cavity length measurement: bias from misalignment and mismatching

Cavity length measurement: bias frommisalignment and mismatching

Robert D. ReasenbergSmithsonian Astrophysical Observatory, Harvard-Smithsonian Center for Astrophysics Cambridge,

Massachusetts 02138, USA ([email protected])

Received 31 July 2013; revised 22 October 2013; accepted 22 October 2013;posted 24 October 2013 (Doc. ID 194882); published 20 November 2013

SR-POEM, the sounding rocket principle of equivalence measurement, uses a set of six tracking-frequency laser gauges operating in Fabry–Perot cavities to determine the relative acceleration oftwo test masses (TMs) that are chemically different. One end of each cavity is a flat mirror on a TM;the other end is a concave coupling mirror mounted to a common reference plate. The tracking-frequencylaser gauges work by locking a variable frequency laser to the cavity by the method of Pound, Drever, andHall. Because the TMs are unconstrained, they are expected to rotate slightly during measurement.Although the distance measurements are intended to be based on the TEM00 cavity mode, any misalign-ment will couple into higher-order transverse modes, particularly the TEM10 and TEM01. Light thuscoupled will contribute a spurious signal to the cavity locking servo that causes a bias (i.e., a systematicerror) in the length determination. The spurious signal proportional to the misalignment has an anti-symmetric distribution at the detector and thus has a zero average, but causes a distance bias because ofthe inhomogeneity of the detector responsivity. To prevent such bias, SR-POEM includes a servo to keepthe incoming laser beam aligned with the cavity. The required performance of that alignment servo is lessstringent than has already been achieved by other projects. There is also a spurious signal proportional tothe square of the misalignment that produces a symmetric distribution at the detector. This signal is alsomade unimportant by the operation of an alignment servo, even when operating well above the shot noiselimit. We also look at the locking of a laser to a high finesse cavity and conclude that the alignmentquality sets a bound on the ratio of measurement accuracy to cavity linewidth.OCIS codes: (120.2230) Fabry-Perot; (120.3940) Metrology; (120.6085) Space instrumentation;

(140.0140) Lasers and laser optics.http://dx.doi.org/10.1364/AO.52.008154

1. Introduction

Precision measurements using optical cavities areimportant in many fields of research from atomicand molecular physics, to metrology, to experimentalgeneral relativity. In the sounding rocket principle ofequivalence measurement (SR-POEM) [1] a set of sixlaser gauges [2,3] measures the displacement of thetest masses (TMs) during eight 120 s “drops.” Eachlaser gauge will have a precision of 0.1 pmHz−1∕2for averaging times under 300 s. Each of the mirrorsmounted on top of the TM is used as one end of thelaser gauge measurement cavity. The SR-POEM goalis to measure the equivalence principle parameter ηwith an uncertainty of 1 × 10−17, a four order of mag-nitude advance over the current best result [4,5].

During the drop, the TMs are in free fall, with nodegree of freedom constrained. This removes theinexorable influence on TM motion in the measure-ment direction of constraints on the other TMdegreesof freedom. However, because the TMs are uncon-strained, they will rotate slightly with respect tothe instrument during a drop. Such rotation can comefrom an error in the pre-drop setup or from theEarth’s gravity gradient and the unequal principalmoments of inertia of the payload producing a torqueon the payload. The resulting misalignment of thecavity with respect to the incoming beam must notcontribute importantly to total measurement error.

As with most experiments that measure smallforces, an important error component for SR-POEM

8154 APPLIED OPTICS / Vol. 52, No. 33 / 20 November 2013

http://dx.doi.org/10.1364/AO.52.008154

is surface potential of the TMs and the surroundingenclosure. The resulting “patch effect” forces must besmall and stable. These surfaces are specified asbeing gold, which is often described as havingsmaller patch potentials than most materials [6].Since there are mirrors on the surface of the TMs,they too must have gold surfaces, and that deter-mines the finesse of the measurement cavity, nomi-nally about 130.

