In most classical measurement schemes1 birefrin-gence measurements dealing with low retardationvalues are either performed at a single location on thesample or if a large probe wave front is used only theaveraged birefringence properties are obtained. Onthe other hand, birefringence imaging is limited tolarge birefringence values because static bencheswithout birefringence modulation are used. Tradi-tionally, to realize birefringence maps of low-birefringence materials, one has to perform localmeasurements on an array of points by moving eitherthe sample or the detection bench.2 This method israther time-consuming if one wants to obtain goodspatial resolution and might be sensitive to thermaland mechanical drifts.
Here, we present a way to perform the measure-ment of a whole birefringence map without movingthe sample and with a sensitivity close to the shot-noise limit. This measurement can be done by use ofstandard birefringence bench geometry, some beam-shaping optics ~i.e., a beam expander!, and an arrayof detectors that is followed by multiplexed lock-in
The authors are with the Laboratoire d’Optique, Ecole Su-perieure de Physique et de Chimie Industrielles, UPR5 CentreNational de la Recherche Scientifique, 10 Rue Vauquelin, 75005Paris, France. V. Loriette’s e-mail address is [email protected].
Received 6 January 2000; revised manuscript received 4 April2000.
3864 APPLIED OPTICS y Vol. 39, No. 22 y 1 August 2000
detection on each of the detector elements. Theain difference between a single-beam scanning
echnique and the use of an imaging array is that, ifhere are imperfections of the bench and either de-ects of the optical components or simply poor orien-ations of the polarization elements, a spurious signals the same at each point in the first case but in theecond case this property is not a priori valid. Whensing large-aperture optical elements, one cannot ex-ect that the beam-polarization state is homogeneousver its whole. If the birefringence measurement iserformed on highly birefringent samples, this prob-em can simply be neglected, but this is not the casehen the birefringence to be measured is close to zeror is at least of comparable magnitude with the spu-ious signals.
In a naive perspective the two-dimensional bire-ringence measurement can be seen as the simulta-eous measurement of local birefringences by use ofimilar measurement benches that have slightly dif-erent defects. The difference with the classicalcheme is that the bench defects vary from point tooint. The following calculations show that, underome hypothesis concerning the relative amplitude ofhe spurious signal and the true signal, the measure-ent can still be performed even without the knowl-
dge of the bench imperfections over the beamection.In Section 2, we derive a mathematical represen-
ation of an imperfect birefringence bench. In Sec-ion 3, we present an actual realization of the imagingystem, whereas, in Section 4, we present some re-ults that were obtained and discuss the performancend limitations of this bench.
nt
w
Tn
oatetbccc
wD
s
wb
2. Light-Ray Representation of the Probe Beam:Elementary Representation of the Bench
To perform our calculations we use a simple repre-sentation of the global measurement bench: Eachpoint of the sample is analyzed by a measurementbench that has its own unknown defects, but all theideal elementary benches have well-defined proper-ties, i.e., they can be represented by sets of idealoptical components that are not necessarily identi-cal over the whole beam section but that haveknown properties and orientations. Each elemen-tary bench can be represented by a set of two Jonesmatrices MA and MB; the first one represents theoptical components placed between the light sourceand the sample, and the second one represents theoptics placed between the sample and the detectorelement. The Jones matrices representing ele-mentary optical components like polarizers or re-tardation plates are not derived here, as they can befound in any textbook on the subject ~see, for exam-ple, Ref. 5!.
A. Jones Matrix Representation of the ElementaryBenches’ Defects
The Jones matrices MA and MB represent real com-ponent sets that exhibit defects and misorienta-tions. In any case they can be written as the sumof a Jones matrix MA@,B#
P representing ideal compo-ents in their correct orientation and a correctiveerm DA@,B#MA@,B#
D .
MA 5 MAP 1 DA MA
D,
MB 5 MBP 1 DB MB
D, (1)
here DA@,B# are scalars, possibly complex, that givethe order of magnitude of the correction and MA@,B#
D
are ~possibly! unknown Jones matrices with elementsof the order of unity. We make the following twohypotheses:
• Hypothesis 1: The Jones matrices DA,@B#MA,@B#D
representing the correctiveterms have elements that aresmall compared to the Jonesmatrices representing the idealcomponents.
• Hypothesis 2: The corrective terms make acontribution to the birefrin-gence signal that is, at most,comparable in magnitude withor lower than the signal givenby the sample.
The first hypothesis simply states that we have anearly perfect knowledge of the bench and that thedefects are small, which can be written as
DA,@B# ,, 1. (2)
he second hypothesis is necessary because we willeed to perform Taylor expansion up to the first
rder in DA,@B# and D, where D is the birefringencemplitude of the sample as seen by a given elemen-ary bench. Hypothesis 2 states that the differ-nce between the expected beam-polarization stateransmitted by the bench and the real one inducedy the uncertainties of the bench properties, areomparable with or smaller than the polarizationhange induced by the sample. This hypothesisan be written as
DADB ,, D. (3)
Using these two hypotheses @expressions ~1! and ~2!#,e perform all the calculations up to the first order inand DA,@B# simultaneously. Experimentally, we
can check the two hypotheses by performing and an-alyzing measurements without any sample, thus ob-taining the order of magnitude of DA,@B#.
