental slow-light mode
was observed in both gratings, for the first time. The fundamental mode of the first grating has
a measured Purcell factor of 38.7. The intensity enhancement of the second grating is 1525.
Both values establish new records for an all-fiber device.
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104
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105
Chapter 7: Conclusions and future work.
In this thesis we studied slow-light FBGs mainly to develop a strain sensor with
unprecedented resolution, and to observe and quantify thermal phase noise in a passive FBG
for the first time. In order to meet these goals we first studied theoretically the magnitude and
frequency dependence of the thermal phase noise in Fabry-Perot resonances. This study
showed that the phase noise is proportional to the group index of the resonance, and inversely
proportional to the square root of the FBG length (when expressed in units of strain). Second,
we showed how we can increase the group index theoretically and experimentally in an FBG
in order to achieve high strain sensitivity in short FBGs, a needed condition to observe thermal
phase noise. Specifically, we were able to increase the group delay by a factor of ~8, to 42 ns
in a 12.5-mm grating, by using a strongly apodized FBG written with a near-infrared
femtosecond laser in a deuterium-loaded fiber, then thermally annealed. Third, we reduced
several sources of noise in our experimental sensor and setup: we probed the sensor with a
laser that had a much narrower linewidth (<200 Hz) and therefore a significantly lower
frequency noise; we used a low-noise photodetector; and we isolated our sensor from the
environment by placing it in an anechoic enclosure.
These various improvements, along with our theoretical study of thermal phase noise in FP
resonances, enabled us to observe for the first time thermal phase noise in a passive FBG. The
device was 5-mm long, and it was dominated by phase noise between 1 kHz and 10 kHz. The
phase noise was found to be exceedingly small, of the order of ~70 fε/√Hz at 5 kHz, or
equivalently a single-pass phase noise of only 1.6 nrad/√Hz, in perfect agreement with the
magnitude predicted by our model. This measurement helped resolve a longstanding
controversy about the magnitude of the phase noise in a multi-pass interferometer such as a
Fabry-Perot.
To further validate this observation, and to demonstrate a slow-light FBG significantly more
sensitive to strain than reported earlier by our research group, we developed a second FBG
sensor with a longer length (20 mm) so that its phase noise would be much lower, as
anticipated by our model. As a result of this noise reduction, this FBG was found to be the
most sensitive passive FBG strain sensor ever reported, with a measured MDS of 110 fε/√Hz
106
at 2 kHz and 30 fε/√Hz at 30 kHz. Its noise was limited by the frequency noise of our laser,
like its predecessor, but at a much lower level. This is about one order of magnitude smaller
than the previous reported MDS at high frequencies, and two times smaller at 2 kHz. Both
figures represent new world records. Figure 7.1 shows the evolution of MDS over the years; it
is the same as Fig. 1.2 but with this new MDS data point included (far left blue arrow). When
this work started the record MDS in a single FBG was 280 fε/√Hz. Also, prior to this thesis
strain resolutions at the level reported in this work (around or smaller than 100 fε/√Hz) had
been attained only in active fiber sensors. One of the main goals of this thesis—to match
and/or exceed the resolution of active FBG sensors—has been met. Strain sensors based on
strong FBGs and ultra-narrow lasers now outperform active fiber strain sensors.
Figure 7.1 Evolution of the MDS of selected strain sensors using FBGs reported over the years.
Such sensitive fiber sensors can be used to sense other quantities, such as humidity or acoustic
waves, that have very weak but measurable effects on the fiber’s properties and hence on the
spectrum of the FBG. To demonstrate this potential, we tested the performance of the same
20-mm FBG as an acoustic sensor by placing it in an acoustically isolated chamber and
subjecting it to sound waves from a speaker. The FBG acoustic sensor was found to have a flat
band between roughly 500 Hz and 2 kHz, with an average minimum detectable pressure
(MDP) of 600 µPa/√Hz over this range. Between 3 kHz and 6 kHz the average MDP was as
low as ~50 µPa/√Hz. These values are about one order of magnitude lower than the best
previously reported values for an FBG acoustic sensor. They are still on average about a factor
of ~100 higher than reported in diaphragm-based fiber FP sensors, but this result was achieved
in a much more robust device that does not involve a fragile diaphragm as a transducer. These
preliminary results are very promising. They can readily be improved significantly in the
107
future, for example by specifically designing the fiber in which the FBG is written to be more
sensitive to acoustic waves.
Finally, we showed that with such FBGs a high Purcell factor and intensity enhancement can
be achieved as a result of the strong temporal and spatial confinement that these devices offer.
Specifically, after optimizing the length of our FBG, we demonstrated a Purcell factor of 38.7
(the highest in an all-fiber device) and an intensity enhancement of 1545, which is the highest
in an FBG. Both numbers are in good agreement with the values predicted from our model.
