mechanics and experiments
TRANSCRIPT
Distributed Bragg Grating Sensing-Strain Transfer
Mechanics and Experiments
Graham Ian Duck
December 14, 2000
Thesis submitted in conformity with the requirements for the Degree of Doctor of Philosophy, Graduate Department of Aerospace Studies in the University of Toronto
Copyright O Graham Ian Duck, 2001
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Abstract Distributed Bragg Grating Sensing-Strain Transfer Mechanics and Experi-
ments, Ph. D Thesis by Graham Ian Duck.
The University of Toronto Institute for Aerospace Studies, 200 1.
The optical fiber sensor is a tool by which a measurement of a thermo-mechanical field
inside a mechanical host can be made. It is possible to interrogate the optical fiber sensor
with light- The parameters of the in-fiber sensor determine the response of the sensor to
Lightwave interrogation. Having the sensor response, a reconstruction of the in-fiber structure
can be attempted. Successful reconstruction of the in-fiber structure allows an inference
of the behaviour of the mechanical substrate. The substrate behaviour is ultimately the
information of interest. Extraction of such a measurement may be seen as two sub-problems.
The first problem addresses the ability of the mechanical system to transfer surrounding
material information into the optical fiber sensor. It is not the field values inside the optical
fiber sensor that are particularly interesting, instead, it is the degree to which the in-fiber
field values accurately represent the field values of the surrounding material. It is important
to examine the mechanical system's ability to transfer strain from surrounding material to
optical fiber sensor. Certainly, if strain information is not transferred into the optical fiber,
even an ideal interrogation of the in-fiber sensor cannot report the full strain field of the
mechanical host.
The second issue concerns the sensor itself. The reflectivity characteristics of the loaded
sensor must first be recovered experimentally, then interpreted in meaningful way. The
experimental arrangement to recover the sensor response and interpretation of this data
constitute the optical measurement problem.
This thesis examines the two problems.
Acknowledgments
First, I would like to thank my supervisor Dr. Raymond Measures for including me in his
dynamic group and allowing me the freedom to grow both aicademicdy and professionally
in the field of fiber optic sensing, and for providing me a stepping stone into the important
field of telecommunications.
My senior colleagues Sharon Huang, Michel Leblanc and Myo Ohn offered advice and
time without limit. Without their input and support, this work would not Have been possible.
I would also like to extend sincere thanks to colleagues Tino Alavie, Rob Maaskant, Shai
Berger, Jason Bigue, Murray Lowery, Ning-Yao Fan, Trent Coroy: and Paul kldvihill.
I extend deepest gratitude to my new colleagues, Henry Postolek, David Moss, Martin
Matthews, Dan Grobnic, Jeffrey KalbBeisch, Dominique Brichard, Michael Lamont and
David Markin for their expertise and support.
I would like to thank professors Dr. Rod Tennyson, Dr. Jorn Hansen and Dr. David
Zingg for their friendly guidance and technical tips.
Wholehearted thanks go out to my dear fiiends Rob C. and Carol-lynn D. for their
consistent and humour-filled support - Thanks Eddie V. and Kurt C. for the many hours of musical accompaniment during the
preparation of this document.
Mom and dad, I am not sufficiently articulate to express how lucky I am that I have
parents like you.
Finally, I acknowledge the generous financial support of fSIS Canada for making investi-
gation of this technology possible.
Contents
1 Introduction 1
I . 1 Problem definition and document organization . . . . . . . . . . . . . . . . . 3
. . . . . . . . . . . . . . . . . . . . . . . . . . 12 A note on co-ordinate systems 6
2 Optical description of the fiber Bragg grating 8
2.1 TheBragggrating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Coupled mode equation 10
2.3 T-matrix formulation for grating simulation . . . . . . . . . . . . . . . . . . 17
2.4 Inverse Fourier transform method for low reflectivity gratings . . . . . . . . . 23
2.5 Hypothetical infinitely long and infinitely short grating discussion . . . . . . 31
3 Optical Response of fiber Bragg grating to strain and temperature 33
3.1 f hase sensitivity of fiber optic sensors . . . . . . . . . . . . . . . . . . . . . . 34
3.2 Experimental evaluation of sensitivities . . . . . . . . . . . . . . . . . . . . . 44
4 Interpretation of mechanical interactions: emphasis on strain transfer be-
tween host and fiber sensor 53
4.1 Strain transfer considerations for point sensors . . . . . . . . . . . . . . . . . 54
4.1.1 Shear lag formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.2 Axial strain transfer fiom host to fiber: Finite element models . . . . . . . . 61
4.2.1 Surface bonded sensor model . . . . . . . . . . . . . . . . . . . . . . . 61
4-22 Embedded sensor model . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.2.3 Finite element analysis of linear strain gradient . . . . . . . . . . . . 76
4.3 Transfer of arbitrary strain distribution . . . . . . . . . . . . . . . . . . . . . 92
4.4 Equivalent embedded geometric ratio for s d w e bonded sensor . . . . . . . 98
4.5 Conclusions of mechanical strain transfer analysis . . . . . . . . . . . . . . . 103
5 Demonstration of method and measurements 105
. . . 5.1 Review of methods for interrogating gratings with non-uniform profiles 106
5.1.1 Phase-based spectrum (PSB) strategy . . . . . . . . . . . . . . . . . . 106
5.1.2 Intensity based spectrum (ISB) strategy . . . . . . . . . . . . . . . . 108
5.1.3 Low coherence interferometry . . . . . . . . . . . . . . . . . . . . . . 110
5.2 Implementation of interferometric phase recovery experiment . . . . . . . . . 112
5.2.1 DFB tunable source . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.2.2 Representative grating interrogation . . . . . . . . . . . . . . . . . . . 119
. . . . . . . 5.3 Interferometry-based phase recovery distributed sensing strategy 125
5.3.1 Uneven thermal apparent loading: experiment a1 demonstration of spa-
tial resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
5.3.2 Grating bonded in the vicinity of a circular hole . . . . . . . . . . . . 130
5.4 Delay-based phase recovery distributed sensing strategy . . . . . . . . . . . . 134
5.4.1 Measurement principle . . . . . . . . . . . . . . . . . . . . . . . . . 134
5.5 Conclusions of measurements . . . . . . . . . . . . . . . . . . . . . . . . . . 141
6 Conclusions and contribution summary 146
6.1 Summary of contributions and conclusions . . . . . . . . . . . . . . . . . . . 147
6.1.1 Contributions in understanding the optical response . . . . . . . . . . 147
. . . 6.1.2 Contributions towards understanding the strain transfer problem 147
. . . . . 6.1.3 Contributions in the area of distributed strain measurements 148
6.2 Suggested future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
List of Figures
. . . . . . . . . . . . . . . . . . . . . 1.1 Conceptual flow of optical fiber sensing 5
1.2 Cartesianandcyllndricalcoordinatesusedinthisdocument . . . . . . . . . . 7
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Fiber Bragg grating 9
2.2 Uniform grating reflectivity and phase response (bn = 2 x lo-', L = 5cm.
no = 1.5. h, = 500nm) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3 Uniform grating reflectivity and delay response (bn = 2x10-'. L = 5cm.
no = 1.5. A, = 500nm) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.4 Cascaded uniform gratings model a complex grating . . . . . . . . . . . . . . 24
2.5 Incremental increase of reflectivity linearly related to local coupling strength 25
2.6 Range of reflectivities as predicted by the coupled mode solution and the FT
approximation-d gratings modeled with L = 5cm . . . . . . . . . . . . . . . . 28
2.7 Ratio of maximum reflectivity as predicted by the coupled mode solution and
the FT approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
. . . 2.8 Quality of coupling distribution recovery using Born (FT) approximation 30
3.1 Butter and Hocker Experiment . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2 Change in effective index and periodicity of a grating subject to strain field . 41
3.3 Pull test experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.4 Pinch test experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.5 Pinch test load apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.6 Apparatus to measure fiber stiffness . . . . . . . . . . . . . . . . . . . . . . . 50
4.1 Fiber segment in infinite host under uni-axial tension . . . . . . . . . . . . . . 55
. . . . . . . . . . . . . . . . . . . . . . 4.2 Sum of axial acting forces in the fiber 55
4.3 Swn of axial acting forces in the surrounding material . . . . . . . . . . . . . 56
4.4 Shear lag plots for N = 0.1. N = 0.2. a n d N = 0.3. . . . . . . . . . . . . . . 59
. . . . . . . . . . . . . . . . . . . . 4.5 Common optical fiber sensor installations 60
. . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Model for surface mounted fiber 62
. . . . . . . . . . . . . . . 4.7 Node distribution across Reissner/Mindlin element 66
4.8 Quasi-t hree-dimensional adhesive element node posit ions . . . . . . . . . . . . 68
. . . . . . . . . . . . . . . . . . . . 4.9 Quasi-three-dimensional adhesive element 69
. . . . . . . . . . . . . . . . . . . . . . . 4.10 Axi-symmetric finite element model 73
. . . . . . . . . . . . . . . . . . . . . . . 4.11 Imposed strains and displacements 77
4.12 Equivalent transferred linear strain gradient . . . . . . . . . . . . . . . . . . 78
4.13 Location of lamina modeled as having no stifiess . . . . . . . . . . . . . . . 81
4.14 Shear stress distribution at fiber/surrounding material interface for embedded
. . . . . . . . . . . . . . . . . . . . L - O 82 installation . " = 1.05, 5 = 70, ;; - f Em
4.15 Quality of transfer for surface mounted installation (Vertical axis represents
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . & x 1000) 86
4.16 Gradient elongation. 5. for stiffness ratio of 2 = 14 (Surface mounted) . . 87
4.17 Gradient elongation. 5. for stiffness ratio of = 35 (Surface mounted) . . 88
4.18 Quality of transfer for embedded installation (Vertical axis represents 3 x 1000) 89
4.19 Gradient elongation. u; for stiffness ratio of 2 = 14 (Embedded) . . . . . 90 rf
4.20 Gradient elongation. e. for stiffness ratio of 2 = 35 (Embedded) . . . . . 91
4.21 Illustration of problem: axial strain transfer from host to sensor . . . . . . . 92
4.22 Transfer function. H ( k ) dependence on spatial frequency k . . . . . . . . . . 95
4.23 Transfer of far field sinusoids . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.24 Transfer o f far field Gaussians . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.25 Peak suppression and broadening of Gaussian far field strain profile . . . . . 100
4.26 Surface bonded installation modelled as embebbed installation with equivalent
radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.1 Phase accumulation due to reflection at z of length dz . . . . . . . . . . . . . 107
5.2 Closed loop schematic of[l] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Open loop schematic of[l]
. . . . . . . . . . . 5.4 DBR Source interferometric experiment with lock-in phase
. . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Phase sbifted grating as DFB
. . . . . . . . . . . . . . . . . . . . . . . . 5.6 DFB laser wavelength calibration
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Interferometric experiment
5 -8 Cosine of inferred phase against interferometric reflectivity . . . . . . . . . .
5.9 Reflectivity and coupling distribution experiment and simulation comparison
5.10 Grating of which one half is bonded to an alllminum substrate . . . . . . . . 5.11 Measured reflectivities from sensor partially bonded to an aluminum substrate
subject to thermal load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.12 Apparent strains reported from sensor bonded on alumhum substrate subject
tothermalload . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.13 Grating bonded to an aluminum plate in the vicinity of a circular hole . . .
5.14 Experimental and simulated reflectivities for different loadings . . . . . . . . .
5.15 Measured and analytical strain profiles for grating bonded near circular hole
for different axial loads (upper plot) . Deviation between theory and experi-
ment (lower plot) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.16 Repeated measurements of two different mountings of the same load (left) and
deviations horn elasticity solution (right) . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . 5.17 Phase shift modulation method
5.18 Delay-based phase recovery experiment . . . . . . . . . . . . . . . . . . . . .
5.19 Beam loaded in bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.20 Reflectivity and delay of unloaded and loaded samples . . . . . . . . . . . . .
5.21 Strain profiles recovered from delay based interrogation . . . . . . . . . . . .
5.22 Comparison of two gratings lcm and 5cm of like peak reflectivity subject to
Like localized strain perturbation . . . . . . . . . . . . . . . . . . . . . . . . .
Nomenclature
A: area (FEM context)
A,: cross sectional area of optical fiber coating
Af: cross sectional area of optical fiber
aij: individual elements of stress, moment, and transverse resultant matrix
a,: m-th Fourier co-efficient of 672 series
A,: Amplitude envelope of m-th optical mode
a (2): backward-propagating envelope with electric field phase dependence
A (2): spatially varying envelope of backward moving wave
a: angle of polarization interrogation (in context of Jone's calculus)
a f : fiber thermal expansion co-efficient
cr j: thermal eG-ion co-efficient tensor
g: magnetic flux density vector
[B]: shorthand for [L] [a] matrix product (FEM context)
bij: individual elements of stress, moment, and transverse resultant matrix
b (2): forward propagating envelope with electric field phase dependence
B (2) : spatially varying envelope of forward moving wave
P,: propagation constant of m-th optical mode
C: stress-optic co-efficient
[Dl : stiffness matrix
5: electric flux density vector
df: fiber diameter
Djj: stiffness tensor
dij : individual elements of stress, moment, and transverse result ant matrix
(6): generalized displacement vector (FEM context)
{6Ie: discrete generalized element nodal displacement vector (FEM context)
6,B: detuning between optical wave and Bragg resonance
&: Kronecker delta function
672: index modulation
t?: electric field vector em, ,. . - t o td strain tensor
eh-i: mechanical strain tensor
{e } : generalized strain vector (FEM context)
Ec: elastic modulus of optical fiber coating
Ef: elastic modulus of optical fiber
Em: elastic modulus of surrounding material
~ f : electric permittivity of optical fiber
€9: electric pennit tivity due to grating corrugation
~ o : vacuum electric permittivity
Ei j : electric permittivity tensor
€ f i j : electric permittivity tensor of optical fiber
c g i j : electic permittivity tensor due to grating corrugation
q: transform plane co-ordinate (FEM context)
{Fig: global nodal force vector (FEM context)
f : superscript indicating an in-fiber quantity
{F)': element nodal force vector (FEM context)
f-: RF modulation frequency
[a]: interpolation function matrix (FEM context)
(@] ': element interpolation function matrix (FEM context)
& i-th interpolation function (FEM context)
g: accelearation due to gravity
G,: surrounding material shear modulus
y: phase offset (interferometric and polarimetric experiment contexts)
g: magnetic field vector
H (k): transfer function describing strain transfer of spatial frequency k
i: drive current
I: optical intesity
lij : identity tensor
[J]: Jacobian matrix (FEM context)
k: spatial frequency (multiple contexts)
K: complex coupling strength
[K] e: element st f i e s s matrix (FEM context )
[KIg: global stiffness matrix (FEM context)
K : coupling strength
L: length (used in multiple contexts)
[L] : operator matrix (FEM context)
A: optical wavelength
A,: nominal optical wavelength
A,: eigenvalue of shear lag equation-physically interpreted as spatial wavelength
A,: spatial period of m-th Fourier component used in reconstruction of coupling distri-
bution
A: period of fiber Bragg grating index modulation
rn: mass
m: superscript indicating a surrounding material quantity
[MI: moment resultant matrix (FEM context)
n: index of refraction
n: effective index of refraction
N: shear lag parameter
[N]: stress resultant matrix (FEM context)
qij: impermeability tensor
u: Poisson's ratio
urn: surrounding material Poisson's ratio
vf: fiber Poisson's ratio
p: subscript indicating transverse polarization
P: mechanical line load in force per meter
p: polarization density vector
[PI : lower virtual node specification matrix (FEM context)
pijkl: full Pockei's strain-optic tensor
pl l : reduced index Pockel's co-efficient
p12: reduced index Pockel's co-efficient
$: phase (used in several contexts)
Q,: m-th modal solution of wave equation
[Q]: transverse shear resultant matrix (FEM context)
r : radial position (cylindrical co-ordinates)
r': position vector
r (bp) : electric field proportional reflectivity
R (w) : power proportional reflectivity
r f : fiber radius -
[R]: upper virtual node specification matrix (FEM context)
s: eigenvalue of coupled mode equation
S.: phase sensitivity to axial strain (gauge factor)
ST: phase sensitivity to temperature
ui: stress tensor
in-fiber axial normal stress
in-fiber normal stress averaged over cross section
in-fiber axial radial stress
in-fiber radial stress averaged over cross section
in-fiber azimuthal normal stress
in-fiber azimuthal normal stress average over cross section
surrounding material axial normal stress
surrounding material normal stress averaged over cross section
(0): generalized stress vector (FEM context)
t: time
T: temperature
zj: T-matrix entries - xj: complex conjugate of T-matrix entries
8: angle of polarizer (in context of Jone's calculus)
8: azimuthal position (cylindrical co-ordinates)
8: m-th transverse mode shape of optical wave
Oi: i-th rotation-like mid-plane variable of Reissner/Mindlin formulation
{ T ) : generalized surface traction vector (FEM context)
r: time delay
rh: in-fiber shear stress
TZ: surro~~lding material shear stress
q-: i-th cartesian displacement
ii$ i-th midplane cartesian displacement
u: radial displacement (cylindrical co-ordinates)
po: vacuum magnetic permeability
v: azimuthal displacement (cylindrical cc>-ordinates)
V: fringe visibility
V: volume (FEM context)
w : axial displacement (cylindrical cuordinat es )
W (A): wavelength dependent window function
w : optical frequency (in radians/second)
x, : m-t h cartesian dimension
Cij: normalized sensitivity of refkactive index to temperature
X i j : electric susceptibility tensor
<: transform plane co-ordinate (FEM context)
z: axial position (cylindrical co-ordinates)
Chapter 1
Introduction
Fiber optic structural sensing is now widely accepted as a high resolution, versatile strat-
egy for extracting therm+mechanical information kom mechanical structures[2] [3]. Butter
and Hocker[4] first derived and demonstrated a phase strain relationship between the load
state of a Michelson interferometer and an optical phase response seen as a n interference
pattern on a screen. Since this initial demonstration the field has evolved considerably. This
work intends to contr ib~te to the evolution of fiber optic sensing, in the particular area of
distributed Bragg grating sensing.
The availability of components for the field of optical fiber sensing is largely due to
the telecommunications community. The confluence of technological advances in optimizing
the attenuation and dispersion properties of optical fibers and the advent of a laser source
to provide coherent light at the appropriate wavelength range supported, and continues to
suppoa, a worldwide research interest in teletommunications [5]. The materials and devices
employed in thesis for purposes of structural sensing: optical fiber, the in-fiber Bragg grating,
and the various light sources owe their development to telecommunications research.
Fiber sensors deployed as structural sensors offer several advantages over conventional
resistive foil strain gauges[6]. The small diameter of the sensor lends itself well to structural
integration-in particulax in the field of composite materials. Within fiber optic sensing,
the in-fiber Bragg grating has emerged as a leading alternative for structural sensing[7] [8].
The dominant advantage of the fiber Bragg sensor is that its response to thermo-mechanical
perturbations is seen in a n absolute wavelength change; this contrasts with interferometric
sensors whereby relative int erferometric phase excursions do not distinguish between, for
example, tensile or compressive strain perturbations of the same magnitude. The wavelength
encoded shift has been demodulated[9] by a filter having a near-linear wavelength response.
This technique provides far successful point measurements.
A second, very significant, advantage of the Bragg grating sensor is demonstrated later:
the full spectral response (including phase) of the Bragg grating encodes the full distributed
strain field inside the sensar[lO]. Where the point sensing strategy reports a measurement in-
dicating the average thermo-mechanical perturbation over the sensor's length, the distributed
approach attempts to recover the t herm~mechanical perturbation as spatially varying func-
tion over the spatial extent of the sensor. The potential of this implementation of distributed
Bragg grating sensing has been investigated in Msious modalities through the 1990's a t the
Fiber Optic Smart Structures laboratory at the University of Toronto Institute for Aerospace
Studies and at other institutions.
This work continues the exploration of the distributed Bragg grating sensing and seeks
to make a contribution t o this field.
1.1 Problem definition and document organization
The optical fiber sensor is a tool by which a measurement of a thermo-mechanical field
inside a mechanical- substrate or host can be made. It is possible to interrogate the optical
fiber sensor with Light. The parameters of the in-fiber sensor determine the response of the
sensor to lightwave interrogation [ll] [12]. Having the sensor response, a reconstruction of
the in-fiber structure can be attempted. Successful reconstruction of the in-fiber structure
allows an inference of t he behaviour of the mechanical substrate. The substrate behaviour is
ultimately the information of interest. The progression of these relationships constitutes the
fundamental problem addressed by this thesis. In this document: this problem is organized
as two broad subproblems:
0 the mechanical strain transfer problem and?
the optical measurement problem.