Traditional distance measurement is subject tomisalignment error. The measured distance betweenthe apices of a pair of hollow retroreflectors is biasedlow by a factor of cos�θ� where θ is the angle ofmisalignment. Similarly, the measured distancebetween two parallel plates is biased high by1∕ cos�θ�. One advantage of using a cavity is that,to lowest order, the distance measured is forgivingof alignment error as long as the TEM00 mode is wellexcited. However, for a cavity-based measurement,the excitation of a higher-order transverse modecan introduce a bias even though the laser is tunedfar from the unintended mode. Here we examine thebiased distance measurement in reflected light. Wefind the largest component of the bias is inverselyproportional to finesse.

2. Alignment Analysis

In the alignment of an incoming beam with a fixedcavity, there are four degrees of freedom: two posi-tions and two angles. Mode matching introducesan additional two degrees of freedom (or four if thecavity is astigmatic). The analysis of alignment hereis later extended to stigmatic mode matching.

Consider a two-mirror cavity with optical axisalong the Z axis of a coordinate system having Xand Y transverse axes. Assume that: (1) the incominglight is mode matched and moderately well alignedwith the cavity; (2) the carrier (with angular fre-quency ω) is locked to a TEM00 cavity mode, sayby the Pound–Drever–Hall (PDH) method [7];(3) the PDH phase modulation is applied to the car-rier at an angular frequency Ω:

Ω � υΦ2F

; (1)

where Φ is the free spectral range of the cavity, F isits finesse, and υ is a free parameter; (4) the couplingmirror is lossless; and (5) the mirrors have equalreflectivity. Recall that Φ∕�2F � is the half-width athalf-maximum of the cavity.

The wave functions associated with the cavity andwith the incoming beam can be decomposed intothe cavity eigenmodes. We are concerned with theCartesian modes, which have Hermite–Gaussianeigenfunctions. Further, these modes can be sepa-rated along the transverse axes:

Um;n � Vm�x�Vn�y�; �2�

where the first two modes in x are

V0�x� ��

2

πw2x

�14

e−� xwx�2e−ix2πλR ;

V1�x� ��

2

πw2x

�14 2xwx

e−� xwx�2e−ix2πλR : (3)

In Eq. (3), λ is the optical wavelength, R is the wavefront radius of curvature, and wx is the beam spotsize. At the waist, R is infinite, which makes thelast factor in both parts of Eq. (3) equal to 1.The modes described by Eq. (3) are orthogonal andnormalized:

Z∞

−∞ViV�

j dx � δi;j; (4)

where V�j is the complex conjugate of Vj. Without loss

of generality, we will consider the misalignments inonly one direction, X , which permits the use of theabbreviated notation:

U0 � U0;0; U1 � U1;0: (5)

Also, following Anderson [8], we choose to performthe analyses at the waist, where the expressions taketheir simplest form.

The incoming beam is phase modulated andassumed to be slightly misaligned but modematched [8]:

Ψin�E0eiωt×�

C0U0�J0�m��J1�m��eiΩt−e−iΩt��p�iq�U1�J0�m��J1�m��eiΩt−e−iΩt��

�;

(6)

where Jk(m) is the kth Bessel function of the firstkind, its argument is the modulation index, m (notshown hereafter), C0 ≈ 1 conserves energy and issecond order in p and q, p � ax∕wx, q � παxwx∕λ,as described by Anderson [8], ax and ax are the dis-placement and rotation, respectively, and we ignoreterms containing Un with n > 1. The expansion isparticularly well justified when p and q are keptsmall, for example, by a null-seeking servo. The elec-tric field is the real part of Ψ. We assume m to besmall enough that we can neglect terms proportionalto Jk for k > 1.

In a reflection cavity, the complex response,reflected field divided by incident field, is given byYariv [9, Section 4.1]:

ΓR � �1 − eiδ��Rc

p1 − Rceiδ

; (7)

where Rc is the coefficient of power reflection of eachof the two cavity mirrors, δ � 4πϵ∕λ is the round-tripoptical phase (modulo 2π), and ϵ is the deviation ofthe cavity length from the resonance length.

