July 1968

JOURNAL OF THE OPTICAL SOCIETY OF AMERICA VOLUME 58, NUMBER 7

Intensity Method for Stress-Optical Measurements

A. J. MICHAELDepartment of Physics, Indian Institute of Science, Bangalore 12, India

(Received 11 January 1968)

The formalisms of the Mueller calculus and the PoincarE sphere are applied to the photoelastic problemand a general intensity method for measurement of stress-optical constants is derived. This general methodovercomes certain limitations of some of the existing methods and hence is about ten times more sensitive.The power of the new method is indicated by utilizing the frozen stress in a specimen for dispersion measure-ments without externally stressing it. We have measured the stress-optical dispersion of fused silica andcalcium fluoride by this technique, over the range from 250 to 580 my".INDEX HEADINGS: Photoelasticity; Polarization; Dispersion.

INTENSITY methods for the determination of photo-elastic constants have been adopted successfully by

Brown and Hickson,' Poindexter,2 Giardini,3 andRobertson4, though Brown and Hickson and Robertsonwere concerned with engineering photoelasticity. Allof these photometric methods are of the integratingtype in the sense that the retardances over a finitesurface area of the specimen are averaged out. I havedevised a general photoelectric photometric methodwhich is not of the integrating type. Retardance acrossa specimen section as thin as 0.02 mm and 0.5 to 1 mmlong is measured directly in degrees. The method is sosensitive that measurements can be made at very lowstress levels.

THEORY

Mueller-Calculus Method

When matrix calculus5-7 is applied to the photo-elastic problem, the principle of a general intensitymethod emerges in a simple manner. The optical

I A. F. C. Brown and V. M. Hickson, Brit. J. Appl. Phys. 1, 39(1950).

2 E. Poindexter, Am. Mineralogist 40, 1032 (1955).3 A. A. Giardini, J. Opt. Soc. Am. 47, 726 (1957).4G. Robertson, Brit. J. Appl. Phys. 16, 207 (1965).M. J. Walker, Am. J. Phys. 22, 170 (1954).

6 W. H. McMaster, Rev. Mod. Phys. 33, 8 (1961).7 W. A. Shurcliff, Polarized Light (Harvard Univ. Press,

Cambridge, Mass., 1962).

arrangement to which the calculus is applied consistsof a polarizer and an analyzer with azimuths set at±450 to the horizontal (which is taken as zero azimuth);a specimen is placed between them in such a way thatthe light beam is horizontal and normal to one of thefaces of the specimen and the stress is vertical. Onbeing stressed, the specimen becomes a retarder (usuallylinear).8

When the Stokes vector of the output of the polarizer(which is normalized9 ) is multiplied by the properMueller matrices, in order, the Stokes vector of thefinal beam is obtained,

1+sin2a cos.6 cos'a

2 sin2a

0 -

where a is the azimuth of the analyzer and +450 thatof the polarizer and a is the retardance of the specimendue to a vertical stress. The fast eigenvector of theretarder (stressed specimen) is assumed to be hori-zontal,10 i.e., the specimen behaves as a positive uniaxial

8 The cases of the specimen becoming biaxial and of showing atilt of the axes are not treated here.

I If the intensity of the unpolarized beam is set equal to unity,an additional factor of 2 will appear in the final Stokes vector.

10 For the fast eigenvector vertical, there will be a change of sign.

JULY 1968

889

A. J. MICHAFV.

135 180oC IN DEGREES

FIG. 1. Variation of intensity with azimuth a of the analyzerand retardancea due to the specimen. o-o-o, 5 zero; A-A-A,8 small; *-e-*, S large.

crystal. This Stokes vector will be

l+sinf2 a cosb o

2 1

if the analyzer is fixed at azimuth ±450 and thepolarizer set at variable value a, showing that theintensity is the same as before. It varies periodicallywith a and a (Fig. 1).

METHOD

Several cases arise. When the specimen is stressed,we get the maximum and the minimum intensities,Imax and Imin,

I ma= (l+cos8)/2,

Cain= (1-cosS)/2.

