JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

Dichroic Ratio Measurements in the Infrared RegionELLIOT CHARNEY

Department of Chemistry, Columbia University, New York, New York

(Received December 6, 1954)

An analysis of conventional optical arrangements for the measurement of dichroic ratios in the infraredregion reveals that such measurements may be in error by factors up to 3 or more. These errors result fromfailing to account for the effects of prism polarization and of the imperfection of polarization in the trans-mitted beam of multiple plate AgCl polarizers. Elimination of the effect of prism polarization by a methodproposed by Elliot, Ambrose, and Temple is shown to be relatively ineffectual when a transmission polarizeris employed. The magnitude of the error varies with the true dichroic ratio and with the optical density ofthe sample for polarized radiation parallel to one of the optic axes. A frequency dependent parameter isfound to account for relatively isotropic intensity losses resulting from absorption and scattering by theAgCl in the polarizers; observed values of this parameter are given for three wavelengths, 1.67 ,u, 3.85ui,and 4.80p.

T wosources of error have been frequently ignoredin the measurements of dichroic ratios of solids

in the infrared region. One of these errors arises fromthe imperfect polarization of the incident beam result-ing from the use of the transmitted beam from a pile ofsilver chloride plates1 2 inclined at Brewster's angle.The other error is a result of the difference in Fresnelreflection of the components of the radiation incidentparallel and perpendicular to the prism face. Theseerrors have previously been recognized3 -7 but appearoften to be brushed aside as of little significance. Aphenomenological analysis of the effects of both im-perfection of polarization and prism polarization ismade here and it is shown that the errors resulting fromdisregarding these effects can sometimes be quite large.

Consider the most common experimental arrange-ment in which the significant (from the point of viewof this analysis) optical elements in the radiation pathbetween the source and the detector are (1) a trans-mission polarizer consisting of a pile of m plates of silverchloride inclined at the polarizing angle to the axis ofpropagation, (2) a prism set with its triangular basehorizontal, and (3) the sample, which may be placedanywhere in the path, but for simplicity in this analysisis considered to be set between the polarizer and theprism. All of the elements are oriented relative to a setof space-fixed orthogonal axes x, y, z; the z axis is theaxis of propagation of the radiation beam, and the xand y axes are respectively the horizontal and verticalaxes. With respect to these axes, the orientation of thepolarizer may be described in terms of the plane ofincidence for the radiation and the polarizer. Thus thepolarizer may be oriented by rotation about the z axisso that either the x or the y axis lies in the plane ofincidence or it may be oriented at any angle in between.Similarly, the sample orientation about the z axis is

1 Elliot, Ambrose, and Temple, J. Opt. Soc. Am. 38, 212 (1948).2 R. Newman, and R. S. Halford, Rcv. Sci. Inst. 19, 270 (1948).3N. Wright, J. Opt. Soc. Am. 38, 69 (1948).4 R. D. B. Frazer, J. Chem. Phys. 21, 1511 (1953).6H. S. Gutowsky, J. Chem. Phys. 19, 438 (1951).6 W. L._Hyde, J. Opt. Soc. Am. 38, 663 (1948).7 Ivan Simon, J. Opt. Soc. Am. 41, 336 (1951).

describable in terms of the angle the sample-fixed jand k orthogonal axes make with the x or y axis. Finally,the prism is permanently fixed with its triangular basehorizontal, i.e., with the base parallel to the x axis.

Several methods of measuring the dichroic ratio ofsolid samples with this type of optical arrangement arepossible, including the use of split beam or double-beamratio recording in which transmission is measured rela-tive to air. We shall consider three methods of directsingle-beam measurement; the results are identical fordirect split-beam ratio measurements in which thepolarizer is in the path of both portions of the beamand for double-beam ratio measurements in which twopolarizers are employed, one in each beam. Extensionof the analysis to other methods will be obvious. Impliedin this analysis are Stokes' theorems regarding the com-position of natural and partially polarized light. Thesepermit the treatment of the electric vector of naturalor unpolarized radiation by means of independentorthogonal components of equal average amplitude.

