JULY 15, i936 PH YSI CAL REVIEW voLUM E 50

Mechanical Detection and Measurement of the Angular Momentum of Light

RIcHARD A. BETH, Worcester Polytechnic Institlte, Worcester, Mass. and Palmer Physical Laboratory, Princeton University

(Received May 8, 1936)

The electromagnetic theory of the torque exerted by abeam of polarized light on a doubly refracting plate whichalters its state of polarization is summarized. The samequantitative result is obtained by assigning an angularmomentum of )I'i ( —k) to each quantum of left (right)circularly polarized light in a vacuum, and assuming theconservation of angular momentum holds at the face of theplate. The apparatus used to detect and measure this effect

was designed to enhance the moment of force to bemeasured by an appropriate arrangement of quartz waveplates, and to reduce interferences. The results of about 120determinations by two observers working independentlyshow the magnitude and sign of the effect to be correct,and show that it varies as predicted by the theory witheach of three - experimental variables which could beindependently adj usted.

ELEcTRQMAGNETIc FIELD THEoRY

HE moment of force or torque exerted on adoubly refracting medium by a light wave

passing through it arises from the fact that thedielectric constant K is a tensor. Consequentlythe electric intensity E is, in general, not parallelto the electric polarization P or to the electricdisplacement

D =KE =E+4+Pin the medium. The torque per unit volumeproduced by the action of the electric held onthe polarization of the medium is

1=PXE= (DXE)/4~.Following Sadowsky and Epstein' we may

calculate this torque for a simple case as follows:Assume a doubly refracting medium of perme-ability unity and take as x, y, and s axes theprincipal axes of the tensor K for the light fre-quency in question. Denote the principal valuesof K by n, ', ri„'and nP so that n, n„and n, will

be the principal indices of refraction. Let theelectric components of a plane light wave

* Member of the Physics Department, WorcesterPolytechnic Institute. Preliminary work and final calcu-lations were carried on at Worcester. Experimental workwas done on leave-of-absence from Worcester as ResearchAssociate at Palmer Physical Laboratory, PrincetonUniversity. Additional measurements were made by Mr.W. Harris at Princeton, as will be explained later.

,' A. Sadowsky, Acta et Commentationes Imp. Uni-

versitatis Jurievensis /, No. 1—3 (1899);8, No. 1—2 (1900).P. S. Epstein, Ann. d. Physik 44, 593 (1914).The deriva-tion is repeated here because the first articles are inRussian and relatively inaccessible, while in the last thereseems to be a misprint in the result. I wish to thankDr. Boris Podolsky for translating parts of Sadowsky'sarticles for me from the Russian. I am also deeply indebtedto Professor A, Einstein not only for his advice in checkingProfessor Epstein's calculation but also for several inter-esting discussions about the experimental part of thework here reported.

propagated in the +s direction be given by

E=A cos 8 cos (Zy+6),A sin 0 cos (Z~ —6), 0,

D=KE=n, 'A cos 0 cos (Zr+6),n„'A sin 0 cos (Z~ —6), 0, (2)

where Z& ~(t ns/c); n =—(n„+n,)/2,

s(n, . n, ) /—X; cv = 2 ~c/X;

For any value of s (in particular s=0) for whichthe wave plate thickness 6/m is a whole numberthe light is plane polarized and Z makes anangle +0 with the x axis.

The torque per unit volume at any point inthe crystal is calculated from (2) according to (1)1=0, 0,

(A'/8~)(n, ' —I„')sin 28 cos (Z~+6) cos (Z~ —5)

and the time average value of the s component is

(A/4m)'(e, ' n') s—in 20 cos 2A.

Integrating from s& to s2 we get the torque perunit area on the crystal between these values:

L = —(A/47r)'nX sin 20(sin 262 —sin 2A&). (3)'

This torque, which has hitherto been generallyconsidered too small for experimental detection,was detected and measured in the present ex-periment. '

' J. H. Poynting, Proc. Roy. Soc. A82, 560 (1909), infersfrom an ingenious mechanical analogy that circularlypolarized light should exert a torque equal to the lightenergy per unit volume times X/2' on unit area of aquarter-wave plate which makes the light plane polarized.This result, which neglects surface reflections, is containedin (3}.