As an example, we calculate the first-order correc-tive term to add to a Jones matrix that represents ahorizontal linear polarizer when it is slightly misori-ented by a small angle up:
Mpol 5 F1 00 0GÇ
ideal polarizer
1 upF0 11 0GÇ
correction matrix
1 o~up2!. (4)
We see from this basic example that the amplitude ofthe corrective term is simply the misorientation an-gle, Dpol 5 up.
B. Jones Matrix Representation of a Low-BirefringenceSample
The birefringence of a sample is characterized by thelocal retardation value D~x, y! and the local orienta-tion g~x, y! of the principal axes in a basis ~eX, eY!linked to the local measurement bench. If the prin-cipal axes are aligned with the bench axes the Jonesmatrix representing the sample ~locally! is
MS 5 3expS2iD
2D 0
0 expSiD
2D4 . (5)
In the general case, the orientation of the sample isgiven by g Þ 0, and the general Jones matrix repre-enting the sample in the ~eX, eY! basis is simply
MS 5 5~2g!3expS2iD
2D 0
0 expSiD
2D45~g!, (6)
here R~g! is the rotation matrix of the angle g, giveny
5~g! ; Fcos~g! sin~g!2sin~g! cos~g!G , (7)
1 August 2000 y Vol. 39, No. 22 y APPLIED OPTICS 3865
w
renwfst
3
so that
When the sample birefringence is low, we can developMS by using a Taylor series up to the first order in D,as given by
MS 5 F1 00 1G 2 i
D
2 Fcos~2g! sin~2g!sin~2g! 2cos~2g!G 1 o~D2!, (9)
hich we can write in a more compact form as
MS 5 1 2 iD
21~2g! 1 o~D2!. (10)
One can recognize 1~2g! as the Jones matrix for arotator of angle 2g.
3. Measurement Procedure
The elements of the ~Dy2!1~2g! matrix of Eq. ~10!have to be measured without our knowing the valuesof the DA@,B# and the MA@,B#
D matrices. To solve thisproblem, one needs to perform two measurements:one without the sample and one with the sample.This double-measurement scheme introduces a newdifficulty: In the naive picture presented in Section1, each elementary detector has to be seen as part ofone single elementary bench, or, if one needs to takeinto account the transfer function of the optical ele-ments, of a known combination of elementarybenches ~by means of Fourier and inverse Fouriertransforms!. This relation between the detector ar-ay and the bench must not be modified by the pres-nce of the sample. It means that the sample mustot deform the incident wave front, or, in otherords, that its Fourier representation must be of the
orm of a complex scalar, constant through the beamection. The measurement scheme is thus limited tohe study of plane-parallel transparent plates.
A. Choice of the Bench Geometry
There are many classical schemes that one can usefor making a birefringence bench ~see, for example,
Fig. 1. Polarization elements in the instrument that measures bplate with its fast axis at 134°; ~S!, the sample; ~Q2!, a quarter-waits axis at 0°; ~A!, an analyzer with its axis at 145°.
866 APPLIED OPTICS y Vol. 39, No. 22 y 1 August 2000
Ref. 1!. We use a particular one that allows thesimultaneous measurement of D and g; however, it isimportant to emphasize that the validity of the cal-culation is completely independent of the particularscheme that is used. In our scheme one of the opti-cal elements placed between the sample and the de-tector is a birefringence modulator. This meansthat a priori the correction term DBMB
D is time de-pendent just like the MB
P matrix. The succession ofpolarization components is represented in Fig. 1.With this configuration it is possible to perform mea-surements without rotating any element, which sim-plifies the setup procedure a great deal and makesthe bench much more stable.
B. Measurement without a Sample
1. Jones Matrix of the Empty BenchFirst one has to perform a measurement withoutsample. The Jones matrix representing the com-plete bench without a sample is
M 5 MB MA 5 ~MBP 1 DB MB
D!~MAP 1 DA MA
D!
5 MBPMA
P 1 DB MBDMA
P 1 DA MBPMA
D 1 o~DADB!.(11)
The MAP matrix represents a horizontal linear polar-
izer followed by a quarter-wave plate with axes at145°, and it is thus written as
MAP 5
1
Î2F1 i
i 1GF1 00 0G 5
1
Î2F1 0
i 0G . (12)
The MBP matrix represents a quarter-wave plate
with axes at 245° and is followed by a modulator with
ngence: ~P!, a polarizer with its axis at 0°; ~Q1!, a quarter-waveate with its fast axis at 245°; ~M!, a birefringence modulator with
irefrive pl
MS 5 3cos2~g!expS2iD
2D 1 sin2~g!expSiD
2D 2i sin~2g!sinSD
2D2i sin~2g!sinSD
2D sin2~g!expS2iD
2D 1 cos2~g!expSiD
2D4 . (8)
wtD
axes at 0° and an analyzer with axes at 145°; it isthus written as
MBP 5
1
2Î2 F1 11 1G3expS2i
c
2D 0
0 expSic
2D4F 1 2i2i 1 G
51
2Î2 HexpS2ic
2DF1 2i1 2iG 1 expSi
c
2DF2i 12i 1GJ ,
(13)
where c 5 c0 sin~vt!, with v being the modulatedangular frequency. We now write MB
P in a morecompact form:
MBP 5
1
2Î2 FexpS2ic
2DMB,2P 1 expSi c
2DMB,1PG . (14)
In the most general case the MBP matrix has three
components ~i.e., those in the square brackets!:
MBP 5
1
2Î2 FMB,0P 1 expS2i
c
2DMB,2P
1 expSic
2DMB,1PG . (15)
MB,0P is not present in our particular configuration.