The development of ultrasensitive strain sensors reported in this thesis could be extended in at
least two directions, namely reducing the MDS of slow-light FBGs at high frequency beyond
what we have achieved, and demonstrating MDSs at the fε/√Hz level at much lower
frequencies (below ~10 Hz). In order to achieve these goals, the frequency noise of the laser
needs to be further reduced. This can be done by switching to a laser with an even smaller
frequency noise (recently new laser from OEwaves claims to have developed a commercial
laser with a sub-Hz linewidth), or by locking our current laser to an ultra-stable reference
frequency, i.e., an ultra-stable cavity, a frequency comb, etc., using the Pound-Drever-Hall
technique. With this technique, it has been demonstrated that sub-Hz linewidths are
achievable. With such ultra-narrow linewidths, the FBG phase noise will become dominant, as
we have demonstrated, and it will have to be reduced. This can be accomplished in one of at
least three ways: (1) decreasing the FBG temperature, at least for a proof of concept; (2)
increasing the FBG length; and (3) canceling out the thermal phase noise using signal
processing techniques. The first two methods take advantage of the physics of the thermal
phase noise and reduce it directly. The third one would involve measuring the thermal phase
noise independently, then subtracting it from the signal. This can be achieved for example by
having two identical FBGs and only one subjected to the strain. Then by subtracting their
outputs either optically or electronically the signal can be isolated from the noise. More
sophisticated techniques can be used as well, such as encoding the output signal in such a way
that it can be separated from the noise, e.g., the sensor is susceptible to strain only for a small
period of time whereas the phase noise is always present. Again the phase noise would be
reduced by using two measured signals, one with and one without the strain, and subtracting
them.
108
Last but not least, we can take advantage of these exceptional properties to enable whole new
applications for these FBGs by writing FBGs in fibers doped with active materials such as a
rare-earth ion or quantum dots to produce ultra-narrowband lasers. These FBGs can also be
written in highly nonlinear fibers to enhance nonlinear effects such as four-wave mixing, self-
phase modulation, and cross-phase modulation. The devices with high Purcell factors that we
have developed can be used to fabricate low-threshold lasers as well. These new FBGs can be
very useful in optical communication and in all-optical information processing. Finally, these
FBGs can be used as a platform to study more fundamental physics involving light-matter
interactions or weak phenomena that could not be observed before. For example, in FBGs
cooled to cryogenic temperature one can investigate whether the thermodynamic description
of thermal-phase noise developed by Wanser and in this work breaks down at low
temperatures, and how the incident light affects the thermodynamic state of the fiber.
109
Appendix: Power spectral density on resonance
In this appendix we derive the PSD of the phase noise for the signal transmitted by a Fabry
Perot interferometer on resonance (Eq. 3.15), discussed in Chapter 3, starting from the
expression of the intra-cavity field Ec2 (Eq. 3.9) and assuming that the autocorrelation function
of the phase noise φ(z, t) is described by Eq. 3.5. Making the generally valid assumption that
the phase noise terms are small, the exponential terms in Eq. 3.9 can be expanded to first order
in the phase noise to give
Ec2
E0
= t1e− iΦ0 e
−i φ (z,t− zv
0
L
∫ )dz
{1+ Rn
n=1
∞
∑ [1− i fmm=1
n
∑ ]} (A1)
where fm = (φ(z, t − 2mL − zv
)+φ(z, t − 2mL + zv
))dz0
L
∫ and R = r1r2e− i2Φ1 . Summing the first two
