Determination of the Optical Verdet Coefficientin Semiconductors and Insulators

Cedric J. Gabriel and Herbert Piller

For many applications in research and development, it is of importance to know accurately the Verdetcoefficients of semiconductors and insulators. The Verdet coefficient is defined as the single pass volumeFaraday rotation per unit magnetic field and unit thickness. By applying the Voigt approximation formultiple internal reflections for coherent and incoherent radiation, one can determine the Verdet co-efficient more accurately. The optical constants of the material must be known. The approximationcan be applied to single and multiple layers of semiconductors or insulators. The advantage of this typeof treatment is that it can be applied to different energy regions in semiconductors and insulators withoutdirectly employing a special microscopic model. Measurements of interband Faraday rotation in silicon

and gallium arsenide, as well as measurements of the free carrier Faraday rotation in gallium arsenideand the changes in amplitude and phase of the measured rotation in a lead sulfide film on a substrate, arediscussed.

Introduction

In the presence of a magnetic field, the plane ofpolarization of light is rotated upon transmissionthrough a substance. The total angular rotation ofthe plane of polarization of the light beam can berepresented as the sum of three angles which are associ-ated with surface-, volume-, and multiple-internal re-flection rotation, respectively.' The angle of rotation0 in a sample for which surface rotation and multiplereflection rotation are neglected is given by 0 = VdH,where V is the Verdet coefficient, d is the thickness, andH is the magnetic field. The development of tech-niques for obtaining high magnetic fields and the in-creased accuracy and precision in the determination ofsmall angles of rotation make it possible to investigateFaraday rotation over a wide absorption region. Inthe strong absorbing region, it is usually correct toapply the formula given above to interpret experi-mental results; but for thin absorbing samples, surfacerotation can be important. In regions of small ab-sorption, surface rotation can be neglected'; however,the expression must be modified in order to account forthe effects of multiple internal reflections.2 For coher-ent radiation, laser radiation, for example, the Faradayrotation shows oscillatory behavior as a function ofwavelength and thickness.', 4 A general treatment ofthis problem was done by Donovan and Medcalf.5 Un-fortunately, for most materials a microscopic theory

The authors are with the U. S. Naval Ordnance Laboratory,Corona, California 91720.

Received 30 December 1966.

does not exist which would enable ca lculation of thenecessary parameters. Only in the ca ses of monovalentmetals and degenerate semiconduct ors can one makeuse of the free electron model which, to a first approxi-mation, describes the spectral vari ation of the opticalconstants. Yet, for some degener ate semiconductors,significant deviations from the t heoretical frequencydependence of the absorption have been reported. 6

In order to solve the proble m more accurately, theVoigt approximation 2 for multiple internal reflectionscan be used in combination with the experimental valuesof the optical constants for the interpretation of themeasured Faraday rotation.7 Also, a more precisemicroscopic model6 can be easily introduced into theanalytical expressions for the measured rotations. Ithas been shown theoretically that, for the free carriercase and for negligible absorption, the rotations calcu-lated according to Donovan and Medcalf and accord-ing to the Voigt approximation agree i n a wide region ofmagnetic field and Faraday rotation.' By using theVoigt approximation, it is also poss ible to derive ex-pressions for the incoherent radiation c ase,7 which hasnot yet been done with the Donova n and Medcalfcalculations. For this calculation, it is a ssumed that theoptical constants n,k satisfy the condition k/n < 0.1.It should be pointed out that, in materials with largerefractive indices, the measured rotation, which is modi-fied by multiple internal reflection, can be quite largecompared with the computed single transmission rota-tion. In germanium, the correction can be as large as26% in the incoherent case and more than 50% in thecoherent case.