Equations ~12! and ~13! give the first term in Eq. ~11!as
MBPMA
P 5
expS2ic
2D2 F1 0
1 0G . (16)
The only information we have concerning the twoother terms of Eq. ~11! is that DA and DB are smalland that MB
D shares the same time dependence asMB
P. We can write the MBD matrix as
MBD 5 MB,0
D 1 expS2ic
2DMB,2D 1 expSi c
2DMB,1D, (17)
where MB,@0,1,2#D are unknown matrices. Using
Eqs. ~12!, ~14!, and ~17! in Eq. ~11!, one obtains
M 5
expS2ic
2D2 F1 0
1 0G 1 DB MB,0DMA
P
1 expS2ic
2DSDB MB,2DMA
P 1DA
2Î2MB,2
PMADD
1 expSic
2DSDB MB,1DMA
P 1DA
2Î2MB,1
PMADD .
(18)
We can write Eq. ~18! in a more compact form as
M 5
expS2ic
2D2 F1 0
1 0G 1 DBP0 1 expS2ic
2DD2P2
1 expSic
2DD1P1, (19)
where P0,2,1 are unknown matrices and D2,1 areunknown scalars with amplitudes comparable withDA,B.
2. Transmitted FieldLet Ei be the incident field, Ei 5 ~Ei,x, Ei,y!. Thetransmitted field is
Et 5
expS2ic
2D2
Ei, x 1 DBE0P 1 expS2i
c
2DD2E2P
1 expSic
2DD1E1P, (20)
where
E0,@2,1#P ; P0,@2,1#Ei, Ei, x ; Ei, xS1
1D .
The intensity on a detector element is the sum of a dcterm and modulated terms. At the zeroth order, thedc term is
It,dc0 5
12
Ei, xEi, x* 1 o~DIi!, (21)
and modulated components are
It,v0 5
expS2ic
2D2
DBEi, xE0P* 1 . . . 1 , (22)
here all these modulated components have ampli-udes that are proportional to linear combinations ofA@,B#.
C. Measurement with the Sample
The Jones matrix representing the complete benchwith the sample is ~to the first order in D!
Q 5 MBF1 2 iD
21~2g!GMA (23)
5 M 2 iD
2MB
P1~2g!MAP. (24)
1 August 2000 y Vol. 39, No. 22 y APPLIED OPTICS 3867
aa
s
sn
3
The second term of Eqs. ~23! and ~24! is
2iD
2MB
P1~g!MAP
5 2iD
4Î2 HexpS2ic
2DF1 2i1 2iG 1 expSi
c
2DF2i 12i 1GJ
3 Fcos~2g! sin~2g!sin~2g! 2cos~2g!G 1
Î2 F1 0i 0G (25)
5 2D
4exp~2ig!expSi
c
2DF1 01 0G , (26)
so the matrix Q can be written as
Q 5 3expS2ic
2D2
2D
4exp~2ig!expSi
c
2D4F1 01 0G 1 DBP0
1 expS2ic
2DD2P2 1 expSic
2DD1P1. (27)
1. Transmitted FieldOnce again, the intensity on the detector is the sumof a dc component of order zero
It,dc 5 It,dc0 5
12 Ei, xEi, x* 1 o~DIi!, (28)
nd the modulated and the dc components of order 1re
It,D 5 2D
2cos~c 1 2g!Ei, xEi, x*
1
expS2ic
2D2
DBEi, xE0P* 1 . . . 1 (29)
5 2D cos~c 1 2g!It,dc 1 It,D0. (30)
We see that, because of the first order approximationin D, we can separate the contribution of the samplefrom the spurious signal It,D
0 in the modulated inten-ity. By measuring It,D
0 without the sample, we cansubtract and cancel the effects of the bench defectsfrom the signal measured with the sample.
The modulated signal can be developed in terms ofBessel functions by use of the relations
cos c~t! < J0~c0! 1 2J2~c0!cos~2vt!,
sin c~t! < 2J1~c0!sin~vt!. (31)
By using expressions ~31! in Eq. ~30!, one obtains
It,D 5 2DIt,dc@J0~c0!cos~2g!
2 2J1~c0!sin~2g!sin~vt!
1 2J2~c0!cos~2g!cos~2vt!# 1 It,D0. (32)
868 APPLIED OPTICS y Vol. 39, No. 22 y 1 August 2000
The first term in Eq. ~32! is a dc term of the order Dthat we can neglect in front of It,dc. If we were usingtandard locked-in detection the three measured sig-als with the sample would be
S0 512 Ei, xEi, x* 5 It,dc,
Sv 5 2DJ1~c0!sin~2g!It,dc 1 Sv,0,
S2v 5 22DJ2~c0!cos~2g!It,dc 1 S2v,0, (33)
where Sv,0 and S2v,0 are the v and 2v components ofIt,D
0, respectively. We would then obtain the valuesof D and g from
D sin~2g! 5Sv 2 Sv,0
2J1~c0!S0, (34)
D cos~2g! 52S2v 1 S2v,0
2J2~c0!S0. (35)
It is important to emphasize at this point that, forEqs. ~34! and ~35! to be used, it is not necessary noreven useful to know where and what the bench de-fects are. The only hypothesis that makes Eqs. ~34!and ~35! valid are that all the birefringence terms aresmall ~i.e., DA,@B# ,, 1, D ,, 1! and that all calcula-tions can be performed up to the first order simulta-neously in DA,@B# and D ~i.e., DADB ,, D!. Moreover,Eqs. ~34! and ~35! hold even if the bench is not ho-mogeneous over the incident wave front. We canrewrite Eqs. ~34! and ~35! as
D~x, y!sin@2g~x, y!# 5Sv~x, y! 2 Sv,0~x, y!
