plane waves of light iii

JOURNALo1 the

OPTICAL SOCIETYOF AMERICA

VOLUME 22

No. 6

a-- - .JUNE1932

PLANE WAVES OF LIGHT III

ABSORPTION BY METALS

By THORNTON C. FRY

[BELL TELEPHONE LABORATORIES, NEW YORK, N. Y. RECEIVED MARCH 15, 19321

§21. Introduction: The study which follows has two different aspects.In one sense it is a continuation of two earlier papers1 bearing the samegeneral title; and as we shall make many cross-references it has seemedwise to run the reference numbers of the present paper consecutivelywith the former ones. In another sense it is a detail of a more generalinvestigation, the purpose of which is to study the relationship betweenthe optical and photoelectric properties of metals. We begin by indicat-ing its connection with this latter study.

On the experimental side, the general investigation makes much useof thin films of alkali metal deposited either on glass or on a metallicbase, frequently platinum. The study of such films, rather than bulkmaterials, recommends itself for three reasons: first, because they showremarkable variations in photo-sensitivity as the plane of polarizationand angle of incidence of the light is varied; second, because the photo-electrons released from bulk material by ordinary light must originatein a very thin layer near the surface, since otherwise they would soonlose their energy by collision; and third, because by working with bases

1 Plane Waves of Light, I, J.O.S.A. 15, 137-161; 1927; and II, J.O.S.A. 16, 1-25; 1928

307

I

THORNTON C. FRY

having widely different optical properties, we can distinguish the effectswhich are due to the photo-sensitive film itself from those due to the na-ture of the base. This we cannot do if bulk materials are used, thoughthe base (the underlying bulk material) is always present as a possibledisturbing factor.

To interpret such experiments adequately, however, we need com-plete knowledge, of the energy distribution within the photo-sensitivelayer, and of its relation to the optical properties of the base. This is apurely optical problem, capable of mathematical solution when the op-tical constants of the materials are known. It is the mathematical studyof this optical problem, and in particular of the amount of energy ab-sorbed by the film, with which the present paper deals.

§22. Physical Background: The following broad statement of physicalbackground may serve to motivate our mathematical presentation:

When a beam of light shines upon a metallic plate, the electric inten-sity inside the plate is proportional to the intensity in the incident beam.The factor of proportionality, however, is not an absolute constant. Thisis obvious since it differs for different metals, and depends on the me-dium (air or glass, for example) through which the light reaches it.It is not even an intrinsic constant of the metal, for if the intensity ofthe incident beam is kept fixed the electric intensity inside the metalwill be different for different planes of polarization, or for different an-gles of incidence, or for different wave lengths. Moreover, even if theintensity, color and plane of polarization of the incident light are allkept fixed, and only the angle of incidence is allowed to change, thecurves which relate the intensity inside the metal to the angle of inci-dence will be of different shapes at different depths below the surface,due to the fact that the light is extinguished more rapidly the moreoblique its angle of incidence. This last point is illustrated by Fig. 21,in which the electric intensity is plotted against angle of incidence atvarious depths below the surface of a plate of rubidium.

The rate at which energy is absorbed by a metal is, according to theMaxwell theory, directly proportional to the electric intensity at theplace where the absorption takes place; that is, inside the metal. The con-stant of proportionality is, in this case, an intrinsic property- of themetal. That is, it does not vary with the plane of polarization or angleof incidence of the incident light, nor does it vary from place to placewithin the metal. It may vary from wave length to wave length, butonly if the optical constants also vary.

308 [J.O.S.A., 22

ABSORPTION OF LIGHT BY METALS

It must be emphasized that this "intrinsic absorbing power," as weshall hereafter call it, is based upon the electric intensity inside themetal. It is not what is commonly called the "absorbing power" of themetal. The latter is the ratio of the rate of absorption to the intensityof the incident beam, and contains not only the intrinsic factor, but alsothe other factor of proportionality to which we referred in the firstparagraph. It therefore varies with almost every conceivable change inthe incident beam. As we shall see in §24, this common "absorbing

U(a (b)

L ZL0 0~~~~~~~~~~~~~~

L ~~~~~~~~~~~L

L ~~~~~~~~~~~L

0 30 50 9 3 0 9

U) U)~a)(b

FIG. 21. ELECTRIC INTENSITY VS. ANGLE OF INCIDENCE

The electric vector in (a) is normal to, and in (b) parallel to, the plane of incidence. The opticalconstants used are those of rubidium at Ao=4893 A: N=0.134, Ko=0.868. The numbers on thecurves denote depths (in c X 0-f) below the surface of the plate.