Associated with multiple internal reflections and sur-

April 1967 / Vol. 6, No. 4 / APPLIED OPTICS 66

face rotations of coherent and incoherent light are el-lipticity and depolarization, respectively, in the trans-mitted light. The ellipticity and depolarization, pro-duced by multiple internal reflections in an insulator orsemiconductor in the low absorbing region, are muchlarger than the single pass volume ellipticity and de-polarization.' The surface ellipticity is also muchsmaller by many orders of magnitude than that result-ing from multiple internal reflections. For these rea-sons, the ellipticity and depolarization resulting frommultiple reflections plays the essential role. In thecase of incoherent radiation, some discrepancies be-tween theory and experiment have been interpreted byconsidering these effects.9

In this paper, we make a comparison of the Voigtapproximation and the experimental results4 for a thinfilm of lead sulfide. Good agreement is obtained forthe modification in phase and amplitude for a thin film ona substrate. In this way, accurate information on bandparameters and optical constants can be obtained.Measurements of interband Faraday rotation in siliconand gallium arsenide demonstrate the power of themeasurement technique. In addition, measurements onfree carrier Faraday rotation in gallium arsenide are re-ported. The interband and free carrier Faraday rota-tion is interpreted in terms of modern band theory.On the basis of the similarity to the dispersion of theinterband Faraday rotation in other semiconductors,conclusions are drawn with respect to the energy gap insilicon.

Instrumentation

The study of interband Faraday rotation in semicon-ductors requires the transmission of electromagneticenergy through highly absorbing materials. In orderto have sufficient transmitted energy to make themeasurements, often samples must be used havingthicknesses of the order of 1/X, where -q is the absorp-tion coefficient. The rotations induced by such thinsamples can be of the order of minutes. A double beamtechnique based on an apparatus used by Ingersoll'0allows the measurement of small rotations in the wave-length region of 0.3-2.7 .

Briefly, the apparatus consists of a high pressurexenon arc, focusing optics, a 510-c/s light chopper,monochromator, Glan-Thompson polarizer, samplecell, Wollaston prism beam splitter, two photodetec-tors, a bridge circuit, and suitable amplifiers. Thesystem is conventional, through the sample cell, andhas been described elsewhere." The Wollaston prismis oriented so the planes of polarization of the trans-mitted beams are 450 to the plane of polarization of theincident beam in zero magnetic field. The two trans-mitted beams are each focused on a photodetector suchas a PbS photoconductor or a silicon photovoltaic diode,and the system is adjusted so that either the sum or thedifference between the two beam intensities can bemeasured. Phase sensitive detection is used to improvelinearity of the system response. The principal ad-vantage of this system is that fluctuations in sourceintensity introduce only a relative error in the measure-

ment of the rotation, thus allowing measurement ofsmall angles.

In the wavelength region beyond 2.7 ji, a standardsingle beam measuring technique utilizing stacked silverchloride plate polarizers and a gold-doped germaniumdetector cooled to liquid nitrogen temperature wasused."I

Measurement Technique

The instruments described above can be used tomeasure Faraday rotation in two ways: by compensa-tion or by intensity measurements. In the compen-satory method, the instruments are balanced with themagnetic field in forward direction (the balance signalneed not be zero); then the field is reversed and thepolarizer is rotated to obtain the balanced condition.The amount of rotation of the polarizer necessary toachieve the balanced condition is twice the Faradayrotation. Errors are introduced by this method ofmeasurement if the intensity or intensity distribution ofthe light transmitted by the polarizer depends on theorientation of the polarizer. The double beam systemis less sensitive to this error than the single beam system.

The rotation is obtained from intensity measure-ments as follows. For the double beam system, let theintensities of the two beams be

I = I,,_ cos2(a - Py) + Imm in'2(a - )

12 = max sin2(ac - y) + Immi cOs2(a - y)

(1)

(2)

where Imax and Imin are, respectively, the maximum andminimum intensities observed as the analyzer is rotated1800, a is the relative orientation of the polarizer andanalyzer, and y is desired rotation. The rotation y isthen obtained from the following expression:

II -I2(+ ) II(-) -2(-)II(+) + I2(+) I,(-) + 12() 2p sin2a sin2,

(3)

where p = (max - Imin)/(Imax + Imin), the +, - argu-ments of I indicate forward and reversed magneticfield, i.e., + y or - y. Similarly, for the single beammethod, one obtains

I(+) - I(-) _ p sin2a sin2,yI(+) + I(-) 1 + p cos2a cos2y

(4)

Thus, if the intensity measurement technique is usedto obtain 0 from y, corrections for deviations from pre-cise linear polarization must be applied before the cor-rections for surface and multiple reflection rotations canbe made. The angle a must also be known. Usually,the experiment is done with a set to 450 to simplify theapplication of the expressions.