The &st problem addresses the ability of the mechanical system to transfer surrounding
material information into the optical fiber sensor. It is not the field values inside the optical
fiber sensor that are particularly interesting, instead, it is the degree to which the in-fiber
field values accurately represent the field values of the surrounding material. It is important
to examine the mechanical system's ability to transfer strain from surrounding material to
optical fiber sensor. Certainly, if strain information is not transferred into the optical fiber,
even an ideal interrogation of the in-fiber sensor cannot report the full strain field of the
mechanical host. This question is addressed in literature for the case of a single ended point
sensor by LeBlanc 1131. The reference brings attention to the fact that strain mismatch is
expected between sensor and substrate.
The second issue concerns the sensor itself. The reflectivity characteristics of the loaded
sensor must first be recovered experimentally, then interpreted in meaningful way The
experimental arrangement to recover the sensor response and interpretation of this data
constitute the optical measurement problem. The background of this problem is provided in
large part by the candidate's predecessors at the University of Toronto. Huang et. al. [lo] describe the dependence of spectral characteristic on the internal strain state of the fiber
B r a g grating. The origins of the particular modality of distributed Bragg studied in this
document are found in two initially distinct approaches, both pursued a t the University of
Toronto. Leblanc [14] proposes an intensity based spectrum analysis, while, concurrently,
Huang [15] identifies the potential of a phase based spectrum approach to distributed sensing.
Further, Ohn [16] consolidates the intensity and phase based approaches. The work of O h
is the direct precursor to the work presented in this thesis. As well, Volanthen [I?] [18] [I], at
the University of Southampton, develops an alternative experimental approach for recovering
in-grating parameters with optical interrogation.
With a view to examining these problems, the following document structure, sketched
schematically in fiewe 1.1, is adopted.
Chapter 2 relates the background necessary to understand the Bragg grating sensor's
response in terms of its in-fiber structure. Starting with Maxwell's equations, the governing
coupled mode equations are derived in detail. An exact reflectivity prediction for a uniform
Bragg grating is discussed. Arrows of fiewe 1.1 describe either 'forward' or 'inverse' scatter-
ing directions. The prediction of spectral response given knowledge of the physical structure
of the grating is an example of the forward scattering problem. The forward scattering
solution is outlined for the case of the arbitrary (norr-uniform) grating with the numeri-
cal T-Matrix formulation. The inverse scattering prob!em refers to recovery of the in-fiber
parameters given optical properties, reflectivity and phase of the grating. The inverse strat-
egy employed in this work is derived and the consequences of simplifving assumptions are -
discussed.
The expected response of in-fiber parameters to thenno-mechanical perturbations is the
subject of chapter 3. The phase strain relationships for the distributed optical fiber sensor
are detailed. Additionally, contributions are made in the experimental evaluation of the
Optical Data Representing Thermo-mechanical Field
Fiber Bragg grating optical d A response
Chapter 3: e of in-fiber parameters
to in-fiber thermo-mechanical
Chapter 4: ship between
host material strain and in fiber strain
Thermo-mechanical Field in Su bstrate/Host Material
Figure 1.1: Conceptual Row of optical fiber sensing
sensitivity parameters required for the measurements that follow in the document. Experi-
mental reports of the strain-optic coefficients, fiber stiffness, thermal and strain sensitivities
are reported.
Chapter 4 investigates the mechanical load transfer relationship between the in-fiber
field and the host substrate field. This content is particularly relevant to the new field of
distributed sensing where fields are expected to vary in space along the sensor length. In
collaboration with co-authour G d a u r n e Renaud contributions are made in the area of pre-
dicting the in-fiber strain field given the substrate/host field. As many optical fibers are
surface mounted to structures, a finite element modeling approach is described to estimate
the strain transfer behaviour kom host/substrate to surface mounted optical fiber. Further,
with colleague and c-authour Michel LeBlanc, a novel theoretical relationship between in-
fiber field and the substrate/host field is presented for the case of an axi-symmetric embedded
fiber sensor. A strategy to represent the surface mounted fiber as a geometrically equiva-
lent embedded fiber is proposed so that the aforementioned theoretical predictions can be
extended to the surface mounted case.
Chapter 5 puts into application the content of chapters 2 through 4. Experimental results
of a demonstrated experimental strategy[lb] are reported in further detail, with particular
emphasis on specifying the resolution of the sensor against theoretical elasticity solutions.
Additional data treatment techniques are discussed. With colleague and co-authour &My0
0 hn, an alternative experimental strategy is demonstrated and characterized.
Conclusions of this work are made in the final chapter. Included is a summary of contri-
butions.
1.2 A note on co-ordinate systems
In this document, Cartesian, (XI, 12, x3), and cylindrical co-ordinates (2, r, 6) are used where
appropriate. In the Cartesian case, xl lies parallel to the optical axis. In the cylindrical
system, z lies parallel the optical axis. Co-ordinate systems are shown in figure 1.2.
Figure 1.2: Cartesian and cylindrical coordinates used in this document.
Chapter 2
Optical description of the fiber Bragg
grating -
Figure 2.1: Fiber Bragg grating
2.1 The Bragg grating
The in-fiber Bragg grating is an all fiber device that has the property of reflecting a narrow
band spectrum related to its in-fiber parameters. Figure 2.1 illustrates a fiber B r a g grating
interacting with forward and backward propagating waves. In the context of structural
sensing the grating is typically interrogated in reflection. The all fiber device employs a
spatially periodic index variation to couple incident light from mode to mode. The period of
the index variation, A, is typically sub-micron. The index variation, hn, is small compared
to the nominal effective index, the order of
To induce a grating in an optical fiber, one takes advantage of the phenomenom of
photosensitivity, first observed by Hill[19]. Photosensitivity refers to a fiber's potential to
have its index of refraction modified by light exposure. The writing technique of Hill is
not widely used due to its self-written nature. Self written gratings arise from standing
wave pattern inside the fiber core: generated by reflection of in-fiber propagating Light-
This method accommodates little control of either the spatial period of the grating, as it
was determined by the launch light, and offered little spatial control of the distribution
of the index modulation strength-apodization-an issue critical to tailoring the reflection
spectrum-
The breakthrough contribution of Meltz et. aZ.[20] resolves some of these difEculties by
proposing a new transverse writing method. The authors indicate how gratings can be ex-
ternally impressed into an optical fiber by introducing the fiber into a standing wave pattern
generated by two hee space interfering beams. Called the transverse holographic method,
this st rate,^ permits wavelength control, and some apodization control. A further improve-
ment is reported by Anderson et. aZ.[21]. This group describes a difiactive optical element
(now commonly called a phase mask) through which a transmitted beam emerges to produce
a standing intensity field whose parameters are determined by the physical characteristics of
only the mask and the writing beam. The gratings used in this work are written with the
phase mask technique.
2.2 Coupled mode equation
The fundamental description of the optical response of a fiber Bragg grating is provided
by coupled mode theory- Coupled mode theory allows the prediction of an optical wave's
behaviour in the presence of a periodic or near periodic variation in the index of refraction
of the waveguide.
Before embarking on the mathematical derivation of the coupled mode relationship, a
qualitative description is useful. The physical key to :understanding the Bragg reflection
is the interaction between the lightwave and its medium. The optical wave oscillates in
space with a period defined by its wavelength and t&e effective index of the waveguide.
This wave interacts with the guiding material by way cf the medium's polarization density.
For sufficiently non-dispersive media, the polarization density is driven to oscillate in phase
by the optical wave. Should the medium be dispersive, the magnitude of the polarization
density's oscillation response will depend on the frequency of the optical wave. This in turn
induces a frequency dependent phase velocity for that particular optical fkeque~cy. If the
polarization densie itself is spatially periodic, the optical wave has opportunity to resonate
with the medium. An optical mode is able to transfer energy to the medium, while the
medium, through accelerating charges implicit in the resonating of the polarization density,
is able to couple energy back into optical modes. Given a precise knowledge of the make-
up of the periodic structure, coupled mode theory allows the prediction of power transfer,
through the medium, between optical modes.
The natural starting point is Maxwell's equations,
The fields in equations 2.1 through 2.4 are the electric field, @', t ) ; the magnetic field,
<(F, t ) ; the electric flux density, 5(F, t ) ; and the magnetic flux density, @F, t).
Two constitutive relationships describe the role of the medium in interacting with an
electromagnetic field,
where p(F, t) is the polarization density of the medium. The free space electric permittivity
appears as E, and p0 denotes the magnetic permeability of the vacuum.
The wave equation, is derived as equation 2.6 is substituted into equation 2.1, then
d x 3? (F, t ) is replaced in the resuiting expression with the relationship 2.2. a - 5 (7, t) is
expanded as in equation 2.5 and the curl operation, v x , is performed on the result. The
vector identity x 9 x Z(F, t ) = d (d - - f2g(F, t ) is employed to finally achieve
A more detailed development of the wave equation many can be found in most optics text-
books, Saleh and Teich[22] for example.
The medium polarization, induced by the electric field is most generallyt anisotropic,
non-linear, inhomogeneous and dispersive,
in which X i j represents the electric susceptibility of the medium. For purposes of arriving
at the coupled mode equations, it is sufficient to model the grating as a linear, spatially
inhomogeneous, isotropic structure, where equation 2.8 reduces to
In terms of the medium permittivity, equation 2.9 is recast as
where E (3 indicates the full spatially v w g permittivity of the medium. Equation 2.10
can be rewritten to explicitly reflect the permittivity contributions from the unperturbed
fiber, q (3 : and from the periodicity of the Bragg grating, E , (3,
In terms of the effective index of the waveguide and the index modulation of the grating,
expanding the square as a two term binomial, equation 2.11 appears as
The A quantity indicates the effective index of the waveguide. For a guided wave, a spatial
distribution of index of refraction exists for purposes of binding light propagation in the
waveguide. After solving the appropriate waveguide equation, one may find the propagation
behaviour of the bound modes. From the speed of the modes, the effective index, E is found.
Typically. lies between the maximum index of the core and the minimum index of the
cladding.
In this document, the fiber will be considered weakly guiding[23]]. This assumption allows
the reduction of equation 2.7 from a full vector expression to a scalar equation, for which
an approximation to the transverse component of the electric fieltd will be solved. Thus the
wave equation to be solved is
Note that the left hand side of equation 2.13 is the governing relationship born which the
permissible mode-field solutions in the unperturbed waveguide cam be found. The right hand
side of the equation accounts for the periodic structure in the wa~eguide.
Following Yariv's normal mode analysis[24], the electric field i s expanded in terms of the
m possible waveguide modes at frequency w ,
In equation 2.14, O;" (r, 0) represents the transverse shape functzion of the m-th mode of
propagation, with forward propagation constant, Dm. The transvezrse shape functions can be
found by solving the wave equation for an unperturbed waveguide The expression
represents a solution to the equation
Further details of such a solution: and the precise form of O;L (r, 0) can be found in Okoshi[25].
As well, the modes of the unperturbed waveguide solution are orthogonal and normalized in
the sense that
To account for coupling between modes, the envelope of the electric field is included as
A, (2). The latter term describes, as a function of z; the growth or decay of a mode along
the waveguide as energy is transfered from mode to mode.
The mode field expansion of equation 2.14 is substituted into t h e wave equation 2.13. Af-
ter some manipulation, including eliminating the time dependence, the following expression
is recovered
1 = -2n (3 (I) w2 C ZAm ( r ) 8;L (r, 0) e-i8.1z.
m
Because Q, (c t) is a solution to equation 2.16, BF (r, 8) homogeneously solves the Helmholtz
equation, thus the content of the first bracketed group of terms is zero. Equation 2.18 reduces
At this point, equation 2.19 is multiplied by 8; (r, 8) and integration is performed over the
cross section: resulting in
xe-"' /Lo lIo o: (T, 8 ) OF (r, 0) rdrdi3
1 = - E / I ~ E ~ S ~ (I) w2 ;Am ( z )
In view of equation 2.17, on the left hand side of equation 2.20 only two possible values of m
will result in non-trivial values after integration, m = +s. The reduced equation will appear
The indices f s indicate modes of the same order, with identical transverse shape functions,
but with propagation constants of opposite sign. The opposite sign of the propagation
constants indicate forward and backward propagating waves.
Further reduction of equation 2.21 merits some comment. In the original work of Yariv[24]
two assumptions are made. The first is that the envelope of the electric field is slowly varying.
This assumption is argued to provide the basis for the following simplifying assumption
This assumption is used to justify the elimination of the second order derivative terms relative
to the &st order derivative terms in equation 2.21. The second assumption is that only terms
on the right hand side of equation 2.21 that are in spatial resonance may drive any terms
appearing on the left hand side of equation 2.21. The latter assumption provides for the
separation of equation 2.21 into the two coupled mode equations relating the forward to
backward propagating waves. Sipe[26] has shown that both these assumptions are incorrect
for the general periodic structure. Using a total field analysis, Sipe explicitly indicates that
contradicting the second assumption of Yariv. Kashyap [27] argues that the assumptions of
Yariv are admissible in the case of a weakly coupled interaction and preserves the steps of
the original derivation.
It occurs that the result of Yariv is not at all an approximation, but exact, due to a coin-
cidental cancelation errors. To arrive at the final coupled mode equations, the assumptions
of normal mode analysis will be used.
The index of refraction perturbation will be expanded as a periodic spatial function,
written as
q=-a0
where A indicates the period of the cormgation. The index modulation is a real quantity,
thus the sum of equation 2.24 is re-expressed as w
671 ( z ) = 2 a, cos (q?) . q=O
Equation 2.24 is inserted into equation 2 -2 1 and the weakly coupled assumption is invoked
yielding
/d /,a Qp (r, 0 ) 8; (r, 0 ) l-drde-
In equation 2.26, there can be interaction between terms on the right side and terms
on the left side only if there is spatial phase matching between the terms. Consider the
M e - i f i 8 z a2 expression. the only terms capable of resonating with this term are those with
a phase matched spatial period. SpeciGcally, those terms that satisfy
In equation 2.27, q- indicates the order of the Fourier component of the corrugation that
couples the modes. Equation 2.27 is referred to as the phase matched condition or the Bragg
condition. Thus for m = s in the right hand side sum of equation 2.26, it is found
d A-, zEe,w -- - -As ( 2 ) &%-%) dz 2 oq JW T-=o 1% e=o [OF (r, 6)12 rdrde.
The coupling constant, K is introduced to simphfy the notation. K represents
Using equations 2-17 and 2.25, K is rewritten as
In equation 2.30 An represents the Fourier co-efficient of the resonant spatial period of the
index variation due to the grating. Thus, the coupling constant is seen to be a measure of the
periodic structure's ability to couple power between optical modes. The detuning, 60, which
is a measure of the mismatch between the period of the grating and twice the propagation
constant of the optical wave, is now introduced
A periodic, but non-sinusoidal corrugation would yield resonances at q values other than
unity. The second order resonance occurs at a resonant wavelength half that of the first
order. In practical interrogations; this second order resonant wavelength will be outside the
range of sources or detectors-approximately 775727~ for typical gratings used in this thesis.
It is appropriate to study only the first resonance where q = 1.
The notation of equations 2-30 and 2.31 are invoked and the first coupled mode relation-
ship is found
With rn = -s in the right hand s u m of equation 2.26 the second coupled mode equation is
analogously derived. Finally, the notation is clarified by replacing the &s subscripts with
the following notation: A (I) represents the backwards propagating wave and B ( z ) denotes
the forward propagating wave. This notation will be used in the entirety of this document.
The h a l coupled mode equations are thus presented as
dl4 (4 = ~ K B (2) exp [-2i6flzl and, dz
where the E denotes the complex conjugate.
2.3 T-matrix formulation for grating simulation
Equations 2.33 can be solved analytically. The analytical solution of the uniform grating
will be seen to be an important building block in the prediction of the optical response of
arbitrarily complex intra-grating structure. Equations 2.33 can be reduced to two uncoupled
second order differential equations,
d2 A (z) dA (2.1 + 2i6fl- - KKA ( z ) = 0 dz2 dz
d2B (2) d B (2) dz2
- 2ibP- - KEB ( 2 ) = 0 , dz
admitting the general solutions
where s = is introduced. At this point, no restrictions will be imposed on the
boundary conditions of the grating and the derivation will proceed with the most general
boundary conditions: A (0) = Ao, B (0) = B,, A (L) = AL: and B (L) = BL. These
boundary conditions can be substituted into equations 2.35 and the envelope functions are
re-expressed
and r = L,
in terms of the magnitudes of the input and output envelopes at both z = 0
To arrive efficiently at a solution the derivative of A (2) f?om 2.36 is computed and substituted
into the first of equations 2.33 and the expression is evaluated at z = L resulting in
s cosh (sL) - ibp sinh (sL) ei6p A0 = AL [
s I + B ~ [
4 6 sinh (s L) -i6pL e
S I -
The analogous operation is performed for B, to recover the following equality,
iri sinh (sL) i6pL
S I + BL[
s cosh (sL) + is0 sinh (sL) S I
It is convenient to add the spatial dependence on the propagation constant to the envelope;
this includes the full wave dependence of the forward and backward propagating waves.
Specifically, a ( z ) = A ( r ) e-iBL and b (2) = B (z j eiBL. Equations 2 -37 and 2.38 express a
useful relationship that may be better seen by casting the equations in matrix form. They
where
s cosh (sL) - 260 sinh (s L) z1 = s
exp [-iAL]
- & and denote complex conjugates of zl and z2. It is seen that
the waveforms at z = 0 to the waveforms at z = L. Equation 2.39 is
equation 2.39 relates
referred to as the T-
Ma tnx relationship and provides the basis for the description of complex grating structures.
The T-matrix formalism is based on the work of Yamada and Sakuda[28], originally used to
describe the reflectivity of distributed feedback laser and applied and verified in an intra-
grating sensing context by Huang[lO] .
Typic-, the proportiofid reflectivity is sought from the grating. This is simply the
ratio
In the simple case of a spatially uniform grating, subject to single end interrogation, i.e. the
backward traveling is set as A (L) = 0, equation 2.41 reduces to
Equation 2.42 describes the ratio of electric fields incident and reflected from the grating.
The power proportional reflectivity is the square of the electric field proportional reflectivity,
in which T denotes the complex conjugate of r . The grating's full power reflectivity is h d y
written as ri2 sinh2 (sL)
R ( 6 4 = s2 cosh2 (sL) + 6P sinh2 (sL) '
The uniform grating response is plotted in figures 2.2 and 2.3 In the plotted example,
the following parameters were used: 6n = 2 x lo-'; L = 5cm, A. = 500n.m and = 1.5. It
is seen that the maximum reflectivity as a function of detuning occurs at the phase matched
condition, 6P = 0
R (60 = 0) = tanh2 (sL) . (2.45)
Photonic Bandgap
.-.-. -.-I_.__ ----. ---.
Wavelength Offset M [nml
Figure 2.2: Uniform grating reflectivity and phase response (6n = 2 x lo-', L = 5cm, rz, =
1.5, A, = 500nm)
Photonic Bandgap
Wavelength Offset AA [nml
Figure 2.3: Uniform grating reflectivity and delay- response (672 = ~ X I O - ~ , L = 5cm, no =
1.5, A, = 500nm)
In terms of wavelength, the center of the reflection spectrum given by 6D = 0 is
The time delay characteristic indicates the differential delays between different probe wave-
lengths. At larger detunings the value of delay approaches 250ps. Consider a simple time
of fight argument. A reflection from the physical mid-point of the grating requires 250ps
to make the return trip from the front of the grating. Effectively, detuned wavelengths are
reflected from the grating center. The lowest value of delay is seen at the phase matched
condition. The wavelength that best resonates with the structure is effectively reflected from
a point physically closer to the front of the grating. For this example, the lowest value of de-
lay is approximately 120ps. This indicates that the penetration depth for the phase matched
wavelength is shortened to approximately a quarter of the grating length. A stronger grat-
ing will display a further decreased penetration depth for the phase matched wavelength.
A third observation can be made regarding the periodicity of the oscillations in the delay
characteristic a t det uned wavelengths. The maxima of the delay characteristic coincide with
the minima in amplitude response. Consider a Fabry-Perot cavity having the same length
as the grating, 0.05m. The free spectral range of such a cavity- is 15pm. This spacing is seen
between successive maxima in delay on both the lower and higher wavelength sides of the
delay spectrum. At detuned wavelengths, the grating shows a Fabry-Perot cavity behaviour
where the maxima in delay indicate wavelengths that are trapped in the cavity.