(2) Filon's banded-spectrum method"' and all of itsvariations make use of the location of intensity minimaover a range of wavelengths which correspond to posi-tions of 2a= 2700 and 3 = 0 or even multiplies of 27r(crossed positions and integral orders of band) or of2a=900 and 5=odd multiples of 7r (parallel positionand half-integral orders of band).

(3) The photometric methods of Poindexter2 andGiardini 3 are also variations of the banded-spectrummethod. A wavelength is first chosen and the stress isvaried until the intensity becomes zero in the parallelposition of the polarizer and analyzer at the wavelength.

GEOMETRICAL REPRESENTATION ONTHE POINCARE SPHERE12"13

In the simple arrangement of crossed polarizer andanalyzer for extinction, the initial and final beams withrespect to the specimen are represented by point A(Fig. 2) on the Poincar6 sphere' since, for a band ofintegral order, the sphere is rotated about OH or OVthrough 27r to arrive at the point representing the final

(1)

(2)

The ratio Imax/Imin can be used to evaluate &.At azimuth a of the analyzer or polarizer, the ratio

of intensities when the specimen is stressed and notstressed gives

(3)

When the specimen is stressed, the ratio of intensitiesat two azimuths ai and a2 is given by

12/11= (1+sin2a 2 cos6)/(1+sin2a, cos5), (4)

where a is variable over a wide range, permitting use ofEqs. (3) and (4). Equation (4) was adopted in order toavoid changing the stress on the specimen duringmeasurement. Further, Eqs. (1) and (2) are useful onlywhen S is sufficiently large.

SPECIAL CASES

(1) Robertson's method4 corresponds to

(Imax -Imin)/(Imax+Imin) = cOs6, (5)

which is identical with Robertson's equation,

(w1c- I2) Ifr= cosm,

which was derived in a diff erent manner.

FIG. 2. Stress-optical behavior of a specimen under stress,shown on Poincar6 sphere. With A representing the initial beam,A or B may represent the final beam, depending on whether thereis an initial retardance of zero of 7r and for a setting at extinction.The final beam is represented by D or C, respectively, in thosetwo cases when the specimen introduces a retardance S. Rotationof the polarizer leads to motion of the initial point A along theequatorial circle to new positions like A', for example, with corre-sponding motion of B, C, and D to B', C', and D' respectively. (O' isthe center of the section A'D'B'C'.) So the final point moves simul-taneously along a circle inclined at a to the equator. Alternatively,in half cycles of rotation of the polarizer, the section through thegreat circle ALBR shrinks and expands between the limitingpoints V and H which represent vertical and horizontal linearpolarizations, respectively.

11 L. N. G. Filon and F. C. Harris, Proc. Roy. Soc. (London)A130, 410 (1931).

12 H. G. Jerrard, J. Opt. Soc. Am. 44, 634 (1954), gives a de-tailed discussion of the use of Poincar6 sphere in treating thepassage of light through birefringent and optically active media.

13 R. Plechata, Acta Tech. (Prague) 2, 230 (1957), gives ageometrical description of birefringence induced by stress on thePoincar6 sphere. This paper was not available to the author.

Vol. 58

11/12 = (I + sin2a cos5)/ (I + sin2a) -

METHOD FOR STRESS-OPTICAL MEASUREMENTS

beam.' 4 (O is the center of the sphere and H and V arepoints representing the horizontal and the verticalpolarizations, respectively). With parallel polarizerand analyzer, the final point is B, for extinction.

Let the initial point be A and let the system be inthe crossed position; when the specimen is unstressed(so that the final point is also A) the detector shows zerointensity. When the specimen is stressed, a retardance a

appears and the final point shifts, for example, to D,which represents the new polarization form.'5 (If thesystem is in the parallel position, B shifts to C.) Theazimuth of the major axis of the ellipse at D is 450since all polarization forms on the half LAR of the greatcircle ALBR have the same azimuth. Since the analyzeris set in the crossed position relative to the polarizer,the minor axis of the ellipse at D is parallel to the majoreigenvector (transmission axis) of the analyzer.