Figure 1 shows the relations among the cavity res-onant frequencies and the laser frequencies (carrier

20 November 2013 / Vol. 52, No. 33 / APPLIED OPTICS 8155

plus first sidebands). Under PDH, the laser is nomi-nally locked to a single mode, assumed here to be theTEM00 mode. The goal is to find the bias δ0 in the lockdue to misalignment. The misalignment puts a por-tion of the light into the TEM10 mode and some ofthat light subsequently reaches the detector.

Since the carrier will be very close to the frequencyof a TEM00 mode, in analyzing the cavity response tothe carrier, ΓR can be expanded around δ � 0:ΓR ∼G0�δ0�, and we discard terms above first orderin δ0. In that expansion, by using ϵ � −λΔf∕�2Φ�,we can write δ0 � −2πΔf ∕Φ, where Δf is thedeviation of the carrier from the TEM00 modalfrequency. Similarly, for the sidebands, ΓR canbe expanded around δ � β, where β � 2πΩ∕Φ∶ΓR ∼G�m�δ0�, where we again discard terms abovefirst order in δ0.

To find the response of the higher-order transversemodes (say TEM10), we expand ΓR around δ � π,discarding terms above first order in δ1, where

δ1 � 2π�f 00 − f 10

Φ� 1

2

�; (8)

and f 00 and f 10 are the frequencies of the TEM00mode to which the laser is locked and the nearestTEM10 mode. Thus, for the carrier, ΓR ∼G1�δ1�,and for the sidebands, ΓR ∼G1�δ1 � β�. The expan-sions are given in Appendix A. Expanding to firstorder in δ1 provides a reasonable representation ofΓR for signals at least a few linewidths from f 10.

We can now describe the beam emerging from thecavity:

Ψout � E0eiωtψout; (9)

where

ψout �

0BBB@

C0U0�J0G0�δ0��J1�Gm�δ0�eiΩt −G−m�δ0�e−iΩt��

�p� iq�U1�J0G1�δ1��J1�G1�δ1 � β�eiΩt −G1�δ1 − β�e−iΩt��

1CCCA: (10)

From Eq. (10) we find the detectable power

Pout � h�Re�Ψout��2iω � E20

2ψoutψ

�out; (11)

where h�iω describes averaging over the optical fre-quency. By expanding Pout and keeping only termscontaining sin�Ωt� and cos�Ωt�, we find the light sig-nals that would cause the bias. From the coefficientof sin�Ωt� we get

δ0U0U0 � Δ � Δ1U0U1 � Δ2U1U1 �…; (12)

and

Δ1 � 21 − Rc

1�Rcq� δ1

�1 − Rc

1� Rc

�2p;Δ2

� −�1 − Rc�4�1�R2

c − 2Rc cos�β��1� Rc�6

× δ1 csc2�β��p2 � q2�β2: (13)

From the coefficient of cos�Ωt� we get a similarexpression except that

Δ2 � 2�1 − Rc�2�1� R2c − 2Rc cos�2β��

�1�Rc�4× δ1 csc�2β��p2 � q2�β: (14)

We note that the sensitivity to displacement param-eter p is smaller than the sensitivity to angular offsetparameter q by a factor of δ1�1 − Rc�∕�2�1�Rc��.(Much of the analysis in this paper was performedusing Mathematica version 9. Terms were kept tofirst order in δ1 and second order in p and q (i.e.,p2; pq; q2). The cross term was found to be zero,although nonzero terms like p2q were found and dis-carded.)

From Eq. (12), the bias is

δ0 � D�Δ�D�U0U0�

; (15)

whereD is an operator that maps the brightness pat-tern to the detected signal:

D�UαUβ� �ZZ

G�x; y�UαU�βdA: (16)

In the above equation, G�x; y� is the detectorresponsivity, dA is the differential of area, and theintegral extends over the detector. We define threedetection coefficients, γ � D�U0U0�, ζ � D�U1U0�,and ξ � D�U1U1�. Then, Eq. (15) becomes

δ0 � ζΔ1 � ξΔ2

γ: (17)

The PDH detection will yield a linear combinationof the above biases [Eqs. (13) and (14)]. The combina-tion depends on the phase of the modulation signalused for the PDH locking. The nominal detection