2J1~c0!S0~x, y!p R~x, y!,
(36)
D~x, y!cos@2g~x, y!# 52S2v~x, y! 1 S2v,0~x, y!
2J2~c0!S0~x, y!p R~x, y!,
(37)
respectively, where R~x, y! is the imaging-systemtransfer function. The only extra hypothesis here isthat the presence of the sample does not affect thewave-front shape. The validity of this assumptiondepends on the optical setup’s geometry and is justi-fied in Section 6.
2. Case of Large Birefringence ValuesEquations ~28!–~37! are valid when all the birefrin-gence terms are small. By performing a measure-ment on an empty bench, we showed that theirvalidity domain is limited, for measurements of low-birefringence samples, by the bench itself. In thecase in which the sample is highly birefringent, thislimitation can be neglected, but, on the other hand,the previous equations do not hold because of thehypothesis that D ,, 1 and the first-order Taylordevelopments that followed. It is easy to show that,by neglect of the bench defects and use the general
Ws
Coaod
t
ppprs
mr
ptut
0
Jones matrix representation of the sample as in Eq.~8!, the matrix Q of Eq. ~27! becomes
Q 512 FcosSD
2DexpS2ic
2D2 sinSD
2Dexp~2ig!expS2ic
2DGF1 01 0G . (38)
ith standard locked-in detection the three mea-ured signals would be
S0 5 @1 2 sin~D!J0~c0!cos~2g!#It,dc,
Sv 5 2 sin~D!J1~c0!sin~2g!It,dc,
S2v 5 22 sin~D!J2~c0!cos~2g!It,dc. (39)
omparison of Eqs. ~39! with Eqs. ~33! shows that thenly differences between the low-birefringence casend the high-birefringence case are the replacementf D by sin~D! in all the equations and the signalependence of the dc term S0 because the J0 term in
the development of Q cannot be neglected. Equa-ions ~34! and ~35! thus become in this case
sin~D!sin~2g! 5Sv
2J1~c0!S0@1 2 J0~c0!sin~D!cos~2g!#,
(40)
sin~D!cos~2g! 52S2v
2J2~c0!S0F1 2
J0~c0!S2v
2J2~c0!S0G21
, (41)
respectively. This set of equations ~40! and ~42! isreferred when D is large and sin~D! cannot be ap-roximated accurately by its first-order Taylor ex-ansion. A limit value can be set arbitrarily to 0.5ad, in which case the error on D reaches 5%. If theample birefringence exceeds this value Eqs. ~40! and
~41! must be used instead of Eqs. ~34! and ~35!. Thisay be the case, for example, when samples are bi-
efringent crystals.
4. Multichannel Detection
The principle of multichannel detection can be foundin Refs. 6–8. The method is well known in phase-shifting interferometry as the four-phase algorithm.We follow here an elegant presentation that was pro-posed in Ref. 9, and we only adapt this presentationto our particular problem. When the detector is aCCD array, and so an integrator, we cannot filter thesignal and independently extract the v and 2v com-
onents, as is the case when using standard locked-inechnique. All the terms in the development of mod-lated signals by use of Bessel functions must beaken into account, and Eq. ~32! has to be written in
its most complete form, as
It,D 5 2It,dcHJ0~c0!cos~2g!
2 2 sin~2g! (n50
1`
J2n11~c0!sin@~2n 1 1!vt#
1 2 cos~2g! (1`
J2n~c0!cos~2nvt!J 1 It,D0. (42)
n51
Because the term It,D can be developed in the sameway, we represent its contribution in the form of twounknown parameters D0 and g0 and write
It,D~t! 5 2It,dcHJ0~c0!@D cos~2g!# 1 D0 cos~2g0!
2 2@D sin~2g! 1 D0 sin~2g0!# (n51
1`
J2n11~c0!
3 sin@~2n 1 1!vt# 1 2@D cos~2g!
1 D0 cos~2g0!# (n51
1`
J2n~c0!cos~2nvt!J . (43)
To mimic lock-in detection, we use a modulatedsource with a time-dependent intensity of the form
Cp~t! 5 IiH14
1 2 (n51
1` sin~npy4!
npcosFnvSt 1
p4f
1f
vDGJ ,
(44)
where f 5 vy~2p! and f is a possibly unknown phaseoffset between the modulator and the light sourcethat is induced by the electronic chain ~see Fig. 2!.The modulation can be phase shifted by an integernumber of a quarter period, depending on the value ofthe parameter p. Four phases are enough to recoverthe amplitude and the phase of the modulated signal.The phase offset in this experiment is similar to thequarter-wave translation of a reference surface inphase-shifting interferometry. The intensity inci-dent on each pixel is the product Cp~t!@It,dc 1 It,D~t!#.Because the modulation frequency induced by thebirefringence modulator’s being much higher than acamera readout frequency ~50 kHz compared with,
Fig. 2. Four signals are acquired, and the light source Cp is phaseshifted by a quarter of a modulation period at each acquisition.