Recalling s = J-=, it is seen that as the detuning becomes larger than the mag-
nitude of the coupling coefficient: s becomes a complex-quantity. The region in which the s
parameter is real is referred to as the photonic bandgap. This region includes a significant
part of the reflectivity spectrum and is used as an indication of the grating's bandwidth.
In figures 2.2 and 2.3, vertical bars indicate the width of the photonic bandgap. Setting
the coupling constant equal to the detuning, (expressing detuning in terms of wavelength % E M excursion, 60 = T), the half width of the photonic bandgap is
In the present example, the half-bandwidth is 0.ln.m. The first zeros occur in outside the pho-
tonic bandgap where the solution becomes oscillatory. Setting R (6P) = 0 in equation 2.44,
the zeroes are found as sL = *irn.ir where m is any positive integer. The locations of the
zeroes are approximated by
In the current example, they occur at k0.019nm fkom the nominal wavelength.
The availability of an analytical solution to the coupled mode equations for a uniform
grating allows a numerical solution to more complex grating structures. Consider figure 2.4.
Equation 2 -39 describes the relationship between the forward and backward propagating
waves at either side of the grating. A complex grating is modeled as a series of cascaded
uniform gratings, each having an arbitrary pitch, index modulation, and effective index. The
relationship between sequential gratings is natural. The (n - 1)-th subgrating segment is
related to the n-th segment by
where [TIn denotes the n-th uniform T-matrix of equation 2.39. The same relationship
holds between the n-th segment and the (n + 1)-th. Thus, multiplied over the entirety of
subgrating segments, the relationship between the left side of the first segment and the right
hand side of the last segment appears as
2.4 Inverse Fourier transform method for low reflec-
tivity gratings
There are several theoretical ways to attempt to invert the coupled mode equations. The
following derivation was proposed by Kogelnik(291 for thin-film optical waveguides in 1975. It
remains valid for periodic and near periodic structures in cylindrical waveguides. To arrive
at an inverse scattering approximation, first, the electric field proportional reflectivity of
Figure 2.4: Cascaded uniform gratings model a complex grating
equation 2.41 is written as a function of 2:
then the derivative with z is taken,
Expressions 2 -33 are substituted into equation 2.52 to yield
dr (2) = [i~e*''~~ + nr2 (t) e-2i6Bz] d,z (2.53)
Expression 2.53 indicates the rate of change of reflectivity over an infinitesimal length of
uniform grating, dz. Consider a concatenation of infinitesimal elements, where a variation
in the magnitude of K and A are possible along the grating. Equation 2.53 is approximated
by
dr (2) = i ~ e ~ ~ ~ ~ d z . (2.54)
In equation 2.54 the term proportional to r2 ( z ) has been neglected. This low reflectivity
assumption limits the accuracy in exchange for computational efficiency. The simplification
Figure 2 -5: Incremental increase of reflectivity linearly related to local coupling strength
is also equivalent to the application of the Born approximation. The Born approximation
is a conventional 'first order' approach to solving the non-linear wave equation. Specifically,
when an optical field is incident on a non-linear medium, the incident optical field excites the
non-linear medium and the medium responds by resonating, thus emitting a second order
optical field. The second order optical field, in turn, will also interact with the medium and
induce a further higher order field. The Born approximation neglects all but the &st order
field originating from the interaction from the incident field on the medium. Approximate
equation 2 -54 expresses the Born simplification: the incremental induced optical field, dr (2)
is seen to be dependent on the local medium alone-independent of the presence of the field
or the medium at other points.
Equation 2.54 indicates the Born approximated inversion of the coupled mode equation.
Specifically. as drawn in figure 2.5, the local increase of reflectivity is related only to the
local value of the coupling strength. The general case of the non-uniform grating will have
a spatial distribution of coupling strength. Thus, the coupling strength is expressed as a
spatially varying function,
K + K (z) . (2.55)
As well! the phase term needs to be refined to reflect the variation with axial position. It is
re-expressed as [30]
where the detuning is now written as a distributed function: reflecting any axial variation
of the propagation constant and pitch- The axially varying detuning is written as the sum
of a spatially invariant nominal detlrning and a term representing the deviation from the
nominal state,
bp (2) = m a + (W (2)) . (2.57)
qb is introduced to clarify the notation
Thus, the refmed phase term appears as
26,~ -+ 26P,2 + $J (z) .
K ( z ) is introduced to simplify the writing of equation 2.54. K (2) appears as
K ( z ) = i~ (2) e-iJI(z). (2.60)
Thus: substituting equations 2.55 through equation 2.60 into equation 2.54, and integrat-
ing the result from r = -00 to z = oo and considering only a grating where no backwards
traveling optical power is incident on the RHS of the grating, an expression for the reflectivity
at the LHS side of the grating is found as
Equation 2.61 is a Fourier relationship, and can be inverted to yield[29]
Expressing detuning explicitly in terms of wavelength, it is seen
Oa K (2) = --
A: L, r (A, 0) exp [laid - d ~ . A:
Equations 2 -61 and 2.62 approximate the relationship between the in-fiber structure
and the complex reflectivity. The relationship improves with lower reflectivities. The low
reflectivity approximation for a uniform
with I
gratimg can be obtained by solving equation 2.61
KO Eor 0 < z < L
0 elsewhere -
It is found that r (AP) = KO sin (b/3L) /bp. Thrus, the power-proportional reflectivity is
Equation 2.65 has an obvious physically in&d attribute-potentially: it predicts a peak
reflectivity larger than unity, for a [ K , I L prodmct larger than unity.
The reflectivity -expressed in equation 2.65 Es compared with the exact solution of equa-
tion 21.44 for a 5cm uniform- grating. The agreement between the exact solutions and the
approximate solutions: for gratings of differend strengths are plotted in figure 2.6. From
strongest to weakest gratings, with peak refleceivities of 99%, 50%: 25%, and 10% are con-
sidered (corresponding values of the product l K l L are 3.0, 0.88, 0.54, and 0.32 respectively).
For the strongest grating the approximation is : clearly invalid. The deviation between the
exact and approximate solution are extreme. The approximate solution erroneously pre-
dicts a peak reflectivity of 9. As reflectivity decreases the approximation improves. At 10%
reflectivity agreement is strong, with the Fouriier transform approximation overstating the
reflectivity somewhat at its peak: but otherwise capturing the oscillatory portion of the so-
lution outside the photonic bandgap. Figure 2 .7 indicates the decrease in integrity of the
approximation as the product IKIL increases. I:t is seen that with a IK[L as low as 0.4, the
approximation is more than 10% in error,
The strength of the inverse relationship, equation 2.62 is tested as well. Again, only
uniform gratings are considered. A uniform gratEng's reflectivity is found using equation 2.42,
this reflectivity, r (6p) is then used in equation 2.62 and the deviation between the imposed
coupling distributions and the prediction are pPotted in figure 2.8. The same four gratings
are analyzed as in the previous paragraph. The .agreement is very poor in the case of a I KI L
product of 3. Note that the magnitude of K is well represented at the front of the grating but
deviates dramatically towards the far end. T h i s is a consequence of power that is reflected
from the fiont of the grating is better approximated by the Born approximation. Deeper
Figure 2.6: Range of reflectivities as predicted by thi coupled mode solution and the FT
approximation-all gratings modeled with L = 5cn
0.0 0.2 0.4 0.6 0.8 1 .O Grating Strength [KL]
Figure 2.7: Ratio of maximum reflectivity as predicted by the coupled mode solution and
the FT approximation.
Figure 2.8: Quality of coupling distribution recovery using Born (FT) approximation.
into the structure, for stronger coupling, the interaction between higher perturbations begins
to build. These observations suggest that gratings with peak reflectivities of approximately
10% are best suited to deployment as distributed sensors using the FT inversion strategy
2 -5 Hypothetical infinitely long and infinitely short grat-
ing discussion
Two extreme examples are given to further an intuitive understanding of the relationship
between a grating's structure and its optical response. Consider an infinitely long, infinitely
weak sating: with a coupling distribution described by, K (2) = KO. Equation 2.61 will
appear as
Completing the integral, it is found that r (6,B) is proportional to a delta function, 6 (6P).
Thus, an infinitely long grating would reflect only a single wavelength at the phase-matched
condition.
On the other extreme, consider a measurement by which the experimenter found a re-
flectivity, constant in wavelength, with a constant accumulation of phase as one tuned the
interrogation source,
where z0 is introduced to indicate the rate of phase accumulation. Inserting equation 2.67
it is found that the coupling distribution is proportional to another delta function, this
time in length space, 5 (G - 2). This narrow band reflector is in fact a mirror, reflecting
all wavelengths. The fringe density seen by the experimenter is related to the mismatch
between interferometer arms, this mismatch distance is 2,.
These two admittedly extreme examples of uniform gratings can be useful in providing
some intuitive guidance in the more complex structures discussed later in this document.
Following the above examples, it is noticed that spatial features over a short length tend to
have wideband repercussions in the reflectivity, and narrowband features are generated from
long spatial features.
Chapter 3
Optical Response of fiber Bragg
grating to strain and temperature
3.1
Section
given a
Phase sensitivity of fiber optic sensors
2-2 describes how the optical properties of a periodic structure can be predicted
knowledge of the physical structure itself. This section outlines how environmental
changes in the fiber properties, namely mechanical strains and temperature alter the grating
structure.
Butter and Hocker demonstrated the first optical-fiber strain sensor in 1978[4]. The
authors constructed an optical fiber Mach-Zehnder interferometer, shown in figure 3.1 in
which both arms of the interferometer are bonded to opposite sides of a cantilever beam
loaded in bending. One arm is loaded in tension and the other in compression. The formula
describing the intensity issuing from the Mach-Zehnder given b ~ ' ,
((£1 + £2) (£1 + &)) = E: + E: + 2&£2 cos ((Y + 2A4) . (3-1)
The parameter y represents the constant phase offset resulting from m y initial mismatch
between the optical path lengths in the two arms, while 44, the term of interest, is the strain
induced relative phase shift between the optical path lengths of the two interferometer arms.
In the confi,wation of Butter and Hocker, the two arms are subject to strains, thus phase
changes: of equal magnitude but opposite sign. The mounting on either side of the beam
explains the sign inversion as well as the factor of 2 in equation 3.1.
The phase change of the optical fiber can be expanded as the following
In equation 3.2, P denotes the propagation constant, and df the fiber core diameter. Equa-
tion 3-2 describes the phase change of a fiber subject to uni-axial mechanical stress as the
sum of three effects: the physical elongation of the path length, the index change induced by
the mechanical strain field, and a change in the propagation constant due to contraction of
the fiber core through the Poisson effect. The latter mode dispersion effect is several orders
of magnitude less important than the former two terms and is dropped from the expansion.
Dividing equation 3.2 with the nominal phase, 4, = PL, a normalized phase change is found
as
Be- Lenses He-Ne Source
L
Load
Screen
Figure 3.1: Butter and Hocker Experiment
As will be seen, this sum of an induced index change and a physical length change is also
appropriate to describe the response of Bragg gratings to mechanical and thermal loads.
The complete phase/strain relationship for the Bragg grating is derived in the following
paragraphs .
First: a simplification of section 2.2 is reconsidered. Specifically, equation 2.9 limits
the susceptibility to an isotropic constant, X, where more generally it is a tensor quantity
dependent on the polarization of the optical field. Preserving the assumption that the sus-
cep tibility is linear and non-dispersive, and adopting tensor notation: equation 2.9 describing
the polarization density induced by the field is recast as
Expressed in terms of permittivities, equation 3.4 becomes
where lij denotes the identity tensor. -4nalogous to the simplification performed in section 2.2
the permittivity tensor is rewritten as a sum of two contributions
The first contribution is the permittivity tensor of the fiber. This quantity will reflect any
birefringence, either nominal or induced, in the optical fiber. A birefringence free fiber (lo-
bi fiber) would be modeled by dii with like permittivities seen by any polarization. The
general case remains a full tensor. The second term includes the spatial variation of index
of the Bragg grating. As well, it is appropriate to leave the tensor subscript in the second
term. This provides for the possibility of strength birefringence of the Bragg grating in which
one polarization is preferentially reflected. Equation 3-6 is normalized with the free space
Finallyz the tensor wave equation is written as
For any tensor quantity a coordinate choice can be made such that E,, becomes diagonal.
This set of axes is called the principal axes of the medium. In the present context, the
principal axes of the fiber will be chosen as the principal axes of the problem. Once cast in
a principal coordinate system, the permittivity tensor of the fiber can be written in terms
of effective index,
It is noted that there is no assurance that the principal axes of a grating strength coincide
with the principal axes of the fiber, thus while the tensor & is diagonalized, the tensor $ (zl) may continue to have non-zero off diagonal entries. The correct expression of the grating's
permittivity tensor would require a tensor transformation to convert the principal grating
axes to the axes of the fiber. In this document, the grating is assumed to have coincident
principal axis with the fiber. Analogously to section 2.2 the index modulation is expressed
Thus: again invoking the weakly guided assumption for both polarizations it is found that
equation 3.8 reduces to two polarization uncoupled scalar equations of the following form
where p = 2 ,3 indicates the transverse polarization index.
Solving equation 3.11 for either polarization, a Bragg condition will be found as
When such a grating is subject to mechanical or thermal loading, uniform over the e x t e ~ t of
the grating, the center wavelength will undergo a shift. Expanding equation 3.12 in a Taylor
series about the nominal center wavelength, the normalized wavelength shift is found to be
Ax, AE.. AA -=- - +h- (3.13) x, n p
The relationship in equation 3.13 will be seen to be linear with axial strain. It is this
effect that drew attention to the Bragg grating as an effective strain sensor[3l]. As only
the reflected center wavelength of the sensor is of interest-i-e. the loading information is
wavelength and not intensity encoded-the Bragg grating sensor can be made insensitive to
optical intensity effects such as power fluctuations of the interrogation source and coupling
changes in input and output leads.
Naturally the ratio is equivalent to the total strain component along the fiber axis,
el; as is the ratio of equation 3.2. Thus the phase change derived by Butter and Hocker
is identical to the normalized wavelength shift of the Bragg grating,
For a nominally uniform grating subject to a load field that is spatially varying along the
grating, equation 2.58 is examined. 11 (xl) is the term indicating the accumulated excursion
in detuning away from the nominal detuning 6P,. The p subscript is added to qb to indicate
the different detuning in each polarization axis. The spatial rate of change of accumulated
detuning is the local perturbation of the detuning fiom the unloaded condition, specifically,
A (6& (XI)) is first expanded to
The propagation constant of polarization p can be expanded as
With equation 3 - 17, equation 3.16 becomes
2xAKp (xl) 2rfiPAA (xl) A ( 6 P p (4) = -I-
A0 A:
Recalling the Bragg condition 3.12, equation 3.18 is recast as
Thus, for an axially varying load across an originally uniform Bragg grating, the recovered
detuning excursion is related to perturbations in the optical fiber sensor as
Again, the RHS of equation 3.20 is a distributed representation of the simple formula p r e
sented by Butter and Hocker for the Mach-Zenhder interferometer,
Equations 3.13, and 3.21 indicate that two phenomena determine the optical response
of either a point or distributed Bragg grating sensor: the physical length change and the
induced index change. In the upcoming paragraphs, the dependence of index of refraction
and of pitch on strain and temperature will be indicated. It is emphasized that the strain
sensitivities of the index and pitch presented in this document, refer to the sensitivities to
total strains, ei , in the sense that total strains indicate the total deformation of the fiber,
regardless of the cause of the deformation. Mechanical strains relate to stresses by Hooke's
where Di,- is the stiffness tensor. As mechanical strains are induced by mechanical stresses,
the total strains are the sum of the mechanical and thermally apparent contributions
where crf is the tensor of thenno-elastic constants of the optical fiber. It is important to
appreciate this distinction in deriving the phase/strain/temperature sensitivities.
The variation of the pitch length is naturally
recalIing that the total axial strain, e:d-'f, includes the possibility of a combination of me-
chanical and thermal apparent strains, as indicated in equation 3.23.
To describe the change in effective index, the relationship between the electric field and
the electric flux density
'oi = eijEj = eE (I*, + xij ) = c o v j z2-E (3.25)
is re-expressed as 1
for purposes of constructing the index ellipsoid. In equation 3.26, vij denotes the imper-
meability tensor. The sensitivity of the impermeability tensor to strain is described by the
For the most general case: it appears that 81 individual strain optic coefficients are
required. Symmetry arguments[32] reduce the number of Pockels coefficients needed to
completely describe the isotropic fiber core to only two values: pll and pl2. Along the
principal axes
thus
The constant ,F denotes the normalized sensitivity of the index to temperature,
Thus the index change as a function of environmental perturbations can be written as the
following sum -4 6
A% = -2 xmefd- + E ~ ~ A T . (3.31) i=1
Equations 3.24 and 3.31 are substituted into either equation 3.13 for the point sensing
phase sensitivity or equation 3.21 for the distributed phase sensitivity. The expansion of the
point sensitivity appears as
In terms of distributed sensing
An illustration of the deformation of the grating's pitch and effective index over the length
of the sensor subject to an axial strain distribution is shown in figure 3.2.
The sensitivity of several common loading conditions are now outlined. In deriving these
relationships, it is useful to adopt a common procedure:
Figure the total strains: ejd.lf, usually functions of both strain and temperature, using
equation 3.23.
a Substitute the total strains into the appropriate phase sensitivity expression, either
equation 3.32 or 3.33.
The simplest (and most common) strain situation is the that of uniaxial tension with no
temperature variation, ol ( x l ) = qo. The strain field is
Axial strain
I I I 1
0 L Xl
n + Deflected
Figure 3.2: Change in effective index and periodicity of a grating subject to strain field.
Substituting 3.34 into equation 3.32
In a point sensing modality, the bracketed term of equation 3.35 is referred to as the gauge
factor of the sensor, S,,
In practice the gauge factor is evaluated by a one step strain versus wavelength shift cali-
bration of the sensor instead of a calculation based on the estimated value of the individual
parmeters.
A temperature perturbation on the optical fiber results in a total strain field of ey*~~ = etot-,f - et%f - 2 - - af AT. Substituting into equation 3.32 the sensitivity is
Equation of 3.37 can be written in the form of 3.36 as
where -2
(3 -39)
A fiber subject to simultaneous uni-axial stress and temperature sees a total strain field of
= 8. Substituting into where e;"echyf is related to the lone axial stress component by el
equation 3.32 the sum of individual sensitivities is found,
Where a fiber is bonded to or embedded into a host material of different thermal expansion
constant ah the sensitivity is computed by recobgizing that the total strain in the host
material is equal to the in-fiber strain
Assuming that it is the host's mechanical strain and temperature that are of interest, in
detail: the phase sensitivity is
Simplif_iring
tivities free
equation 3.43, the sensitivity of the surface mounted fiber in terms of the sensi-
fiber is written as
in which sensitivity sTfed is
Given that the host material thermal expansion is known, equation 3.45 is useM because it
permits the evaluation of the thermal sensitivity of the embedded or surface mounted sensor
in terms of the calibration of the free fiber sensor.
There are many more configurations for which sensitivities can be derived. In all cases the
procedure remains the same- Sirkis has derived the sensitivities in a unified manner for Mach-
Zehnder, IvIichelson, Fabry-Perot , polarimetric, dual-mode and Bragg grating sensors [1 l] [l2].
The latter references detail six thermo-elastic loading cases. Kim, KollL and Springer[33]
and Kim, Ismail, and Springer[33] use the method of Lekhnitskii[34] to describe the strain
transfer from a general anisotropic host to the fiber core of a Fabry-Perot sensor due to
a general six component far field material stress field and a spatially inmriant thermal
perturbation.
3.2 Experiment a1 evaluation of sensitivities
In the course of this work, experiments are devised to recover the Pockels strain optic coef-
ficients that appear in the sensitivity expressions. Much of the literature uses values of the
PockeIs const ants as reported by Bertholds and Diindliker [35]. The latter reference evaluates
a fiber having a pure silica core and a 8 2 0 3 doped cladding at 633nm interrogation. The
strain measurements that constitute a large part of this thesis are performed in range of
1500n.m to 1560nm. Thus the Pockels constants are evaluated in this range.
Borrelli and Miller[36] provide a procedure to measure the Pockels constants in bulk
silica and evaluate the coefficients pll and p12 simultaneously. The authors construct a
Mach-Zehnder interferometer which includes: in one arm, an element of bulk fused silica.