1. ABSORPTION MEASUREMENTS WITHSAMPLE ORIENTATION FIXED

Consider an unpolarized beam of effective intensityo, incident on the polarizer. The polarizer transmits

a partially polarized beam of intensity I1. Part of thisbeam is absorbed* by the solid samples. The beamtransmitted by the sample has the intensity I2. Partialreflection of 12 by the prism results in (possibly after

* Reflection by the solid sample can be disregarded when thesurface of the sample is normal to the axis of propagation. Whilesuch reflections do distinguish components of the beam orientedat different angles in the plane of the surface of dichroic samples,even in the extreme case of large dichroic ratio and large absorp-tion coefficient the error which results from ignoring this effect isnegligible; the errors in dichroic ratio are about an order of mag-nitude larger because of failure to consider the imperfections inthe polarizers. In this analysis therefore the reflections are treatedby assuming that they contribute to an isotropic diminution ofthe incident intensity. Similarly, any planar or symmetric elementin the optical path oriented normal to the axis of propagation hasno effect on the dichroic ratio. This is also true, of course, fortotally reflecting elements such as front surfaced metallic mirrors.The effect of beam convergence is not accounted for in this an-alysis because observations indicate that it is small, confirmingan analysis by Frazer.4

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November1955 . DICHROIC RATIO MEASUREMENTS IN REGION

two or more passes through the prism) a transmittedbeam of intensity I3 to be measured by the detector.The additional subscripts x and y on 13 will be used todesignate the polarizer orientation; the subscripts xand y indicate that the plane of incidence is parallel tothe x and y axes, respectively.

Analysis of the beam transmitted by the polarizer:If Im is the transmission coefficient of the polarizer forthe perpendicular component of the radiation, the beamIi consists of two parts I1' and I1" where

I,' = tj o/2 (1)is the intensity of the component of I1 transmittedperpendicular to the plane of incidence and

I1" = Io/2 (2)

is the intensity of the component transmitted by thepolarizer parallel to the plane of incidence.

The transmission coefficient t for a pile of dielectricplates oriented at Brewster's angle represents the im-perfection of polarization of the transmitted beam. It isa variable coefficient dependent on the number ofplates. It is defined as the fraction of the component ofthe incident beam parallel to the plane of incidencewhich is transmitted by the polarizer. It may be cal-

TABLE I. Transmission coefficients for silver chloride polarizers.

m 1 2 3 4 5 6 7 8 9 10

tm 0.501 0.294 0.190 0.128 0.090 0.067 0.050 0.040 0.035 0.031

culated from the known index of refraction and Fresnel'sequations using the relation given by Tuckerman.5 Wehave found, however, that while Tuckerman's relationrepresents the transmission coefficient better than asimple exponential relation, it does not accuratelyrepresent the observed coefficient corrected for absorp-tion and scattering. Part of the difference undoubtedlyresults from convergence of the beam and part fromvariation in thickness of the silver chloride plates.Values of the transmission coefficient calculated frommeasurements made in this laboratory using polarizersconstructed from clean plates of silver chloride 0.264-0.02 mm thick (Harshaw Chemical Company) aregiven in Table I.

Analysis of the beam transmitted by the sample:The expressions for the intensities of the componentstransmitted by the solid sample serve to define thefractions of the radiation absorbed and transmitted.Thus,

orI21 = I11= flk o/2,

12' =#$,' =jtjol2,

(3a)

Define a = -Aj and ak = 1-iflk as the fractions of radia-tion absorbed by the fixed sample from the componentsof the beam oscillating parallel to the sample-fixed j andk axes, respectively. In this part of the analysis the jand k axes are fixed parallel to the space-fixed x and yaxes, respectively. Equations (3a) and (4b) give theintensities of the components of the beam for thepolarizer orientation for which the plane of incidenceis parallel to the x axis. Equations (3b) and (4a) aresimilar expressions for the polarizer orientation forwhich the plane of incidence is parallel to the y axis.

Analysis of the beam transmitted by the prism:Without detailing the action of the prism at this point,it is sufficient to note that the propagation axis of theradiation beam makes an angle other than 900 withboth prism faces. Consequently the prism acts as apolarizer. Unless the angle which the prism faces makewith this axis is Brewster's angle, two transmissioncoefficients with values less than one are required todefine the loss of radiation by reflection from the prism-air interfaces. Consolidating all such reflection coeffi-cients parallel to each axis, we define the coefficientst. and t, as the fractions of the components of the radia-tion transmitted by the prism in the x and y directions.Thus the component intensities of 13 are given by:

I"3/= = t22 "= t1,,jlo/2,or

(5a)

13'= tz,12' = tykI o/2, (Sb)

(6a)or

13 I = tI2' = tsytm jI 0/2, (6b)so that

13.=tl2"+tyI2'=Io/2- (txoj+ttj~k), (7)

I 3y = tzI2" = t1I2' = I o/2 (tyank+ tj) (8)

depending on the orientation of the polarizer. The caseof polarizer orientations other than parallel and per-pendicular to the prism base will be considered below.