power" is not primarily governed by the ability of a substance to absorbenergy. In fact, it might more appropriately be called a "refractingpower," particularly since it depends upon the material through whichthe light arrives,' as well as that in which absorption takes place. How-ever, in order not to depart too widely from established usage, we shallcall it the "bulk absorbing power" of the material, expanding thisphrase occasionally to such forms as the "bulk absorbing power of plati-num from air," or " . . . from glass," when it is necessary to call particu-lar attention to the medium of incidence.

2 The term "absorbing power" usually implies that the incident beam is in free space.

June, 19321 309

THORNTON C. FRY

It does not seem unreasonable to assume that the relationship betweenthe rate at which a metal absorbs light energy and the rate at which itgives off photoelectrons is also an intrinsic property of the metal: thatis, that it does not depend at all upon plane of polarization or angle ofincidence and that its relation to wave length is not a very complicatedone. This hypothesis has neither been proved nor disproved. It merelybelongs to the class of things which, if experimentally demonstrated,would be characterized as "what was to have been expected." If itis true, the highly complicated way in which photo-emission varieswith color, plane of polarization and angle of incidence must be attri-buted primarily to the fact that the absorbing power of the metal varieswith all these factors, not to any peculiarities of the mechanism of photo-emission. It is even possible that part of the variation of emissivity withwave length may be capable of a similar explanation.

On the other hand, even if the hypothesis is false, the ratio of photo-emission to incident intensity is still the product of three factors: onehaving to do with the refractive peculiarities of the metal, a second withits intrinsic absorbing power, and only the third with the mechanism ofphoto-emission per se. Before we can effectively study the third, there-fore, it seems highly desirable to determine the first two independently;by doing this, we shall be able to correct our observations for purelyoptical influences, and thus arrive at results which can properly be saidto represent the intrinsic photo-emissivdity of the metal.

Such a program is being carried out in the Bell Laboratories by Dr.H. E. Ives and his associates. Some of the experimental results have al-ready been published,3 but the study has so far been handicapped bythe lack of reliable optical constants in the spectral region where thework should be done, and by the need for an adequate theory of thepurely optical effects. As was said above, the present paper aims tosupply the latter need.

For simplicity, we have so far spoken as if we were to deal with onlytwo media. But as the experimental set-up involves three media moreoften than two, our mathematical study must also include three media:the slab or film in which we are principally interested, the base uponwhich it rests, and the medium through which the light reaches it.Naturally, the presence of the base does not affect the "intrinsic ab-

3 The Vectorial Photoelectric Effect in Thin Films of Alkali Metals, H. E. Ives, Phys. Rev.38,1209-1218; 1931; The Photoelectric Effect From Thin Films of Alkali Metal on Silver, H. E.

Ives and 1. B. hrlggs, PLhys. Rev. A9, 147-1489; 1931.

310 [J.O.S.A., 22


sorbing power" of the film. It may, however, profoundly affect the lightintensity within the film, and therefore cause the amount of energy ab-sorbed to vary with wave length, plane of polarization, and angle ofincidence in a way quite different from that observed with only twomedia. This is particularly true if the layer is so thin that the resultsof multiple internal reflection are important. In such cases, we shallhave to distinguish between the "bulk absorbing power" of an opaquelayer and the "lamellar absorbing power" of a thin layer.

We shall find (§24) that when the film is neither nearly transparentnor nearly opaque, the mathematical formulae for this lamellar absorb-ing power are so involved that their physical interpretation is rathervague. But when the film is almost completely transparent, they sepa-rate naturally into factors which are quite easy to interpret; and of coursethey degenerate into those appropriate to two media when the filmbecomes opaque.

§23. Intrinsic Absorbing Power: The instantaneous rate at whichenergy is absorbed per unit time per unit volume is given by the formula

Q = o-E2 /47r.

Here E means the instantaneous value of the physical electric force, and is,of course, to be identified with the real component of the complex ex-pressions with which the present paper deals. The question which weraise, therefore, is this:

How much heat energy is absorbed by unit volume of a given sub-stance, in unit time, from a plane wave of unit intensity?