The silica sample is rnechdcally attached to a piezo-electric crystai which is driven to
induce a mechanical resonance. The mechanical resonance induces an acoustic wave in
the silica sample for which the resulting strain field is computed. Upon interrogation with
Linearly- polarized Light the output intensity of the of the interferometer is dependent on the
birefringence induced in the crystal and the path length through the crystal. Separation of
the two coeEcients depends on the different sensitivities of these effects resulting in different
frequencies of the time response for each of the two effects for a given piezo driving frequency.
Barlow and Payne[37] propose an experiment to determine the stress-optic constant, C,
in optical fibers. The stress-optic coefficient is a combination of the Pockels coefficients
emressed as
where vf is the fiber's Poisson ratio and Ef the Young's modulus of the fiber. The constant
is normalized with the acceleration due to gravity, g. Barlow and Payne [37] apply a twist
to the fiber and measure the birefringence with a polarizer and an analyzer. The authors
also fit the stress optic constant over a wavelength range, l O O O n m to 1600nm assuming the
relative shape of wavelength curve of optical fiber is the same as that of bulk silica, save for
a constant; offset.
Namihira[38] measures the stress-optic coefficient by impressing a birefringent strain state
on the fiber. The birefringence is generated by mechanically loading the fiber with a unidirec-
tional transverse line load over a length of fiber. Using several wavelengths of interrogation-
different lasers-t he authors generate a curve of the st ress-opt ic constmt over a wavelength
range spanning 633nm to 1580nm. The wavelength dependence is fit to a fourth order
polynomial and compared to[37].
An evaluation of the individual Pockels strain-optic coefficients in an optical fiber is done
by Bertholds and Diindliker[39]. Two experiments are required to lrncouple the specific
dependence of pll and p12- A mechanical stretching of the fiber resulks in a phase change
with one dependence on pll and plz , while a twisting of the fiber, as used in[37], results in a
phase change with a merent strain-optic dependence. These two experiments are performed
on fused silica fiber. described as LT-F1506B, a t 633n.m interrogation.
The majority of grating experiments of the FOSS group employ gratings with center
wavelengths in the approximate range of 1500nm to 1560nm. As such, a n evaluation of
the Pockels coefficients is performed in the wavelength range of interest. To decouple the
independent Pockels coefficients, two experiments need to be employed for which the strain-
optic dependence in uncoupled. Drawing from the literature reviewed above, the stretch test
and the pinch test are selected.
The pull test is applied in the configuration of an all-fiber Michelson, figure 3.3. One arm
of the Michelson is stretched between two stages. The translation is done by an actuator
with a displacement resolution of a single micron. There exists some uncertainly due to
deformation of the fiber/epoxy bond on the translation stage. Thus, it is not true that the
fiber extension is precisely the translation of the stage. A long length of diber, approximately
one meter, is suspended between the translation stages to minimize this effect. A modest
strain extension of 0.5% strain is measured with resolution of I part in approximately 5000.
As one arm of the interferometer is extended, a power fluctuation dependent on the
relative phase difference between the two path lengths is expected as
where V is the fringe visibility. The phase sensitivity to extension is derived to be
Coherent Source
Coupler r L
Photodetector
Mirror
Fixed Stage w Translating Stage
Figure 3.3: Pull test experiment
Equation 3.48 is different from equation 3-2 by a factor of 2 because of the return trip
of the Michelson interferometer not present in the Mach-Zehnder. As the photodetector's
photocurrent is amplified and measured on a digital oscilloscope. The number of fringes is
manually counted for a measured excursion of the actuator to acquire 3. The second experiment, shown in figure 3.4, consists of a linearly polarized source, a
polarization rotater, a length of fiber on which a birefringence is mechanicalIy impressed,
a polarizer and finally a photodetector. The polarization rotator consists of two paddles
each having a loop of optical fiber. Rotating the paddles relative to each other provides
torsional birefringence to each of the fiber loops, thus controlling the state of polarization.
The input polarization is not i n t idy known. However, the response of the sytem can be
sampled by viewing the transmitted intensity variation on the oscilloscope. The polarization
controller is manipulated so that the fringe visibility is maximized. This adjustment ensures
equal lauch power into each to the principle polarization axis of the portion of fiber beneath
the pressing block. The birefringence of the fiber is ihduced by a mechanical load. The
apparatus employed is shown in figure 3.5. A glass bottomed pressing block diametrically
loads two fibers against a glass plate. Only one fiber is interrogated, the other is a dummy-
A mass a t the end of a lever arm provides force. To restrict forces to the downward direction
only, a ball bearing is used between the lever arm and the top of the pressing block. This
Polarization
Figure 3.4: Pinch test experiment
loading apparatus is described by Sirkis[40]. The degree of induced birefiingence alters the
distribution of power between polarizations. The polarization distribution is monitored by
insertion of the polarizer preceeding the photodetector.
The Jones calculus is used to derive the intensity variation at the photodetector as a
function of impressed transverse load. A representation of the electric field is made as
where E,; indicates the electric field strength, and cr indicates the axis of interrogation of
the linearly polarized source relative to principal axis of the fiber. The electric fields, Era
and Ef3, beyond the birefringent element are found by mdtipiplying equation 3.49 with an
appropriate transfer matrix describing the phase difference between polarizations given by
the birefringent element. This is written as
where A0 = ,B2 - ,B3 indicates the mechanically induced birefringence. It is useful to re-
express primed field strengths into the principal coordinates of the polarizer (indicated with
Load ures Downward
Fibers Under Diametric Load
Figure 3.5: Pinch test load apparatus
a double prime). This is done with a conventional rotation of angle 0,
The polarizer allows only one polarization component pass through to the detector, E1'2.
Thus, multiplying Elf2 with its complex conjugate gives the power transmitted to the pho-
todetector. Performing this operation yields
= cos2 a sin2 9 + sin2 cr cos2 B - 2 cos B sin B cos cr sin a cos (APL) . E O 2
(3.52)
Both €J and n are adjusted to maximize the sensitivity of the experiment. This occurs when
8 = a = 45", and equation 3.52, simplifies to
E ' I ~ E ~ 1 = - (1 - cos (A@)) a2 2
in which
4 4 = a p L
is introduced as the phase response of the experiment. In terms of strains, equation 3.54
becomes
The strain components of the fiber core can be related to P, the load in force per unit length
applied diametrically to the fiber as [U],
Combining equations 3.55 and 3.56 The final phase sensitivity is written as
Load is applied in increments of roughly 10 grams, measured on a balance with a res-
olution of better than a milligram. Output intensity is recorded manually. The choice of
weight increment yields approximately 180 points per 27i phase excursion of the experiment
to assess the sensitivity g. 6
Equations 3.48 and 3.55 are cast as to extract the Pockels coefficients as
The experiment is performed using SMF-28 fiber at two wavelengths, 1506n.m and 1560nm,
using a tunable laser, manufactured by New Focus as a high coherence Linearly polarized
source. The experiment is repeated using a helium-neon source to assess a York fiber (Sm600)
with a numerical aperture suitable to be single moded at 633nm. The results are presented
in table 3.1.
Three parameters are required to perform the computation of equation 3.58, the fibers
Young's modulus of the fiber: Ef , Poisson's ratio, uf , and index of refraction, n. An attempt
is made to evaluate the Young's modulus of SMF fiber. Figure 3.6 shows the experiment in
which a fiber is suspended vertically from a mandrel. The fiber hangs through two guiding
devices to constrain the swinging of the fiber. The guiding devices are simply small holes
drilled into metal plates. Two microscopes are mounted adjacent to each of the guiding
Microscope
Fiber Suspended fkom Mandrel on Translation Stage
T Guide Hole and Refence Pin
Fiber Gauge Length - lm.
- 4 Guide Hole and Refence Pin on Translation Stage
Clamp
Figure 3.6: Apparatus to measure fiber stiffness
Table 3.1: Strain and Stress Optic Constant Strmmary
Authour
Borrelli & MilIer[36] Bertholds & Diindliker(39l
Namihira [3 81 Barlow & Payne[37J
This work This work This work
device. The microscopes are-used to determine the fiber deflection under load. *4t the lower
end of the fiber, a specially designed clamp allows a mass to be suspended from the fiber. A
receptacle is attached to this clasp to load the fiber.
The measurement procedure is as follows: Two reference positions are defined, visible
through the upper and lower microscopes respectively. For a coated fiber, two small inden-
tations in the coating are made with a knife. For an uncoated fiber, the coated/uncoated
interfaces are used as the reference positions. The fiber is then loaded. The upper reference
mark on the fiber translated downward due to extension of the fiber portion above the top
mark, and some deformation of the mandrel material. To compensate, the mandrel is trans-
lated upwards until the reference mark coincides with the original position relative to the
guide hole. The lower guide hole is translated downwards to find the lower reference position
on the fiber. The translation of the lower guide pin defines the fiber extension under load.
The resolution of the micrometers that translates the lower guide pin is 5 microns. Coated
and uncoated SMF fibers are tested in this way. Each set of measurements includes ten
individual loading and unloading. The standard deviations of the normalized extensions are
respectively &0.47%; f 0.7 9%, and f 0.37%. The constant of proportionality between strain
and load for the uncoated fiber is 10.75pc/gm &0.06&grn; for the stripped length of fiber
a value of 1 1 . 2 6 ~ ~ * O.OGp~/grn is found. Through Hooke's law,
P11
0.126 0.113
N/A N/A 0.127 0.146 0.150
P12
0.26 0.252
N/A N/A 0.243 0.269 0.272
C bm2/kl
N/A N/A 3.34 3.17 3.09 3.05 3.03
X [=I 633 633 633 1300 633 1506 1560
Material
Code 7940 Silica Silica FiberfLT-FI506Bj GeOz Doped Fiber GeOa Doped Fibre York Sm600 Fibre
SMF-28 SMF-28
the Young's modulus of the uncoated fiber is found as 70.96GPa f 0-SGPa. Comparing this
result with that of the coated fiber, an estimate of the coating rnoddus , Ec: can be found.
The distribution of force carried in each region can be written as
Assuming that equal strains exist in the coating and fiber, an expression for the coating
modulus can be found as
where A, is the area of the coating. The coating modulus is foun& to be Ec = L18GPa
f 0-5GPa. The error in the coating modulus is, proportionately, much higher than that of
the fiber and can be considered only as an order of magnitude estimate.
In the computation of the Pockels constants, the fiber's Poisson ratio, vf = 0.17[42] and
the index, ii = 1.4682[43], are taken from literature.
Chapter 4
Interpret at ion of mechanical
interactions: emphasis on strain
transfer between host and fiber sensor
4.1 Strain transfer considerations for point sensors
The role of the optical fiber sensor is to extract thermc+mechanical information from a
structural host. As indicated in the previous chapter, the optical fiber's spectral and phase
variation from some nominal condition are determined by the thermo-mechanical field vari-
ation inside the sensor. For a measurement to be meaningfully interpreted, the field values
in the sensor must be properly related to the underlying host field values.
4.1.1 Shear lag formulation
The simplest implementation of the Bragg sensor is its deployment as a point sensing tool-
in other words, a sensor reporthg an average axial strain value over the typically short
sensor length. For this configuration to be valid the axial strain in the sensor must closely
approximate the st rain in the material under interrogation. A derivation, originally presented
by Cox[44] is useful as a starting point. This Cox article is originally presented in the context
of the study of the strength of fibrous materials such as paper. The geometry of Cox's
elasticity model is appropriate so that the model can be transferred to the field of optical
fiber strain sensing. Cox indicates the axial stress and strain fields that are expected when
a fiber of finite length is surrounded by an infinite material subject to uni-axial far field
tension: figure 4.1. Both materials are considered isotropic. Figure 4.2 shows a summation
of axial acting forces over an incremental element of the optical fiber sensor. c i indicates in-
fiber normal stress. Over the fiber cross-section this quantity is averaged and taken radially
constant, T{ (z). The in-fiber radial and azimuthal stresses are assumed to be negligible,
Summing the forces acting on the element and s impwng , the following relationship is
found between the mean normal stress and the shear acting on the outer surface,
Figure 4.3 shows the forces acting in the z direction on an infinitesimally thin element of
the surrounding material. Summing yields the following equilibrium expression
Figure 4.1: Fiber segment in infinite host under uni-axial tension.
Figure 4.2: Sum of axial acting forces in the fiber
Figure 4.3: Sum of axial acting forces in the surrounding material
in which a,-, again, denotes the mean normal stress over the cross section.
Equations 4-1 and 4.2 are related through the equivalence of shear forces at the sen-
sor/surrounding material boundary. Specifically. TZ (rf, z) = T:= (rf, z)_ The interfacial
shear can be eliminated, and the expression recast as
m 3 a: ( z ) ( z ) r2 - r; rTZ (T: z) = -- -
27- dz dz 2r . Assuming negligible transverse stresses, F: = F{ = 0, equation 4.3 can be written directly
in terms of mean axial strains as:
where Em and Ef indicate the stiffnesses of the surrounding material and the fiber sensor
respectively.
At this point a simplifymg assumption is invoked. Consider the bracketed terms of
equation 4.4. The axial strain gradients are anticipated to be of the same order,
thus the strongest factor determining the relative magnitude of the two bracketed terms is
the ratio of stihesses: present in the second term. Optical fiber coatings, in the case of a
coated fiber, or typical epoxies: in the instance of a bare embedded fiber are between one
and two orders of magnitude less stiff than the fiber itself, thus
Therefore, the second bracketed term of equation 4.4 can be neglected.
The shear stress is then expressed in terms of the shear modulus and material displace-
Equation 4.7 can be recast &
At this point it is necessary to make a second simplifying assumption. The term is
neglected in comparison with the term. The u-displacements are due, principally: to
Poisson contraction and are less significant than the w-displacements,
The resulting expression is integrated, over r: from the fiber/material boundary, to the
surrounding material radius r,.
The shear modulus is written in terms of the stiffness, G, = 2(5;m1 , resulting in the following
expression.
Equation 4.11 is differentiated with respect to z yielding
Ef e y (rm, Z) - ef (z) = - (1 + v,) - In Em
The shear lag parameter, AT, encapsulates both the effects of geometry and relative s t i fbas
of the system components [45]. N is written as
E m @=- 1
J% (1 + urn) Zn (T) ' and thus equation 4.12 simplifies to
For the case of a host subject to uniform axial tension, (R, z) = e , . m . The stress
transfer to the fiber happens through shear transfer at the radial interface. Any tensile axial
effects at the fiber host interface can be neglected[46]. Equation 4.14 admits the general
solution
i$ = Aexp [A,r] + B exp [X.r] + co. The eigenvalue of the solution, A, is determined by inspection of equation 4.14
Boundary conditiolzs are invoked that reflect the assumption that there is no axial strain
transfer on the end surfaces of the fiber: Z: (-L) = F: (-L) = 0. Solving for A and B the
strain distribution along the fiber is
Fiewe 4.4
host to fiber.
-m ez (4 = ey',
cosh (7) shows a family of curves indicating the transfer
N values of 0.1, 0.2, and 0.3 are considered.
behaviour of axial strain from
These numbers are chosen to
represent realistic values. (For example, a geometric ratio of = 2: and a stiffness ratio rf
of 3 = % G P ~ / G P ~ corresponds to N = 0.29: while a geometric ratio of = 5 , and a rf
stifhess ratio of E~ = ~ G P ~ / G P ~ 70 corresponds to N = 0.12.) It is clear that the spatial
rate of transfer increases with stiffer surrounding materials. In the worst case presented,
N = 0.1: after an adhesion distance of 50 fiber radii, the ratio of host axial strain to fiber
axial strain is better than 99%. For a 62.5prn radius fiber the required build up distance,
to overcome shear lag, is approximately 3mm. Thus, it is reasonable to assume that the
10 20 3 0 40
Axial Position zlr,
Figure 4.4: Shear lag plots for N = 0.1, N = 0.2, and N = 0.3.
Figure 4.5: Common optical fiber sensor installations
axial strains inside the fiber sensor are sufkiently close to host strains if the sensor bonding
configuration accommodates a modest development length the order of several &meters.
Also, figure 4.4 suggests an important phenomenon: the study of which constitutes a
significant portion of this document. The in-fiber strain does not necessarily equal the host
strain. In the case of a finite length sensor: the effect is seen at the end of the sensor? as
a certain development length is required for the sensor to accurately approximate the host
strain- Later in this document, the effect of strain mismatch between mechanical host and
fiber sensor due a spatially varying field is investigated in some depth.
For the case of the distributed sensor: the strain transfer relationship requires rethinking.
Equation 4.17 indicates that the strain field in an embedded sensor does not necessarily
replicate the strain field in the material host. The distributed sensor has been shown to
be capable of measuring arbitrary, sharply varying spatial strain distributions[lO] [47] [16] [17]
[18] [I] [48] [49] [14] [50] with high resolution. The host/sensor interactions are the most relevant
for this type of sensor when high resolution strain measurements are attempted. With this
in mind, the following question naturally arises: What type of strain variations can be
transferred mechanically from host to sensor? And once transferred to the fiber, what are
the optical limits on the extraction of sharp in-fiber elasticity field variations? This chapter
addresses the former part of this question by pursuing three objectives:
1. establish criteria to define the quality of axial strain transfer between a substrate and
an optical fiber sensor,
2. describe and quantify the parameters influencing this traasfer, and,
3. for both the embedded and surface bonded case, present the expected performance of
the sensor.
4.2 Axial strain transfer from host to fiber: Finite el-
ement -models
The initial work in this thesis towards addressing the question of the quality of strain transfer
is performed with two finite element models; one for the surface bonded case and one for the
embedded case. The two models will be described in turn.
4.2.1 Surface bonded sensor model
The surface bonded model is a hybrid model of Reissner/Mindlin and quasi-three-dimensional
elements. Reissner/Mindlin elements are typically used to model plates of composite ma-
terials having several laminates of different material properties. Conventionally, different
material properties for the different laminates are due to different ply orientations of the
same anisotropic composite material. Figure 4.6 shows the position of the Reissner/Mindlin
elements in the surface bonded model. In this implementation, different laminates, within
elements are assigned different mechanical properties, not due to ply orientation, but due
to individual laminates representing either optical fiber sensor or surrounding adhesive. It
is common for composite materials to be anisotropic such that the stiffness in the direction
parallel to the reinforcing fibers is approximately two orders of magnitude greater than the
stiffness perpendicular to the reinforcements. When comparing the optkd fiber sensor stiff-
ness (70GPa) with that of typical structural epoxies; (range of 2 to 5 GPa[51] [52]) a similar
disparity of mechanical properties is found. Reissner/Mindlin elements also model the un-
derlying substrate. Figure 4.6 illustrates the co-ordinate system used for Reissner/Mindlin
Reissner1Midli.n plate elements model stacking of fiber and adhesive
Reis sner/Midlin plate elements model surrounding adhesive
Quasi-three-dimensional elements model thin
I Reissner/Midlin plate elements model underlying sub strate
Figure 4.6: Model for surface mounted fiber.
analysis. A quasi-three-dimensional element is used to represent the thin layer of adhesive
that lies between the fiber and the substrate l.
The following assumptions describe Reissner-Mindlin plate theory:
The model can accommodate structures of moderate thickness
The midplane displacements ul: up; and u3 are small compared to structure thickness.
The in-plane strains el, e2, and el*, strains are small compared with unity.
The transverse normal strain, es, is neglected.
h he computer code used to generate results for the surface bonded model was written by colleague and
co-authour[53J Guillaume Renaud.
The transverse shear stresses, a13 and 023, me parabolic in x3.
The Reissner/Mindlin elements are two dimensional, having nodes along the mid-plane
only. Five degrees of freedom are sought in the Reissner/Mindlin formulation: mid-plane dis-
placements, ul (xl, x2), u2 (51, x2), and 113 (xi, x2), and two variables 91 (xi, ~ 2 ) and 02 ( ~ 1 ~ x 2 )
that provide an indication of rotation. 6 indicates the vector of midplane variables
Displacements as functions of three dimensions, (indicated with an overbar, .Ltl (xl, x2, x3)), - u2 (xl. 52: x3) and us (xl: x2, x3) are recovered from the mid-plane strains and the two
rotation-like terms, and 02, by the following relationships:
Strains, in turn are derived from the displacements of equation 4.19
To facilitate the notation of the stress resultants it is useful to introduce the following
not ation:
Equation 4.21 is further condensed to
The strain vector, {el, can be expressed in terms of the displacements, 6,
The matrix of operators in equation 4.23 is labeled [L]. As will be seen, the matrix [L] acts
on the interpolation functions of the discretized dement.