Let I be the intensity transmitted in this position.Since the maximum intensity transmitted when thespecimen is unstressed is taken as unity, the ratio of Ito this intensity gives

I= b2/a2+b2, (6)

where a and b are the semi-major and semi-minor axesof the ellipse, respectively. When the ellipticity

tan16/2 1= b/a,

is substituted in Eq. (6), a and b are eliminated and

I= (1-cos6)/2, (7)which is Eq. (2).

1. Rotation of Analyzer

I varies cyclically with a. Figure 3 shows the dis-position of the ellipse with respect to the major eigen-vector of the analyzer. The amplitudes transmitted' 6

area cos(45 0 -a), b cos(45-+a)

14 In the Mueller-calculus derivation of the Stokes vector of thefinal beam it has been assumed that the fast eigenvector of the(stressed) specimen-retarder is horizontal, so that the axis ofrotation of the Poincar6 sphere is OH.

15 We follow the modern terminology for description of polarizedlight [Shurcliff (Ref. 7)]. For example, linear polarization has aninfinite variety of forms, depending on azimuth, and ellipticalpolarization has an infinite variety of forms depending on azimuth,ellipticity, and handedness (which, of course, can be either left orright only). Circular polarization has only two forms, left andright circular. The poles L and R (Fig. 2) of the Poincare sphererepresent the circular-polarization forms. Points on the equatorrepresent linear-polarization forms. All other points on the sphererepresent elliptical-polarization forms.

1' Though the ellipse here is a composition of two equal andperpendicular vibrations differing in phase by 6, it is treated as acomposition of two unequal (a and b) perpendicular linear vibra-tions differing in phase by 7r/2 . Hence, if components of the vibra-tions a and b are taken along the major eigenvector of the analyzer,the resultant vector is

A sin(wt+e) =A, sinwt+A 2 sin (t+7r/2),

so that the ratio of the intensities as described in the text is

I= (A '+A 22)/aa+b.

and

1=a2 cos2 (45 0-a)+b2 cos2(450 +a)

When a and b are eliminated as before, we get

1= (1+sin2a cosa)/2, (9)

which compares with the intensity part of the Stokesvector described above.

2. Rotation of Polarizer

While the ellipse is fixed for a rotating analyzer, itgoes on changing for a rotating polarizer. Resolutionof the vector and calculation of intensity follow thesame method as before. When the azimuth of thepolarizer is rotated through 0, A shifts to A' (Fig. 2),where the angle AOA' = 20. The retardance 6 is un-affected, so that the Poincar6 sphere is rotated about OHthrough the same angle 8 and D shifts to D'. Similarly,B and C shift to B' and C'. In a half cycle of a, thesection of the sphere passing through the great circleALBR alternately shrinks and expands between limitingpoints V and H. Alternatively, as the initial point movesalong the equatorial circle, the final point moves alonga circle inclined to the equator at 8, the sense of motionbeing the same if the final point (D) is on the half LARand opposite if the final point (C) is on the orthogonalhalf LBR.

If 8 is measured differentially, relative to a standardgiving a known retardance, the above description re-mains the same except that there will be a point inter-mediate between A and D (not shown in Fig. 2) whichalso moves on a circle of its own.

VERTICAL

MAJOR EIGENVECTOROF ANALYZER

45/

HO-'IRIZONTAL

S

FIG. 3. Relative dispositions of the elliptical vibration emergingfrom the stressed specimen and the major eigenvector of theanalyzer.

July 1968

A. J. MICHAEL

EXPERIMENTAL ARRANGEMENT

The entire system is made of quartz, because dis-persion has to be measured in the ultraviolet also. Anintermediate quartz spectroscope (type E 486 of AdamHilger, London) gives a line spectrum, the source beinga quartz mercury arc. Wollaston double-image prismsare used as polarizer and analyzer, one of the beams fromeach prism being rejected by a suitable shutter. Amirror system turns the vertical image of the slit atthe source end into the horizontal position on the slit ofthe spectroscope, so that a horizontal section of thespecimen is selected for observation. 1 7

The light beam is interrupted by a mechanicalchopper at 400 cps, immediately before entering thespectroscope. An IP28 multiplier phototube is mountedon a precision micrometer drive in such a way that a slitin front of the tube is in the spectral plane. Movementof the beam away from the photocathode, caused bydispersion, is overcome by placing two small strips ofplane mirror at a suitable angle within the metal shieldof the multiplier tube and ihrmediately behind the slit.Reflections in the mirror system guide the beam ontothe photocathode when the multiplier tube is removedto extreme positions in the spectrum. This system isfound to work satisfactorily over a wavelength range of250-600 myA.