3 R. A. Beth, Phys. Rev. 48, 471 (1935); also AnnualMeeting of the Am. Phys. Soc., St. Louis, Missouri,January 1, 1936.A. H. S. Holbourn, Nature 13'7, 31 (1936),also reports successful measurement of the effect.

ii6 RICHARD A. BETH

QUANTUM THEORY

It is worth observing that (3) may be derivedfrom the quantum theory by assigning an angularmomentum or spin of Ii( fi)4 t—o each photon ofleft (right) circularly polarized light, the spinaxis being in the direction of propagation of thelight. '

Consider an elliptically polarized light wave,propagated in the +s direction in a vacuum,whose components in the directions of the prin-cipal axes of the ellipse (phase difference= ir/2)have amplitudes Xo and I'0. Let the componentsof the same wave, when resolved along anarbitrary pair of perpendicular x and y axes inthe plane of the ellipse, be given by

E=X cos (Z+6), Ycos (Z —6), 0, (4)

where Z= co(t z/c) a—nd 2A is the phase angle bywhich the y component lags behind the x com-ponent of the wave. It may be shown that'

XYsin 25=XOYO, X'+ Y'=Xo'+ Yo'. (5)

If we now regard the wave (4) as the super-position of left and right circularly polarizedcomponents of amplitudes L and R, respectively,calculation shows that

L' R'=XY sin—2A L'+R'= (X'+ Y')/2.

The number of left circularly polarized photonstransmitted per unit area per second is thePoynting energy flow cL'/47r divided by theenergy per photon 27rfiv=2irfic/'A, or XL'/Sir%.Multiplying by 5 gives XL2/8ir2, the angularmomentum transmitted per unit area per secondby the left circularly polarized component.Taking account of the similar expression for theright circularly polarized component, the angularmomentum transmitted per unit area per secondby the wave (4) is

DE= X(L' —R')/8m'= XXY sin 2A/Sir'=XXn Yo/8x' (6)

It is well known that the linear momentum

4 A =h/27r.'A. E. Ruark and H. C. Urey, Proc. Nat. Acad. Sci.

13, 763 (1927); Harnwell and Livingood, ExperimentalAtomic Physics (1933), p. 81; Brillouin, les StctistiquesQucntigues, Chap. III (1930).

'See, e,g. , Schuster and Nicholson, Theory of Optics,third edition, pp. 14, 15 (1924), especially Eqs. (15) and(17).

transmitted per unit area per second is theenergy How divided by c, in this case:

(L'+R')/4' = (X'+ Y')/8n. = (Xo'+ Yo')/8ir.

Thus the angular and linear momenta are pro-portional to the natural invariants (5) of thewave (4). For Xo——Yo their ratio is Ii=X/2ir,which is another form of Poynting's result. '

The value (6) of the angular momentum, whichis independent of the quantum constant k, canbe derived from the linear momentum (Poyntingvector divided by c) in a finite beam, but theelementary method given is more descriptive inbringing out the fact that the quantum theoryleads to the same value (3) as the wave theoryfor the torque on a section of a crystal.

To show this equivalence we have now toobtain the angular momentum for a wave (2)in the crystal from the expression (6) for a wavein a vacuum, by placing the face of the crystalat z=zi (vacuum for z(z&) and using the con-servation of angular momentum and Fresnel'sexpressions for the amplitudes of the, reHectedand transmitted waves. In the reHected wavethe y component of amplitude Y(n„1)/(n—„+&)lags behind the x component of amplitq. deX(n 1)/(n—+1) by the phase angle 2A. Weget the angular momentum transmitted throughthe face z=zi by calculating (6) for the directand reflected waves in the vacuum (takingaccount of direction of transmission of each) andassuming the conservation of angular momentumat the face

M, =XY(ii,+nv) X sin 2A/4''(n, +1)(nv+1)

Using Fresnel's expressions for the amplitudes,we identify the transmitted wave in the crystalwith (2) at z=zi by choosing

A cos 0 = 2X/(n, +1); A sin 0 = 2 Y/(n v+1);~=a, =~z,(n„—n, )/l .

Hence the angular momentum transmitted by(2) at z =zi is

3Ei ——(A/4ir)'n) sin 28 sin 2hi. (7)

It can now be seen directly that the torque (3)is equal to the excess of the angular momentum

(7) per unit area per second flowing into asection of the crystal at s=s& over that flowingout at s =z2. Hence both the field and the

ANGULAR MOMENTUM OF LIGHT 117

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stray heating effects, radiometer and gas effects,and variations in light pressure, the apparatuswas constructed as shown in Figs. 1, 2, 3, and 4.Some of the precautions taken to avoid theseinterferences may have been unnecessary, butthey were sufficient.

The whole apparatus is supported on theheavy cast iron brackets I in Fig. 1, each'12"&(12"., which are bolted to a brick pier toavoid mechanical disturbances, especially tor-sional vibrations of a period comparable to that

FIG. 1. Diagram of apparatus.

quantum theories predict the same value forthe effect which has been measured in the presentexperiment.