Relevant stress components appear as
I 4 =
Stress, N , moment, M, and transverse shear, Q, resultants are obtained by sllmming stresses
over the faces of the element,
The transverse integration of equation 4.26 is expressed in terms of the stresses of equa-
tion 4.22 as:
The inner sub-matrices, G,, bij: 6, and Sj are representation of the aggregate behaviour
of a multi-layered composite material element. The constructions of Gj; bij, dij7 and Eij
accommodate an element having an arbitrary number of plies, each with generally anisotropic
mechanical properties. Explicit descriptions of how to compute the entries in these matrices
can be found in composite materials tex*s such as[54]. Equation 4.27 is rewritten as
The governing expression for a single element is found by considering a virtual displace-
ment field, d (61, imposed on the element. The work done against the element boundary by
external forces, moments? and transverse shears must be stored inside the element as internal
strain energy. This equivalency, in terms of the virtual displacement, is written as
where {r} represents the stresses acting on the element boundary. {r} includes the same
content of normal stresses, moments: and transverse shears as the vector, {a). Inserting
Figure 4.7: Node distribution across Reissner/Mindlin element.
the stress resultants of equation 4.26 into equation 4.29 reduces the right h a d side of the
equation to a two dimensional integral,
Then, expressing the stress resultants in terms of strains, and strains in terms of displace-
ments, the following is found
At this point it is necessary to discretize the element. This is done by assigning discrete
nodes to the element. Consider first the square element in the E-v plane bounded by -1 < < < l a n d - 1 < q < 1 .
Figure 4.7 shows the distribution of nodes across this element. The coordinates XI and
x2 are mapped into the <-9 plane through interpolation functions &
The continuous displacements are approximated by a discrete set of vector of nodal displace-
ments, {6Ie and a matrix of interpolation functions, [@]. Specifically:
where,
for which each of the sixteen interpolations functions is given as
The k-th interpolation function has the property that it has the value of unity at the k-th
node and zero at all other nodes-
The physical problem requires casting from the XI-xa plane to the J-r) plane. This is
accomplished with the coordinate transform,
The two by two matrix of derivatives in equation 4.36 is called the Jacobian of the transfor-
mation, [J] . Area transformations scale between the two coordinate systems as
Line integrals
coordinates is
are required to sum boundary loads. An infinitesimal length ds in Cartesian
expressed as d s = J ( d z l ) 2 + ( d ~ ~ ) ~ . In the <-) space this length becomes
Thus, combining equations 4.29 through 4.36, and introducing [B] = [L] [a], a discrete
system of equations for the single element is written as
A c
a Reis sner/Mindlin Element
2 @ Quasi-three- ,E dimensional
3 , element
ReissnerMindlin Element
Fiowe 4.8: Quasi-t hree-dimensional adhesive element node positions.
Canceling the virtual displacement from either side of equation 4.40 the governing equation
for the single Reissner/Edlin is found as
with
and
{F}' = /' [@lT {/f -- {r) dZ3} d ~ . 2
The adhesive layer is modeled as a quasi-threedimensional element with virtual nodes
as proposed by Renaud and Hansen[55] [56]. Such an element is shown in figure 4.9. The
nodes are described as 'virtual' because their displacements can be fully determined with
Fiewe 4.9 : Quasi- three-dimensional adhesive element.
knowledge of the nodal degrees of fkeedom of The Reissner/Mindlin elements both above and
below the adhesive layer elements. Ultimately, the addition of the adhesive element does not
increase the size of the global stifhess matrix. Assuring continuity, displacements on the
top surface of the adhesive element are assigned values corresponding to the displacement
of the bottom surface of the upper Reissner/Mindlin element. Analogously, the adhesive's
lower surface displacements are assigned values determined by the lower Reissner/&hdlin
element. Figure 4.8 illustrates the relationship.
Three degrees of heedom are accommodated in the quasi-three-dimensional formulation;
displacements, u1: u2; and 213,
The relationship between displacements on the top surface of the adhesive element are ex-
pressed in terms of the degrees of freedom of the upper Reissner/Mindlin element through
Analogously, the displacements of the bottom surface of the adhesive element are related to
the underlyimg Reissner/Mindlin element through
and
Expressioms 4.45 and 4-46 are re-cast in matrix form as
respectively.
Strains ape expressed in terms of displacements in the conventional manner:
Again, a mat ~ i x of operators relates st rains and displacements,
The stress vector includes five relevant components,
Stresses relate to s t r e through an isotropic stiffness matrix,
To determine the governing equation for the element, a virtual displacement is imagined
to act on the element and the work done at the element boundary is equated to the internal
energy,
(-2.53)
With a view to discretization: the element is transformed into the <: q, and < space
as indicated in figure 4.8. The three dimensional represent at ion requires three-dimensional
interpolation functions. An interpolation function is assigned to each of the 33 nodes of the
adhesive element. The interpolation functions are constructed as Lagrange polynomials and
are cubic in the J-77 plane and linear through the thickness c. The interpolation functions
appear as
The interpolation functions relate the discrete to the continuous representation of mriables.
For example, dispiacements are converted through
Equation 4.53 is discretized to read:
It is desired to write the contributions of the adhesive nodes in terms of the Reissner/Mindlin
nodes. This is performed by invoking the transformations of equations 4.45 and 4.46. Equa-
tion 4.56 is rewritten as
4 7 3 Eightnode,isoparametric - axi-symmetric elements
1
Figure 4.10: Axi-symmetric finite element model.
The
and
displacements are shortened in vector form to
the strains are related through the operator matrix [L] to the displacements, {e} =
[L] (6). [L] is written as
[Ll
Stresses in axisymmetric coordinates appear as
The stresses are related to strains through an isotropic stiffness matrix; [Dl; analogous to
equation 4-52.
The governing element equation is derived: as for the surface bonded model, by consid-
ering the effect of a virtual displacement. The redefinitions of [L]; [Dl; { T } : m d (6) for
the cylindrical element permit the governing energy balance to be written analogously to
equations 4.29 and 4-53.
The discretization is done with an eight node isoparametric serendipity element. The
nodal enumeration is also show in figure 4.10. r and z are mapped to the J-7 space as
As before, the k-th interpolation function, pk, equals unity at the k-th node and is zero at
a,ll other nodes. The individual interpolation functions are
Again ( 6 ) = [a] The coordinate transform between the two systems is described by
where
Line integrals are transformed in the same manner as indicated in equation 4.38.
Rewriting ?? in discretized form with [B] = [L] [O]:
is obtained. The element stiffness matrix is computed as
and the element force vector is resolved as
{Fie = { [a] {r} rds. s
As before, a global set of equations is found by summing the nodal contributions in the
global indexing scheme
[KIg {5)g =
in which
and
4.2.3 Finite element analysis of linear strain gradient
To gain preliminary appreciation for the strain transfer problem, the behaviour of a fiber/host
structure to which a linear strain gradient is applied is investigated with the two finite element
configurations described in section 4.2 and the degree
sensor reflects the strain gradient in the substrate is
described mathematically as-
where ey indicates the maximum strain
to which the strain gradient in the
assessed. This axial strain field is
attained by the gradient, and L denotes the length
of the gradient. The axial dimension, XI: is normalized with the fiber radius, rf.
For both embedded and surface bonded descriptions, boundary conditions are most easily
imposed in terms of displacements in the substrate or surrounding material. Such displace-
ments, uy (xl), can be found by integrating the strains of equation 4.78. In this instance
where the axial displacement has been non-dimensionalized with rf. The strain and dis-
placement relationships are illustrated in figure 4.11. -
Given an axial strain gradient in the substrate or-surrounding material, analysis can
recover the axial strain profile in the fiber optic sensor. A typical example of the strain
transfer is presented qualitatively in figure 4.12. Compared are the impressed axial strain
field as a function of axial position in the substrate, ey (xl) and the resultant axial strain
Axial Strain Imposed in Substrate
Axial Displacement Imposed in Substrate
Figure 4.11 : Imposed st rains and displacements
Figure 4-12: Equivalent transferred linear strain gradient
field in the sensor core; e{ (~1). The impressed gradient in the host; following equation 4.78,
is sharp and well defined; the strain distribution in the sensor core cannot be assumed to
have the same degree of 'sharpness.'
As axial strain in the sensor core does not necessarily vary linearly with position, it
is useful to describe an equivalent linear slope for purposes of comparing the slope of the
substrate gradient to the gradient transmitted to the sensor core. The equivalent slope
defined by the line in figure 4.12 is selected by demanding that the area beneath the line - AB and the area beneath the ef (zl) curve, from -oo < rf < - L. 2rf are equal. Mathematically,
On a plot in which the maximum strain excursion is normalized with ey = 1, the d u e of 6
will be *. Further; a can be extracted from the solution of the elasticity field in the following
manner. The left hand side of equation 4.80 can be related to the axial displacements of the
sensor core, u{ (r l). Specifically,
Far to the left of the gradient,
So the function of integration is zero, 3(x2, x3) = 0. Therefore, the value of a can be written
The quality of transfer will be described as follows. The length by which the transferred
equivalent gradient in the fiber core exceeds the length of the imposed gradient in the host
structure, normalized with the fiber radius is indicated as Referring again to figure 4-12? rf
it is seen that
which: considering equation 4.83, reduces to
Equation 4.85 proposes a simple formula to estimate the gradient 'smoothing' in terms of
parameters readily available from the finite element solution.
Edge displacements as described by equation 4.79 are set at r,. r, is taken to be the
location at which the host strain field can be considered unaffected by the presence of the
optical fiber sensor.
At the left face of the bod% the asial displacements are held at zero,
and the axial displacement on the right face is held constant in the 22-23 plane at XI,-,
No other loads or displacements are imposed. Thus, the appropriate Poisson contraction of
the specimen is expected.
The length of either the surface mounted or embedded model, delimited between xlvmin
and x1,-, is chosen such that the elasticity state a t either end of the body is uncoupled
kern the presence of the gradient. In particular, it is asserted that axial strain in the fiber
core matches axial strain in the substrate boundary to better than 0.1%. SpecScally.
The condition expressed in equation 4.88 ensures that the region which is included in the
model is sufficient to properly describe the entire effect of the imposed strain gradient.
The surface mounted cod?guration, as described in sectian 4.2.1, is viewed as two plates
bonded together with a thin adhesive layer. The top section comprising the optical fiber
sensor and some surrounding adhesive is modeled to vary linearly &om its maximum height
to zero at a distance r, from the a of symmetry. This group of elements is laminated in
order to approximate the properties of the fiber/adhesive set. The circular cross-section is
approximated by different stackings of fiber and bonding materials for each element. The
relative thickness of each ply is chosen in order to best represent the cross-sectional area of
the fiber relative to the cross-sectional area of adhesive on which it sits. For a given element,
the thickness of the adhesive layer is selected so that the cross-sectional area of the adhesive
matches the cross sectional area of the adhesive that would have been present below a circular
fiber over the width of the element. The lateral tapering of t he bond is modeled with varying
element thicknesses imposed at the Gauss points. The thin adhesive layer is assigned a
thickness of l p m . The underlying substrate, in which the boundary conditions are enforced
is modeled as another group of Reissner/bIindlin plate elements. The fiber, considered
isotropic, has radius rf, Young's modulus Ef and, Poisson's ratio, vf . Equivalently, the
adhesive material has Em, and v,. Recalling equation 4.46, the displacements imposed
in the substrate define fully the displacements of the bottom surface of the adhesive, the
material properties of the lower Reissner/Mindlin plate are not parameters of the problem
and are assigned arbitrary values.
Given the thickness distribution of the adhesive material, the most likely breach of the
linear elastic and perfectly adhered assumptions will occur as shear failure in the adhesive
as high intensity shear strains develop in the region where contact is made between the
Lamina assigned zero structural stiffness
Figure 4.13: Location of lamina modeled as having no stiffness
substrate and the fiber in the presence of a strain gradient. This potentially jeopardizes the
integrity of the bond and the assumption that perfect adhesion is present. A high gradient
test case is evaluated to assess the assumption. The substrate is typically a structural epoxy L having a range of shear stress limit between 10 and 30MPa[51] [52j. A step gradient, ;T = 0
climbing from 0 to 1000 pe, is imposed in the substrate. This load case results in an adhesive
shear stress above its strength limit. As such, for the surface model, it is assumed that an
adhesive failure of the width of the fiber occurs over the whole length of the model. Figure 4.13
indicates the location of the lamina assigned no structural stifiess. The load transfer from
the substrate to the fiber is therefore exclusively through the lateral portion of the bond
cross-sections. This is incorporated into the model by assuming no strength of the adhesive
in the region directly beneath the fiber. This modification helps assure that the quality of
transfer is conservatively estimated.
The embedded geometry is modeled as an axisymmetric system consisting of two concen-
tric regions, representing the optical fiber sensor surrounded by the substrate material. The
fiber, considered isotropic, has radius r f : Young's modulus Ef and, Poisson's ratio, vf. The
substrate is described by rm, Em, and v,. The finite element formulation of section 4.2.2 is
employed.
Figure 4.14: Shear stress distribution at fiber/surrounding material interface for embedded
installation. = 1.05, 2 = 70: $ = 0
To evaluate the potential shear failure of the surrounding material for the embedded
model, the same high gradient load case is tested as was done for the surface bonded case.
-4 l 0 0 0 p ~ step gradient is imposed in close proximity.to the fiber, = 1.05. The same
compliant surrounding material is chosen with = 70. Figure 4.14 depicts the shear
stress, in the substrate, at the substrate fiber/host interface for the previously described
case. It can be seen that the highest stress found in -:that substrate is 8MPa. This case
indicates that shear failure for the embedded installation requires high gradients, compliant
bonds, and very small values: and is unlikely with typical structural strain gradients.
The parameters that impact the quality of transfer are
a the relative stifbesses of the fiber and the surrounding material,
the geometry of the surrounding materials relative to the fiber, and
the degree to which the gradient can be called 'steep.'
For the embedded installation, the geometric parameter of interest is simply the ratio of
the radii of the cylindrical host region and fiber regions. In the surface bonded installation
the geometry is characterized by the half-bond length, divided by the fiber radius. In both
cases this ratio is expressed as k. Material parameters are also easily described by the ratio rf
of sensor s t i5ess to bond stifhess 2. The gradient is given some consideration. The degree to which a gradient can be consid-
ered 'steep', at first, would seem a combination of two factors: the maximum strain achieved
by the gradient and the length, relative to some characteristic length, over which the max-
imum strain is achieved. The rate of change of axial strain with axial position, normalized
using the fiber radius is rf 2. Perhaps surprisingly! for the assessment of the strain transfer
the value of the parameter ey has no impact on the degree of strain quality of transfer as
defined by equation 4.85.
This can be argued by considering the general finite element formulation. In both surface
mounted and embedded formulations, a finite element analysis generates a global system of
equations such as equation 4.61 and 4.75,
where [KIg is dependent on material properties and geometry, (6)' is a vector of nodal
displacements for which equation 4.89 is solved, and { F j g is a load vector containing the
nodal forces acting on the system.
Only displacement boundary conditions are imposed in either the surface mounted or
embedded study, no explicit nodal forces are applied. So initially a system such as
is found. To leave the expression as such is clearly inadequate because some of the dis-
placements have been explicitly prescribed. The prescribed displacements at the boundary
are removed from the (6)' vector and shifted to the right han~d side of equation 4.90 after
having been multiplied by the appropriate entry of the stiffness matrix. This will result in a
system with fewer degrees of freedom and with a new force vectoor, derived from the imposed
displacements
[K'l{6') = {F'} * (4.9 1)
where [Kr] is the reduced s t f i e s s matrix, {6'} represents the vector of unknown displace-
ments and {F') denotes the force vector equivalent to the prescriibed displacement conditions
at the nodes.
The force vector {Ft ) is constructed from imposed displacements and corresponding
stiffness matrix elements. For example, in a system with N degrees of freedom with boundary
displacements imposed at nodes ni, a typical force vector would be written as
It can be seen from equation 4.92 that the implicit load vectorr is directly proportional to
the magnitude of the imposed displacements. Recalling equatiom 4.79 the displacements are
directly proportional to the ey term of the prescribed gradient. Thus, equation 4.91 can be
arbitrarily scaled, by choice of e$ Thus, a potent i d y counter-imtuitive conclusion is found:
The maximum height of the gradient does not affect the quality- of transfer. A larger ey will
scale the vertical axis of figure 4.12 without changing the qualitzy of transfer. Thus, for two
situations both with a gradient of length L and a fiber radius rf, and different maximum
strain excursions: both will show the same degree of quality of -transfer, to the elastic limit
of the structure.
The treatment of all further gradients in this study will be forr unit e;. Thus, the measure
of gradient steepness is simply T. The finite element analysis is performed for a range of values of the parameters z, p,
and 3. For each case, a value of is recovered to assess the equality of transfer. Typical
epoxy stifhesses range between 2GPa and 5GPa[52] [56]. Thus a t the two extremes, two
values of 2 are analyzed, 14 and 35. The range of gradients considered are
For a typical 62.5pm radius fiber this represents a range of gradient lengths between 0 (a
step function) and lcm. The range of bond sizes is
Fiewe 4-15 shows an indication of quality of gradient transfer over the parameter space.
The plot is given in terms of rf / 4L so that higher values indicate better quality of transfer.
It is seen that stiffer substrate materials, less steep gradients, and larger wider bond profiles
lead to better quality of trimsfer. The effect of increasing bond width is limited. For
geometric ratios larger than % > 16 the quality of transfer does not increase for further
increases in geometric ratio. This result indicates that, when surface bonding optical fibers
to a substrate: there is little advantage to increasing the epoxy coverage over the fiber to a
larger ratio than > 16. rf
Individual contour plots for the surface mounted installation are presented in fi,o;ures 4.16
and 4.17 for stiffness ratios of = 114 and = 35 respectively.
The quality of transfer over the parameter space for the embedded model is plotted in
figure 4.18. The trend is different than seen with the surface mounted model. The transfer
quality decreases monotonically with geometric ratio. This is expected considering that
larger and larger geometric ratios indicate that the imposed gradient exists at farther and
farther distances from the sensor itself. Similarly to the surface mounted configuration,
stiffer substrate materials and less steep gradients lead to better strain transfer.
The two-dimensional contour plots corresponding to the two stiffness ratios are presented
in figures 4.19 and 4.20. These charts can be used as design tools for determining an adequate
bond width for an expected transfer efficiency. For example, consider a situation in which a
gradient of length 5mm exists in an epoxy of stifFness 2GPa 0.5mm away from an embedded
fiber sensor of radius 62.5pm and stifhess 7OGpa. This corresponds to values of = 35
and = 8- Figure 4.20 indicates a normalized elogation value of approximately 12. Thus, a
gradient of length 5mm in the substrate is reflected in the fiber as a gradient of length 5.8mm.
Figure 4.15: Quality of transfer for surface mounted installation (Vertical axis represents
5 x 1000)
5.0 10.0 15.0 20.0 25.0 30.0 35.0 Geometric Ratio r ,/rf
Figure 4.16: Gradient elongation, e, for stiffness ratio of 2 = 11 (Surface mounted)
5.0 10.0 15.0 20.0 25.0 30.0 35.0 Geometric Ratio r ,/r,
Figure 4.17: Gradient elongation, u, for stiffness ratio of = 35 (Surface mounted) rf
Figure 4.18: Quality of transfer for embedded installation (Vertical axis represents & x 1000)
5.0 10.0 15.0 20.0 25.0 30-0 35.0 Geometric Ratio r ,/r,
Figure 4.19: Gradient elongation, 5, for stiffness ratio of 2 = 14 (Embedded)
5.0 10.0 15.0 20.0 25.0 30.0 35.0 Geometric Ratio r ,/rf
Figure 4.20: Gradient elongation, e; for stiffness ratio of = 35 (Embedded)
ez
Figure 4.21: Illustration of problem: axial strain transfer from host to sensor
If a stiffer epoxy is selected, 2 = 14; it is found that the gradient is better transfered such
that an equivalent gradient of 5.4- is found in the fiber.
4.3 Transfer of arbitrary strain distribution
The finite element method of section 4.2 is a stable md proven method. However, pro-
gramming the finite element code requires recoding boundary conditions on a case by case
basis. The matrix inverison required to implement the solution is computationally extensive.
Naturally. an analytical approach would be desirable to. illuminate parameter dependencies
and to provide a 'feel' as to the behaviour of the system. Figure 4.21 illustrates the problem
to which an analytical description is sought. In cylindrical co-ordinates, (r: z, 0) an infinite
fiber is bounded by an axi-symmetric region of surrounding material. At some radius, r,, in
the surrounding material and arbitrary strain distribution is specified. In view of the results
of section 4.2.3 it is expected that the fiber will be subject to different axial field than the
surrounding material.