1 August 2000 y Vol. 39, No. 22 y APPLIED OPTICS 3869
pc
w
c
3
typically, 200 Hz!, we can detect the time average ofthis quantity by
Sp 5 ^Cp~t!@It,dc 1 It,D~t!#&
5 Ii$14 1 2@D cos~2g! 1 D0 cos~2g0!#Kp~c0, f!
1 2@D sin~2g! 1 D0 sin~2g0!#Qp~c0, f!%, (45)
where
Kp~c0, f! 5 (n51
1`
J2n~c0!sin~npy2!
2npcos@n~pp 1 2f!#,
(46)
QP~c0, f! 5 (n50
1`
J2n11~c0!
sinF~2n 1 1!p
4G~2n 1 1!p
3 sinF~2n 1 1!Spp
21 fDG , (47)
and we neglected, in the dc signal, the birefringence-induced contribution, i.e., the first term in Eq. ~43!.
can take one of the values $0, 1, 2, 3%, and theorresponding signals are $S0, S1, S2, S3%. Linear
combinations of those four signals allow us to extractthe data. The complete derivation of the final re-sults are given in Appendix A. The final set of equa-tions is
w <S0 2 S2
S1 2 S3
Q1~c0!
Q2~c0!, (48)
D sin~2g! 1 D0 sin~2g0! <S0 1 S1 2 S2 2 S3
4Ii@Q1~c0! 1 wQ09~c0!#, (49)
D cos~2g! 1 D0 cos~2g0! <S0 2 S1 1 S2 2 S3
8IiK0~c0!, (50)
here
K0~c0! 51p (
n50
1`
~21!nJ2~2n11!~c0!
2~2n 1 1!, (51)
Q1~c0! 5 (n50
1`
J2n11~c0!sin@~2n 1 1!py4#
~2n 1 1!p~21!n, (52)
Q09~c0! 5Î22p (
n50
N
~21!nJ2n11~c0!. (53)
If the phase f is close to zero the best choice for c0 isto obtain 8K0~c0! 5 4Q1~c0! so that neither projectionof the birefringence @D sin~2g! or D cos~2g!# is favored.This occurs for c0 ' 2.1 rad and for c0 ' 4.3 rad, asan be seen from Fig. 3.
5. Optical Setup
The main considerations that led us to choose theoptical setup we are presenting were to be able to testsamples with diameters as long as 20 cm and to loweras much as possible the spurious birefringence signalinduced by the bench components. The first require-
870 APPLIED OPTICS y Vol. 39, No. 22 y 1 August 2000
ment makes it necessary to use beam-expanding op-tics somewhere, whereas the second implies that weshould use optical components that are as small aspossible to obtain polarization properties that are ashomogeneous as possible. Those two a priori oppo-site requirements led us to choose a reflector scheme.The main drawback of this approach over the onethat uses a beam expander is that the wave frontincident upon the sample is spherical. The mainadvantage is that no large transmissive optics areused.
The bench is made of two parts: an imaging sys-tem and a birefringence bench. The imaging systemis made of a light source, an 830-nm light-emittingdiode ~LED!, and a 53 microscope objective giving animage of the LED at the concave-mirror center ofcurvature. The mirror radius of curvature is r 52 m. To obtain better polarization effects and goodprecision of the wavelength value, we placed an830-nm interference filter with a FWHM of 10 nmafter the objective. The use of a LED instead of alaser source eliminates the eventual problems ofFabry–Perot effects in any component, especially thebirefringence modulator. The camera @DALSA,Model CA-D1 ~R01!, with 8-bit ~256-level! analog-to-digital converters# images the sample that is placedin front of the concave mirror, whereas the sourceimage is rejected at infinity, thus giving uniform il-lumination across the image field. The depth of fieldis greater than 30 cm. This optical scheme is repre-sented in Fig. 4. The camera works at a maximumspeed of 200 framesys, and, as we need a four-phasesequence to perform the calculation, the birefrin-gence field is acquired at 50 framesys. The data aresent to a frame grabber ~EPIX, Model PIXCI-D!.The images captured by the frame grabber are storedand treated on a PC.
The birefringence bench is the one that was pre-sented in Section 1. As can be seen from Fig. 5, thepolarization axes are defined not by the polarizer butby the small, flat, silicon mirror. The polarizer axesmust be aligned with this flat mirror; otherwise, the
Fig. 3. Absolute values of the functions 4Q1 and 8K0 at c 5 2.1rad and c 5 4.3 rad, respectively.
ppiTwpuOis
niosim~ni6mTwc
pLa
field incident upon the quarter-wave plate would beslightly elliptically polarized instead of linearly po-larized. The polarizer is aligned with the flat mirrorby optimization of the extinction between the crossedpolarizer and the analyzer without a quarter-waveplate or a modulator. The polarizer and the ana-lyzer are of the Glan–Thomson type with an extinc-tion factor of 1024. The two quarter-wave plates
resent in the mathematical representation are re-laced with a single one that is crossed twice to min-mize the off-axis incidence on the concave mirror.his makes the precision of centering of this quarter-ave plate a crucial parameter. The quarter-wavelate is a zero-order plate. The birefringence mod-lator is a Hinds Model PEM-90 working at 50 kHz.ptimization of the orientations of the various polar-
zation components is done by minimization of theignals arriving on the camera, but, as the spurious
Fig. 4. Each pixel ~P! of the CCD array images a 470 mm 3 470mm region ~R! of the sample ~S!, while the image ~O2! of the source~O1! illuminates the CCD array uniformly. Q, quarter-waveplate; M, modulator; A, analyzer; Obj., objective.
