REVIEW OF SCIENTIFIC INSTRUMENTS 83, 043111 (2012)

Measurement and control of residual amplitude modulationin optical phase modulation

Liufeng Li,1,2 Fang Liu,1,2 Chun Wang,1,2 and Lisheng Chen1,3,4,a)

1Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China2Graduate School of the Chinese Academy of Sciences, Beijing 100080, China3State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan 430071, China4Laboratory of Atomic frequency Standards, Chinese Academy of Sciences, Wuhan 430071, China

(Received 3 June 2011; accepted 1 April 2012; published online 19 April 2012)

Residual amplitude modulation is one of the major sources of instability in ultra-sensitive opticaldetections based on frequency modulation. Using a MgO · LiNbO3 electro-optic crystal, we sys-tematically measure the temperature and polarization dependence of residual amplitude modulationand our experimental results are in good agreement with a previous theoretical analysis. After op-tical phase modulation, two independent arms including optical detection and frequency demodula-tion are employed to closely examine the instability of the residual amplitude modulation. Residualamplitude modulation below 25 ppm is obtained with an active cancellation scheme in which thecrystal temperature is varied so as to zero the baseline drifts with different origins. Possible im-provements for better suppression and stability are discussed. © 2012 American Institute of Physics.[http://dx.doi.org/10.1063/1.4704084]

I. INTRODUCTION

Because of their high sensitivity and potentials to reachthe quantum detection limit,1 frequency modulation (FM)detection schemes2, 3 have been applied to many fieldssuch as precision laser spectroscopy,4–8 optical frequencystandards,9–11 and laser interferometers of various scales.12

Optical phase modulation is employed in FM techniquesto extract the line-center information, a task that is thecentral practice of the precision laser spectroscopy. In thisprocess, many mechanisms reduce the purity of the phasemodulation, producing an amplitude modulation (AM) thatis often referred to as residual amplitude modulation (RAM).RAM results in a systematic and drifting frequency offset,which degrades the frequency stabilization and is especiallydetrimental in laser interferometers13–17 where the stabilityof the frequency locking directly affects the accuracy of themeasurement.

Mechanisms of RAM in optical phase modulation andsubsequent frequency demodulation have been analyzedin great detail. Wong and Hall18 related RAM to the bire-fringence of the electro-optic (EO) crystal and adopted anactive servo control to suppress RAM. Whittaker et al.19

analyzed the effect of F-P cavities formed by the EO crystaland other optical elements. RAM in a modulation transferspectroscopy was investigated in Ref. 20. In addition to theelimination of interference fringes,21–23 various methodshave been adopted to suppress RAM, such as two-tone fre-quency modulation24, 25 and harmonic frequency modulationspectroscopy.26, 27 Meanwhile, RAM compensation by meansof feedback control was also pursued.28–32

RAM is often indirectly inferred by using optical hetero-dyne beat or atomic/molecular transitions. However, a direct

a)Author to whom correspondence should be addressed. Electronic mail:lchen@wipm.ac.cn.

measurement of RAM and its long-term stability is of impor-tance in that this approach separates RAM from other stabilityissues such as the drift of the optical cavity in a Pound-Drever-Hall (PDH) frequency locking system. A special but equallyimportant case can be found in Fabry-Perot laser interferom-eters where the differential frequency stability between twointerference arms is of greater interest.

We conduct a detailed characterization of RAM in an ex-perimental setup that includes the optical phase modulationand subsequent demodulation. By varying the crystal temper-ature and optical polarization we experimentally verify thetheoretical analysis based on the birefringence of the EO crys-tal. After optical phase modulation two independent detec-tion arms are used to closely examine the instability causedby RAM. Active cancellation of RAM is also implementedand its performance is evaluated by measurements both in-side and outside the control loop. The investigation providesuseful information for improving the frequency stabilities ofFabry-Perot laser interferometers as well as ultra-stable localoscillators in optical frequency standards.

This paper is organized as follows. Section II briefly out-lines the theoretical model that describes the temperature andpolarization dependence of RAM at the frequency of phasemodulation. Characterization of RAM based on a modifiedPDH frequency locking system and its comparison with the-ory are detailed in Sec. III. Section IV concentrates on theactive cancellation of RAM and the verifications of its per-formance. Section V discusses possible improvements thatcan be explored in follow-up investigations. Conclusions aregiven in Sec. VI.