The electronic circuitry consists of a preamplifiera Twin-T amplifier turned to 400 cps, a phase-inverterdetector, and a millivolt chart recorder. Provision isalso made for simultaneous projection of the output ofthe Twin-T amplifier on the screen of an oscilloscopeto ensure that the signal is being handled in the linearrange of the amplifier system.

The conventional lever type of stressing is used forcalibration, at a single wavelength. All possible pre-cautions are taken to ensure uniformity of stress withinthe specimen. Retardance is measured for small weightscalibrated correct to 0.1 g, using a differential method sothat the effect of inaccurate knowledge of the weight ofthe lever is eliminated. Measurements at all other wave-lengths are made by placing the specimen in a screwpress in which the stress can be adjusted to any desiredvalue by re-establishing the corresponding retardanceat the calibration wavelength. Uniformity of stress canbe achieved easily in the press.

SOURCES OF ERROR

The following are the sources of error in the systemdescribed. (1) On account of divergence of the orthog-onal beams inside the Wollaston prisms, the emergentbeams show a small ellipticity, which can be detected

17 A horizontal section is preferred because uniformity of stressin a horizontal section is achieved much more easily than in avertical section, as can be judged by inspection between crossedpolarizers. However, this distinction becomes unimportant whena very short and thin section is chosen for observation, as is thecase in the investigation reported here.

because the equipment is quite sensitive. (2) Themultiplier photocathode is sensitive to the azimuth ofthe light polarization. (3) The crystalline-quartz lensbeyond the analyzer rotates the azimuth of the finallight vector to different extents at different wavelengths.(4) Polarizer and analyzer settings may be inexact onthe circular scales. (5) The image on the slit of thespectroscope moves to some extent when the analyzer(a Wollaston prism) is rotated. (6) The output of themercury arc is dependent on the line voltage. (7) Theresidual stress in the specimen may not be uniform.

(1) The error arising from the ellipticity introducedby the Wollaston prisms can be eliminated in thefollowing manner. We can suppose that a virtualretarder in the system introduces a retardance (whichis fixed for the combination of the two Wallastonprisms) and converts the linear beam into an ellipticalbeam. Equations (1) and (2) or Eq. (4) can be used toevaluate this retardance, when the specimen is notstressed. Alternatively, the error can be eliminated bysubtraction, if 8 is measured differentially, eitherusing different stresses on the specimen or using astandard whose retardance (including the effect of theWollaston prisms) is separately measured. The effect oferrors (2) and (3) can be eliminated either by keepingthe azimuth of the final beam fixed or in some mannereliminating the sensitivity of the photocathode toazimuth. If the Wollaston polarizer were rotated,while the analyzer is kept fixed, the initial beam wouldshift relative to the specimen so that the same sectionof the specimen could not be used for measurement atall settings of the polarizer. Further, at certain azimuths,the beam would move partially out of the specimen.So the first of the alternatives mentioned was notpossible, since the polarizer could not be rotated forthese reasons and since the crystalline-quartz lensbeyond the analyzer rotated the azimuth of the finalbeam. So, thin mica films of arbitrary thicknesses andorientation of axes were fixed to the windows of thechopper which made the polarization of the beam inci-dent on the photocathode elliptical; the resultingazimuth and ellipticity changed at the rate of 400 cps.This was found to work quite satisfactorily. (4)Mueller's calculation shows that the errors of setting ofpolarizer and analyzer cancel out in the crossed aridparallel positions. In other positions, the contribution