Fig. 5. Schematic diagram of the birefringence bench: The up-per part of the figure shows in detail the section of the lower partof the figure that is enclosed in the dashed box. The sample ~S! is
laced in front of a concave mirror ~CM!. The light source is aED. The basic birefringence measurement setup is made up ofpolarizer ~P!, a quarter-wave plate ~Q! that is crossed twice or two
quarter-wave plates, a birefringence modulator ~Mod!, and an an-alyzer ~A!. The imaging system is made up of a microscope ob-jective ~MO!, which creates an image of the source at a distance ofr 5 2 m from the concave mirror, and a 256 3 256 pixel CCDcamera with an f 5 80 mm objective ~O!.
birefringence field is not homogeneous over the cam-era field, the choice is made to minimize the averagespurious signal and not its value on a particular re-gion. It takes no more than 10 min to align thepolarization components, and the same amount oftime is needed to set up the imaging system. Themain advantage of the absence of any moving parts isthat the bench settings do not drift at all duringweeks, so after the setup has been optimized once andfor all, it takes less than 2 min to make the systemoperational.
6. Performance
A spherical wave front is used to illuminate the sam-ple, and a plane-parallel sample is not stigmatic inthis case. However, because of the low focal ratio ofthe spherical mirror ~ fy# 5 0.1!, the spherical aber-ration introduced by the sample is negligible. If thefollowing equation10 for the longitudinal spherical ab-erration ~LSA! by a plane-parallel plate
LSA 5 tn2 2 1
2n3 SrrD
2
, (54)
where r is the sample radius, n is its index of refrac-tion, and t is its thickness, is used a typical 10-cm-thick, 5-cm-radius silica sample thus gives a LSA of12 mm, which is completely negligible in our setup.
The accuracy of the bench is limited mainly by theorientations of the various polarization elements, butthis limitation is mostly cancelled by the fact thatmisorientations appear as the spurious signal @D0sin~2g0!, D0 cos~2g0!#, which is subtracted from thetotal signal. As was emphasized in the misoriented-polarizer example, the spurious signal’s amplitude isproportional to the various misorientation angles.During the preparations for measurement, the accu-racy with which the components can be oriented isapproximately 10 arc min. The measurement pre-cision is limited by the accuracy with which the mod-ulator is calibrated. The spectral width of the lightsource is not a key issue because both polarizers arewide band and in the modulator and the quarter-wave plate the effects cancel at first order, thanks tothe symmetrically shaped interference-filter trans-mission band.
Because the shot-noise value is close to the CCDresolution ~0.6 bit!, the sensitivity is limited chiefly by
oise and not by digitization. This allows one toncrease the sensitivity by averaging over a numberf sequences. Figure 6 shows experimental mea-urements of the instrument sensitivity. The max-mum achievable sensitivity depends on the
aximum amount of memory usable for storing dataeach sequence is 1 Mo large! and on the presence ofoise sources that may become dominant for long
ntegration times. When we averaged as many as00 sequences no discrepancy was found between theeasured noise and the theoretical shot-noise limit.he noise present in addition to that of the shot noise,hich increases the total noise from 0.6 to 0.74 bit,
omes from electronic noise in the camera and should
1 August 2000 y Vol. 39, No. 22 y APPLIED OPTICS 3871
ttTtofg
tgtes
ppup
stop
3
be lowered by suitable cooling. Sensitivity can alsobe improved by use of converters with more than 8bits together with a CCD array capable of collectingmore charges. The sensitivity is worse by an orderof magnitude than what can be obtained with asingle-element detector; this is the main drawback ofthe use of a CCD array.
7. Birefringence Maps of Transparent Samples
We tested the bench on various samples of fused sil-ica and CaF2. As the bench was designed primarilyo test low-birefringence samples, the numericalreatment of the data makes use of the first-orderaylor expansions detailed in Section 2. We presentwo measurement examples that show the usefulnessf such a bench for measuring inhomogeneous bire-ringence and studying dynamic aspects of birefrin-ence.The first example shows the birefringence ampli-
tude of a CaF2 24 mm 3 30 mm section that is 38 mmhick and parallel piped. The fact that the birefrin-ence pattern exhibits large values ~mostly becausehe sample is rather thick! allows us to look at theffects of rotation of the sample. The two imageshown in Fig. 7 are of two consecutive measurements
872 APPLIED OPTICS y Vol. 39, No. 22 y 1 August 2000
erformed before and after a 90° rotation of the sam-le on its holder. The birefringence pattern is leftnchanged, showing that no principal direction isromoted by the bench, thanks to the choice of c0.