II. THEORY

Wong and Hall18 modeled the RAM based on thebirefringence of a lithium tantalate (LiTaO3) EO crystal. We

0034-6748/2012/83(4)/043111/10/$30.00 © 2012 American Institute of Physics83, 043111-1

043111-2 Li et al. Rev. Sci. Instrum. 83, 043111 (2012)

Y

Z

X

Z

Front view Side view

ne nolaser beamd

l

P1

P2β

γ

FIG. 1. Cutting and orientation of MgO · LiNbO3 crystal. The crystal axis isalong the z direction. Light propagates in the x direction, and the RF electricfield is applied along the z direction. ne and no label the refractive indicesof extraordinary and ordinary lights, respectively. Two polarizers, P1 and P2,are placed before and after the crystal, forming angles of β and γ relative tothe z axis.

follow their theoretical model to experimentally investigatethe RAM in optical phase modulation using a magnesiumdoped lithium niobate (MgO · LiNbO3) crystal. Here, the rel-evant theory is briefly summarized and for detailed derivationreaders are referred to Ref. 18.

Figure 1 shows the coordinate system and the orientationof a uniaxial crystal with length of l and height of d. The z andy axes are the principal axes of the refractive indices ne andno, respectively. The laser propagates along the x axis. To havea clear definition of light polarization, two polarizers, P1 andP2, are placed before and after the crystal, having polarizationangles of β and γ with respect to z axis. A RF modulationfield parallel to the z axis is applied to the crystal. RAM inoptical phase modulation appears in the form of an amplitude-modulated photo current at ωm, which is

I (ωm) = −4ab |ε0|2 J1 (M) sin (ωmt) sin (�φ) , (1)

where a = sin βsin γ and b = cos βcos γ , ε0 is the amplitudeof the input optical field that is polarized along the direction ofP1, J1(M) is the first-order Bessel function with its variable Mthe difference in the modulation index between the extraordi-nary and ordinary lights whose polarizations are along the twoprincipal axes of the refractive index ne and no, respectively,and �φ is the phase shift induced by the natural birefringenceof the crystal, i.e.,

�φ = 2πl

λ(ne − no) , (2)

where l is the length of the crystal, λ is the optical wavelength,ne and no are the refractive indices of extraordinary and ordi-nary lights, respectively. The temperature of the crystal canbe tuned to vary the phase shift �φ such that sin (�φ) is zero,resulting in a pure phase modulation at the modulation fre-quency ωm. In Sec. III, a detailed examination of the temper-ature and polarization dependence of RAM is given and theresults are compared with the theory.

III. CHARACTERIZATION OF RAM

A. Experimental setup for FM detection

Figure 2(a) shows the experimental setup for mapping thetemperature and polarization dependence of RAM. This setupis modified from a standard PDH frequency locking systemby replacing the optical cavity with a flat mirror. Alternatively,the light after the second polarizer (P2) can be directly sent to

λ

Laser

ND P1 EOM P2 PBS

PD

IF

LO

DBMPS

2λ−

4λ−

Aspheric lens

To voltage meter

2−

(a) Experimental layout

LN crystalCopper blockTEC

Heat sink

(b) EOM construction

FIG. 2. (a) Experimental setup for the characterization of RAM. (b)EOM construction. Radiation from a ND:YAG laser passes a EOM crystal(MgO · LiNbO3) with a modulation frequency of 10 MHz and a modulationindex of 0.52. Here, the optical cavity used in a PDH frequency locking isreplaced by a flat mirror. The RAM is demodulated by a doubly balancedmixer. ND, neutral density filter; λ/2, half wave plate; λ/4, quarter waveplate; P, Glan Taylor prism; EOM, electro-optic modulator; PBS, polariza-tion beam splitter; PD, photodetector; LO, local oscillator; PS, phase shifter;DBM, doubly balanced mixer.

a photodetector (PD) for this purpose. A 1064-nm Nd:YAGlaser with an output power of 100 mW is used and after atten-uation the power reaching PD is about 350 μW. The polar-izations of the laser before and after the phase modulation arecontrolled by two polarizers (P1 and P2, Glan Taylor prisms).After phase modulation, the laser passes a polarization beamsplitter (PBS) and is reflected by a flat mirror. The reflectedbeam is guided by the PBS toward a PD and then demod-ulated by a doubly balanced mixer (DBM). The dc voltagefrom the DBM is measured by a digital multimeter (DMM).