2. ABSORPTION MEASUREMENTS WITHPOLARIZER ORIENTATION FIXED

The analysis of this case is straightforward and en-tirely analogous to the previous one. There are twopossible polarizer orientations; considering only theorientation for which the plane of incidence is parallelto the y axis, the intensities of the beam transmittedby the prism are

I3j=Io/2 - ( j+ t-tm)I3k = o/2 X

(9)

(10)

where the subscripts j and k on 13 are used to designate(3b) the sample orientation with respect to the space-fixedIA-\ y axis.

3. CALCULATION OF THE DICHROIC RATIOSI_2_ _ =k1 =J tmjo/2.

8 L. B. Tuckerman, J. Opt. Soc. Am. 37, 818 (1947).(4b) The calculation of the optical densities in each of the

configurations considered requires a measurement of

981

12" =flili" =,6j1'o12,

ka)

ELLIOT CHARNEY

intensity incident on the sample. The measurement ismade by removing the sample, which corresponds tosetting j=0k=1; that is, the absorptions aj=ak=O-The corresponding intensities are:

IX°=1o/2- (,+Itt(113v0=1o/2* (t,+txtm), (12)1, .0= Io/2 (,+ t 1), (13)

Ik0=0o/2* (t,+tztm). (14)

Note that these differ from the true incident intensitiesby the inclusion of the prism transmission coefficientst., and t, neglecting symmetric reflections which do notdisturb the result. The dichroic ratios calculated fromthe measured intensities are therefore:

log[I3y/I3y] log[(t+txtm)/ (tzlj+tmlBk)

log[I3j0 /I3j] log[(I+tztm)/(tVj+txtmIk)] )R8= = 16

logE132°/13k] logE(t,+1x1,X)(1,0k+1.1tm3j)]

where R, is the ratio for the case in which the polarizerorientation is varied and R8 is the ratio for the case inwhich the sample orientation is varied. These are to becompared with the corresponding true dichroic ratiowhich is

log(1/0j)RT= . (17)

log (1/k)

Equations (15) and (16) reduce to (17) for the case of aperfect polarizer, (ti= 0), and identical prism reflectionsin both directions, (t= 4,). The former condition canbe satisfied by the use of the reflected beam from aninclined plate polarizer but the latter condition is neversatisfied for a prism instrument. The effect of the prismmay be cancelled out by the use of a scheme first sug-

gested by Elliot, Ambrose, and Temple in which themeasurements are made with the electric vector of thelinearly polarized portion of the beam at 450 to theprism base. For example, if the solid sample is orientedso that the j axis is parallel to the I" oscillation andmakes an angle of 450 with the prism base, the intensi-ties of the prism transmitted components are:

13 2 2

131/= f- tj2

'2"+ -) ly12f2/

= (tx+t,)I2"/2= (t1+t )0jIo/ 4

13 2

13 = 1.1tz2+2

= (+tv)12'/2= (t,+1)0jt,,jjo/4.

Similarly, if the polarizer orientation is changed by 900,

13"= (t.+t,)#k1o/4I3'= (t,+t)0jtmIo14

so thatI1,4 = (tz+ ty) ( j+0kt)I o/4I 3 = (to+ ty) (k+B jt.)1 /4,

where the subscripts + and - are used to designate 45°orientations 900 apart. The corresponding measured in-tensities of the radiation incident on the sample are

1,+° = (tzA+ to) ( + t) Io/4

I ,_° = (tS+ ty) ( 1 + t)/4

so that the dichroic ratio is:

log[I 0+0/I3+] log:j(1 +t.)/(j+1nk)]

R, 10[13_°/13- 10g: (I1 e,)/ (ok,+ tm$7) I

R± differs from RT, therefore, only by the polarizerinefficiency factor t and obviously RE-PRT as t>O.From the point of view of this analysis, tm is quite largefor m= 6 and decreases only very slowly for m> 6.