To answer this, we must notice that in the most general case, whereE is a rotating vector, it has two components at right angles to oneanother with amplitudes Eou and Eou, respectively. (See §7). Thesecomponents are 900 out of phase with one another: hence they can bewritten as Eou cos pt and Eou sin pt, respectively. As these componentsadd vectorially, we have

0,Q = -Eo 2 (u 2 cos2 pt + u2 sin2 pt).

4~r

This is the instantaneous rate of absorption. The average rate (timeaverage) is obviously

p 27r/p AQ K =-f Qdt -E 2(u2 + U2),27r 8er

since the average value of either cos2 pt or sin2pt is 1/2. But, upon recal-ling the meanings which were attached to the symbols u, u and Eo in §7,we readily see that u 2 +u 2 = 112 + Im2 I + 1n2 1; hence that

June, 1932] 311

THORNTON C. FRY

Q |Et + E +lE2 8'r

Finally we may note that, by (8), (10) and footnote 7 (§6) ,'

a= 2NKopb8whence we have

NKop 2

Q =- ( E.2 + E ,2 | + I E72 I ).(9(42r+Iul li l.(59)

Now, the mean square value of the electric force (the "electric inten-sity," or simply the "intensity") is

2 1(j E.2 + I ES2I + IE 2 1 )

and since we have defined the term "intrinsic absorbing power" to bethe ratio Q we have

AI = o/4ir = NKoP/27r.l. (60)

It should be said in passing that the result (59) may not be so obviousas it seems. Its truth for plane or elliptically polarized light is self-ap-parent; but not for hybrid light in which the phase relations among thecomponents Ez, E,, and E, are rather peculiar.

We must also note that the results contained in this section are ab-solutely general. We are at liberty to apply them to any medium, pro-vided we use the optical constants appropriate to that medium.

§24. Case of Two Media: Bulk Absorbing Power: The study of re-flection and refraction in the case of two media has already been carriedout in §§13-15, the final results being presented in Tables 1 and 2. Toapply these results to our present purposes we need only recall that ourupper medium is a dielectric. The lower one may be either conductor ordielectric, but it must be "semi-infinite." Mathematically this meansthat it must have no other boundary than the plane z = 0 which sepa-rates it from the upper medium. Physically, it implies only that thepossibility of light getting back to this plane by internal reflection canbe ignored: hence it is satisfied by any opaque body, and also by trans-parent bodies whose shapes are such as to assure that the refracted beameventually emerges from other surfaces than the one through which itentered.

We define the "bulk absorbing power" of any such medium by theformula

AB = 1- E2 /E12; (61)In this and many equations which follow, we replace the velocity of light, which was de-

noted by c in §11, by its equivalent pAo/27r, in order to avoid confusion with the direction

cosine c that appears constantly throughout this part of our work.

[J.O.S.A., 22312


that is, one minus its "reflecting power." From Table 1 we readily findthis to be

k- k 2 2k 2 k(ABI Re-i ~~~~~(62 1)

AR± = 1- k k+ki k + k k

when the electric vector is normal to the plane of incidence; and fromTable 2

AB1K = 1 - K 2 2K 2 RK- (62 ||

K + 1 |K + Ki K

when the electric vector is in the plane of incidence.5

To get a physical insight into the meaning of (61) and (62), we goback to the equations listed in §2, and to the long footnote in §6. Wethere find that6

h = 27ri/Ao, (65)

q = (27r/Ao) (N + iKo). (66)

Hence it follows from the definitions of k and K (Tables 1 and 2) that

k = (N + iKo)c/lA (67)and, since q/g = hq,

K = - c/(N + iKo). (68)We also note that the total electric force e just above the bound-

ing plane z = 0 is the vector sum of the forces in the incident andreflected beams. In the I case the x and z components are zero, andthe y components are [by equations (2)] Emeeiq(ax+by+cz)-iPt andE2M2 eiq(a2 x+b2y+c2z)-iPt, which, upon putting z = 0 and taking the otherquantities from Table 1, add up to

8 = (E + E2)eiqrx-ipt;hence

6 The third member of each of these equations is obtained from the second by means of thefundamental identity

I 12- b 12=| a + b 12 R be (63)

which is satisfied by any complex numbers a and b. Re x means the real part of x.There is another identity of a similar kind which we shall have occasion to use later, and

which may best be listed here for reference. It is

Re (1/a) = I 1/a Re a. (64)

It need hardly be-mentioned that these equations are quite general and therefore applyto any medium provided suitable subscripts are attached. That is, in the lower medium wehave q=27r(N1+iKo)/1Ao, etc., the "sub 1" being affixed to the N and K0 which vary frommaterial to material, but not to Ao, the "ether wave length" of the light, which does not.