Figure 2(b) illustrates the structure of the EO modulator(EOM). The LiNbO3 crystal (35 × 4 × 2 mm) has a nomi-nal 5-mol.% doping of MgO. Each end surface of the crys-tal has a flatness of λ/10 and is anti-reflection (AR) coatedwith residual power reflectance less than 0.5% at 1064 nm.As indicated in Fig. 1, the laser propagates along the x axiswith its polarization along the z axis and the RF modulationfield is along the z axis, which is also the principal axis ofthe refractive index ne. A modulation frequency of 10 MHzand a modulation index of 0.52 are used in the measurements.The crystal is mounted on a copper block that is temperaturecontrolled via two thermoelectric coolers (TECs) underneaththe copper block. With temperature control, the measuredstep response of the temperature of the copper block can befitted to a second-order model G (s) = 1/(τ 2s2 + 2ξτ s + 1)with a time constant τ = 0.7 s and a damping coefficientξ = 0.5.

043111-3 Li et al. Rev. Sci. Instrum. 83, 043111 (2012)

FIG. 3. RAM versus the temperature of the EO crystal. Solid circles are dcvoltages measured at the IF port of the DBM. The solid curve is a sinusoidalfit, which gives a local period of 0.7685(2) ◦C. Data obtained with P1 rotatedby 20◦ from the z axis and P2 rotated by a small angle (1◦–2◦) from itsoptimized position.

To avoid parasitic etalons, optical components areslightly tilted and their transmitting surfaces are AR coated.The length of the optical path is shortened to reduce theinfluence of the drift of the optical alignment. Because ofthe inhomogeneous distribution of RAM across the laserwavefront,18, 33 aperture stops that truncate the laser beamwill produce rather large RAM and thus should be avoided.In particular, the small detecting area of photodiode is apotential aperture stop and we use aspherical lens with ashort focal length (11 mm) to focus the laser beam onto thephotodetector.

B. Temperature and polarization dependences of RAM

To measure the temperature dependence of the RAM, theGlan-Taylor prism (P1) in front of the crystal is rotated by20◦ from the z axis while the prism (P2) after the crystal isrotated by a small angle (1◦–2◦) from its optimized position.The power level and the phase of the RF local oscillator areadjusted to maximize the demodulated dc level at the interme-diate frequency (IF) port of the DBM. Then the crystal tem-perature is varied and the demodulated signal is measured.Figure 3 shows the result and a sinusoidal fit of the experi-mental data.

When the variation of the temperature is small, the phaseshift �φ in Eq. (2) can be treated as a linear function of tem-perature, i.e.,

�φ (T ) = 2πl

λ(αe − αo) (T − T0) , (3)

where αo and αe are the temperature coefficients of refrac-tive indices no and ne, respectively. Equations (1) and (3) in-dicate that RAM has a rather simple sinusoidal dependenceon the temperature. The experimental data fit well to a si-nusoidal dependence y = A1 + A2sin [A3(x − A4)] and thefitted parameters are A1 = 2.2(2) mV, A2 = 54.7(3) mV, A3

= 8.176(2) ◦C-1 with A4 fixed to 26.67 ◦C. From Eq. (3) andthe fitted parameter A3, the differential temperature coefficient(αe − αo) is determined as 3.956(1) × 10−5 ◦C−1 at tempera-tures 27 − 31 ◦C.

Our value of (αe − αo) can be compared to 4.85× 10−5 ◦C−1 that was independently determined from the

-90 -60 -30 0 30 60 90

-200

0

200

Am

plitu

de (

mV

)

Polarization angle (degree)

Experimental data Theoretical fit

FIG. 4. Polarization dependence of RAM. P2 is first rotated by a small angle(1◦–2◦) from its optimized position. P1 is then rotated by various angles. Toisolate this measurement from temperature effect, at each angle of P1 thetemperature of the EO crystal is scanned to search for the maximum RAM.The solid curve is a theoretical fit of the data. See text for detail.

principal refractive indices and Sellmeier equations.34, 35

There is a 20% difference in (αe − αo) between two methods.We note that these two methods are quite different and asystematic offset could exist in the measurement of refractiveindices, especially of its temperature dependence. Also therefractive indices vary with the concentration of the MgO,which can potentially modify the temperature dependence of(αe − αo).

The polarization dependence of RAM is also measuredin current system. The factor ab in Eq. (1) is csin βcos β fora fixed γ (polarizer P2 fixed), where c is a constant. Here, anextra angular dependence of cos 2β enters because the inputlaser before P1 is polarized along the z axis (cf. Fig. 1) andthus the laser field has to be first projected to the directionof P1. Together with this extra factor, the total angular de-pendence of the RAM is csin βcos 3β. In the measurement,we keep the polarizer (P2) after the crystal fixed and varythe angle of the first polarizer (P1). To isolate this measure-ment from the temperature effect, at each angle the temper-ature of the EO crystal is scanned to search for a maximumin RAM. Figure 4 shows the experimental data and the theo-retical fit using y = A1 + A2sin [π (x + A3)/180]cos 3[π (x +A3)/180], which yields A1 = 5(1) mV, A2 = 585(5) mV, andA3 = 3.4(2). That the experimental polarization dependenceis well described by the theory further confirms the crystalbirefringence as a major contributor of RAM.