In situations where the errors introduced by inac-curate knowledge of p and a cannot be tolerated, analternate method is available. In this method, in-tensity measurements are made for a series of incre-mented values of a. Let these values be writteno = (p + 1 - 2i)(Aa/2) + 7r/4 - , where i =1, 2, 3, ..., p, and represents the deviation from the de-sired 450 balance condition. From Eqs. (1) and (2),we obtain the relationship

662 APPLIED OPTICS / Vol. 6, No. 4 / April 1967

2(p tanAa - tan pAce)2' tan pace tanAct

(5)

-12where

P2 = E (i -ri),

i= 1

1 = E (p + 1 - 2)(1i + ri),i = 1

(6)

(7)

in which li = I'(+) - I2i(+) and r = I(-) -I2i(-). Equation (5) is valid for both the single beamand double beam instruments. For use with thesingle beam instrument, 12 is set to zero. The quanti-ties r and p do not appear in Eq. (5); however, theprecision of measurement increases as approacheszero and p approaches unity.

Free Carrier Faraday Rotation in LeadSulfide and Gallium Arsenide

There are two examples to be discussed: the firstone is where there are no coherent effects observed be-cause the coherence conditions are not fulfilled. Inter-ference effects can only be observed directly if thewavelength X and the thickness d are well defined, theoptical path length is shorter than the coherencelength, and the dimension A of the emitting element ofthe source and the solid angle of the beam Q fulfill thecondition A sins << X/2. We measured free carrierFaraday rotation in n-type gallium arsenide. For-mer Faraday rotation, reflection, and absorption datagive a range for the gallium arsenide effective mass atthe bottom of the Pic conduction band between 0.046 mne

and 0.072 me.'2 Those discrepancies could be inter-

preted in terms of multiple internal reflection effectsin the measured rotation according to Eq. (AS) ofthe appendix.9 Equation (AS) describes the Faradayrotation for perfectly incoherent radiation. At amagnetic field of 20 kG, the application of Eq. (AS)shows that, as the absorption coefficient changes withwavelength, the correction of the rotation varies be-tween 11% and 15%. The optical constants weretaken from published work.9 Calculations show thatthe effect of depolarization for the intensity measure-ment technique [see Eqs. (3) and (4)] is of the order of afew percent. This contribution was determined bymaking compensation and intensity measurements inthe wavelength region where the double beam Wollas-ton prism technique can be used. (In this case, thepolarizer can be rotated without largely effecting theaccuracy.) The study of the gallium arsenide n-typesamples from ours and various published Faraday rota-tion measurements leads to the conclusion that the ef-fective mass at the bottom of the F,, conduction bandis m*/m = 0.066 0.002 at room temperature. 9

This value is in good agreement with theory' 3 , as wellas with experiments: Vrehen' 4 determined from in-terband magnetooptical absorption the effective massm*/ine = 0.067 At 0.002 at 770K. DeMeis and Paul 5

proposed from free carrier Faraday rotation measure-ments on wedged samples (10 wedge, to reject multipleinternal reflected rays) an effective mass n*/me =

-10

- (

w -8

I-

o

-2

WAVELENGTH X ( )20 25 30

0 100 200 300 400 500 600 700 800 900WAVELENGTH SQUARE X

2(

Fig. 1. Faraday rotation in a 4.45 X 10-4 cm thick, n-typeepitaxial PbS film measured at room temperature in a field of 132kG. The data points and single pass rotation are from Palik et al.4

The observed rotation is calculated using the Voigt approxima-tion. NaCl substrate. A No substrate. --- Single pass

rotation. -Total rotation.