APPARATUS

The basic idea used in detecting and measuringthe effect~ is to observe the deflection of a quartzwave plate hung from a fine quartz fiber whensuitably polarized light is sent through the plate.In order to increase the effect to be measured andto avoid as far as possible interfering effects dueto mechanical and electrostatic disturbances,

~ See Poynting, reference 2, A. Kastler, Societe desSciences physiques et naturelles de Bordeaux, Jan. 28,1932. R, A. Beth, abstract, Boston Meeting, AmericanPhysical Society, Phys. Rev. 45, 296 (1934).

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FIG. 3. Wave plate arrangement.

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FrG. 4. Detail showing method by which light beam couldbe given lateral adjustment.

of the torsional pendulum formed by the one-inch circular wave plate 3II hanging at the bottomof the quartz fiber (about ten minutes). The fiberis about 25 centimeters long and hangs in aquarter inch hole bored lengthwise through acopper cylinder two inches in diameter. Theupper end of the fiber is attached with a bit offlake shellac to a conical copper plug which wascarefully ground in place and sealed withApiezon grease. Thus the zero position as wellas the amplitude of the torsional pendulum canbe changed while the chamber is evacuated.

The hanging plate and the fixed plate aboveit, Fig. 3, are located in the cylindrical vacuumchamber C which is of copper, three inches indiameter with walls one-half inch thick. Theheavy copper chamber is intended to eliminateunequal heating effects from the room as well asto shield off stray light and electrostatic dis-turbances. 8

The chamber was evacuated through widetubing by a high speed diffusion pump and threeliquid-air traps. A two-stage diffusion pump andforepump provided the forevacuum for the highspeed pump. According to the rated air calibra-tion of the Western Electric ionization gauge,pressures below 10 ' mm of mercury existed inthe chamber while the effect was being measured;no noticeable fluctuations appeared in the actual

' I wish to thank Professor G. P. Harnwell for this andmany other helpful suggestions concerning the mech@nica]and optical design of the apparatus.

measurements which might be ascribed to radi-ometer or gas effects.

Light from a three-millimeter tungsten ribbonfilament Ii is focused by the fused quartz lens I,10.7 centimeters in diameter, through a largeNicol prism N, the lower plate 8 (Fig. 3), thefused quartz window W, and the hanging plateM, on the reflecting layer of aluminum on thetop of the upper plate T in the chamber. Themaximum deviation of the light from the verticalwas about 1.0'. W has a circular aperture one inchin diameter. A round one-quarter inch copperdisk d fastened in the middle of W prevents lightfrom passing through the hole II in the top plate'r. With this arrangement no light energyreaches the fiber and most of the energy isreflected out of the vacuum chamber altogether,thus minimizing fluctuations in the position ofthe pendulum due to unequal heating of thefiber and light pressure on the small mirror m.These undesirable effects are found to be presentif the shield d is removed.

The lower plate 8 is placed outside the vacuumchamber in a brass ring bushing held so it is freeto turn in a brass frame attached to the uppercast iron bracket. The position of the lower waveplate may be read on a circular scale by meansof a pointer attached to the bushing. A circularbrass plate P attached to the under side of thebushing carries adjustable stops S, and the platemay be rotated from one stop to the other whileobserving the swing of the pendulum (with atelescope placed in front of the apparatus) bymeans of cords running in grooves on the edgeof the plate P'. Frame, plate P,. cords, stops,pointer and circular scale are visible at the topof Fig. 4.

The ideas underlying the wave plate arrange-ment, Fig. 3, may be qualitatively understoodas follows. The light coming upward through theplates is reflected by the aluminum player on thetop, side of T, passing downward again throughthe plates. For a certain wave-length ) o, 8 andT are quarter wave plates with their axes asshown at 90' to those of the hanging half-wave

plate 3f. If light of wave-length Xo from theNicol, plane polarized at 0 =45', enters thebottom plate 8, both direct and reflected beamswill be circularly polarized in each space betweenthe plates in the directions indicated by the

ANGULAR MOMENTUM OF LIGHT

curved arrows. Thus the angular momentumdelivered per second to the half-wave plate 3fis almost four times what could be obtained bysimply absorbing the same amount of circularlypolarized light.

For wave-lengths ) different from )0 it willbe seen that, because of the alternate advancingand retarding of each component with respectto the other, light in both direct and reflectedbeams is plane polarized at the initial angle 0 atthe bottom of 8, the middle of M, and the topof T. If the plates vary somewhat from the exactspecifications these levels of plane polarizationwill be correspondingly shifted. In general thelight will be elliptically polarized in each spacebetween the plates. Careful consideration showsthat the angular momentum delivered to M is inthe same sense as that for ) 0 for all wave-lengths) for which

(n„—n, )g/k(2(n„—n, ))„/Xpor, approximately, X)Xo/2.