Figures 1, 2, and 3 show the variation of the ratiosRp/RT, Rs/RT, and R±JRT with optical density of thesolid sample for the radiation component parallel tothe j axis; each set of curves is plotted for three widelyseparated values of the optical density of the sample forthe orthogonal k component. These span the usefulrange for infrared dichroism measurements. The prismtransmission coefficients used in these plots are for a60° CaF2 prism mounted for minimum deviation. Com-parison of the curves for the same value of transmissionin the k direction shows that the scheme suggested byElliot, Ambrose, and Temple for eliminating the effectof the prism is of relatively little value when transmis-sion polarizers are employed.

4. CALCULATION OF THE PRISM TRANSMISSIONCOEFFICIENTS

The calculation of the prism transmission coefficientsinvolves only determining the angle of incidence of thebeam for each prism-air interface. From this angle thetransmission coefficients for each such interface may bedetermined using the index of refraction of the prismmaterial and Fresnel's equations. The over-all prismcoefficients are then the products of the individualcoefficients, neglecting multiple internal reflections; forexample, if the beam traverses the prism twice, fourinterface reflections are involved and

t== t- t42 t3 t4.

Similarly, for t. If the prism is used at minimumdeviation, then two of the coefficients will be almostidentical with the other two and average values squaredmay be used. For the 60° CaF2 prism used in this labora-tory in a Perkin Elmer Model 12b monochromator, thetransmission coefficients for the horizontal and verticalcomponents are respectively t,=0.96 and t,=0.78 inthe neighborhood of 2 .4 g. The coefficients for a 600

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DICHROIC RATIO MEASUREMENTS IN REGION

NaCl prism are t=0.94 and t,=0.75 in the neighbor-hood of 8.3y.

5. INTENSITY LOSSES FROM ABSORPTIONAND SCATTERING

Losses of radiation intensity resulting from absorp-tion and scattering by the polarizer seem to be relativelyisotropic and, therefore, have little or no effect on themeasurements of dichroism. It is important, however,in the analysis of the data from single-beam measure-ments on the polarizers themselves to take account ofsuch losses. For silver chloride the losses may be ac-counted for by including a reduction factor ty- which isa variable coefficient dependent on frequency and tosome extent on the age and condition of the silver chlo-ride ( is, as before, the number of plates). Values of yobtained for clean unscratched plates were 0.94 atX= 1.671t, 0.96 at X=3.85/t, and 0.97 at X= 4.80u.

6. THE EXPERIMENTAL DETERMINATION OFPOLARIZER TRANSMISSION COEFFICIENTS

The transmission coefficients for silver chloride polar-izers tabulated in Table I were calculated from intensitymeasurements made with carefully aligned polarizerand polarizer-analyzer combinations. For example,using a phenomenological analysis of the type describedhere, it is possible to show that the measured intensityof the radiation transmitted by "crossed" polarizersvaries with the number of plates m according to

JIL= y2t(t.,+t1,)12

where the symbol l is used to designate the conditionof "crossed" polarizers. Similarly, using the ratio ofthe intensity measured with "crossed" polarizers to theintensity measured with "parallel" polarizers, y iseliminated:

IL tm t+I)

The same ratio for the case in which for both configura-tions the planes of incidence are at 450 to the x and yaxes is

± 2itm

I,, 45° t.2+ 1

The ratio of intensities measured with a single polarizeroriented with the plane of incidence parallel alternatelyto the x and y axes is

I tmtzy+tX

ly tmta+ty

The calculation of tm from these ratios using measuredvalues of I, Ih, 1, and I,, yielded the values tabulatedin Table I. Values of y calculated from the expression for1 were given in Sec. 5. Furthr measurements made

2.0[

F IN~

L.

I

1.0 2.0Rp/IR-

FIG. 1.

2.0

1-0 2.0

RFG/R2FIG. 2.

X o

Rs/RrFIG. 3.

FIGS. 1-3. The dependence of the ratios (measured dichroicratio/true dichroic ratio) on the optical density of a solid samplefor the radiation component parallel to the sample-fixed j axis.The dependence is plotted in each case for three values: (a),0k=0.95, (b) flk=0.50, (c), fk=0.05 of the transmission of theradiation component parallel to the sample-fixed k axis. Thesecurves are plotted for 6 plate AgCl polarizers. The square bracketson the left of these plots indicate the useful range for dichroismmeasurements in the infrared region.

with other polarizer configurations served to test theself-consistency of these results. Values of tm were ob-tained in this way to a precision of better than L5%.

7. ACKNOWLEDGMENTS

The author wishes to thank Professor Ralph S.Halford and Martin F. Gellert for valuable discussionson this problem.

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983November 1955

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