Investigations in Ref. 18 and here provide guidance foran optimization of the polarization and crystal temperature.To minimize RAM, the polarization angles of two polarizers(P1 and P2) are first adjusted. During this process the crystaltemperature and the phase of the local oscillator (LO) arevaried to maximize the sensitivity of the adjustment. Thenthe crystal temperature is optimized and stabilized to furthersuppress the RAM. Moreover, as shown in Fig. 3, at a temper-ature range of ±0.2 ◦C the demodulated signal has the correctslope and sign for a feedback control of RAM by varyingcrystal temperature. In this case the two polarizers are rotatedby 1◦–2◦ away from their optimized polarization angles in

043111-4 Li et al. Rev. Sci. Instrum. 83, 043111 (2012)

(a)

(b)

FIG. 5. Spatial RAM distributions probed by partially blocking the laserbeam with a razor blade. (a) and (b) are two scans with a difference in inci-dent angles about 1◦ in horizontal. The transverse displacement of the bladeis normalized by the beam size w at the cutting position. The beam waistw0 is 0.2 mm and located at the front surface of the EO crystal. The bladeand PD are placed 150 mm and 230 mm downstream from the beam waist,respectively.

order to achieve an enough discriminator gain for thefeedback control.

C. Inhomogeneous spatial RAM distribution

With facilities shown in Fig. 2, the spatial distributionof the modulation depth is probed by transversely scanning arazor blade across the phase-modulated laser beam and simul-taneously recording the variations of the demodulated signal.The beam waist w0 is 0.2 mm and located at the front surfaceof the EO crystal. The optical power of the partially blockedbeam is also measured by the same PD using its dc outputand is used for determining the relative position of the cuttingedge to the center of the beam. The blade and PD are placed150 mm and 230 mm downstream from the beam waist, re-spectively. Figure 5 gives the detected spatial profiles in twoscans with a difference in incident angles about 1◦ in hori-zontal. The two profiles show opposite symmetries and aremost representative in that they transform gradually from oneto the other when the incident angle is gradually varied in onedirection. While the current result is obtained using a NewFo-cus 4002 EOM (MgO · LiNbO3), similar results are also seenwith our home-built EOM.

We note that the results of RAM suppression describedin Sec. IV do not indicate a uniform spatial distribution ofRAM. What is detected in our experiments and many otherapplications is a spatially averaged RAM that can be zeroedby adjusting the polarization of the incoming beam and thecrystal temperature. However, unstable or newly introducedapertures will again induce a nonzero average and hence ad-ditional RAM. In practice, we use apertures whose diametersare at least 8 times larger than local beam size and supportoptical components with solid stainless steel posts 1 in. in di-ameter to ensure their stable relative positions with respect tothe optical axis.

Previous works18, 33 have already pointed out manypossible mechanisms such as the beam quality, edge effectof the EOM electrodes, local variations of the refractive in-dex, and divergence/convergence of Gaussian beams. Nev-ertheless, well-controlled experiments and theoretical anal-ysis are still needed to verify each potential cause quan-titatively. We are conducting a systematical mapping ofthe spatial RAM distribution in terms of the beam inci-dent angle, the translation/rotation of the crystal, and thelocations of the cutting blade and photodetector. SpatialRAM distributions in these measurements will be com-pared with theoretical calculations that take into accountthe wavefront curvatures of Gaussian beams. A detaileddescription of the theory, experimental data, and compar-ison among them are underway and will be publishedelsewhere.

IV. ACTIVE CANCELLATION OF RAM

A. Modified experimental setup with twoFM detection arms

Figure 6 illustrates the experimental setup for the activecancellation and long-term monitoring of RAM. The phase

Laser

ND P1 EOM

PD2

LO

DBM1PS

2λ−

Aspheric lens 1

DBM2

ISO1

ISO2

EOMtemp. control

To voltage meter

To voltage meterError sig.

2λ− S1

PD1

Aspheric lens 2

S2P2

Loopfilter

FIG. 6. Experimental setup for active RAM cancellation. After phase mod-ulation, the laser power is equally split into two paths. One is used for activeRAM cancellation and the other for out-of-loop analysis. ISO, optical iso-lator; ND, neutral density filter; λ/2, half wave plate; P, Glan Taylor prism;EOM, electro-optic modulator; S, mirror substrate; PD, photodetector; LO,local oscillator; PS, phase shifter; DBM, doubly balanced mixer.