The resolution depends on the imaging system.With our 256 3 256 pixel array one pixel images aurface of 470 mm 3 470 mm on the sample. Theotal field is limited by the concave-mirror diameterf 10 cm. Figure 8 shows the local direction of therincipal axes in the same CaF2 sample.The second example shows the variations of stress-
induced birefringence in a birefringence modulator.The stress is localized where the ceramic transducers
Fig. 6. Sensitivity ~in radians! plotted as a function of the numberof sequences averaged; the acquisition time of a single sequence of$S0, S1, S2, S3% is 20 ms.
Fig. 7. Birefringence-amplitude maps of a CaF2 sample. Thegray scales give the birefringence values in radians. The scales ofthe coordinates are given in pixels ~1 pixel corresponds to 470 mmon the sample!. Shown are two consecutive measurements per-formed ~a! before and ~b! after a 90° rotation of the sample in itsholder.
Fig. 8. Principal-axes directions in a CaF2 sample. The gradu-ations of the coordinates are in pixels ~470 mm 3 470 mm!. Thelength of each arrow is proportional to the birefringence amplitude.
sasoiawuteamtanni
w
are glued onto the silica rod, and the stress amplitudedepends on the applied voltage. Figure 9 shows thatthe birefringence variations versus voltage do notdemonstrate the possibility of carrying this study outin quasi-real time. The acquisition of four framesand the data treatment take approximately 0.1 swithout any kind of data-handling optimization.Thus this bench can be used to study dynamic behav-iors with frequencies as high as a few hertz withreal-time data treatment. To perform the calcula-tion, we represented the data as floating-point num-bers. Optimizing the numerical data-treatmentstep, which uses more than 70% of the 0.1-s of theframe-acquisition and the data-treatment time, i.e.,because the process works with an integer-numberrepresentation, should make it possible to work atrates of as much as 40 Hz. If data treatment iscarried out off-line the frequency limit is 50 Hz.
8. Conclusions
We have presented an instrument capable of per-forming fast measurements of birefringence fieldsin large transparent samples. The sensitivity,which is limited by the performance of the CCDarray, is 3 3 1024 rad for a 2-s data accumulation.The accuracy of 1% is limited by the birefringence-modulator calibration. The spatial resolution ofthe bench is 470 mm, and the imaged-field dimen-ions are 12 cm 3 12 cm. The minimum measur-ble signal, which is limited by the presence of thepurious signals induced by misorientations of theptical elements, is 1024 rad. This value shouldmprove by use of better quality optical componentsnd by the finding of a geometric configuration thatould make the presence of the small, flat mirrornnecessary. Although it may seem difficult to ob-ain a much better accuracy, the sensitivity could benhanced by use of a more efficient imaging device,nd it should be possible to reach the sensitivity ofonodetector benches, while keeping all the advan-
ages of the imaging system. This instrument isdapted to the measurement of time-evolving sig-als with frequencies as high as 50 Hz. As there iso constraint on the sample-fixing method, holder-
nduced birefringences can be investigated. The
Fig. 9. Effect of the stress-induced birefringence between the twoceramic transducers that are glued onto a photoelastic modulatorfor different values of applied voltage. The coordinates are inunits of pixels. The gray scales give the birefringence amplitudein radians.
presented instrument’s speed makes it a good can-didate for production-line measurements of largecomponents.
Appendix A: Derivation of the Signal Equations
The mathematical expression for the four signals is
Sp 5 ^Cp~t!@It,dc 1 It,D~t!#&
5 Ii$14 1 2@D cos~2g! 1 D0 cos~2g0!#Kp~c0, f!
1 2@D sin~2g! 1 D0 sin~2g0!#Qp~c0, f!%, (A1)
here
Kp~c0, w! 5 (n51
1`
J2n~c0!sin~npy2!
2np
3 cos@n~pp 1 2w!#, (A2)
Qp~c0, w! 5 (n50
1`
J2n11~c0!sin@~2n 1 1!~py4!#
~2n 1 1!p
3 sinF~2n 1 1!Spp
21 wDG (A3)
It is easy to show that
K0 5 K2 5 (n51
1`
J2n~c0!sin~npy2!
2npcos~2nf!, (A4)
K1 5 K3 5 (n51
1`
J2n~c0!sin~npy2!
2np~21!n cos~2nf!, (A5)
Q0 5 2Q2 5 (n50
1`
J2n11~c0!sin@~2n 1 1!py4#
~2n 1 1!p
3 sin@~2n 1 1!f#, (A6)
Q1 5 2Q3 5 (n50
1`
J2n11~c0!sin@~2n 1 1!py4#
~2n 1 1!p~21!n
3 cos@~2n 1 1!f#. (A7)
Because sin~npy2! 5 0 for n even, we have K0 5 K2 52K1 5 2K3 so that, by inserting Eqs. ~A4!–~A7! intoEq. ~A1!, we can write the four signals as
S0 5 Ii$14 1 2@D cos~2g! 1 D0 cos~2g0!#K0~c0, f!
1 2@D sin~2g! 1 D0 sin~2g0!#Q0~c0, f!%, (A8)
S1 5 Ii$14 2 2@D cos~2g! 1 D0 cos~2g0!#K0~c0, f!