If the lower plate 8 is rotated 90' from theposition shown, the torque due to light of wave-length Xo will be just reversed. For this positionthe contributions to the torque of other wave-lengths is not easily considered qualitatively, butdetailed calculation shows, as might be expected,that the integrated torque for all values of X

used is smaller in magnitude and opposite ihdirection compared to the torque similarly cal-culated for the position shown in Fig. 3. Thetorque is a continuous function of the positionangle of the bottom plate, and its value can, withsome labor, be calculated for each position. .

The design wave-length ) 0 is determined'onthe basis of (3) by noting that A' is practicallyproportional to the energy in the light at eachwave-length, and that ) enters as a factor. Theenergy distribution has a rather sharp peak inthe region just above 1.0 p for the tungstenfilament temperatures to be considered. By aseries of trial calculations ) ~ ——1.2 p was chosento make the light torque a maximum.

The actual plates used, of course, vary some-what from the ideal specifications. The mostaccurate plates obtained were made to. within afew percent of the specified thickness by theBryden Company of Waltham, Massachusetts,whom I wish to thank for this excellent coopera-

tion. For the purposes of the exact calculationsthe plates were measured by means of a Babinettype compensator with sodium light. From theknown values of the refractive indices of quartz'the retardation was then calculated for eachplate for all wave-lengths contributing to theeffect.

Besides almost quadrupling the torque com-pared to what would be obtained if the lightwere simply absorbed, and furthermore reducingheating, radiometer, and gas effects in thevacuum chamber, this plate arrangement greatlyreduced difficulties due to radiation pressure.The disk 3I cannot be mounted exactly at rightangles to the fiber axis, nor can the light beprojected exactly in a vertical direction. Thusthe resultant light pressure integrated over thedisk will not be quite vertical and will produce a"light pressure torque" unless its line of actionlies in a vertical plane containing the fiber axis.Without special precautions this torque mightvery easily mask the effect to be detected andmeasured. '0

A tungsten filament light source was chosen,pt a sacrifice of light intensity compared to anarc, in order to keep the resultant light pressureon the disk 3II as steady as possible, both inposition and magnitude, and to give a steadyand reproducible spectral energy distribution forpurposes of calculation. The undesirable result-ant light pressure on the disk M is decreased inthe plate arrangement described, first, becausemost of the energy is allowed to pass through thedisk and, secondly, because the pressure due tosurface reflections and absorption for the upwardbeam is largely cancelled by the similar pressurefor the beam reflected downward.

Finally, the moment arm for the residual lightpressure can be varied by shifting the entirelight beam parallel to itself. This is done bymounting the entire optical system up to A —Ain Fig. 1 on the double slide D, which in turn is

supported in the lower of the two cast ironbrackets I. The detail view, Fig. 4, shows thetwo micrometer screws by which the lateral

International Critical TabLes, VI, 341—342."Besides the smallness of the eEect itself, the light

pressure torque is perhaps the greatest difficulty in thisexperiment. I wish to thank Professor A. Kastler of theUniversity of Bordeaux for a personal communicationemphasizing this fact.

120 RI CHARD A. BETH

motion in two coordinates is produced. Theseare calibrated in thousandths of an inch so thatany particular setting is easily repeated at alater time. By using this arrangement and leavingthe lower wave plate 8 in a fixed position, thechange in the sum of light torque, and lightpressure, radiometer, and gas effect torqueproduced by a ten-percent change in filamentcurrent was measured for various settings ofone of the double slide screws, the other beingkept fixed. The results when plotted give asmooth curve crossing the zero torque line whenthe filament image lies across the middle of thewindow t/t/' as nearly as the eye can judge. Thisindicates that the spurious torques are probablyof the nature anticipated and, in any event, canbe held steady and made very small by appro-priate setting of the double slide. In making themeasurements of the light torque effect itself,the filament current was held constant to betterthan one percent, and the filament image wasplaced across the middle of the window. Carewas also taken to use the same lengthwiseportion of the filament in all cases.

The entire optical system can be rotated in itssupport on the double slide. Changes in theangle 0 so produced are read on the circular scalevisible in Fig. 4 on top of the double slide andbelow the Nicol housing N.

As described below, changes in the light torqueeffect corresponding to definite rotations of thebottom plate were measured and found to bereproducible for the same settings to within afew percent. When the bottom plate was re-moved, . or when it was replaced by a fused quartzplate, rotation in the same manner produced nomeasurable effect. Hence it may be assumed thatrotating the lower plate leaves the light energy,light pressure, heating, radiometer and gaseffects sensibly unchanged. In other words, theapparatus used successfully eliminates the inter-fering effects, and enhances the light torqueeffect to be measured.