043111-5 Li et al. Rev. Sci. Instrum. 83, 043111 (2012)

modulated light is divided into two nearly identical arms; onearm is used for the active cancellation and in-loop measure-ment of RAM while the other is used for the out-of-loop ver-ification. This arrangement has the benefits that (1) it avoidsthe possible drift caused by the frequency references such asan optical cavity and (2) the differential drift of two detec-tion arms gives a good estimation of the RAM-induced in-stability in Fabry-Perot laser interferometers that rely on FMtechnique.

Here, the laser beam after the EOM is divided by twomirror substrates (S1 and S2, 7-mm thickness) to avoid theinterference among various reflecting surfaces. To matchthe optical powers in the two arms, the back surface of theS1 and the front surface of S2 are used to reflect the twobeams toward the photodetectors. The optical powers at thetwo photodetectors are 120 μW (PD1) and 140 μW (PD2).Measurements with strong environmental perturbations(local airflows and temperature cycles) reveals that this ar-rangement improves the power and polarization stabilitiesin both arms. RF signal used for optical phase modula-tion is divided and sent to two mixers and the lengthsof the electric cables are matched. Back reflections fromtwo arms can hit upon various parasitic etalons along theirway to the laser, generating additional drifting RAM. Tosuppress this instability, two 30-dB optical isolators havebeen added, respectively, at the laser output and after theEOM. BNC-type connectors, whenever possible, are replacedby soldering and local temperature cycles are used to lo-cate and replace electrical contacts with large temperaturecoefficients.

B. Passive temperature stabilization, activecancellation, and performance evaluation

Three types of measurement are made to investigate theperformance of the active RAM cancellation. First, we com-pare two situations in which the crystal temperature is onlypassively stabilized and an active RAM cancellation is imple-mented. Second, both in-loop and out-of-loop stabilities areexamined when the active cancellation is engaged. Additionalmeasurements are performed to expose drifts of electronic ori-gin. In current system (Fig. 6), the demodulated signal fromDBM 1 is used for the measurement without active cancel-lation and also for the out-of-loop measurement in the caseof active cancellation. The output from DBM 2 is split intotwo paths. One is used for in-loop measurement and the otheris amplified, integrated, and then fed to the temperature con-troller for the cancellation of the RAM. RAM is always min-imized before each measurement according to the proceduredetailed at the end of Sec. III B, and the EO crystal is tem-perature stabilized throughout the measurement. In the caseof active cancellation, the polarizers P1 and P2 are slightlydeviated from their optimized positions to allow for enoughdiscriminator gain.

Figure 7(a) compares the drifts of the demodulatedsignals obtained without and with active RAM cancellation.With the stabilization of crystal temperature but no activecancellation, in 20 h the demodulated signal has changedabout 0.18 mV, corresponding to an AM of 180 ppm. Inanother 24-h measurement, active cancellation has reducedthe out-of-loop drift to within 25 μV (25 ppm, AM), which

0 2 4 6 8 10 12 14

-0.08

-0.04

0.00

0.04

0.08

0.12

-80

-40

0

40

80

120

In loop

Modulation depth (ppm

)A

mpl

itude

(mV

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Time (hour)

Out of loop

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0 5 10 15 20

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0.08

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0

80

Modulation depth (ppm

)A

mpl

itude

(mV

)

Time (hour)

Active control on

Active control off

(a)

FIG. 7. Demodulated signal after optical phase modulation. (a) Out-of-loop signals with and without active RAM control. (b) In-loop and out-of-loop signalswhen active control is engaged. Vertical scales in the left and right are the dc levels of the demodulated signals and the corresponding modulation depth,respectively. For the case of no active cancellation, the RAM is minimized before the measurement and the EO crystal is then temperature stabilized throughoutthe measurement.

043111-6 Li et al. Rev. Sci. Instrum. 83, 043111 (2012)

0.02

0.0 0.5 1.0 1.5 2.0-0.02

-0.01

0.00

0.01DBM1DBM2

(b)

Ampl

itude

(mV)

Time (hour)

0 1 2 3

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-124

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-120

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-68

-66

-64

-62

-60

Amplitude

(μV)Ampl

itude

(μV)

Time (hour)

DBM 1

(a)

DBM 2

FIG. 8. The dc levels at IF ports of two doubly balanced mixers with two configurations: (a) The optical phase modulation is switched off, but two independentarms are still used for optical detection and frequency demodulation. (b) Only one detection arm is used and the output of the photodetector is split and sent totwo doubly balanced mixers. One of the demodulated signals is used for active RAM control and the other is sent to a digital multimeter.

is roughly 1/7 of the drift with only temperature stabilizationof the crystal.