Consider the original derivation of Cox, presented in section 4.1.1. Continuing &om
equation 4.14, the . At r = rm an axially varying strain field exists, e? (r,, 2 ) . In the instance
of an optical fiber sensor with a relatively compliant coating? rm naturally corresponds to the
outer coating radius, where ep (r,; z) denotes the strain field present at that boundary[l3].
For the more general embedded situation, r, is interpreted as the distance from the fiber
core at which er (r,, z ) is known. T o determine r, it is necessary to first estimate the
effective shear lag parameter N of the fibre/surrounding material geometry and stfiesses.
This can be performed by comparison to finite element results or experimental data[45].
Equation 4.14, for the case of an arbitrary far field strain is solved by Fourier decompe
sition. The solution of equation 4.14, E: ( r ) , will be composed of a weighted distribution of
spatial wavelengths: A,. The spatial wavenumber is introduced to describe a single Fourier
component of the solution where k = $. The Fourier transform relates the strain descrip-
tions horn the z to the k space as
00
E! ( k ) = /__ < ( 2 ) exp ( -2r ikz ) d z , and roo
a: (rm, k ) = J- e y (T,, Z ) exp ( -2r ikz ) dz- 00
The inverse relationships appear as
00 < ( 2 ) = 1- e! (k) exp (2rrikr) dk, and - -
e: (T,, Z) = /-_ &r (r,, k ) exp ( k i k r ) dk.
The Fourier transform of equation 4.14 yields the expression
From equation 4.97, the transfer function of the system, H ( k ) , is recovered,
6: (k) - - 1 H (k) = e y ( r m j k ) I+ ( .Y , )~ '
Using equation 4.13 to
The transfer function,
distribution, of spatial
expand 4.98 in t e n of the shear lag parameter yields 1 I H (k) =
1 + 47r2k2r; ((I +- L..) 1D. (T) (z) X ( k ) , indicates the proportion by which a purely sinusoidal strain
frequency k, will be transferred horn surrounding material to fiber
as function of N and k. As k increases, indicating shorter spatial wavelength content, the
transfer gets progressively weaker. This supports the intuitive notion that quickly varying
strain fields with high wavenumber content, are more poorly transfered than slowly varying
strain fields. At the extreme limits, a constant strain field, ey (r,, z ) = c, is perfectly trans-
ferred, H (k = 0) =-I, while a very quickly varying harmonic strain field in the surrounding
material (high k) is fully attenuated H (k --+ oo) = 0. Further, sharp strain variations in
the surrounding material will be 'smoothed7 as they are transferred to the fiber sensor. The
high wavenumber components will be strongly attenuated relative to the low wavenumber
components.
In the section 4-2.3 individual cases are made to suggest the parameters on which lin-
ear gradient transfer is dependent. It is argued that they are material stiffness ratio, g; geometric ratio; and an indication of gradient steepness, s. For the arbitrary case the
rf
parametric dependence is now revealed in equation 4.99. The stiffness and geometric ratios
appear explicitly, and the spatial wavelength content is used to more precisely define the
degree to which a strain field varies in space.
Finally, the expression by which the in-fiber strain function can be found is written as
( z ) = Jm H (k) C (rm, k) exp (2x ik t ) dk, -M
where Ey (r,, k) is found from equation 4.95. Equation 4.100 is written in detail as a function
of the e," (r,, z),
x exp (27rikz) dk. (4.101)
Figure 4.22 shows the transfer function H (k) . Also included in figure 4.22 are the results
from a the conventional FEM analysis of the same physical configuration. FEM calculations
Figure 4.22: Transfer function, H ( k ) dependence on spatial frequency k
were made using the
boundary conditions
T m 7
geometry of figure 4.21 and the model of section 4.2.2. DispIacement
are imposed at r = r,. A strain of the following form is specified a t
e," ( z ) = eEo cos (27rkz) .
Integrating equation 4.102 the displacements to impose at the model boundaq are found as
z m
w m ( z ) = / e y (2') dzr = % sin (27rkz) . 27ik
In this study, N values of 0.1: 0.2, and 0.3 are considered. The trend towards poor
transfer with increasing wavenumber is clear, and the plots also indicate that higher N values increase the quality of transfer.
There exists significant agreement between this work and the FEM results, but limitations
originating from the simplifjying assumptions are noted. Specifically, disagreement increases
as N increases at high wavenumber. For example, a t k = 1000m-', and N = 0.3 an
attenuation of 0.37 is predicted by this work, while the finite element analysis predicts
0.31. At the same k, for the smaller Af = 0.1: the predicted peak suppression is 0.06
(equation 4.101) compared to 0.05 (FEM) . A s the wavenumber increases, the transfered
profile is considerably attenuated, compromising the assertion that the strain gradients are
of the same order (Recall assumption of equation 4.5). As N increases, the assumption that
the material stifiess ratio is much smaller than unity becomes less robust.
A family of curves is presented in figure 4.23. This set of plots compares the transfer of
a single period of the function e? (R, z ) = cos(27ikz) for different k with the induced strain
field in the fiber core E{ ( 2 ) . Seven plots appear in each graph: the plot of er., (R, Z ) and
the solution of equation 4.100 for the three N and the three corresponding FEM solutions.
Over the z range, there is agreement between this work and FEM. The weak transfer of high
k surrounding material strain fields (short As) is apparent. The tendency of higher N to
improve the transfer is also seen. The attenuation in peak to peak magnitudes of e F (R, Z )
compared with E: ( z ) is precisely the physical meaning of the transfer function, H (k) : a t a
specific wavenumber k.
Figure 4.24 compares the transfer of a family of Gaussian strain profiles of e? (R, z). The
Gaussian shape is studied because many realistic applications of fiber optic sensing attempt
Fibwe 4.23: Transfer of far field sinusoids 97
to extract information from strain distributions that, Like Gaussians, can be characterized
in terms of a peak strain and a full width at half maximum ( F W B ) , 4zm. The strain
distributions due to holes: advancing cracks, or other structural imperfections can be de-
scribed in terms of a maximum strain and a E m . A range of F W M are investigated
from 4 z m = 0.lm.m to 42" = 5mm. The imposed strain field at the r = r, boundary is,
I d
In this case, a numerical integration of equation 4.104 is employed to compute the required
boundary displacements, z
w m (rm,z) = / ey (r,, 2') dr'.
4 and the The transfer can be described in terms of the degree of peak suppression eF(rm,c proportion by which the FWHM broadens, g.
For very narrow Arm, the peak suppression is significant. For example at Azm = 0. lmm
the peak is suppressed from unity to 0-02, while the in-fiber FWHM is estimated to be
~ z f = Smm, an expansion by a factor of 90. The latter result indicates that sharp, narrow,
strain 'spikes' due to structurd imperfections axe poorly transferred to the fiber sensor core.
The results of figure 4.24 are synthesized in figure 4.25. The peak suppression and Azm
increase are shown as functions of the imposed FWHM. for the three values of Af. The solution of equation 4.100 is performed with a fast Fourier transform (FFT) alge
rithm. The FFT is considerably faster than an analogous FEM solution, the latter requiring
a matrix inversion. Arbitrary functions of eF (r,, r ) , either periodic and aperiodic are cal-
culated with the same computer code. The proposed f6rmulation allows the calculation of
the axial strain field in the fiber from an arbitrary sti& profile in the host provided the
shear-lag parameter, N, can be determined.
4.4 Equivalent embedded geometric ratio for surface
bonded sensor
The mathematical representation of equation 4.101 is useful for the embedded case. An
extension to the case of the surface mounted fiber is attempted in this section. As was
Figure 4.24: Transfer of far field Gaussians 99
Profile F'WHM Azm (mm)
0.0 0.5 1 .O 1.5 2.0 2.5 ProfiIe FWHM A Z ~ (mm)
Figure 4.25: Peak suppression and broadening of Gaussian far field strain profile
Adhesive layer of thickness "t"
Figure 4.26: Surface bonded installation modelled as embebbed installation with equivalent
radius
argued in section 4.2 and explicitly derived in section 4.3 the shear transfer from host to
fiber in the embedded case is dependent on three factors-stiffness ratio between host and
fiber, the geometric ratio indicating the location of the far field relative to the fiber, and the
degree to which. the field varies rapidly in space. The first two factors, stifbess and geometry
are included in the shear lag parameter N. The quality of transfer is expected to vary with
the degree of far field variation. To determine an estimate for the behaviour of the surface
mounted sensor, the following approach is used: The model of section 4.2.1 is employed to
model the transfer of periodic strain variations (specifk k values) from the substrate to the
optical fiber. Then, the attenuation incured during mechanical transfer is extracted from
the finite element solution. Recalling that the transfer function H (k) of equation 4.99 is
precisely the ratio of the maximum amplitudes of the far field and transferred profiles, the
transfer function is directly computed kom the results of the surface bonded model. Further
an equivalent embedded far field radius is estimated such that the surface mounted behaviour
emulates the behaviour of an embedded situation with a fictitious equivalent far-field radius.
Thus in equation 4.99 the only unknown variable is the geometric ratio % which is
estimated from the transfer efficiency. Table 4.1 indicates the results obtained in this analysis.
For a sensible range of stiffness ratios, a range of transfer efficiencies between near unity and
50% peak suppression is found. Figure 4.26 illustrates the idea of an equivalent geometric
ratio. The surface bonded model is considered to be equivalent in strain transfer performance
to an embedded sensor with the equivalent geometric ratio. The results of table 4.1 report
an equivalent geometric ratio of between 1.16 and 1.26. For a fiber of radius 62.5pm the
ratio period transfer efficiency Estimated equivalent
geometric ratio 5
rf
Table 4.1: Equaivalent radius estimate for surface mounted situation
equivalent radius is estimated to be between 73pm and 79pm-a distance of between 10pm
and 17pm from the outer surface of the fiber. The finite element model of the surface bonded
configuration considers the fiber to sit on a layer of adhesive one micron thick. The equivalent
embedded radius is estimated to be between ten and seventeen times this value.
For the balance of this work, the surface bonded case is assigned the most conserva-
tive equivalent geometric ratio found kom the finite element simulations, = 1.26. The +f
Poisson's ratio of the adhesive is taken to be 0.25. The modulus of the adhesive is also as-
signed a conservative value of 1GPa. Thus the shear lag parameter for the surface mounted
configuration is computed to be
For the remainder of this document, N is taken as 0.22.
It is noted that the analysis of this section is limited to axial strain transfer. There will
be an amount of bi-axial stress induced in the fiber due to the asymmetry of the bonding
configuration. For example, when the substrate/fiber sensor system is under axial tension,
transverse Poisson contraction is expected. The fiber admits a low Poisson ratio and has
small a5ni ty for contraction. The host/adhesive set has a stronger desire for contraction. As
displacements must be continuous, the relative contractions are equilibrated by an internal
bi-axial transverse stress field-
4.5 Conclusions of mechanical strain transfer analysis
This section presents several approaches by which one can estimate the ability of a sensor
to mechanically transfer spatially varying strains from a substrate to an optical fiber. Sec-
tion 4.2 details finite element methodologies by which both an embedded fiber and a surface
mounted fiber can be analyzed in terms of strain transfer behaviour. The embedded analysis
is straightforward, using cyclindrical elements. The surface mounted approach is novel, using
Reissner/Mindlin plate elements to model the tapered profile of an optical fiber &ed to
the surface of a substrate. The parameters contributing to the strain transfer behaviour are
considered. Geometry of the fiber/bond set, ratio of stifhesses clearly contribute. In the
limit of elastic behaviour, strain gradients are defined in terms only of their spatial extent:
not the strain excursion. Design charts are provided to make an evaluation of the strain
transfer response of either an embedded or surface mounted optical fiber sensor subject to a
linear gradient.
In section 4.3 a more general and analytical approach to the arbitrary strain transfer
problem is explored. For an embedded fiber, an analytical derivation predicts strain transfer
behaviour when the substrate strain field is expanded into its spatial harmonic components.
In this description, a transfer function determining the attenuation of each harmonic com-
ponent is derived as a function of the shear lag parameter N. The results of this technique
are compared against conventional finite element solutions of the same physical problem and
agreement is found to be strong. As well, the parameters argued in section 4.2 to determine
the strain transfer appear explicitly in the transfer function.
Finally, to extend the transfer function analysis to the surface mounted situation, an
equivalent N is estimated. The estimation is done by comparing the surface bonded sen-
sor's transfer function (evaluated by the finite element method) to the transfer function of
the embedded sensor and inferring an equivalent shear lag parameter. The latter analysis
provides a method by which the surface mounted sensor's ability to transfer a strain field
can be quickly evaluated.
Chapter 5
Demonstration of method and
measurements
This section presents optical measurements of B r a g gratings subject to a variety of load
conditions. From this optical data, the physical structure of the grating is reconstructed
using the theory of chapters 2 and 3. The strain transfer mismatch issues of chapter 4 are
considered in the comparison to the resolution of the sensor. The processing of the recovered
optical data requires some consideration which is also discussed in this section-
5.1 Review of methods for interrogating gratings with
non-uniform profiles
Several methodologies exist for recovering distributed information from a non-uniform grat-
ing. Two similar strategies-, phase spectmm based (PSB) and intensity spectrum based
(ISB) interrogation were developed in parallel at the University of Toronto Fiber Optic
Smart Structures (FOSS) group in the mid to late 1990's. The strategies are simila because
of their dependence on an estimate of penetration depth of a single wavelength into the
sensor grating. In both cases an explicit wavelength-position relationship is derived- These
relationships hold for the case of monotonically increasing or decreasing in-fiber strain fields.
5.1.1 Phase-based spectrum (PSB) strategy
The concept of local resonant wavelength is introduced,
Equation 5.1 is simply 2.46 expressed as a function of position. In other words: at position
z: wavelength X(z) is locally reflected. As well, equation 3.35 of chapter 3 is used to specify
the relationship between local wavelength shift and local axid strain
The PSB is derived by considering the phase accumulation of a wavelength, A, reflected at
z; over an interaction length dz7 figure 5.1. In terms of the wavelength of the propagating
wave, and the local index of refraction: the incremental phase accumulation is written as
Figure 5.1: Phase accumulation due to reflection at z of length dz.
The rate of change of expression 5.3 with wavelength is
Summing 5.4, d+P (A,) -- - 47mp (2) z
~ X P % (5.5)
is obtained. Equation 5.5 is re-arranged to solve for the axial position of the specific wave-
length A,; as a function of the grating's phase, @p,
Equation 5.6 indicates the position of each local wavelength, A,. Thus, using equation 5.2
the local Bragg wavelength shift can be used to find the local strain contribution at every
2. The si,dficance of expression 5.6 can be understood with an analogy to an interferom-
eter. Imagine two wavelengths, between which a large phase excursion is found. The large
phase excursion suggests a large physical distance between the penetration depths of the
two wavelengths. The converse analogy can also be imagined: two wavelengths with a small
phase excursion between them are reflected fiom physically proximate reflection centers. The
shorter the physical distance between subsequent wavelengths, the larger the spatial chirp,
thus the strain gradient. PSB measurements are reported in[15] and[47] - This type of ex-
periment requires a high coherence continuously tunable source for narrow band wavelength
discrimination. The local coupling strength of the grating is of no consequence provided that
there is significant reflectivity to recover the phase.
5.1.2 Intensity based spectrum (ISB) strategy
The reflected intensity approach, again, derives a specific one-to-one relationship between
physical space and local reflected wavelength. Matsuhara and Hill[57] study a monotonically
chirped grating. The authors describe the chirped grating as having a distribution of local
wavelengths as
where A, is the central wavelength of the spectrum. The aforementioned reference provides
a relationship between the reflectivity at the center wavelength as a function of the chirp
and grating strengths. In the context of sensing, the relationship between the reflectivity of
a certain wavelength, Ap is extended to the case where the gradient is not necessarily linear,
and the grating coupling is not necessarily constant,
R (Ap (2)) = 1 - exp X' (An (2))
Equation 5.8 can be rearranged, as the spatial rate of change of local B r a g wavelength is
sought,
Equation 5.9 can be integrated as it appears, or, if a grating of uniform coupling strength is
employed, An (I) = An,, an explicit expression for z can be found,
The last equation indicates the specific one-to-one relation between spatial position and local
wavelength. As can be seen born equation 5.9 the reflectivity contribution from a specific
physical location is dependent not only on the local chirp, but also the local coupling strength.
This method provides an advantage as there is no requirement that a high coherence tunable
source need be employed-a broadband light emitting diode is sficient-but an estimate of
the original grating strength distribution must be made to discriminate between the effects
of strong local coupling and weak local chirp. Published accounts of results of the ISB cite
spatial resolution of approximately 3mm[47] and[l5].
For the case of an arbitraq strain field, the single penetration depth approximation ceases
to hold. The physical reason for this may be seen by considering the following example: A
sensor subject to a field in which a strain field increases to some value then, with increasing
position, decreases mew to the original nominal strain value. In practice, this type of field
is common, as strain concentrations in the vicinity of holes or advancing crack tend to show
a high localized strain field surrounded by a nominal far field strain value on either side
of the imperfection. Thus, the same local wavelength exists at two distinct points in both
the increasing and decreasing branches of the strain field. The intuitive ascription of a
single physical location to a single reflected wavelength fails to describe the field. Which
location should be chosen to represent the reflected wavelength? As well, multiple reflection
centers for the same wavelength produce Fabry-Perot cavity interactions incompatible with
a one-to-one relationship between position and reflected wavelength.
To overcome this problem two solutions are proposed. The first suggests the continued
use of the ISB method, but with a p r e - w e d grating as the sensing device. The sensor
is designed such that the nominal chirp is sufficiently strong so that with the additional
imposition of the non-monotonic field due to structural perturbations of the substrate, the
resulting in-fiber field remains monotonically increasing or decreasing and the interrogation
methods for a monotonically strained sensor can be used. The substrate measurement is
recovered by subtracting the nominal pre-chq from the total in-fiber chirp- An improvement
in spatial resolution to lmm is reported using a pre-chirped grating[58].
The second strategy, demonstrated in the late nineteen nineties a t the University of
Toronto FOSS group, can be seen as the confluence of the individual phase and reflectivity
strategies- As derived in section 2.4, for low reflectivity cases, a Fourier relationship relates
the complex reflectivity (of which constituent parts are the reflectivity and optical phase)
ELED Delay line Bragg Grating
? stretcher
VCO I
[ counter , I
Figwe 5.2: Closed loop schematic of[l]
and the physical structure of the grating for the arbitrary strain field. Beginning with both
reflectivity and phase, the inverse transform technique uses two pieces of information in
one inverting algorithm to recover arbitraxy distributed in-fiber strain fields. The inverse
transform method is used in this work.
5.1.3 Low coherence interferometry .
Before outlining in detail the inverse transform experiment, an alternative for interrogating
non-uniform Bragg gratings is discussed. The University of Southampton's Opteoelectronic
Research Center has proposed a low coherence strategy fcr performing this measurement [l8] [17]
111 PSI The proposed approach uses a low coherence source to interrogate the Bragg grating in
two configurations. Both employ Michelson interferometers of which one arm leads to the
VCO
m 1 ELED
Optical Amplifier \ I
Lens coupling to Gree space mirror
/ Acquisition
Figure 5.3: Open loop schematic of[l]
grating under interrogation. The key to the low coherence approach is that interference
h g e s appear only when the interferometric path mismatch of the two Michelson arms
are balanced to within the sub-millimeter coherence length of t b e ELED source. Thus, by
displacement of the reference arm either by stretching the fiber or by displacing a bulk
mirror, a short resonant position is interrogated in the Bragg grating. All other positions
being outside the source coherence length-
In the closed loop configuration, a feedback loop tracks the local Bragg wavelength. The
feedback algorithm maximizes the power through an acoust~optic tunable filter (AOTF) by
continuously varying its transmission window. The reference arm is 50m long. A portion
is wrapped around-an oscillating PZT cylinder for purposes of generating several fringes
within the coherence length. Two posts 1.7m apart can be actuated relative to one another
to provide fiber extension. In the open loop configuration, position is scanned by displacing
an actuated hee space mirror coupling into the end of the reference arm. The open loop
configuration characterizes devices as functions of time of flight from the reference mirror.
The results of this technique are strong. Spatial resolution of 342pm in conjuction with
strain resolution of 5 . 4 p ~ / m is reported.