Fig. 1. Cavity response for a finesse of 30: TEM00 mode, solid line;TEM01 mode, dashed line (shifted down for clarity). Spectrum attop shows laser line and first sidebands. Phases shown (e.g., δ1) arefor light of the indicated frequency after a single round trip refer-enced to the light before the round trip.


phase, ϕ, for PDH locking provides the strongestsignal and depends on the ratio of the modulationfrequency to the cavity free spectral range

ϕ � a tan�1�Rc

1 − Rctan�β∕2�

�: (18)

This phase reaches π∕4 at

Ω ≈Φ2F

��Rc

p 21�Rc

; (19)

where the approximation is good forRc > 0.9 and cor-rections are of orderF−2. Thus, the phase reaches π∕4when the modulation frequency is equal to about thehalf-width at half-maximum of the cavity response[i.e., υ ≈ 1, see Eq. (1)].

We next evaluate the three detection coefficients. Ifwe assume G � 1, then by Eq. (4), γ � 1, ζ � 0, andξ � 1. Thus, there would be no first-order sensitivityto either p or q, and the bias would be proportional top2 � q2. However, if G has a gradient, G � �1� ϵ1x�,then ζ ≠ 0:

ζ �Z

∞

−∞�1� ϵ1x�V1V0dx � ϵ1wx

2: (20)

Note that, if the light were scrambled before reachingthe detector, ζ would likely be reduced.

Larason and Bruce [10] have investigated the spa-tial uniformity of photodiode response for severalkinds of photodiode and at a variety of wavelengths.In particular, they looked at the Telcom Devices35PD5M-TO InGaAs detector at 1000 and 1600 nm.Over a diameter of 5 mm, they found response varia-tion of about �3% and �1% at the two wavelengths,respectively. They described the result they show forone photodiode as typical of the four they measured.In their figure showing the spatial response, themean slope is in different (nearly opposite) directionsat the two wavelengths. If we assume the detectordiameter is 6wx, then, for the 1600 nm responsivity,ζ ≈ 1.7 × 10−3. Since the figures presented by Larasonand Bruce generally showed the spatial variations ofthe photodiodes’ responses not to be linear, the valueof ζ will vary as the illuminated spot moves withrespect to the detector.

3. Matching Analysis

The analysis of the mismatching case is similar tothe above. For the mismatching case, it is naturalto work in polar coordinates so the eigenfunctionsare generalized Laguerre polynomials multipliedby a Gaussian. We designate these as ~Un;l, wheren and l are the radial and angular mode numbers.The first two of these with l � 0 are

~U0;0 ��2π

r1wr

e−� rwr�2e−ir2πλR

~U2;0 ��2π

r1wr

�1–2

�rwr

�2�e−� r

wr�2e−ir2πλR ; (21)

wherewr is the radial scale. Like themodes of Eq. (3),these modes are orthogonal and normalized:

ZZ~Um;0

~U�n;0rdrdθ � δm;n; (22)

where the integration is over the entire plane.The incoming beam is assumed to be aligned and

slightly mismatched. We introduce normalized mis-match coefficients, τ and κ, and assume both aresmall [8]. The waist size is off, w0

0 � �1� τ�w0, asis its position along the Z axis (by distance b),κ � λb∕�πw2

0�. Then, the incoming phase-modulatedbeam is [8]

Ψin � E0eiωt ×

C0

~U0;0�J0 � J1�eiΩt − e−iΩt��τ� iκ� ~U2;0�J0 � J1�eiΩt − e−iΩt��

!;

(23)

which is the analog of Eq. (6).We again assume the carrier is close to the

frequency of one of the TEM00 modes and find thefrequency offset of the ~U2;0 mode from a frequencyhalfway between two adjacent TEM00 modes:

δ2 � 2π�f 00 − f 20

Φ� 1

2

�; (24)

where f 20 is the frequency of the ~U2;0 mode. By usingthe same expansions we found in Section 2, we candescribe the outgoing beam from the cavity:

Ψout � E0eiωtψout; (25)

where

ψout �

0BBBBB@

C0~U0;0�J0G0�δ0��

J1�Gm�δ0�eiΩt −G−m�δ0�e−iΩt��τ� iκ� ~U2;0�J0G1�δ2��

J1�G1�δ2 � β�eiΩt −G1�δ2 − β�e−iΩt�

1CCCCCA; (26)

which is the analog of Eq. (10).By again expanding Pout � h�Re�Ψout��2iω and

keeping only terms at the PDH modulation fre-quency, we find ~δ0, the PDH bias from mismatching.From the coefficient of sin�Ωt� we get


~δ0 ~U0;0~U0;0 � ~Δ � ~Δ1

~U0;0~U2;0 � ~Δ2

~U2;0~U2;0 �…

~Δ1 � 21 − Rc

1�Rcτ� δ2

�1 − Rc

1�Rc

�2κ

~Δ2 � −�1 − Rc�4�1� R2

c − 2Rc cos�β��1�Rc�6

× δ2 csc2�β��τ2 � κ2�β2∕4; (27)

and from the coefficient of cos�Ωt� we get a similarexpression except

~Δ2 � −�1 − Rc�2�1� R2

c − 2Rc cos�β��1�Rc�4

× δ2 csc�β��τ2 � κ2�β: �28�

The PDH bias from mismatching is then

~δ0 � D� ~Δ�D� ~U0;0

~U0;0�; (29)

and we again have the largest terms suppressedto the extent that the detector has uniform detec-tion efficiency. We define three detection coeffi-

cients, ~γ � D� ~U0;0~U0;0�, ~ζ � D� ~U0;0

~U2;0�, and ξ∼�

D�U∼2;0U

∼2;0�. Then, Eq. (29) becomes

~δ0 �~ζ ~Δ1 � ~ξ ~Δ2

~γ: (30)

We know that γ∼ � ξ

∼� 1, but must find ζ

∼based on

the response of the detector. If we assume thatG � �1� ϵ2r2�, then ζ

∼� −e2w2

r∕2. It is harder to esti-mate e2 from the figure given by Larason and Bruce[10] than it was to estimate e1. However, a reasonableguess is that the quadratic term is about 0.1% overthe full diameter and so ~ζ ≈ 6 × 10−5 (for a detectorradius of 3wr).

4. Application to SR-POEM

During the SR-POEM periods of TM free fall, thepayload and both TMs will rotate slowly bothbecause of imperfect setup and because of the torquefrom terrestrial gravity. Although each TM’s princi-pal moments of inertia can be made equal to betterthan 1%, the payload is assumed to have substan-tially unequal principal second mass moments.

We consider the bad but plausible case of q varyingwith the period of the inversion cycles:

q � q0 sin�2π

tP

�; (31)

where P is twice the drop time, Q, plus twice theinversion time, I. The current nominal values areQ � 120 s, I � 30 s, P � 300 s. Then, over a singledrop, the average of q̈ is

hq̈i � −q03.32 × 10−3 s−2: (32)

By combining Eqs. (13) and (17) with the definition ofδ, we find the distance bias

ϵ0 � λζ

2π�1 − Rc��1�Rc�

q; (33)

and, by using Eq. (32), the average acceleration bias

hϵ̈0i ≈ −λζ

4F2

�1�Rc�q03.32 × 10−3 s−2: (34)

For SR-POEM, with F ≈ 130 and λ � 1.55 μm, wefind jhϵ̈0ij ≈ 9.9 × 10−12ζq0 m× s−2.

There are three laser gauges observing each oftwo TMs. We will assume the laser gauge errorsare independent, but are all of the form ofEq. (31). For laser beam mispointing to contributea “minor error” to the mission (under 20% of the mis-sion error), the acceleration error of a single lasergauge must be under 7.84 × 10−18 m × s−2. Then, werequire ζq0 < 8 × 10−7 and, assuming ζ � 2 × 10−3,we further require q0 < 4 × 10−4.