MEASUREMENTS

The apparatus constructed according to theplan described showed the effect in the predicteddirection and of about the predicted magnitudeat the first trial on July 10, 1935. With this, the

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first aim, that of detecting the effect, was accom-plished. The second aim, that of measuring theeffect, was carried out by a resonance method(see reference 7). One of the first series of meas-urements taken is shown in Fig. 5, which illus-trates both the definiteness of the effect observedand the kind of minor fluctuations encountered.

At the top of Fig. 5 are given the readingsa„(v=0,1, 2, ) in centimeters on the scale(338 centimeters from the fiber axis) correspond-ing to the end points of the swing of the torsionalpendulum. Tenths of millimeters on the scalecould be estimated by means of the cross hairsin the telescope. The bottom plate 8 was in theposition shown in Fig. 3 (8=45', right stop 8)while the readings ao to u4 were being taken, butwas turned 90' to the left (to 0= —45', to leftstop I.) at the instant at which the end point a5was read. The change in light torque producedby this rotation of 8 was such as to oppose theswing of the pendulum from c5 to a6. At a6 theplate 8 was turned back to R, the torque changeagain opposing the ensuing swing from a6 to a7.The plate was turned similarly in "antireson-ance" at ay and a8 as indicated, but was thenleft in the position R when a9, a~o and a~~ were

ANGULAR MOMENTUM OF LIGHT 121

read. At a~2 the plate was turned R to L, thechange in torque now aiding the swing of thependulum between @~2 and a~3. At a~3 the platewas turned L to R. This "resonance" turningwas continued at aj4 and a~5, and then the platewas again left in the position R for the driftreadings a~6 to a2O. For most of the readingstaken another "antiresonance" series was addedto the schedule described, and, for the readingstaken by Mr. Harris, the "drift" periods wereshortened while the "resonance" series in themiddle of the schedule was lengthened to sixhalf-cycles.

From the end-point readings taken accordingto such a schedule 4A„=—a„~+2a„—a„+~and4Z, =a, ~+2a„+a,+~ were calculated. The ab-solute values of the "amplitudes" A„areplottedin the upper part of Fig. 5 and the "zeros" Z„(circles) in the lower part. For small values ofdamping (see "drift" slopes) the arithmeticmeans used differ by much less than the experi-mental error from the ideally correct geometricmeans for the quantities calculated.

From each such set of readings we wish todetermine the value of the change in light torqueon the hanging plate 3f when the bottom plate8 is turned R to L or L to R. Since the torsionalamplitude of the pendulum was at most a fewdegrees, we may assume that the light torqueon M was a constant for a given position of thebottom plate B. We may consider a change inthe position of the bottom plate to be equivalentto a change in the equilibrium position aboutwhich the torsional pendulum executes its oscil-lations. Let x be the corresponding change inscale reading. Careful consideration shows that,leaving damping out of account, the change inA„per cycle must then be 2x during "resonance"or "antiresonance" sequences. With a smallamount of damping, x is therefore very nearlyequal to the average of the rates of change of A,per half cycle during -"resonance" and "anti-resonance. " For example, in Fig. 5, the magni-tudes of the slopes A and 8, by least squares,are 0.341 and 0.289 centimeter per half-cycle,respectively. Hence x =0.315 centimeter.

This method of taking readings and calculatingaffords several checks on the self-consistency ofthe sequence of readings a„used to determineone value of x. For example, the damping slopes

a, b, and c should be approximately equal to(A —8)/2 in magnitude. Furthermore Z„+x/2during the "resonance". and "antiresonance"intervals (plotted as crosses) should form asmooth sequence with the values of Z„during thedrift intervals. Slight disturbances, such asbetween a9 and u&o, may be taken into account inevaluating these diagrams.

The quartz fibers were made in a gas-oxygenblast Hame. Of those which seemed straight anduniform over a length of a foot or so, the /nest(smal/ X)"was taken. X for the first pendulum,with which the measurements of Fig. 5 weretaken, was 8.52X10 ' dyne-cm/radian, or 1.26X 10 8 dyne-cm/scale centimeter. Hence thechange in light torque in Fig. 5 was 3.97&10 'dyne-cm. This first fiber was accidentally brokenand most of the measurements were taken withthe second fiber for which X was 10.1&(10 'dyne-cm/radian or 1.50X 10 s dyne-cm/scalecentimeter. The period in the latter case wasabout nine minutes, so that most of the deter-minations of x required consecutive observationof end-points for practically two hours. I madeabout 80 determinations during July, Augustand September, 1935, and Mr. Wilbur Harrismade about 40 determinations during October,November and December, 1935, after I had leftPrinceton. Each of us worked in the absenceof the other, and each made determinations ofthe three types to be described. Mr. Harrisused a different bottom plate from the one