5.2 Implementation of interferometric phase recovery
experiment
O h et al. [16] first describe the interferometric Fourier transform experiment for distributed
strain sensing. A schematic of the experiment is illustrated in figure 5.4
The tunable source is a semiconductor distributed Bragg reflector (DBR) laser. The laser
includes three sections: a Bragg structure lies at one extremity of the lasing cavity, a passive
waveguide provides phase control and gain is achieved in an active section. Three input
currents are required to control the device--excluding thermal stabilization. A first injection
current controls lasing power as it is supplied to the active region. The other two currents are
used together to determine the lasing wavelength. With a goal of preserving single moded
operation a coarse tuning current flows through the Bragg section of the semiconductor. The
center Bragg wavelength, specified by the same Bragg condition seen in the context of fiber
Reflectivity Photodetector Bragg
Grating
Tunable DBR Source
Photodetector A
! Lock-in Ramp Signal / Amplifier
Fiewe 5.4: DBR Source interferometric experiment with lock-in phase.
devices, is determined by period and current (which influences E): X = BiA. The grating
is tuned by varying the carrier density though the Bragg section, which manipulates the
section's index. To maintain single h e behaviour, current is supplied to the phase section
to compensate for the change in optical path in the Bragg section. The DBR is able to tune
continuously over a 3 . 5 ~ range.
Section 2.4 details the derivation of the recovery of in-fiber strain from in-fiber parameters.
For the balance of the experiments, low birefringence fiber is used, thus the polarization
specifying p subscript is dropped from all nomenclature. It is required to construct the
complex electric-field proportional reflectivity r (A) as a function of the tuning wavelength.
It must be assembled from independent measurements of reflected intensity and phase. In
the first set of distributed experiments, the DBR is tuned somewhat slowly, taking nearly
two minutes to scan over the tuning range. A dither and lock in amplifier is used to better
extract phase measurements-
Results reported in reference[l6] cite a theoreticd strain resolution of approximately
25pm and a spatial resolution of 3mm. The latter reference does not include a comparison
with analytical elasticity solutions for purposes of verifying the strain and spatial resolutions.
It is at this point that the current thesis seeks to continue to contribute to the field, following
the developments of the ISB, PSB; and the initial reporting of the potential of the inverse
transform method.
5.2.1 DFB tunable source
The source employed in the bulk of this work is generously provided by Nortel Networks.
The specific fabrication of the device itself is proprietary to Nortel Networks and is not
known to this author. It is known that the device is a distributed feedback (DFB) laser.
A general overview of a DFB laser is given. This type of semiconductor has a single active
section. Inside the active section a phase shifted Bragg grating is impressed[59]. Viewed in
transmission, a notch exists in the center of the grating stop band-where typically reflectivity
is highest. The lasing cavity exists about the mid-point of the T-shifted grating. Figure 5.5
indicates that the phase shifted grating can be imagined as two juxtaposed gratings separated
by the phase shift. Imagine the wave fkom the center traveling to the grating to the right.
Lightwave resonates between the
i penetration depths of the x-shifted
Resonator formed by "two" gratings separated by a spatial mphase shift in DFB structure.
Figure 5.5: Phase shifted grating as DFB
The wave penetrates to a certain effective depth and is reflected. As it passes the phase shift,
the wave, coherent with the right side grating is maximally out of phase with the left side
grating. Directly after the phase shift the coupling between the wave and the LHS grating is
minimal due to the phase mismatch between them. The wave penetrates and is reflected and
repeats the cycle on the other side. Because the Bragg reflectivity is strongly wavelength
dependent, cavity modes that axe detuned from the Bragg condition are strongly suppressed
permitting stable single mode operation.
Tuning is achieved by altering the effective index of the substrate. The physical spacing
between the penetration depths and the index define the cavity modes. Cavity spacing
depends on this an optical path, ZL- As well, the Bragg reflections are wavelength dependent
with the peak reflectivity defined by the Bragg condition. This is an optical path-like term
EA. With injection current, the Bragg condition moves in parallel with the change in optical
path, preserving single line behaviour over a range of lasing wavelengths. Lasing power is
not independently controlled and some variation in output intensity over the tuning range
is expected.
The Nortel device in question is driven with a variable current source capable of ramping
bstween 30mA and 300mA in scans as fast as lms. Temperature control is accomplished
with a commercially available Pelletier thermoelectric cooler. The calibration of t h e device
is performed with a simple all fiber Michelson interferometer- The output intensity of a
Michelson interferometer as a function of wavelength is
in which Il and I2 are dependent on the splitting ratio of the coupler and 4 L is the mismatch
length between interferometer arms. As the wavelength of the laser is tuned a wawelength
dependent fringe pattern is received at a photodetector. The injection current is sirmultane-
ously monitored as the laser is tuned giving phase, as a function of drive current, @ I (2). The
total phase excursion can be expressed in tenns of the beginning and end wavelenmhs: XI
and X2 respectively, as
The group of variables 47rEA L can be solved in terms of the total phase excursion, pelrmitting
to write the calibration of lasing wavelength in terms of injection current, i,
X (i) = A@~otcrlxl A2 (5.13) 4 ( 2 ) (A1 - A2) + &'Tddx2 '
The tuning behaviour of the DFB is plotted in figure 5 -6. A continuous tuning range at nearly
lOnm is characterized. The end wavelengths are measured using an HP optical spectrum
analyzer (OSA) with a resolution of 80 pm. This error may lead to an absolute offset of
the same order. The relative accuracy of the calibration is much better than 80pm -because
any error scales over the lOnm tuning range. Figure 5.6 shows two experimental calibration
curves derived kom two differently mismatched interferameters and a best fit plot. T h e high
wavelength portion of the calibration obtained from the interferometer of several milllimeters
of mismatch deviates slightly horn smooth behaviour. This is corrected by lengtherning the
mismatch of the interferometer and increasing the fringe density at the detector. Higher
fringe density facilitates the fringe counting algorithni as the fringe density increases to
the point where it was no longer required to seek intra-hinge information. A 10-th degree
polynomial is used to fit the calibration of the long interferometer calibration in cmmputer
code used to process experimental data
Calibration of Nortel DFB Laser
. . . . . . . Long Interferometer --.- -.- Curve Fit Used for Code
0.0 0 5 1.0 15 2.0 2 5 3 -0
Ramp Signal [V]
Figure 5.6: DFB laser wavelength calibration
Tuneable DFB Laser (1536nm to 1545nm range) Mirror
(Cleaved end is suflicient) -
A* k -1 Reference Power
. Coupler
Phase Photodetector
Reflectivity Photodetector
Figure 5.7: Interferometric experiment
The interferometric experiment is shown in figure 5.7. Three New Focus transimpedance
photodetectors rated to a bandwidth of IMHz are used to capture the three optical signals.
Data acquisition is done with two PGbased DAC cards. The DAC cards, manufactured by
Gage Corporation, can track data samples at rates of up to 1 million readings per second.
Each DAC card has two analog inputs. Four sets of variables are measured over the course of
a single scan, a ramp voltage-directly related to injection current, the reflectivity response,
the phase response, and a reference power. The scan time is lms. This fast sweep minimizes
the vulnerability of the experiment to thermal crosstalk. The reference power is captured to
track the expected source power fluctuations as the source tunes. A trigger signal issued horn
the current supply synchronizes the data capture of the two boards. The code to control the
boards is written in C++. Figure 5.7 also shows qualitative drawings of the data recovered
horn each of the photodetectors.
5.2 -2 Representative grating interrogation
The goal of the data recovery is to f ist represent the wavelength domain information with
a complex electric field proportional reflectivity, r (A), recalling equation 2.54 of chapter 2 .
The magnitude of the electric field proportional reflectivity is found by taking the square root
of the reflected intensity, accounting for the reference power. The optical phase extraction is
more difiicult. An interference pattern is recovered a t the phase photodetector from which
the phase must be recovered. A formal phase expression for the intensity received at the
phase photodetector can be written as
I, (A) indicates the reflectivity of the silvered mirror termination of the Michelson arm.
I, (A) is a weak function of wavelength. RFBo (A) indicates the power proportional reflec-
tivity originally introduced in equation 2.43. RFBG (A) is a strong function of wavelength.
Equation 5.14 has little practical use. To invert and recover phase explicitly following equa-
tion 5.14, the reflectivity of the mirror would need to be calibrated and the reflectivity of
the grating would need to be divided out. Dividing out the reflectivity envelope proved
unprofitable as the small but fmite noise floor did not permit exact knowledge of the reflec-
tivity envelope. Attempts to self normalize the interferometric response resulted in many
numerical 'division-by-zero7 type problems.
As well, the fringe density is dependent on two factors: the strain content of the sensor
(high strain content increases the fringe density) and the optical path mismatch between the
interferometer arms (large optical path mismatch induces high hinge density). Empirically,
it is found that the experiment performs best when receiving a moderate Mnge density- The
phase accumulation of an interferometer is derived in equation 5.12. Replacing one mirrored
end of the interferometer with a Bragg grating will give a slightly merent phase response,
A - A, 27rc t$ (A) = 45~riaL- + - r (A) d i -
A20 A20
The first term indicates the Michelson phase excursion due to the interferometric mismatch
between the mirrored arm and a distance at the fioont of the Bragg grating. The second
term accounts for differential penetration depths into the grating. Recalling the comments
of section 2.3 relating delay to penetration depth, the grating arm can be considered a
mirrored arm: but with a wavelength dependent arm length. SpecXcally: the delay charac-
teristic defiaes the penetration depth for every wavelength. To manipulate the rate of phase
accumulation, AL can be lengthened or shortened.
As hinge density increases the wavelength resolution required to discriminate phase also
increases. Too high a fringe density can exceed the source's ability to resolve individual
fringes resulting in lost phase information. As well? too low a fringe density requires intra-
fringe phase interpolation. This too is undesirable because the envelope of reflectivity can
vary with the same order as the interferometric phase, making interpretation difEicult. In
other words, if interferometric phase varies too slowly, both the RFBG (A) term and the
cos ( 4 (A)) vary with similar rates making separation of their respective contributions to
I (A) problematic.
In practice, a numerical routine determines the phase from the int erferometric reflectivity-
but a check is performed on every data set to ensure that the recovered phase makes sense
when viewed against the reflected intensity. An example is shown in figure 5.8 where the
cosine of the inferred phase is plotted with the interferometric reflectivity.
The complex reflectivity is assembled fiom the phase and reflected power as
in which the computation of R (A) includes the effect of the varying source power over the
scan. Equation 2.63 relates the spatial distribution of the coupling strength to the complex
reflectivity as a function of wavelength,
2 = K (z) = --
A2 /--- r (A) exp
A uniform grating of length 8mm and peak reflectivity of near 0.6 is interrogated with the
interferometric system. For comparison, the grating behaviour is simulated by the T-Matrix
method of chapter 2.4. The coupling distribution is computed fiom both the experimentally
recovered reflectivity and the T-matrix simulated reflectivity. Knowing the grating length
and its coupling distribution allows accurate synthesis of the of the reflectivity. Figure 5.9
shows the experimental and simulated reflectivities in the upper plot. The lower shows three
6000 6500 7000 7500 8000
Sample Number
Figure 5.8: Cosine of inferred phase against interferometric reflectivity
-0.4 -0.2 0.0 0.2 0.4 Detuning (A-Lo) [nm]
=120 - - I - C 0 1 80 - 3 L Comprud fmm simulued reflectivity
r on which simvlalion is basal
Interferometric Mismatch [m]
Figure 5.9: Reflectivity and coupling distribution experiment and simulation comparison
curves of coupling distribution. The perfectly rectangular c w e is the coupling distribution
used to simulate the reflectivity shown in the upper plot. The two other plots are the results
of equation 5.17 using both the simulated and experiment a1 reflectivities as reflectivity inputs.
Both coupling distributions show a similm behaviour, climbing in magnitude from left to
right. The computed non-uniformity, predicted by simulation and seen by experiment, is
due to the violation of the low-reflectivity assumption used in deriving equation 5.17. The
grating has a peak reflectivity of 0.6 due to a coupling distribution/length product, KL = 0.8.
Figure 2.8 of section 2.4 explores the limitations of the low reflectivity simplification.
Notice as well, that the coupling distribution computed from the experimental data shows
a different magnitude from the coupling distribution computed from the simulation. The
simulated result also shows a ripple on the coupling distribution. The reason for these effects
lies in the data treatment. The integral of equation 5.17 needs to be discretized for numerical
computation. Naturally, the function r ( A ) is represented by a vector of complex numbers
r,: each at A,, and the integral is approximated by a discrete, finite sum:
in which the coupling distribution is also discretized as Kl at y. The I-th z is found in terms
of wavelength by, y
-Z[ = 277 (A* - XI) - (5.19)
In practice equation 5.18 is implemented as a discrete digital Fourier transform. The simu-
lated coupling distribution shown in figure 5.9 is computed using relationship 5.18.
Reconsider equation 5.18 in the following light: Equation 5.18 expands the coupling
distribution in terms of spatial wavelength components. Specifically, it may be rewritten as
where the new variable A,, represents the spatial period of the components used to construct
the coupling distribution. In terms of wavelength span, the m-th spatial period, A,,, is
Naturally, to achieve higher spatial resolution, more and more spatial periods need be in-
cluded in the sum of equation 5.20. In theory, one can specify an arbitrary spatial resolution
by extending the range of the wavelength scan. For example, if one desires a spatial resolution
of l p m the wavelength range required by the scan is f 8OOnm about the center wavelength
of the scan. More reasonably, a 100pm resolution requires a scan width of 8nm- The first
limitation to implementing such a strategy is the Limited bandwidth of source and detectors-
A second experimental limitation is more important: signal to noise ratio. Consider a scan
over ten nanometers. The spectrum of a the Bragg grating is concentrated in a narrow
bandwidth surrounded by a finite noise floor. As equation 5.18 is an integral over the scan
range, the longer the scan, the more background noise will be included in the sum. Thus as
the scan range is increased arbitrarily, an advantage in principle of more spatial resolution
is gained at the expense of including more background noise in the integral.
To moderate this effect, the technique of vtzndowzng is used. Windowing selectively
suppresses the reflectivity data in an attempt to reduce the contribution of background
noise, while enabling a longer scan range. The windowing envelope is superposed on the
reflectivity data, suppressing the end points of the scan to zero. A basic Hanning window is
used in this work. The actual computer coded implementation of equation 5.17 appears as
where the Hanning window, centered
written as
between the beginning and end points of the scan is
+ cos ( 2r ;X - 7r (A2 - XI) A2 - A1 )I -
In figure 5 -9, the experimental reflectivity is inverted using the data window. The success of
the window of reducing the ripple is seen. However, windowing changes the reflectivity data.
In particular, it preferentially suppresses the reflectivity in the extremities of the scan. This
reduction of the reflectivity induces an underreporting of the grating's coupling strength.
The consequences of experimental data processing determine the spatial resolution. For a
typical m~asurement as wide as possible a wavelength range is used. Performing the inversion
of equation 5.18 coupling distributions are recovered with significant high hequency noise.
The high frequency spatial noise is clearly because of the inclusion of a significant amount
of background noise in the wide wavelength range scan. The window function is applied to
the data to reduce the spatial oscillations. The tightening of the window explicitly specifies
Bragg grating, partially
Fibwe 5.10: Grating of which one half is bonded to an aluminum substrate
the spatial frequencies that are included in the reconstruction of the coupling distribution,
and as such, determines the spatial resolution. This is a critical observation: the resolution
of the sensor is determined by the lowest period spatial wavelength that can be included in
the construction of the coaplzng distribution.
5 -3 Interferometry-based phase recovery distributed sens-
ing strategy
The following section describes some experimental results obtained with the interferometric
phase recovery distributed sensing experiment.
5.3.1 Uneven thermal apparent loading: experimental demonstra-
tion of spatial resolution
Figure 5.10 shows a schematic of an experiment designed to test the spatial resolution of
the interferometric distributed sensing technique. An 8mm long uniform Bragg grating is
bonded to an aluminum substrate. The grating is bonded such that approximately one half
of the length of the grating is affixed to the aluminum, while the remaining portion hangs
free, beyond the aluminum substrate. An initial measurement is taken at room temperature
after allowing a sufficient cure time for the epoxy. Then, the specimen is heated b i d e
a laboratory oven. The temperature is increased to 39OC above ambient conditions. The
specimen is held at this temperature for two hours at which point it is removed from the
oven. After cooling, it is interrogated again.
It is expected that the portion of the grating bonded to the alllminum substrate will
behave ditferently than the fkee portion of the grating. The response of either side of the
grating has been derived in chapter 3. Equation 3.39 describes the response of the free portion
of the fiber; a sensitivity of 9.66pm/OC, is predicted. The bonded side sensitivity, taking the
thermal expansion coefficient of the T2024 aluminum as 22.5p~/OC[60]~ is computed with
equation 3.45. The expected bonded response is 35.6pm/OC. The apparent strain is found
as the ratio of thermal and strain sensitivities
For the free side of the grating, an apparent strain sensitivity of 8.2p~/OC is expected. On the
bonded side, a higher sensitivity is expected, 30.1p~/OC, as the thermal expansion of the alu-
minum dominates the response. Figure 5.11 depicts the reflectivity of the partially adhered
grating as it is cycled thermally. As the grating is heated to 14OC above room temperature,
the power distribution of the reflectivity shows two distinct features, one centered roughly
0 . 2 m above the nominal wavelength, the other 0.5nm above A,. These distinct features
correspond to the two halves of the grating. The higher wavelength (higher apparent strain)
portion of the reflectivity corresponds to the bonded portion of the grating with its larger
thermal sensitivity. Figure 5.12 illustrates the apparent strains recovered from the experi-
ment. The apparent strain for the specimen heated to 14OC above room temperature shows
the spatial distribution across the grating. Two distinct regions of different strain are seen
as well, again corresponding to the two halves of the sensor. The grating is then held at 39°C
above room temperature where it undergoes a post cure. Initially, there is relative motion
between the fiber and the epoxy as the epoxy softens. After some time-approximately two
hours-the epoxy re-cures at the higher temperature and the optical fiber is fixed relative
Figure 5.11: Measured reflectivities fkom sensor partially bonded to an aluminum substrate
subject to thermal load.
-
-0.003 -0.002 -0.001 0.000 0.001 0.002 0.003
Position [m]
Figure 5.12: Apparent strains reported from sensor bonded on dllminum substrate subject
to thermal load
to the aluminum at this new position- When the specimen is permitted to return to room
temperature the apparent strain on the bonded side of the fiber is seen to have dropped
appreciably with the thermal contraction of the alllminum. The sensitivity of 35.6pm/*C
predicts an apparent strain drop of 1180y~. This decrease, illustrated in figure 5.12, is very
close to the recorded strain drop of the bonded side of the grating. The reflectivity plot
for the post cured specimen shows the group of wavelengths that, when heated appear on
the long wavelength side of the spectrum, have fallen to the short wavelength side of the
spectrum.
The scans are processed as described in section 5.2.2. Initial inversions of the data reveal
the expected high spatial frequency oscillations in the recovered coupling distributions. The
window function is applied to the data to suppress off band background noise. The results
presented in figure 5.12 are windowed such that spatial periods more narrow than 200pm are
excluded from the inversion. For the group of the first three scans, this level of windowing
is acceptable. The final scan reveals some oscillations in the region of the high spatial
gradient. This result indicates that the spatial resolution of the sensor is being approached.
As the inversion attempts to recreate the narrow spatial feature, more spatial h-equencies
are required. The experimenter may be tempted to include more bandwidth in the inversion
to further increase the spatial resolution, however a point of diminishing returns has been
reached. The experiment is not capable of extracting further real information from the
scan. Increasing the scan range will simply include more of the off band noise floor, not
real reflectivity information from which the resolution of the coupling distribution can be
improved. For the final example, the window is tightened to restrict spatial features to
less than 500pm and the high kequency oscillation is suppressed at the expense of spatial
resolution.
This series of results proves that it is possible to resolve in-fiber spatial features with
high resolution, in this example 500pm- As well: a variation of 6 6 0 p ~ is recovered over the
space of l2mm-a gradient of 550p~lmrn
Figure 5.13: Grating bonded to an alllminllm plate in the vicinity of a circular hole
5.3.2 Grating bonded in the vicinity of a circular hole
A longer grating of 50mm is. affixed to an allrmin~~ll substrate into which a circular hole of
radius 5.4m.m is drilled, figure 5.13. The fiber lies 7.7- from the center of the hole. Axial
tension is applied using a Tinius-Olsen tensile testing machine. The specimen is strained to
several different loads. The circular hole is a desirable test because it admits an analytical
solution with which experimental results can compared[61]. The longer grating has a peak
undeflected reflectivity of 17% from which the coupling strength-length product can be found
as EL = 0.43, which is well within the low reflectivity assumption. The expected reflectivity
for each of the load cases is simulated with the T-matrix formulation.
Individual reflectivities are plotted for the four loadings in figure 5.14. The experimental
reflectivities are consistent with simulation. Some salient features of the spectra are noted.