Sampas and Anderson [11], working in a transmis-sion cavity with a detected power of 160 μW, demon-strated an active alignment system with a tilt errorof 0.1 nradHz−1∕2, corresponding to an error in q of2 × 10−7 Hz−1∕2 in the frequency range from 1 Hzto 1 kHz. The VIRGO gravitational wave antennahas an alignment system that uses informationfrom multiple beam samples to maintain thealignment of five mirrors. Babusci et al. [12] demon-strated alignment with an uncertainty of about10 nradHz−1∕2 using no more than 30 μWof detectedpower and a “wavefront sensing” technique. Thatuncertainty, which corresponds to an error in q of4 × 10−5 Hz−1∕2, is 100-fold smaller (better) thanthe VIRGO requirement. In the narrow bandwidthof an SR-POEM Mission, the Sampas–Andersonresult yields σ�q� � 7.3 × 10−8 when scaled to 1 μWof detected power.

Thus, the SR-POEM requirement that ζq0 <8 × 10−7 could be met entirely by making q small.With that level of suppression of q and a similarsuppression of p, the terms in Eqs. (12) and (13) pro-portional to �p2 � q2� become negligible. However,the Sampas–Anderson results were obtained atfrequencies above 1 Hz and we require that stabil-ity be achieved down to 3.3 mHz. Such an exten-sion to low frequency frequently introduces addeddifficulties—the white noise assumption breaksdown. For this reason, we anticipate making use ofthe factor ζ � 2 × 10−3. The problem of mode match-ing error is less problematic for SR-POEM.

5. Precision Locking to Cavities

When an optical cavity is used to stabilize a laser, afixed bias is usually unimportant. However, a biasthat varies with time will contribute to the Allandeviation of the locked laser. For high-precision


work, using a high finesse cavity, we may write thebias as

δ0 � π

F�ζq� ~ζτ�; (35)

where the terms involving p and κ have been droppedbecause they contain an extra factor of F−1. In termsof a frequency bias f b this becomes

f b � −Φ2F

�ζq� ~ζτ�. (36)

We define the fringe-splitting factor by

ρ�T �≡ ν

σ�f ; T � ; (37)

where σ�f ; T � is the standard deviation (with averag-ing time T ) of the frequency of the locked laserand ν � Φ∕F is the linewidth (full width at half-maximum) of the cavity. To find a bound on ρ�T �,we require that jf bj be smaller than α σ�f ; T �:

ρ�T � < 2α

ζq� ~ζτ: (38)

If σ�f ; T � depends, for example, on a white-noiseprocess

σ�f ; T � � σ�f ; 1��T

p ; (39)

then

ρ�T � � ρ�1��T ;

p(40)

and Eq. (38) allows us to find the maximum validvalue of T .

The value of α will depend on the variability of qand τ. In the absence of servo control and on longtime scales, where ρ�T � is large, one envisions thevariation of q and τ to be due to creep and changingthermal distortion.

Appendix: A

The cavity response function was given in Eq. (7):

ΓR � �1 − eiδ��Rc

p1 −Rceiδ

� �1�Rc��Rc

p�1 − cos�δ��

1� R2c − 2Rc cos�δ�

− i�1 − Rc�

��Rc

psin�δ�

1�R2c − 2Rc cos�δ�

:

(A1)

The first-order series expansions are

G0�δ0� � 0 − i

��Rc

p1 − Rc

δ0; (A2)

G�m�δ0�

� �1�Rc��Rc

p�1− cos�β��

1�R2c −2Rc cos�β�

− i�1−Rc�

��Rc

psin��β�

1�R2c −2Rc cos�β�

�

0BB@

�1�Rc��1−Rc�2��Rc

psin��β�

�1�R2c−2Rc cos�β��2 −

i�1−Rc��Rc

p �cos�β�

1�R2c−2Rc cos�β�−

2Rc sin�β�2�1�R2

c−2Rc cos�β��2�1CCAδ0;(A3)

where β � 2πΩ∕Φ, and to lowest order in β

G�m�δ0� � ∓i

��Rc

pβ

�1 −Rc�

��1�Rc�

��Rc

pβ

�1 − Rc�2− i

��Rc

p�1 − Rc�

�δ0. (A4)

Finally, for the light interacting with the TEM10mode, the series expansions are

G1�δ1� �2

��Rc

p1�Rc

� i�1 − Rc�

��Rc

p�1�Rc�2

δ1; (A5)

G1�δ1 � β� � G1�δ1� � i�1 − Rc�

��Rc

p�1�Rc�2

��β�: (A6)

6. Conclusion

SR-POEM is a proposed sounding-rocket experimentto measure or bound the equivalence principleparameter η with an uncertainty of 1 × 10−17, a fourorder of magnitude advance over the current bestresult. In the instrument, the motion of the TMs ismonitored by a set of laser gauges operating incavities of modest finesse. Because of gravitationaltorques and setup error, the TMs and thus thecavity mirrors will rotate during the measurement.Real-time beam alignment is thus required.