1 2@D sin~2g! 1 D0 sin~2g0!#Q1~c0, f!%, (A9)
S2 5 Ii$14 1 2@D cos~2g! 1 D0 cos~2g0!#K0~c0, f!
2 2@D sin~2g! 1 D0 sin~2g0!#Q0~c0, f!%, (A10)
S3 5 Ii$14 2 2@D cos~2g! 1 D0 cos~2g0!#K0~c0, f!
2 2@D sin~2g! 1 D0 sin~2g0!#Q1~c0, f!%. (A11)
1 August 2000 y Vol. 39, No. 22 y APPLIED OPTICS 3873
Ezs
few
ct
3
Those four signals are all we need to extract D and g:
Ii 5 S0 1 S1 1 S2 1 S3, (A12)
D sin~2g! 1 D0 sin~2g0! 5S0 1 S1 2 S2 2 S3
4Ii@Q0~c0, f! 1 Q1~c0, f!#,
(A13)
D cos~2g! 1 D0 cos~2g0! 5S0 2 S1 1 S2 2 S3
8IiK0~c0, f!. (A14)
This calculation has to be carried for each pixel. Toavoid computation problems, which correspond to acircular permutation of the four signals, as w ap-proaches py2, we apply a corrective phase offset toCp~t! to make f as close as possible to zero. Whenthis is done, we have Q0 ' 0, so S0 ' S2. This canbe easily checked by pixel-by-pixel subtraction ofthose two images. When f 5 0, we have
K0~c0! 5 (n51
1`
J2n~c0!sin~npy2!
2np5
1p (
n50
1`
~21!nJ2~2n11!~c0!
2~2n 1 1!,
(A15)
Q1~c0! 5 (n50
1`
J2n11~c0!sin@~2n 1 1!py4#
~2n 1 1!p~21!n. (A16)
xperimentally, one cannot obtain f exactly equal toero over the whole field, but in the case in which f ismall, typically a few degrees, we can expand Q0 to
the first order in f and keep the first terms in thedevelopment:
Q0~c0, f ,, 1! < fÎ22p (
n50
N
~21!nJ2n11~c0!. (A17)
Then we write
Q0~c0, f ,, 1! < fQ09~c0!. (A18)
This development holds as long as 2fn ,, 1, and for5 5° it is valid to as high as N 5 6, which is
ssentially enough to obtain convergence of the sumhen c0 , 2p. Using expression ~A18! allows Eq.
~A13! to be corrected to the first order in w, and w
874 APPLIED OPTICS y Vol. 39, No. 22 y 1 August 2000
an be determined. The final set of expression ishus
f <S0 2 S2
S1 2 S3
Q1~c0!
Q09~c0!, (A19)
D sin~2g! 1 D0 sin~2g0! <S0 1 S1 2 S2 2 S3
4Ii@Q1~c0! 1 fQ09~c0!#,
(A20)
D cos~2g! 1 D0 cos~2g0! <S0 2 S1 1 S2 2 S3
8IiK0~c0!. (A21)
References and Notes1. J. Badoz, M. Billardon, J. C. Canit, and M. F. Russel, “Sensi-
tive devices to determine the state and degree of polarization ofa light beam using a birefringence modulator,” J. Opt. 8, 373–384 ~1977!.
2. B. Wang and T. C. Oakberg, “A new instrument for measuringboth the magnitude and angle of low level linear birefrin-gence,” Rev. Sci. Instrum. 70, 3847–3854 ~1999!.
3. Y. Zhu, T. Koyama, T. Takada, Y. Murooka, and T. Otsuka,“Two-dimensional measurement technique for birefringencevector distributions: data processing and experimental veri-fication,” Appl. Opt. 38, 2216–2224 ~1999!.
4. Y. Zhu, T. Koyama, T. Takada, and Y. Murooka, “Two-dimensional measurement technique for birefringence vectordistributions: measurement principle,” Appl. Opt. 38, 2225–2231 ~1999!.
5. E. Collett, Polarized Light ~Marcel Dekker, New York, 1993!,Chap. 10.
6. A. C. Boccara, F. Charbonnier, D. Fournier, and P. Gleyzes,“Method and device for multichannel analog detection,” Frenchpatent FR 90.08255 ~5 June 1990! and international exten-sions 266 C US 3221 ~December 1993!.
7. P. Gleyzes, F. Guernet, and A. C. Boccara, “Profilometrie pi-cometrique. II. L’approche multi-detecteur et la detectionsynchrone multiplexee,” J. Opt. ~Paris! 26, 251–265 ~1995!.
8. P. Gleyzes, V. Loriette, H. Saint-Jalmes, and A. C. Boccara,“Roughness measurements in the picometric range using a po-larization interferometer and a multichannel lock-in detectiontechnique,” Int. J. Mach. Tools Manufact. 38, 715–717 ~1998!.
9. A. Dubois, A. C. Boccara, and M. Lebec, “Real-time reflectivityand topography of depth-resolved microscopic surfaces,” Opt.Lett. 24, 309–311 ~1999!.
10. W. J. Smith, “Image formation: geometrical and physical op-tics,” in Handbook of Optics, 1st ed., W. G. Driscoll and W.Vaughan, eds. ~McGraw-Hill, New York, 1978!, Chap. 2, pp.40–41. This reference is not included in the second edition ofthe handbook.