To further investigate the performance of the activecancellation, we measure the long-term drifts of RAM bothinside and outside the control loop. Figure 7(b) shows two14-h records of the demodulated RAM signal measuredrespectively inside and outside the control loop. In the 14-hperiod the in-loop drift is within 5 μV while the out-of-loopdrift is about 20 μV, corresponding to a modulation depth of20 ppm.

The drift in the demodulated signal can be converted tofrequency instability. For instance, the discriminator slope ina typical PDH frequency locking is estimated to be 0.1 mV/Hzusing a 10-cm optical cavity with a finesse of 1,00,000, amodulation index of 0.52, and a 120-μW optical power atthe photodetector. Applying this conversion coefficient tothe drifts shown in Fig. 7, the frequency instability is about1.8 Hz with only temperature stabilization, while it reduces to0.25 Hz when the active cancellation is adopted.

To track the source of the residual drift in the out-of-loopmeasurement, two additional diagnostic tests are performed.First, the demodulated signals in two detection arms are si-multaneously monitored as before, but with the exception thatthe RF signal applied to the crystal is switched off. In the sec-ond test, the optical phase modulation is switched on again,but only one detection arm is used in which the output of thephotodetector is divided and demodulated by two DBMs. Oneof the demodulated signals (DBM2) is used for active cancel-lation and the other (DBM1) is used for out-of-loop measure-ment. Figure 8 plots the experimental results for both tests.For the case of no optical phase modulation, each arm has

a monotonic drift of 8 μV (steps on the black curve are dueto finite resolution in one of the DMMs), but the differen-tial drift between two arms is within 1−2 μV. This stabil-ity confirms that optical detection, demodulation, and voltagemeasurement contribute only a small part (a few μV) to the∼20 μV differential drift of RAM shown in Fig. 7. Similarly,the two demodulated signals exhibit drifts below 10 μV anddifferential drift about 3–5 μV when the RF inputs of the twomixers are from the same photodetector, indicating that theservo-induced drift is below a few μV.

Noise mechanisms such as parasitic etalons result in bothin-phase and quadrature components.18 When related noisesare not properly suppressed, cancelling the in-phase compo-nent does not guaranty a RAM-free baseline in the other com-ponent. This is of significance in applications such as absorp-tion measurement using FM techniques.

With an added second mixer and a phase shifter, experi-mental setup depicted in Fig. 2 is used to detect both in-phaseand quadrature signals. The phase shifter is adjusted to en-sure a 90◦ phase shift with accuracy better than 2◦. Figure 9shows the in-phase and quadrature signals for three experi-mental conditions: (1) the crystal is temperature controlledbut there is no active RAM cancellation, (2) both tempera-ture control and RAM cancellation are implemented, and (3)neither temperature control nor RAM cancellation is used. Inthe case of active RAM cancellation, the in-phase componenthas the smallest instability while its quadrature counterpartshows ∼75 ppm fluctuations. However, the quadrature com-ponent is much smaller than in-phase one when there is noactive RAM control or even no temperature control of thecrystal.

043111-7 Li et al. Rev. Sci. Instrum. 83, 043111 (2012)

FIG. 9. (a) In-phase and (b) quadrature components of RAM detected by two doubly balanced mixers. An in-phase measurement with active RAM cancellationis also added to panel (b) for a close comparison. Data are measured by two DMMs (Th1961 and Keithley 196A) with 4-Hz sampling rate and 20-ms integrationtime.

C. Measurements with unmatched detection arms

The next step is to investigate whether the active cancel-lation scheme can be effectively extended to more realisticexperimental setups where differences in arm length, opticalpower, and others could degrade or destroy the coherence ofthe RAM signals in the two detection arms. To achieve thisgoal, parasitic etalons in both two detection arms should beavoided and the air turbulence be kept as low as possible. Inaddition, an offset of the locking point away from zero willinduce additional error when unequal optical powers are usedin two detection arms.