We have examined the effect of misalignmentbetween the cavity and incoming beam to determinethe bias it produces in the measurement of thecavity length. Because of detector imperfection, thereis a bias that is first order in the misalignmentparameters, with the normalized angle parameterq being the most important. However, it is wellwithin the capability of the alignment servo to reducethe alignment error and resulting distance bias tobelow the level of significance. Alignment error canbe sensed by the method of Anderson [8] as demon-strated in a transmission cavity by Sampas andAnderson [11] and shown to work in reflection byReasenberg [13]. That same alignment makes negli-gible the bias contribution from the term that issecond order in the alignment parameters.

This work was supported by the SmithsonianInstitution. SR-POEM has been supported bythe NASA Astrophysics Division through grant


NNX08AO04G. I thank J.D. Phillips for bringing thework of Larason and Bruce to my attention and forhis thoughtful comments on the manuscript.

References1. R. D. Reasenberg, B. R. Patla, J. D. Phillips, and R. Thapa,

“Design and characteristics of a WEP test in a sounding-rocket payload,” Class. Quantum Grav. 29, 184013 (2012).

2. R. Thapa, J. D. Phillips, E. Rocco, and R. D. Reasenberg,“Subpicometer length measurement using semiconductorlaser tracking frequency gauge,” Opt. Lett. 36, 3759–3761(2011).

3. J. D. Phillips and R. D. Reasenberg, “Semiconductor lasertracking frequency distance gauge,” Proc. SPIE 7436,74360T (2009).

4. S. Schlamminger, K.-Y. Choi, T. A. Wagner, J. H. Gundlach,and E. G. Adelberger, “Test of the equivalence principle usinga rotating torsion balance,” Phys. Rev. Lett. 100, 041101(2008).

5. T. A. Wagner, S. Schlamminger, J. H. Gundlachand, and E. G.Adelberger, “Torsion-balance tests of the weak equivalenceprinciple,” Class. Quantum Grav. 29, 184002 (2012).

6. N. A. Robertson, J. R. Blackwood, S. Buchman, R. L. Byer, J.Camp, D. Gill, J. Hanson, S. Williams, and P. Zhou, “Kelvin

probe measurements: investigations of the patch effect withapplications to ST-7 and LISA,” Class. Quantum Grav. 23,2665–2680 (2006).

7. R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford,A. J. Munley, and H. Ward, “Laser phase, and frequencystabilization using an optical resonator,” Appl. Phys. B 31,97–105 (1983).

8. D. Z. Anderson, “Alignment of resonant optical cavities,” Appl.Opt. 23, 2944–2949 (1984).

9. A. Yariv, Introduction to Optical Electronics (Holt, Rinehartand Winston, 1976).

10. T. C. Larason and S. S. Bruce, “Spatial uniformity of respon-sivity for silicon, gallium nitride, germanium, and indiumgallium arsenide photodiodes,” Metrologia 35, 491–496(1998).

11. N. M. Sampasand and D. Z. Anderson, “Stabilization oflaser beam alignment to an optical resonator by heterodynedetection of off-axis modes,” Appl. Opt. 29, 394–403(1990).

12. D. Babusci, H. Fang, G. Giordano, G. Matone, L. Matone, andV. Sannibale, “Alignment procedure for the VIRGO interfer-ometer: experimental results from the Frascati prototype,”Phys. Lett. A 226, 31–40 (1997).

13. R. D. Reasenberg, “Aligning a reflection cavity by Anderson’smethod,” Appl. Opt. 51, 3132–3136 (2012).


cavity length measurement: bias from misalignment and mismatching

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