0.064 A4 0.002 for room temperature and atmosphericpressure.

Recently, Palik et al.4 investigated the free carrierFaraday rotation in a thin film semiconductor of leadsulfide. This represents the first measurement of thecoherent multiple internal reflection in the ir region.The rotation was measured for magnetic fields ofA- 132 kOe. With the increasing use of epitaxiallygrown thin film semiconductors, the calculation of theeffects of multiple reflections upon Faraday rotationmeasurements becomes increasingly important be-cause of the low absorption of such films and becausethese effects cannot be eliminated experimentally.They have fitted the data with calculated curves ob-tained by applying the theory of Donovan and Med-calf.5 This theory applies for an unbacked sample,that is, a free film. So far, there has been no pub-lished extension of this theory for a multilayer system.In this case, the effect of a substrate on the measuredrotation is important. The measurements on the leadsulfide film were made over the range 3-19 u with thesample on its backing of NaCl; subsequently, thesample was removed from the backing and measuredfrom 14-30 /.e. The experimental results show that therotation at the fringe maxima has been increased byremoving the backing. The contribution to thenonoscillatory rotation from the backing is negligiblebeyond 7 ,t.4 Figure 1 shows the Faraday rotation in a4.45 X 10-4 cm thick n-type epitaxial PbS film. Thedata points, measured at room temperature in a fieldof 132 kOe, and the single pass rotations are from Paliket al.4 The observed rotation is calculated using theVoigt approximation, Eq. (A5). For the computation,essentially the same data were used as those given byPalik et al. in their paper. To fit the data, somesmall changes were made: the carrier concentrationwas increased from 2 X 1018 cm-' to 2.2 X 1018cm-3 (Palik et al. measured 2.3 X 101s cm- 3 ). The mo-bility was changed from 450 cm2/Vsec to 500 cm2/Vsec.

April 1967 / Vol. 6, No. 4 / APPLIED OPTICS 663

40

30

20

10

0

-10

-20 F

-30 -

-40-

-50-

-60

0.4 0.6 0.8 1.0 1.2 1.4PHOTON ENERGY (eV)

oscillation of the rotation. This calculation shows thatthe rotation is about 30% smaller owing to the NaClsubstrate at the peak at 270 2'. The experimentaldata also show a change of that order.

41..104 !F

Oe-

z2

00L.

0

10

1.6 1.8 2.0

Fig. 2. Faraday rotation in 32-M thick n-type (n = 3.2 X 1016cm-') GaAs. The data were taken with H = 20.0 kOe at3000K. The absorption data from Ref. 27 are for n-type (n =

X 10"1 cm3) GaAs at 300 0 K in zero field. -0- Faradayrotation. - - - Absorption coefficient.

In this way the fit of the calculated rotation curve to theexperimental data was improved, as one can see at thepeak at 650 2. The free carrier model'6 was used tocalculate the refractive index n, where en = 17.1,

* = 0.16 mne. The absorption was calculated accordingto Gibson", taking into account the contribution to theabsorption from lattice vibrations of wavenumber 50cm-'. The extrema are mainly determined by the re-fractive index n, the amplitudes of the rotation by theabsorption.

The agreement between interpretations of the data ofPalik et at. on the basis of the Donovan and Medcalftheory, and ours, on the basis of the Voigt approxima-tion, justifies the use of the Voigt approximation for theeffect of multiple internal reflections in semiconductorsand insulators for coherent and incoherent radiation;this was shown theoretically by Palik and Henvis8 forzero absorption. They observed that the deviationbetween the exact model and the Voigt approximationis of the order of the precision of reading the peaks ofoscillations from the computer results, or errors in thecalculated rotation using the free carrier model. Ineffect, the agreement is good below the equivalent of200 kOe or a first pass rotation of 200 (most experi-mental rotation in thin films is much smaller than10°). The advantages of using the Voigt approxima-tion are that (1) it is possible to use experimental datafor the optical constants, (2) one can simply introducea more precise free carrier model' in the analytical ex-pression for the measured rotation, and (3) an expres-sion for incoherent radiation can be easily derived andcalculated. By introducing the optical constants of theNaCl substrate in Eqs. (A2) and (A5), one can calcu-late the effect of the backing on the amplitude of the