1 The precise method is to treat the Wollaston prisms aselliptical polarizers whose Mueller matrices carry the informationregarding the ellipticities and azimuths of the eigenvectors. If weassume that these eigenvectors have the same ellipticity, itshould be possible to extinguish the beam in the crossed position.(There is only one beam, since one of the beams from each prismis rejected in the experimental arrangement). However, this is notso in the experiment. Therefore, the ellipticities of the two eigen-vectors are different, so that a residual intensity is transmitted inthe crossed position. To a first approximation, we can assumethat one of the prisms is a perfect linear polarizer, (for example,the first prism), and that the ellipticity of the final beam isentirely due to the other prism. Mueller calculation shows thatthis approximation gives exactly the same result as the simpleassumption that the effect is due to a virtual retarder.

Vol. 58

METHOD FOR STRESS-OPTICAL MEASUREMENTS

-4.7C

4.50 -

-4.30

-4.10-

-3.90 -

-3.70-

-3.50

220 260 300 340 380 420 460 500 540 580 620 660A? mp

FIG. 4. Dispersion of the stress-optical coefficient Cx of fusedquartz with wavelength X. *- values reported by R. S. Krishnanet al.'9 o- values obtained by the present intensity method.

of this error to the final result depends on the magnitudeof the error. In the measurements reported here,azimuths could in principle be set correct to 0.05 deg,the effect of which would be negligible in the finalresult. In practice, however, this error may be muchgreater, owing to errors of judgment. (5) Movement ofthe image can be completely eliminated by using apolarizer or analyzer that gives a single undeviatedbeam. As this was not possible in the present apparatus,rotation of the analyzer was limited to nearly equalazimuths, in pairs, so that the shift of the image betweenthe two settings was very small. (6) An electronicallystabilized supply capable of giving 3 to 4 A at about200 V would effectively remove this error. As such asupply was not available, we had to depend on thestability of the mains. Measurements were made onlywhen the line voltage was satisfactorily steady. Asmeasurements are very rapid, even short periods ofsteady voltage can be used. In spite of this precaution,the major error in the present measurements is due tothis. (7) The specimen of CaF2 used in this investiga-tion showed a residual stress which was quite uniformand along the [001] direction. An unevenly strainedspecimen has to be rejected unless there is interest inmapping the strain field. The total error due to thesources does not exceed 1%.

RESULTS

A fused-quartz specimen, commercially known asOG-type, obtained from the Thermal Syndicate Ltd.,England, the photoelastic dispersion of which wasstudied by Krishnan et al.'9 was used for an evaluation ofthe method. The results are shown in Fig. 4. Krishnanet al.& had employed stresses of 60 to 130 kg, producingretardances of 1800 to 3600 in Filon's banded-spectrumtechnique." I have duplicated their results on the samespecimen, applying about 5 kg, which20 produced aretardance of about 150 at 546.1 mu.

'9 R. S. Krishnan et al., Nature 181, 692 (1958).20 A stress of 1 kg or less is sufficient for measurement with a

differential method producing retardances of about 30 in the testspecimen and 100 to 200 in a reference specimen.

TABLE I. Relative stress-optical constant (qil-ql2)of CaF2 in Brewsters.

Source (qt-q,2) X in my

Pockels"l -1.45 589.3Ramachandran2 2 -1.393 589.3Iyengar and Bansigir2" -1.478 ...Krishna Murty2 4 -1.47 546.1This investigation (temp. 260 C) -1.44 546.1

The path retardance is

R= X(3/360) Cxf(VV/b),

where X is wavelength in cm, Cx the stress-opticalcoefficient, W the load in dynes, and b the lateraldimension of the specimen normal to both stress direc-tion and light beam.

A specimen of CaF2 , 25X4.682X4.5 mm, was usedfor dispersion measurement of (qiu-q12), the differentialstress-optic constant; the 4.682-mm edge, parallel to[010], was the parameter b in Eq. (10) and the 25-mmedge, parallel to [001] was vertical. Results of previousinvestigations2'-2 4 are shown in Table I for comparison.Dispersion of Co in the ultraviolet, which was studiedfor the first time, is shown in Fig. 5. Refractive indices25

were corrected for temperature.