Common to all load cases, the bulk of the reflectivity is concentrated at the lower wavelength
portion of the spectrum. This is consistent with the shape of the strain profile in the
neighbourhood of a circular hole where a significant portion of the physical length of the
grating is close to the far-field strain level. Oscillations in the reflectivity are seen in all
cases. A trend appears as the oscillations become less rapid at higher wavelengths. This
feature is explained by a Fabry-Perot like reflection inside the strained grating. Lower
wavelength regions exist at two points in the grating, on either side of the hole-perhaps a
few centimeters-and a resonance between these two physical points manifests itself as the
rapid oscillations in the low wavelength portion of the spectrum. The Fabry-Perot cavity
for the higher wavelength group, is physically shorter in length-perhaps sub centimeter-
Figure 5.14: Experimental and simulated reflectivities for different loadings.
Table 5 -1: Summary of measurements at different far field strains
Far Field Tension
e:
k 4 172 534 818 1115 ,
located on either side of the high strain peak.
Using the results from section 4.4, with N = 0.22 it is found that the strain field trans-
ferred into the fiber will be replicated inside the fiber with less than 1 percent attenuation.
Thus, the performance of the measurement is limited only by sensor performance.
The strain results are plotted in figure 5.15 in the upper plot, with the deviation from
Maximum Measured
er" (4 b 1 267 804 1256 1695 ,
elasticity theory with distance along the grating is presented in the figure's lower plot. The
results are compared in terms of the root mean square error as a function of axial position,
% Maximum Error
eFPI ey0' (z1
27 8 7 I1 ,
Maximum Error
eF" (4 [PEI 71 64 90 189
defined as I
Table 5.1 summarizes the results of the measurements. The maximum error over the mea-
RMS Errar
ez RMS
[ P I 9 11 17 3 3 ,
surement in absolute terms increases with the magnitude of the strain content. At high far
% RMS Error
eF- efLMS(x)
3 2
< 2 < 2 .
field strains: the strain content of the region near the hole increases. In turn, the wavelength
content of the reflectivity is increased and a subsequent signal to noise reduction occurs as a
similar total optical power is spread across a larger bandwidth. As well, the phase becomes
increasingly rapidly varying with increasing strain content.
For a single loading repeated measurements are performed to gain an appreciation for
the reproducibility of the experiment. Several scans are made, the specimen is removed horn
the holding jaws of the tensioning apparatus, remounted, and reloaded to the same nominal
tension. Figure 5.16 shows the results of both mountings. Measurements are compared with
elasticity solutions in the left hand side plots and the deviation &om theory is presented in the
right hand side plots. The nature of the error is similar between measurements at the same
Figure 5.15: Measured and analytical strain profiles for grating bonded near circular hole
for different axial loads (upper plot). Deviation between theory and experiment (lower plot)
Table 5.2: Summ;trv of repeated measurements
Far Field Tension
e [wI 765
(First Mounting)
765 (Second Mounting)
Mounting)
loading conditions, suggesting rceasonable repeatability of the measurement system. The
first mounting results consistently over-report strain values on the left side of the maximum,
while the right side is systematiocdy under-reported. This feature is less prominent in the
second mounting attempt. The second mounting more closely reports the theoretical strain
field. Table 5.2 summarizes b o t h the absolute and RVIS errors of the measurements.
The long grating is less successful than its shorter counterpart. Strong windowing is
required to eliminate erroneous ~ g h spatial frequency oscillations in the measurements. No
spatial frequencies are included wi th spatial periods shorter than 2mm. .Attempts to extend
the window to larger scan ranges resulted increased spatial noise and no improvement in
spatial resolution. The reason f a r the variation in sensor performance in spatial resolution
is discussed in section 5.5. The stteepest gradient reported by the longer sensor is 58pelmm.
M b u m Measured
eZ"" (4 [ ~ . 4
1353
1271
5.4 Delay-based
%Y
5.4.1 Measurement
phase recovery distributed sensing strat-
Maximum Error
eF"- (4 [wI 141
91
Experimentally, a drawback of t h e interferometric technique is the requirement that a Michel-
son interferometric path must be included in the optical arrangement to recover phase infor-
% M a w Error r \z) eFor
10
7
WIS Error
RMS e z
[PEI
40
25
% R M S Error
efMdS (2)
eF-
3
2
0 0 0 0 m o m
Figure 5.16: Repeated measurements of two different mountings of the same load (left) and
deviations from elasticity solution (right).
Tunable Source Modulator Grating
Figure 5.17: Phase shift modulation method
mation from an interferornetric kinge pattern. The phase of the reflectivity must be found by
a hinge counting algorithm. From measurement to measurement, the fringe density changes
depending on factors such as the initial bandwidth of the grating, the strain content in the
sensor, and the mirror/grating optical path mismatch. That the fringe density is measure-
ment dependent limits the potential of the interferometric strategy to be implemented as
a field deployable system. An arbitrarily high fringe density requires an arbitrarily high
wavelength resolution of the scanning source to ensure hinge discrimination.
This section proposes a method by which the interferometric arm can be eliminated horn
the experiment: the group delay characteristic of the grating sensor is measured directly.
A proof of concept test of this new strategy involves a more simple optical arrangement
that can be implemented for distributed sensing.
The elimination of the interferometric arm is realized by recognizing the relationship
between the delay of a wavelength component and its phase. The relationship between delay
Tunable Electro-Optic cpoWff Meter Bragg Source Isolator MZ Modulator Coupler Grating
Coupler Signal Amplifier Photodetector
Figure 5.18: Delay-based phase recovery experiment
and optical phase is given as[62] d4 (N
7 (A) = --. a!#
Equation 5.26 is recast as
Equation 5.27 provides the phase for the construction of the function r (A). Measurement
of the group delay is performed by the phase shzft modulation method first proposed by
Rhu[63]. Figure 5.17 helps to explain the phase shift modulation method principle. A
tunable source is modulated at an RF frequency, whose period determines the available
range of continuously measurable group delay. The modulation also affixes an indication of
time of flight to the wave. The time of flight of the wave is continuously compared to a
fixed phase reference (such as the modulator drive signal) . The reflection &om the grating
carries with it the modulation. As certain wavelengths suffer different penetration depths
(i-e. different delays) the relative phase between the fixed reference and the reflected wave
varies. The phase difference is determined by a high speed network analyzer. The phase
difference is translated into time delay through knowledge of the modulation frequency.
An alternative derivation can be performed in the frequency domain. Writing the mod-
ulated signal as the sum of two optical frequencies separated by the modulation frequency,
f2 - fl = fmod., the reflection horn the grating is properly seen as the superposition of reflec-
tions originating from X (fi ) and X ( f2). The phase of the superposition of the reflected waves
is a function of the differential delay between the two closely spaced wavelengths. The mod-
ulation frequency is selected such that the separation of the two probe wavelengths matches
the step resolution of the reflection measurement: effectively matching the resolution of the
delay and amplitude measurements.
The optical arrangement is shown in figure 5.18- An Anritsu tunable source is externally
RF modulated by a JDS-Uniphase Mach-Zehnder modulator. The Bragg grating sensor is
then interrogated with the moddated wave. A low bandwidth power meter detects reflected
optical power, R (A). The modulated signal reflected fiom the Bragg sensor is compared to a
reference signal by an EfP network analyzer to acquire the group delay as the laser is tuned.
The reference signal is obtained electrically. an optical reference, a t the expense of a second
high speed photo-detector, will provide a measure of thermal compensation if the optical
reference arm is proximate to the sensing arm.
The RF network analyzer and the Tinius-Olsen tension testing machine were not simul-
taneously available, thus an alternative loading apparatus is devised based on beam bending,
in lieu of pure tensioo, figure 5-19. Two aluminum beam samples are prepared to demon-
strate the measurement. Into each specimen a circular hole is drilled a smaller hole in one
specimen of diameter 5mm, and a larger hole in the other: with diameter 10m.m. An in-fiber
Bragg grating sensor, of 50mm length, is bonded to each beam, parallel to the long axis of
the sample, at a distance of lmm fiom the hole perimeter. A frequency of 400MHz is used to
modulate the interrogation lightwave. The available differential phase resolution was &0.l0.
This corresponds to a group delay resolution of *0.7ps.
To assess the strain resolution: the strain field was theoretically estimated. A combination
of two simple models is proposed. The elasticity solution in the neighbourhood of a circular
hole[4l] due to a uni-axial tension is linearly scaled to account for the bending beam. In the
absence of aoy hole, the upper surface of the beam on which the sensor is mounted, would be
in linearly varying tension as dictated by conventional beam theory. The strain prediction
is made by scaling the solution of the hole in a plate under uniform tension, a t each point,
by a ratio of the local strain of the equivalent uniform beam (without a hole) to the beam
tensile strain value at the sensor midpoint.
The specimen is loaded by suspending a mass from the end of the beam. The reflected
Bragg grating senso bonded near hole.
Figure 5.19: Beam loaded in bending
Figure 5.20: Reflectivity and delay of unloaded and loaded samples.
optical power and the group delay are plotted in figure 5.20.
The analytical estimate and the experimental measurements are plotted in fiewe 5.21
for the two samples. The evaluation of the strain error is performed by computing the root
mean square over the range of measurement, yielding an error of f 2 0 p ~ for the smaller hole
and & 2 4 p ~ for the larger hole. Normalized with the maximum strain excursion, the RkfS
error is, for both loadings, 7%. Spatial resolution is again limited by the maximum amount
of scan range that can be included in the coupiing distribution integral. The window giving
the best results eliminated spatial frequencies having periods less than 1 -65mm.
The demonstration of the proposed method etiminates the undesirable interferometric
mirrored arm necessary in previous interferometric distributed sensing systems.
5.5 Conclusions of measurements
In this section experiments are undertaken to evaluate the performance of fiber Bragg grating
as a distributed sensor and to attempt to understand factors relating to its performance.
The ability of the sensor to resolve spatial feat-ures-the spatial resolution-is of primary
importance.
Section 5.2.2 shows that the coupling distriburtion computed &om experimental data is in
fact a Fourier sum of spatial frequencies. High spatial resolution depends on the inclusion of
high spatial fkequencies in this summation. More. high spatial frequencies in the z domain
are related to reflectivities at large wavelength excursions horn the center of the scan. Thus
to achieve high spatial resolution, it is required to be able to recover accurate information
over a large bandwidth in reflectiviv
The degree to which a grating offers recoverable reflectivity over a large bandwidth de-
termines the potential spatial resolution. Fibwe 5.22 shows two gratings of equal peak
reflectivi~ 30%; undeflected then subject to a localized strain perturbation. The strain
perturbation is a lmm Gaussian shape with a peak strain of l o o p . One grating is modeled
as having a 5cm length and the other icm. To achieve similar peak reflectivity the lcm
grating requires an index modulation, 672, five times stronger than the 5cm, 5.87 x
versus 29.3 x respectively. The long grating has a much more narrow bandwidth than
, # I * * , , . , Ncar Large Holc (Theordical) Ncar Small Holc (Expcrimcnlal) ..-.---
0.00 Position [m]
- lcm Grating Undefl ected --- With localized permrbarion
-0.2 -0.1 0.0 0.1 0.2
Wavefength Offset fnm]
-0.2 -0.1 0.0 0.1 0.2
Wavelength Offset [nm]
0.4
A
6 0.3 er: h 4 -- 1 0.2 - U 0
C 4 0-1
Fi,we 5.22: Comparison of two gratings lcm and 5cm of like peak reflectivity subject to
Like localized strain perturbation
-
5cm Grating - Undeflected --- With Iocalized perturbation
Z I \ 1 1 I t I I I I I \
0.0 -
the short sensor. When deflected: both gratings behave in a similar manner qualitatively:
the central shape shifts to a higher wavelength and a side lobe grows on the low wavelength
side of the spectrum. The longer, low bandwidth grating is much more difficult to interre
gate. While scanning with a tunable source of fixed step size, the longer, low bandwidth
grating offers fewer points of reflectivity available for interrogation. For example, with a
bandwidth of approximately lOOpm the short grating provides an order of magnitude more
points available for interrogation. Recalling that each point in wavelength space represents
a spatial frequency in the z space, it is clear that more spatial resolution i s available using
the short grating.
For example, using equation 5.21, a wavelength offset of O . l n m corresponds to the spatial
period of 7.5mm. The magnitude of the reflectivity at this position for the short grating is
appro-dmately 1.5%. The corresponding reflectivity of the longer grating is 0.065%. The
reflectivity of the longer grating, at this position, is more than twenty times smaller than
the analogous reflectivity of the short grating. Thus, the spatial kequency of the coupling
distribution represented by this wavelength position is twenty times more difficult to extract
for the case of the narrowband grating. Should the noise floor of the experiment exceed the
reflectivity of the sensor, the information relating to that spatial frequency i s lost. Including
this erroneous information in the inverse transform will lead to a high spatial frequency noise
on the recovered coupling distribution. Windowing is required to suppress these hequencies.
The limit where the recoverable reflectivity falls below the noise floor determines the spatial
resolution.
Experiments support this observation. Section 5.3.1 presents the results of the deploy-
ment of a short high bandwidth sensor. Spatial resolution is very strong, between 200pm and
500pm. The spatial resolution of the sensor determines its ability to extract gradients. With
the high spatial resolution, the recovery of a gradient of more than 500p~fmm is demon-
strated. A short sensor can be deployed for extremely high spatial resolution measurements.
The experiments of section 5.3.2 deploy long, low nominal bandwidth gratings. The ad-
vantage of a using a longer sensor is compromised by the reduction in spatial resolution.
Spatial resolution is limited by the sensor's narrow bandwidth. Only harmonic components
of spatial periods exceeding one or two millimeters are relevantly extracted from the reflec-
tivities. A maximum gradient of 6 0 p ~ / m m is recovered in this configuration. Percentage
errors in recovered strain profiles are presented. Typical strain errors are of the order 10%
when one considers the maximum excursion between theory and measurement. Taken as an
m4S value, the errors are typically of the order of 5% to 10%.
A second, novel, experimental approach is reported in section 5.4. The interferometer
arm is replaced with a delay based strategy for phase recovery. The performance of this
approach is equivalent to the interferometric approach and offers a more direct method by
which to extract the grating phase.
In s u m m a r y 7 the following conclusion are drawn from the experiments of this work.
0 Spatial resolution is determined by the shortest spatial wavelength used in the Fourier
series representation of the coupling distribution.
Wider scan ranges in wavelength enable the recovery of better spatial resolution.
Practical limits due to the finite noise floor limit the degree to which the scan range can
be arbitrarily expanded. The technique of windowing the reflectivity data is required
to moderate the effect of the finite noise floor.
Spatial resolutions of sub millimeter are achieved with a lcm long, high nominal band-
width sensor.
Spatial resolutions the order of millimeters are achieved with a 5cm long, narrow
nominal bandwidth sensor.
The interferometric arm of the Michelson can be replaced with a delay based measure-
ment approach with no performance penalty
The bandwidth of the sensor is the most important factor in determining the spatial
resolution of the sensor.
Chapter 6
Conclusions and contribution
summary
6.1 Summary of contributions and conclusions
The content of this thesis is intended to contribute to the fiber optic sensing community.
in particular, in the area of Bragg grating distributed sensing. The content of chapter 2 is
largely background derivations necessary to describe the optical response of a grating. Tools
employed heavily in later chapters such as the T-Matrix formulation to simulate a grating's
optical response, and the approximate Fourier inversion technique for reconstructing the
coupling distribution based on optical data are reviewed.
6.1.1 Contributions in understanding the optical response
-4 responsible investigation of the Bragg grating's deployment as a distributed sensing device
mandates the investigation of the sensor's response to thermo-mechanical variables. The
phase response is detailed for the distributed Bragg sensor for strain and temperature in
chapter 3. Experimental evaluations of the strain optic coefficients, the fiber stiffness, the
Bragg grating thermal and strain sensitivities are performed.
6.1.2 Contributions towards understanding the strain transfer prob-
lem
Significant work is done to illustrate the relationship between the in-fiber strain field and the
strain field in the host or substrate material in the presence of a spatially varying field in the
host. Chapter 4 details this work. The initial study considers the transfer of a linear strain
gradient from host or substrate material to fiber. This is the &st reporting of the problem
of relating the directly measured in-fiber strain to the desired host or surrounding material
strain in the context of distributed sensing. Results for both the embedded and the surface
mounted case are presented. The finite element modeling of the surface mounted optical
fiber, using appropriate stackings of Reisner/hlindlin plate elements is presented as novel.
A parameter study of the factors affecting axial strain transfer is undertaken. A first author
journal article in Smart Materials and Stmctures[53] supports the latter contributions.
The investigation into the strain transfer problem is further pursued. A novel analytical
derivation is presented, proposing a methodology to relate the in-fiber strain to the host
surrounding material strain field for the case of an arbitrarily varying strain field. A unified
description of the strain transfer problem is outlined, with emphasis on spatial frequency
content of the strain field and the transfer of individual components- Agreement is found
between the proposed theory and finite element analysis. A second, first author article in
Smart Materials and Structures[64] details this derivation. As well, a new strategy for simple
modeling of the three dimensional case of a surface mounted sensor is proposed through the
use of the equivalent geometric ratio.
6.1.3 Contributions in the area of distributed strain measure-
ment s -
Chapter 5 reports distributed Bragg grating measurements. This portion of the thesis con-
tinues in the same direction established by the work of colleagues at the FOSS laboratory.
At the inception of- this thesis, the potential for distributed Bragg measurements was known,
but the resolution of the technique was not strictly defined and presentation of definitive
measurements was not complete. In this work, e-xperiments are undertaken to carefully eval-
uate the performance of the sensor. Comparisons with analytical solutions are employed to
report the experimental error.
The relationships between sensor bandwidth and spatial resolution are defined. Sub
millimeter spatial features are recovered using a short length broad band grating. Longer
g a t ings are used to demonstrate measurements. The best strain resolution obtained with
the longer sensors shows an mC3 error of 25 microstrain over a measurement with a strain
content of more than 1000 microstrain. These results are included in the proceedings of the
1999 CLEO rneeting[49].
Spatial resolution is discussed in terms of grating bandwidth. The data treatment tech-
nique of windowing is proposed and implemented to improve the measurement. The param-
eters determining strain spatial resolution are clearly specified.
A contribution is made in terms of the distributed sensing experiment design. A novel
sensing scheme based on the recovery of group delay in lieu of interferometric phase is
proposed. Similar resolution of twenty or so microstrain is demonstrated. This contribution
is supported by a first author Optics Letters publication[50].
6.2 Suggested future work
There are several departure points for future study implicit in the content of this thesis.
The Fourier transform inverse scattering method and its limitations are outlined in chap
ter 2. The principal limitation of the FT is the low reflectivity assumption included in its
derivation. Poladian [65] and Feced et. al. [66] have recently proposed inversion algorithms
that are not restricted to low reflectivity devices. An attempt to apply improved inverse
scattering algorithms to distributed Bragg grating sensing is suggested.
The question of strain transfer quality to the optical fiber sensor from the mechanical
host holds several unexplored areas. The approach studied in this document is restricted to
the case of a perfectly adhered sensor, performing in the elastic regime. A comprehensive
evaluation of the strain trahsfer problem: including debonding, and non-linear effects is
relevant to the real world deployment of distributed Bragg sensors-
-Also useful, would be a fully three dimensional finite element model of the surface
mounted sensor. First, such an analysis would be expected to improve the estimate of
equivalent transfer performance of the surface mounted sensor. As well, in the telecommuni-
cations industry, the bi-axial strain induced by the surface mounting of a fiber is of interest.
Extension of the surface mounted analysis with a view to suppressing unwanted induced
birefringence would prove a valuable study.
Several experimental improvements require further assessment. A lock-in technique may
be attempted to better extract relevant sensor bandwidth from the noise floor--effectively
increasing the spatial resolution of the sensor. The results of such an experiment might be
useful in specifying a strict condition by which to apply the data window. Miniaturization
and cost reduction of the RF circuitry required to perform the delay measurement are key
to distributed Bragg grating sensing's widespread viability. A further increase in the scan
speed of a single measurement wodd make dynamic distributed measurements possible.
The results of this document argue that larger bandwidth sensors provide better spatial
resolution. A series of spa t idy cascaded shorter sensors of different wavelengths can be
imagined as a first step to optimizing the length of the sensor, while preserving sub-millimeter
spatial resolution. Such a progression may lead to the use of a chirped grating of moderate
strength. In general, the apodization of the sensor; in conjuction with the improved inverse
scattering methods, merits further study to optimize distributed sensing performance.
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