"With regard to the choice of the torsion constant K(in dyne-cemtimeters per radian) for the fiber, the follow-ing is of interest. During a "resonance" interval the ampli-tude (neglecting damping) increases by an amount 2Nx inN cycles. For a given torque change, x is inversely propor-tional to X.The number of cycles in a given time is directlyproportional to QX, assuming the moment of inertia ofthe pendulum is a constant. Hence the change in amplitude2Nx in a given time is inversely proportional to QE. Inother words, the time required to achieve a given increasein amplitude by the resonance method varies directly asQE, other factors being constant. Increasing X in this

experiment would therefore involve an increase in thetime required for each measurement, a disadvantage whichwas not present in the corresponding case of the Einstein-de Haas experiment, in which the impulse per half cyclewas independent of the length of the cycle. Nevertheless,Professor Einstein has pointed out that it would also beworth while in this experiment to increase X considerably,in spite of the increase in time required to achieve a givenchange in amplitude, because then the ratio of the energyin the effect desired to the energy of random disturbanceswould become much more favorable. Furthermore, it isan advantage to have the end-point readings e„comeatshorter intervals insofar as the possible segregation ofthe larger random disturbances is concerned.

122 RI CHARD A. BETH

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which I used, which seems to eliminate thepossibility that the results, which were through-out consistent, could have been influenced byirregular patches of "infrared dirt" on the clearcrystal quartz plates. Mr. Harris also investi-gated very thoroughly whether any effects wereobtainable in the absence of the bottom plate,and found absolutely nothing. These checkswere invaluable in substantiating the results ofthe present experiment, and Mr. Harris deservesmuch credit for the thoroughness and enthusiasmwith which he carried out the tedious observa-tions.

Three types of measurements, presented inFigs. 6, 8 and 9, were taken to show that theeffect varies as required by the theory. Thecrosses represent my determinations and thecircles those of Mr. Harris. Each point is derivedfrom observations over a period of one and one-half to two hours as described in connection withFig. 5. The smooth curves give the correspondingtheoretical variation in each case.

For type I (Fig. 6) the torque change wasmeasured for various filament temperatures asdetermined by the current through the lamp.

The bottom plate was turned between stops at0 =45' and 0 = —45', and the plane of polariza-tion of the Nicol was at 45' to the axes of theplates as in Fig. 3. Mr, Harris' values (circles),taken consecutively, without shifting the posi-tion of the lamp, form the smoothest sequenceobtained. The scattering of my values (crosses)taken with a different bottom plate is partlyexplained-by the fact that I did not make thesedeterminations consecutively, but shifted thelamp position a number of times for otherreadings. The total amount of light entering theplate system changes rather sharply with theposition of the lamp in its socket and the settingof the double slide because of the presence of thesmall light shield d in the middle of the windowW. The measurements previously reported" ofthis type, were made with a different lamp andfiber, and were made with a somewhat largerlight shield on the window 8'. The values weretherefore somewhat smaller than those shownin Fig. 6, but the magnitudes increased in asimilar way with temperature.

The torque to be expected at various filamenttemperatures was calculated from (3) and thespectral energy distributions as follows. Fromthe Poynting vector, the energy flow in the plateM is A'(n, cos' 0+m„sin' 0)c/8m or, practically,A2nc/87r = 2Irc(A/4~)'n ergs/cm2-sec. Hence, ifJ&,'dl is the total energy flow (integrated overthe area of the plate) in M, in the wave-lengthrange X to X+0), the net torque on the platefrom the energy flow in this wave-length rangewill be L~'dX dyne-centimeters, where

I I' = sin 28I,'(sin 2hr —sin 26&)XJI,'/2Irc.

Here tan 0),' is, in general, the ratio of the

amplitudes of the components of the light in thedirections of the axes of the plates T and 3I.0&' = 0 only when the position of the bottom plate8 is such that the directions of its axes coincidewith those for 3f and T. If the total energy. flowentering the plate system in this wave-lengthrange is JI,d'h ergs/sec. , we set JI,'= JI,FI„foreach of the light beams contributing to the effect(enumerated by the index It), and compute FI„using Edison Pettit's values for the reflectivityof aluminum (top surface of T)" and the re-

"R. A. Beth, Phys. Rev. 48, 471 (1935)."Edison Pettit, Pub. Astronom. Soc. PaciAc 46, 27

(1934).