Interband Faraday Rotation inGallium Arsenide and Silicon

The interband Faraday rotation in GaAs was ob-served and discussed by Moss and Walton' 7 , Cardona",Piller", and Thielemann and Rheinlaender. 2 ' Most ofthe prior work had been done on the free carrier Fara-day rotation. The situation is similar to that of sili-con.2 ' An excellent discussion on the subject of inter-band Faraday rotation can be found in the review, "In-terband Magnetooptical Effects", by Lax and Mav-roides.2 2 Theories concerning these effects were givenby Roth23 , Boswarva and Lidiard2 4 , and Halpern etal.2 ' The interband Faraday rotation in intrinsic ma-terial indicates a sign which is opposite to the freecarrier Faraday rotation. Most early interband Fara-day investigations were done in germanium. Similarstructure was observed by Nishina et a.

2 ' in GaSe.Contrary to these measurements, which were done inhigh magnetic fields with high resolution, our measure-ments in the weak field region do not resolve the finestructures in the rotatory dispersion caused by thedirect electronic transitions at the r point. Magneticfields up to 20 kOe were used. The measurementswere made at room temperature. The objective of ourwork was to extend the measurements published be-fore'9 , 2 ' to the high energy region, the region beyondthe band gap. By using the described method with ahigh energy xenon source and highly sensitive rotationdetection methods, we could measure these effects inrelatively low fields.

The GaAs sample is 32 IL thick. The carrier con-centration is 3.2 X 10"1 cm-'. The sample was cutapproximately perpendicular to a (100) axis. Figure 2shows the interband Faraday rotation in GaAs and theabsorption coefficient27 in the region between 0.4 eVand 2 eV. The measurements were made with a Leisssingle monochromator having a resolving power greaterthan or of the order of 20. The spectra observedare similar to the spectrum of the rotary dispersion inGe and GaSe. As in the case of InSb, the observedrotation at the energy gap is mainly determined bytransitions between magnetic Landau levels. TheCoulomb force of the excitons merely perturb thequantized magnetic Landau levels. This effect issmall at high temperatures. Direct transitions athigher bands, at Brillouin zone points such as L and X,are assumed to contribute a positive rotation which isof opposite sign to the contribution from the directenergy gap at r. The line shape observed at the energygap is also partly determined by the background rota-tion which adds to the first singularity caused by thefirst Landau transition. This contribution of thebackground is proportional to (r)-', and seems,therefore, important at small magnetic fields.2 ' Forthe determination of the sign of the interband rotation

664 APPLIED OPTICS / Vol. 6, No. 4 / April 1967

II.1III

III

I

z

z

0

I-

a:0

below the energy gap, the theory by Roth23 can beapplied. 9 In this way, the effective g value was de-termined: geff = -2.6. The theoretical value is-3.7.23 The experimental value is in good agreementwith theory, considering the fact that there is dampingas well as contributions from transitions with energiesdifferent from the gap energy. The effect of multipleinternal reflections in the highly absorbing region of thissample is negligible. No corrections were made for thesurface rotation' because there is not enough informa-tion available to calculate these effects. Preliminarymeasurements on silicon surfaces show that the con-tribution is of the order of a few percent at the gap.The major singularity of the rotation is at (1.43 0.02) eV which corresponds to the energy gap of 1.43 eVat room temperature.'

Figure 3 shows the Faraday rotation in silicon. Thesingle crystal sample is n-type, the thickness is 39 A, theresistivity is 30 Q cm. The data were taken in a mag-netic field of 20.2 kOe at room temperature. Theabsorption coefficient2 ' is large in the region of meas-urement. In this region, the silicon data are in agree-ment with theory, if one assumes that the Faraday ro-tation is due to direct transitions. The oscillatoryrotation associated with indirect transitions at the in-direct gap around 1.1 eV could not be resolved with theavailable field strength of 20 kOe. The Faraday rota-tions attributed to direct, indirect, and direct forbiddentransitions all show at long wavelengths a wavelengthsquare dependence; therefore, it is difficult to dif-ferentiate between these effects. The decrease of theFaraday rotation at 1.65 eV is difficult to explain interms of the direct transition energy r,5-r 25' of 3.4 eV,because in other semiconductors such as Ge, GaAs,and GaSb the same decrease of rotation associated withdirect transitions appears at energies 10% to 20%smaller than the direct energy gap. For an energygap of 3.4 eV, the rotatory dispersion in silicon wouldrepresent an exception to the experimentally observedbehavior. We believe that a value of about 2.8 eV,calculated by Herman et al. , would give better agree-ment with the observation in silicon.