Asymmetric setups are gradually introduced in the twodetection arms. First, the length of one detection arm is var-ied within 2 m and no difference in the differential RAMsignal is detected. Second, the optical power of one arm ischanged by a few mW and the lens (11-mm focal length, as-pherical) before one of the two photodetectors is replaced bya 30-mm spherical lens. With these modifications, the varia-tions of the demodulated RAM signals in two arms are still insync, allowing a robust RAM cancellation. Last, we relax therequirements on the equal mapping of RAM on two detectionarms. Optical components such as PBS, quarter and half waveplates, and mirrors are added to one arm and the demodulatedRAM signals in two arms are compared. The comparison ver-ifies that the correlation between two arms is still preserved.Although weak differential RAM signals can be produced bystrong local temperature cycles applied to the added opticalcomponents, they return to baseline after the perturbation isremoved, albeit with some time delay on the order of a fewtens of seconds.

We perform an independent measurement that in-volves two frequency-stabilized lasers (labeled as EAST andWEST). In each system, a Nd:YAG laser is locked to an ultra-stable optical cavity using PDH method. The laser beam afterphase modulation is split into two beams, one for frequencylocking to the cavity and the other for monitoring RAM.The optical cavity is vertically mounted in a temperature-stabilized vacuum chamber that is installed on a passive vi-bration isolation platform. Placed on the platform are also thephase modulator, photodetectors, and optics for PDH lock-ing and RAM detection. A polarization-maintaining fiber withlength of a few meters is used to send the light to the vibrationisolation platforms. Each system is enclosed in a hermeticallysealed and temperature controlled aluminum box.

First, the two lasers are locked to their corresponding ref-erence cavities. Two RAM signals, together with the beat fre-quency of the lasers, are measured by two DMMs (Th1961and Keithley 196A, 4-Hz sampling rate and 20-ms integra-tion time) and a frequency counter (HP5334B, 0.2-s gatetime), respectively. Whenever the temperature of one crystalis scanned, the crystal in the other system is temperature con-trolled in order to provide a stable frequency reference. Thedrift of each RAM signal fits well to the corresponding beatfrequency with resort to only a constant scaling factor, whichis the RAM-to-frequency sensitivity of the relevant system.Figure 10(a) plots the beat frequency and the frequency ex-cursion deduced from RAM.

Our next step is to simultaneously record the two RAMsignals and the beat frequency between two stabilized lasers.The temperatures of two EO crystals are stabilized duringthe measurement. The two RAM signals are converted to

043111-8 Li et al. Rev. Sci. Instrum. 83, 043111 (2012)

Deduced from RAM

FIG. 10. Comparisions between RAM and beat-frequency fluctuations in two PDH frequency locking systems. (a) Dots are beat frequency of the two lockedlasers when the temperatures of two EO crystals are varied successively. The demodulated signals of RAM in two monitoring arms are converted to frequency andplotted using solid curves. (b) Frequency fluctuations deduced from the out-of-loop detections of RAM in two systems and the directly counted beat frequencywhen the temperature of two EO crystals are stabilized.

frequency fluctuations using predetermined sensitivities C1

and C2. The resultant frequency drift in the beat note is thendetermined by �νR AM (t) = V1 (t) C1 − V2 (t) C2. InFig. 10(b), this new time series is superposed on the directlycounted beat frequency of the two locked lasers. Note that(1) the two time series have been vertically shifted byconstant frequencies and (2) linear drifts both in the beatnote and RAM signals have been removed. For time scaleslonger than ∼2 s, there is again a strong correlation betweenbeat frequency and RAM-induced frequency drift, indicatingthat the monitoring signal in current experimental setupis a reliable sample of the RAM in the locking arm forslow-varying components (<∼0.5 Hz).

V. DISCUSSIONS

Here, we discuss several improvements and alternativeapproaches remain to be explored in follow-up investigations.These improvements are not limited to active cancellationschemes but can also lead to a better stability in situationswhere a feedback control is not available.

EO crystals are closely related to many noise mecha-nisms in FM detection schemes. Crystals with large wavefrontdistortion are one of the potential sources of spatial RAM in-homogeneity and are in general susceptible to environmen-tal perturbations that take effect through unstable interfer-

ence fringes. Note that the wavefront distortion are affectedby many parameters such as surface flatness, inhomogeneity,striae, and stress birefringence. For example, with a lengthof 35 mm, a material inhomogeneity of �n ≈ 2.3 × 10−6 isrequired to achieve a wavefront distortion below λ/6.6. Theetalon formed by the two end surfaces of the crystal is a ma-jor contributor of RAM instability. A small wedge angle be-tween two end surfaces can reduce the etalon effect with per-haps some extra spatial RAM inhomogeneity. Alternatively,the etalon effect can be reduced by using high quality ARcoating and deviating slightly from the normal incidence. Itshould also be pointed out that the stability of beam positioninside the crystal is vital in obtaining a stable baseline, in cir-cumstances without active cancellation.