RESIDUAL STRESS IN PHOTOELASTICMEASUREMENTS

Strained specimens can now be used for photoelasticmeasurement, provided that the strain is uniform. Thisis possible because the residual stress can be preciselycorrected by the method described in this paper. Evenwhen a specimen is apparently strain free, there may be

i2n

-2.50

.2.40- \

.2.30-

.220-

,, .., I360 400 440

7\ mu480 520 560 600

FIG. 5. Dispersion of the stress-optical coefficientCx of CaF2 with wavelength.

21 F. Pockels, Lehrbuch der Kristalloptik (B. G. Teubner,Leipzig and Berlin, 1906).

22 G. N. Ramachandran, Proc. Indian Acad. Sci. A27, 421 (1948).23 K. S. Iyengar and K. G. Bansigir, Current Sci. (India) 27,

436 (1958).04 V. G. Krishna Murty, Ph.D. thesis (Osmania University,

Hyderabad, India, 1964).25 Alexander Smakula, Physical Properties of Optical Crystals

with Special Reference to Infrared (Office of Technical Services,U. S. Department of Commerce, 1952).

July 1968 893

(10)

24 -0 280 320

A. J. MICHAEL

a very small strain which is not detectable by theprevious methods. By adopting a differential techniquewith the present intensity method, we can evaluateresidual stresses that give retardances as small as 0.5deg, with 80% accuracy, 1 deg with 90% accuracy, and2 deg with more than 98% accuracy. The residualretardance measured in the CaF2 specimen used was3.8 deg at 546.1 mgs using a KBr plate as reference,which gave a retardance of ,20° under compression.As the combined retardance fell to -16° it was alsoinferred that the internal stress was compressive.28 Itsmagnitude was estimated as -1.3 kg or -5.6 kg/cm2

(the case of retardance being 360n 0 + 3.80, msz=positiveinteger, being most improbable). Because this residualstrain was along the [001] direction it was used alsofor dispersion measurement without stressing the speci-men externally. Values thus obtained agreed quite satis-factorily with those obtained by direct measurement.

CONCLUSION

The new method is about ten times more sensitivethan the previous methods, both in direct and indifferential measurements, in the sense that the samemeasurements can be made with retardances only onetenth as great. It is also capable of high precisionwhen the necessary precautions are taken, such as usinga stabilized supply for the light source and using

2G KBr behaves under compression as a positive uniaxial crystalat wavelengths greater than 276 my and as a negative uniaxialcrystal at shorter wavelengths. [R. Sreenivasan, Z. Physik 155,281 (1959)].

polarizers of good quality which give undeviated beams.The ability of the method to detect very small retard-ances opens the way to investigate crystals availablein very smiall dimensions. The general formalism of themethod is applicable to all crystal classes, excludingthose which are optically biaxial.27 This means that thefinal point on the Poincar6 sphere has to be somewhereon the great circle ALBR (Fig. 2) regardless of theparticular symmetry of the crystal and that its locationdepends on the retardance only. By a judicious choice ofthe thickness of a specimen which is not of cubic sym-metry, the final point on the sphere can be brought toany desired position on this circle. Further, the sensi-tivity of the measurement of a is independent of X thewavelength, unlike the banded-spectrum method, withwhich the sensitivity decreases rapidly towards shorterwavelengths, where the higher-order bands crowdtogether.

ACKNOWLEDGMENTS

The author wishes to thank Professor Krishnan forhis kind encouragement and valuable criticisms and theUniversity Grants Commission for the award of aFellowship which covered part of the work reported inthis paper.

27 Application of the Mueller calculus to stress-optical problemas described in this paper is valid for noncubic crystals, providedthat one of the principal axes of the optical indicatrix is parallelto the stress direction and another is parallel to the light beam.This condition can be achieved quite easily in orthorhombiccrystals, for a particular orientation of the specimen, though thesecrystals are biaxial.

D. E. Pearson (Bell Labs) at Color Technical Group meeting,13 October,

894 Vol. 58