Conclusions

It has been shown that the measurement of the Verdetcoefficient in the low absorbing region can be done ac-curately without employing a microscopic model. Inthe case of coherent radiation, one measurement ofFaraday rotation enables one to attain information onthe optical constants and the Faraday rotation. Fromthe Faraday rotation and the optical constants, onecan determine the carrier concentration and the ef-fective mass of the carriers. Therefore, the informa-tion can be used to determine electronic band structurevalues. This technique seems to be a powerful toolfor the investigation of thin films like epitaxial films oncertain substrates. With the advent of laser systems,the Faraday rotator is applied for laser experimentsand devices. Many materials like semiconductors,rare earth, doped glasses, etc., can be used for that

2 4C

0- 3C'3(

° 2'

g 2C3:EI'

IC

IC

0310'

I-2

10 ILIL00z0

0

m'3

Fig. 3. Faraday rotation in 39-g thick n-type (30 Q cm) Si. Thedata were taken with H = 20.2 kOe at 300 0K. The absorptiondata from Ref. 28 are for n-type intrinsic Si at 300 0K in zero

field. -- 0- Faraday rotation. - - - Absorption coefficient.

purpose. By applying the approximate equationsgiven in the report in the optical and ir region, one canaccurately calculate the effect of increased reflectivityat the surfaces to enhance Faraday rotation. The freecarrier Faraday effect in GaAs and PbS are used todemonstrate these techniques. The measurements ofinterband Faraday rotation in gallium arsenide andsilicon were extended into the highly absorbing region.These magnetooptical effects are discussed in terms ofthe electron theory of solids.

AppendixThe following model is used to discuss the effects of

multiple reflections (Voigt approximation2' 7'8 ).

(1) A collimated, linearly polarized light beam isnormally incident upon a plane parallel slab.

(2) The amplitude of the transmitted beam of lightis the sum of the amplitudes of the light beamsproduced by partial internal reflection on thesample surfaces.

(3) The effect of the magnetic field is introducedphenomenologically as a rotation of the plane ofpolarization upon each passage through thesample and upon each internal reflection fromthe surfaces. The ellipticity produced in asingle passage through the sample and surround-ing media is neglected.

Only a brief treatment of this model is given here.After passing through an analyzer rotated by an

angle a with respect to the polarizer, the amplitude of theq times reflected beam is proportional to

Tq = tlt2(-rir2 )e-i( 2q + 1)6 cos[(2q + 1)0 + 2qp + s' - al,

(Al)

April 1967 / Vol. 6, No. 4 / APPLIED OPTICS 665

- I1 °I

o 0.40.6 I

0 0.2 0.4 0.6 0.8 1.0 1.2 I.4 I.6 1.8'PHOTON ENERGY (eV)

5

i

i

where tl, t, r,, and r are the Fresnel coefficients for nor-mal incidence:

V, - V

= N, + N2'N2 -N3

1:.-N2 + V3'

2NI

IV + N ̂

2N2

N2 + IV3

in which N, = n - ilc, N = n - ic2, and N =n3 - ic3 are, respectively, the complex indices of re-fraction of the incidence medium, the sample material,and the transmission medium. The complex phaseshift in the sample material a is given by = 27rN2d/X,where d is the sample thickness and X is the wavelengthin vacuum. The angle 0 is the Faraday rotation for asingle passage; o is the polar Faraday rotation thattakes place upon each internal reflection; and so' is anymagneto rotation that might take place on passagethrough both surfaces.