ANGULAR MOMENTUM OF LIGHT 123

where the torque factors I.& are given by

Lq QXF&,„——I sin 28q'(sin 2d ~—sin 2h~) ) q„/2mc

seconds. (9)

The torque factors Iq were calculated for thesum q of the effects of the main beam goingupward through M, the main beam reflecteddownward from the aluminized top surface of T,and for the refIections of these beams at thesurfaces of 8, W, 3I, and T, repeated surface

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FiG. 7'. Torque factor as a function of wave-length.

fractive indices of quartz, which enter intoFresnel's expressions for the transmitted andrefIected intensities at the plate surfaces. Theabsorption in the quartz plates may be neglectedbecause the light has been filtered through thefused quartz lens I. and the calcite and Canadabalsam of the Nicol prism. From the theory forelliptically polarized light, the angles 0&' andthe phases 2A may be calculated for each beamand each wave-length and for any position ofthe bottom plate, because the incoming lightfrom the Nicol is plane-polarized, and the platethicknesses, as measured by the compensator inwave plates for sodium light, were: T=0.649,3E= 1.051, Bz ——0.535 (my torque measurements),82=0.454 (Harris' torque measurements). Thewhole torque on half is then

L = jLq Jqdk dyne-centimeters, (8)

reflections being neglected. These values of (9)are shown in Fig. 7, the angles denoting theazimuthal position of the bottom plate; at 45'the direction of the "fast" axis of 8 coincideswith the direction of the "slow" axis of 3f andthe "fast" axis of' T (see Fig. 3). The full curverefers to the calculations for 82,, the dotted curveto those for B~.

Using a rocksalt spectrometer and thermo-couple arrangement Mr. Harris compared theamount of radiation from a black body at aknown temperature (radiation from the interiorof a furnace made out of an alundum cylinder2-,'inches in diameter and a foot long at 766'Cand 784'C was used) with the amount of radi-ation coming through the Nicol in the samesolid angle at intervals of 0.1 p for each of sixdifferent lamp currents. The energy factors J),were computed in absolute terms from thesemeasurements using calculated values of blackbody radiation, Lambert's law, and constants ofthe optical system to determine the solid angleof the cone of light entering the plate systemfrom the Nicol.

Replacing the integral in (8) by the corre-sponding sum for 0.1 p intervals from 1.0 pto 2.0 p inclusive, the change in total torquefrom 0=45' to 0= —45' for each of the six fila-

ment currents and for the bottom plate 82 wasobtained. These theoretical torque values areplotted against the temperatures, 1948', 2069',2297', 2506', 2701' and 2879' absolute, corre-sponding to the filament currents used, in Fig. 6and connected by straight lines A. The irregu-larity of this broken line A indicates the dif-ficulties encountered in making the energydeterminations. The principal sources of errorare: The spectrometer was not set up especiallyfor this purpose, but was originally designed forsomewhat longer wave-lengths; the solid angle ofthe, light cone is difficult to determine withaccuracy; the neglected effect below 1.0 p, isprobably not exactly compensated by assumingthat the Nicol actually polarizes all the light outto 2.0 p. There is reason to believe, from theenergy measurements themselves, that the ener-

gies computed for the upper two temperaturesare somewhat too low. The root-mean-squaredeviation of the measured torques (circles) from

124 RICHARD A. BETH

the computed values A is 0.29X10 ' dyne-centimeter.

A second attempt was made to get the valuesof the energy factors J), in absolute terms byextrapolating the tungsten emissivities given byForsythe and Worthing" (assuming the emis-sivity to be a linear function of temperature ateach wave-length), multiplying by the blackbody radiation at the temperature of the fila-

ment, taking account of the solid angle of radi-ation collected from each point on the filament

by the lens L, , and making an allowance for thereflections from the surfaces of the bulb of thelamp, the lens, and the Nicol prism. The changein torque at each temperature was calculatedfrom (8) as before, the integral in this case beingreplaced by the corresponding sum for 0.2 p,

intervals from 0.8 p, to 2.0 p inclusive. Theresults must be reduced by about 15 percent togive the curve 8 in Fig. 6. This 15 percent maybe caused by the absorption of the Canadabalsam in the Nicol, by nonpolarization of thelight near 2.0 p, and by inaccuracies in theemissivities used. The root-mean-square devi-ation of the circles from the curve 8 as drawn

is 0.24&(10 ' dyne-centimeter. It should beemphasized that the uncertainties in 8 leave theabsolute vulue in considerably greater doubt thanin the case of A.

An important conclusion from Fig. 6 is,however, that the slope and shape of the theo-retically determined curves A and 8 agreeswithin the limits of error with the trend of themeasured torque values. The trend of 8 relativeto the torque values (measured for certain lampclrremts) depends on the current-temperaturecalibration. of the lamp which was extrapolatedfrom one value at 2941' absolute by comparison'with the calibration for a similar lamp, while

the trend of A relative to the torque valuesmeasured does not depend on this calibration,the energy distributions having been measuredfor given lamp currents also. The dotted smoothcurve C, empirically fitted to the upper nine ofHarris' torque measurements, appears as anatural compromise between the trends of Aand B. Half of the circles lie within one percentof this curve. Their root-mean-square deviationfrom the curve is 0.054&(10 ' dyne-centimeter.