terms (independent of phase noise) in the {} term gives
Ec2
E0
= t1e− iΦ0 e
−i φ (z,t− zv
0
L
∫ )dz
{1
1− R− i Rn
n=1
∞
∑ fmm=1
n
∑ } (A2)
With straightforward manipulation, the product of the two series can be re-written as a double
sum of a product
Ec2
E0
= t1e− iΦ0 e
−i φ (z,t− zv
0
L
∫ )dz
{1
1− R− i Rm fm
m=1
∞
∑ Rn
n=0
∞
∑ } (A3)
Since this last infinite series is equal to (1 - R)-1, it can be factored out of the bracket
Ec2
E0
=t1e
− iΦ0
1−Re−i φ (z,t− z
v0
L
∫ )dz
{1− i Rm fmm=1
∞
∑ } (A4)
Finally, expanding also the exponential term to first order in the phase noise and neglecting
110
higher order terms in the product gives
Ec2
E0
=t1e
− iΦ0
1−R(1− i φ(z, t − z
v0
L
∫ )dz)− i Rm fmm=1
∞
∑ ) (A5)
The phase fluctuations resulting from random temperature fluctuations are given by the
argument of Ec2/E0. In general, because R is complex neither this argument nor its
autocorrelation have a simple closed form, and there is no simple analytical expression for the
PSD. However, in the particular case of a signal on resonance, Φ1 is a multiple of π and
R = r1r2 is real. The argument of Ec2/E0 (see Eq. A5) is then simply
θ (t) = φ(z, t − zv0
L
∫ )dz)+ (r1r2 )mm=1
∞
∑ ⋅ (φ(z, t − 2mL − zv
)+φ(z, t − 2mL + zv
))dz0
L
∫ (A6)
Using the short-hand notation
φm± (z) = φ(z, t −2mL ± z
v)
φm± (z ') = φ(z ', t −2mL ± z '
v+ τ )
(A7)
we can write the autocorrelation < θ (t)θ (t + τ ) > as a sum of nine integrals Ik
< θ (t)θ (t + τ ) >= Ikk=1
9
∑ (A8)
given by:
I1 = < φ0+ (z)φ0+ (z ') > dz dz '0
L
∫0
L
∫ (A9a)
I2 = (r1r2 )n
n=1
∞
∑ < φ0+ (z)φn− (z ') > dz dz '0
L
∫0
L
∫ (A9b)
I3 = (r1r2 )n
n=1
∞
∑ < φ0+ (z)φn+ (z ') > dz dz '0
L
∫0
L
∫ (A9c)
111
I4 = (r1r2 )m
m=1
∞
∑ < φm− (z)φ0+ (z ') > dz dz '0
L
∫0
L
∫ (A9d)
I5 = (r1r2 )m
m=1
∞
∑ < φm+ (z)φ0+ (z ') > dz dz '0
L
∫0
L
∫ (A9e)
I6 = (r1r2 )m
m=1
∞
∑ (r1r2 )n
n=1
∞
∑ < φm− (z)φn+ (z ') > dz dz '0
L
∫0
L
∫ (A9f)
I7 = (r1r2 )m
m=1
∞
∑ (r1r2 )n
n=1
∞
∑ < φm− (z)φn− (z ') > dz dz '0
L
∫0
L
∫ (A9g)
I8 = (r1r2 )m
m=1
∞
∑ (r1r2 )n
n=1
∞
∑ < φm+ (z)φn− (z ') > dz dz '0
L
∫0
L
∫ (A9h)
I9 = (r1r2 )m
m=1
∞
∑ (r1r2 )n
n=1
∞
∑ < φm+ (z)φn+ (z ') > dz dz '0
L
∫0
L
∫
(A9i)
For symmetry reasons, in all integrals with odd index the signs in front of z and z' have the
same sign, and they can all be evaluated the same way. The general form of the solution is
Jm±,n± = < φm± (z)φn± (z ') > dz dz '0
L
∫0
L
∫ = Rφ (τ + 2(m − n)Lv
)δ(z − z ')dz dz '0
L
∫0
L
∫
= Rφ(τ + 2(m − n)L
v)L
(A10)
The remaining four integrals (even index), in which the signs in front of z and z’ are opposite,
also of the same form, given by
Jm∓,n± = < φm∓ (z)φn± (z ') > dz dz '0
L
∫0
L
∫ = Rφ (τ + 2(m − n)L ∓ z
v)δ(z − z ')dz dz '
0
L
∫0
L
∫
= ∓v2
Rφ (y)dyτ+2 ((m−n )L/v )
τ+2 ((m−n∓1)L/v )
∫ (A11)
The phase-noise PSD is the Fourier transform of the sum of all nine Ik integrals (Eq. A8).
These individual Fourier transforms can be calculated more expediently by recognizing that
the Fourier transform of the Jm±,n± and Jm∓,n± terms can be expressed as
112
Sm±,n± = S f e2 jω (m−n)L
v
Sm∓,n± = S f e2 jω (m−n)L
v e∓ jω L
v sincωLv
(A12)
Using these relationships and denoting the Fourier transform of Ik as Sk, we have
S1 =F J0+,0+( ) = S f (A13a)
S3 + S5 = r1r2( )nF J0+,n+( )
n=1
∞
∑ + r1r2( )mF Jm+,0+( )
m=1
∞
∑ = 2Sf
1− r1r2 cos2ωLv
1+ r1r2( )2− 2r1r2 cos
2ωLv
−1
$
%
&&&
'
(
)))
(A13b)
S2 + S4 = r1r2( )nF J0+,n−( )
n=1
∞
∑ + r1r2( )mF Jm−,0+( )
m=1
∞
∑
= 2Sf [1+ r1r2( )sinc 2ωLv − 2r1r2sinc
4ωLv
1+ r1r2( )2− 2r1r2 cos
2ωLv
− sinc 2ωLv]
(A13c)
Skk=6
9
∑ = r1r2( )m
m=1
∞
∑ r1r2( )n
n=1
∞
∑ .F Jm−,n+ + Jm−,n− + Jm+,n− + Jm+,n+( )= 2Sf
1+ sinc 2ωLv
$%&
'() r1r2( )
2
1+ r1r2( )2− 2r1r2 cos
2ωLv
(A13d)
Summing all the Sk (Eqs. A13) yields the final expression for the PSD of the phase noise
SFPt = Skk=1
9
∑ = Sf 1+cos 2ωL
v+ sinc 2ωL
v"#$
%&'2r1r2
1+ r1r2( )2− 2r1r2 cos
2ωLv
"
#
$$$$
%
&
''''
(A14)
which is Eq. 3.15.