The quantities 0, , and ' are related to the opticalproperties of the sample as follows:

0 = (rcl/X)An2,

= - 4(Ao' + A2), (A3)XP = _ (A¾i" + A12'),

where An = n(+) -n2(-), the difference betweenthe indices of refraction for right and left circularlypolarized waves in the sample in the presence of alongitudinal magnetic field, tanq/1 = Imr,/Reri,tan , = Imr2/Rer2, tan,,' = Imt,/Rell, tani2' =Imt2/Ret2, and

AO = dAn + E Aki, (A4)

in which An,, An3, Ak, Ak2, and A/k3 are defined similarlyto An2.

In Eq. (Al), the rotation introduced by the sur-rounding media has not been explicitly introduced be-cause this rotation simply adds to the rotation pro-duced by the sample.

The intensity of the emerging beam will, in general,be a periodic function of the analyzer position a. Thevalue of a for which the intensity is a relative maximumis the observed rotation and is designated by y. Theratio of the minimum intensity to the maximum in-tensity gives the ellipticity in the case of complete co-herence and a measure of the depolarization in the caseof complete incoherence.

Completely Coherent Beams

The transmitted amplitude for coherent beams isgiven by the summation of the amplitudes obtainedfrom Eq. (Al). The condition d(TT*)/da = 0 yieldsthe expression

tan2l' = (1 - c) sin28 (A5)(1 + C2) cos2e + cl

where ' = y + f - ', 0 = + so, cl = 2(a cosD +b sin(g) exp(- d), and C2 = (a' + b) exp(-2nd), in

which the quantities a and b are the real and imaginaryparts, respectively, of rr2; and cJ = 4rnd/X, X =47rk/X.

The transmitted beam becomes elliptically polarizedwhen passing through the sample. The ellipticityarising in this model is, of course, not related to the in-trinsic ellipticity, which has been neglected here, butis caused by the multiple reflections. The ellipticity eis defined by e2 = min/Imax, where Imi. and 'max arethe minimum and maximum intensities obtained whena = y + 7r/2 and a = y, respectively. Using straight-forward but tedious algebraic manipulations, the fol-lowing result is obtained:

p = [ + 42 - 22r 2,(1 - 22F(A6)

where p = (1 - E2)/(1 + ).

Completely Incoherent Beams

In the case of incoherent beams, the intensity of thetransmitted beam is given by

(A7)rTT* = E Tqq*.

q =

The condition d(rr*)/da = 0 yields

tan2e = [(1 - c2)/(1 + C2)] tan2r. (A8)

The transmitted beam is not elliptically polarizedbut is partially depolarized. The polarization factor(Imax -Imi)/(Imax + Imin) has the same form as thequantity p used above to describe the ellipticity. Thesame symbol p will be used for the polarization factor.Further algebraic manipulations yield

[i (I + sin'(1 + C2 )2

(A9)

In general, these results for incoherent beams are agood approximation if the conditions ki/ni < 0.1,where i = 1,2,3, are satisfied. Otherwise, the crossterms neglected in Eq. (A7) can be appreciable.

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666 APPLIED OPTICS / Vol. 6, No. 4 / April 1967

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THE OPTICAL SOCIETY OF AMERICA

INCORPORATED

Purpose and Scope

THE OPTICAL SOCIETY OF AMERICA is an organization devoted to the advancement of optics and theservice of all who are interested in any phase of that science-be it fundamental research, teaching, the manu-facture of optical instruments and products, or the application of optical techniques to any of variouspurposes in science and industry.The activities of the Optical Society, its meetings, and the contents of its publications will be found to be ofinterest and service to an extensive and diverse audience-physicists, chemists, biologists, psychologists,ophthalmologists, optometrists, astonomers, spectroscopists, mineralogists. artists, illuminating engineers,manufacturers, and various technologists who are concerned with the application of optical methods. OSAsolicits the support and membership of all persons interested in optics whatever the specific interest maybe.

Applications for Membership for Regular, Corporation, and Student Members

All persons desiring to join the Society or cooperate with it in any way are invited to communicate with theExecutive Secretary.Detailed information concerning the Society, classes of membership and dues, and membership applicationblanks may be obtained from the Executive Secretary:

MARY E. WARGA

OSA Executive Office1155 16th Street NWWashington, D. C. 20036

April 1967 / Vol. 6, No. 4 / APPLIED OPTICS 667