I4 Forsythe and Worthing, Astrophys. J. 61, 151 (1925).

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/n

0-4o

-50

-/$0-kS-/24' -/0$ «90' "78' -~ ~-M' -/5 O /8 30 45 60'

PLANE W ~4B/ZAP'lON — AiV6LE'8

Fro. 8. Type EI measurements: variation of light torquewith angle between plane of polarization of light and theaxes of the plates.

If we choose to think of C as an empirical cali-bration curve, this indicates that we may deter-mine the temperature of the lamp in this rangewith a probable error of about 7' by measuringthe torque with the accuracy attained by Mr.Harris, the lamp position being left unchanged.

The type I measurements may be regarded asa test of the factors XJ&,/2~c in the calculationof the torque from (8) and (9).

For the type II measurements (Fig. 8) thelamp current, and therefore the light intensity,was maintained constant and the plate B~ wasrotated between the same stops as in type I, butthe. plane of polarization of the light was rotatedto various angles 0 with the axes of the plates.These measurements are intended to test thefactor sin 20 in the formula for the torque.The root-mean-square deviation of the meas-ured values from the computed sine curveshown is 0.34X10 ' dyne-centimeter. A smallsystematic deviation - from the sine curvemay be due to the diAiculty encountered inlining up the plate axes accurately in setting upthe apparatus. In setting the angle 0 the wholelamp housing and optical system, including theNicol, was rotated and the double slide reset asdescribed. Because of the presence of the lightshield d this causes a greater scattering of themeasurements than would be obtained if thelamp could be left fixed in position.

The measurements shown were made by Mr.Harris, .In my measurements of this type only a

ANGULAR MOMENTUM OF LIGHT

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I0 05'h

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0 -/O

y5 0

Pos/ rloN oF Q(5H r STopLEs r Sros ~r -+~~

4Q

FIG. 9. Type III measurements: light torque as a functionof position of the right stop.

quarter cycle of a sine curve was covered, but at15' intervals. These points also form a smoothsegment of a sine curve; though the root-mean-square deviation is considerably less, the test isnot as critical and the result not as interesting asthe full cycle curve shown.

For the type III measurements (Fig. 9) thelight intensity was again maintained constantand the plane of polarization of the opticalsystem was again put at 45' to the directions ofthe axes of plates 3f and T as in type I. The leftstop was fixed at —94-,"and measurements madefor different positions of the right stop. Since thetorque for B~ at the left stop was constant, thecurve shows how the light torque in this platesystem varies with the position of the bottomplate. The elliptical polarization produced by thebottom plate depends on its position, and thesemeasurements are, therefore, in the main a testof the phase factors (sin 2hj —sin 262) in theformula. From the torque factors of Fig. 7 thesmooth curve shown in Fig. 9 was calculatedfrom the energy distribution. The root-mean-square deviation of the measured values of thetorque from this curve is 0.073&(10—' dyne-centimeter. This very good, agreement is againto be expected because the lamp position wasleft fixed. Mr. Harris also made measurements

of this type, but did not cover a whole cycle.They are in similar agreement with the predictedvalues.

CQNcLUsIoN

The change in the angular momentum ofpolarized light in passing through a crystal platehas been detected and measured with the appa-ratus described. This conclusion rests on the factthat the measured effect agrees in sign and,within the limits of error, in magnitude with theeffect as predicted both by the wave and by thequantum theories of light. In particular it hasbeen shown that the effect varies as required bythe theory with each of three experimentalvariables studied: filament temperature (lightintensity), azimuth of the plane of polarization,and position of the bottom plate.

I am deeply indebted to Professor A. W. Duffof the Worcester Polytechnic Institute who firstsuggested this problem to me and who personallyhelped me with the purchase of apparatus in thepreliminary stages of the investigation. I wishto thank the National Research Council for aGrant-in-Aid for the purchase of apparatus andthe Palmer Physical Laboratory of Princetonfor providing all the specially made apparatusand technical assistance as well as for the hos-pitality of the Laboratory. In particular I wishalso to thank Professors H am well, Einstein,Condon, Robertson, Shenstone, Bleakney, andDr. Barnes of Princeton for their willing helpful-ness in discussing the work with me on numerousoccasions. Dr. S. N. Van Voorhis and Messrs.C. W. Curtis and C. W. Lampson of Princetonhelped me both with suggestions about experi-mental details and in actually taking readings.Mr. Curtis shared the work of taking readingswith me for many weeks. The very valuable partof the work done by Mr. W. Harris has beennoted in detail above. Finally, I want to thankMr. A. H. S. Holbourn of the Clarendon Labora-tory, Oxford, for his interesting correspondencewith me about the problem.