The EOM depicted in Fig. 2 has a rather simple struc-ture. Only the bottom surface of the crystal is in thermal con-tact with a copper block that is temperature controlled. Thissimple configuration results in a temperature gradient acrossthe crystal. Housing the crystal with a thermally controlledcooper cylinder with proper insulations and gaps on the sidesof the crystal can reduce this inhomogeneity and lead to a bet-ter thermal stability.

When active cancellation is adopted, a fluctuating RAM,albeit at a smaller scale, may still persist because of mismatchin the two detection arms with various optical or electricalorigins such as residual parasitic etalons, nonuniformity of the

043111-9 Li et al. Rev. Sci. Instrum. 83, 043111 (2012)

RAM distributed across the laser wavefront,18, 33 variation ofoptical alignment, and optical power fluctuation. It is of greatimportance to minimize the abovementioned effects betweenthe monitoring and the main arms, the latter being used forlocking lasers to frequency references such as optical cavitiesor atomic/molecular lines.

In current experimental setup, back reflections in two de-tection arms are highly suspicious to be responsible for theout-of-loop drift observed in Fig. 7. Additional optical iso-lation should clarify this issue. However, introducing an ex-tra commercially available optical isolator may not help be-cause of new reflecting surfaces introduced. Electronic driftsfrom circuits and thermocouple effect in electric contact canbe overcome by chopping the optical phase modulation andlock-in detection after the frequency demodulation.

An actuator that works with both the crystal temperatureand a bias voltage applied on crystal will lead to a better sta-bility for both short and long terms. Although it has a broaddynamic range, a temperature servo is effective only at lowfrequencies that barely go beyond 1 Hz. On the other hand,a servo loop with a bias electric field applied to the crystalis more suitable for suppressing fluctuations at even higherfrequencies.18 Implementing such a control loop is straight-forward for fiber-type EOMs, but high-voltage amplificationmay be needed for EO crystals made of bulk materials be-cause of higher half wave voltage (on the order of a few hun-dred volts). However, the laser frequency shifts induced byfast and large voltage swings should be quantitatively investi-gated.

VI. CONCLUSIONS

We have mapped out the temperature and polarizationdependences of RAM in a typical FM setup in which aMgO · LiNbO3 crystal is used for phase modulation. Theexperimental data are in good agreement with a theoreticalanalysis18 that takes account of the light polarization and tem-perature dependence of the crystal birefringence. In addition,the spatial inhomogeneity of the RAM on a transverse planeis probed by partially blocking the laser beam. This investi-gation allows us to adopt a simple adjustment to minimizethe RAM originated from the birefringence of the crystal.

To further improve the stability of the FM detection, bothpassive stabilization of crystal temperature and active cancel-lation of RAM are investigated. Long-term drifts in these twomethods are monitored and compared. For the case of ac-tive cancellation, both the in-loop and out-of-loop measure-ments are performed to evaluate the potentials of this schemein laser frequency locking and Fabry-Perot laser interferome-ters. The best stability in current system comes from an activecancellation of RAM, which exhibits out-of-loop drift below25 ppm in 24 h. Additional measurements show that the con-tributions of electrical origin are not fully responsible for theresidual drift observed in the out-of-loop measurement. Theinvestigation is extended to unmatched detection arms andRAM-induced frequency fluctuations are experimentally ver-ified with two lasers that are independently locked to ultra-stable optical cavities. These investigations indicate that theactive cancellation scheme through crystal temperature can

effectively offset the common-mode drifts of RAM in two de-tection arms originated from mechanisms of different origins,achieving a better stability compared with passive means.

The frequency instabilities of modern optical oscillatorsare being continuously pushed down to the realm of 10−16

and further improvements are underway. Similar to previousefforts on this subject, a rigorous assessment of the system-atics raised by the impurity of the phase modulation appearsimperative and will be helpful in the development of the nextgeneration optical oscillators whose applications extend fromspectroscopy and frequency standard to many precision mea-surements that probe the fundamental laws of physics.

ACKNOWLEDGMENTS

This work is supported by the Instrument DevelopingProject of the Chinese Academy of Sciences Grant No.Y20638. Equipment loans from Baolong Lu, MingshengZhan, and Jin Wang at Wuhan Institute of Physics andMathematics (WIPM) are highly appreciated. Chen benefitsfrom discussions with Mark Notcutt at Stable Laser Systems,Michael Martin, Jun Ye, and John L. Hall at JILA, Univer-sity of Colorado at Boulder. Li acknowledges the illuminatingsuggestions from Xiaodong He at WIPM.

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