June, 19321 313

314 THORNTON C. FRY [J.O.S.A., 22

A, 2k_= . (69)E k + i

In the || case there are two components of the electric vector. Treatingthem just as the y-component was treated above, we find that

He 2KI-= - V1-r2 , (70)E K + Ki

Fez 2K- r. (71)

E K + Ki

Finally, we may note that the tangential components of the electricvector E1 just below the bounding surface (that is, at z = 0) are identicalwith (69) and (70), as they must be by virtue of Boundary ConditionII, §14. The normal component of E1, however, is not (71) but

E1 2K g=- -r.

E K+K gi

Hence we haveEl,/z = g/gl. (72)

This coefficient g/gl defines the discontinuity between the normalcomponents of electric force on the two sides of the boundary.7 Thoughwe have derived it from a particular example in which there are only twomedia and the bounding surface is a plane, it is nevertheless a perfectlygeneral relation, satisfied at all surfaces where two different media meet.Hence we have the rule that:

At the boundary of any two media reflection and refraction of light takeplace in such a way that the tangential components of the electric vector E,and the normal component of gE, are continuous.

We now return to (62 1), and note that, by virtue of (69), it reducesto

7 When the media are both dielectrics, g/gi reduces to E/fE and (72) becomes the well-known rule that the normal component of displacement is continuous.

If the lower medium is a perfect conductor, gi is infinite, and hence E1z=O, as it should be.In the more general case, in which both conduction and displacement currents exist,

neither of these rules is valid. Instead, the general relation is found from the second of equa-tions (1) to be that aE 2+ eaE/t must be continuous; for the tangential components of H, andtherefore the z-component of curl H, are known to be continuous. As we are dealing withplane waves in which the time factor is eiP', this rule requires continuity of (a-ip)Ez; orwhat amounts to the same thing, of gEe.

Finally, we may remark that if the two media are of like permeability, so that It =pa g/g,is identical with q2

/fq2. This explains the appearance of the factor q2

/lq2 in the paper referred

to in Footnote 3.


A B l V= Re- (73 1)E k

while (6211) can be shown to be equivalent to8

ABII + ( E R (7311)

The interpretation of these equations scarcely needs comment: ineach case the first factor is the ratio of the intensity just below the sur-face to the intensity of the incident beam, while the second factor playsa role somewhat similar to the "intrinsic absorbing power" in equation(59). But it is not the intrinsic absorbing power, nor indeed any intrinsicproperty of the medium, as is evident from the fact that it varies withthe angle of incidence.

In fact, the "bulk absorbing powers" (62), or (73), are not trulynamed at all. They are in reality phenomena of refraction, not of ab-sorption. Whatever succeeds in passing the bounding surface is absorbedsooner or later, if the lower medium is opaque, or passes out of thesystem if it is transparent. The amount "absorbed" is therefore deter-mined solely by the reflecting power of the metal; not by its subsequentdisposal of the refracted portion.

8 To establish this, we first note that, by definition,

i/Kl = (c'ku/ci2k). (74)Also, it follows from Table 2 that

c2/Al2 = 2 + (r2q2c2/q2c12).

We therefore havek / q'c k (75Re-= C

2 Re- + r2Re (q 2 k + ie(75)

the term i having no effect on the real part of the expression as long as e is a real member.But we may write the last term in the form

r2Re LY' qic qc - (76)k q2c qc 1](6

in which form it is easy to give such a value that the term in parenthesis will be real. To dothis, call qc/qicl =u +iv. Then the parenthesis factors into

(a + iv) + i + i -)which will obviously be real and equal to lt+v2= qc/qic1 j2 provided we set e equal to -2vy/pl,as we may do since this value is real. Thus (76) becomes

and hence by (64) and (75), I qc/qic 2(ki/k);

Re-= -|-Re-k + r Rek Kim

June, 1932] 315

[J.O.S.A., 22THORNTON C. FRY

However, as it is this quantity which is spoken of as "absorbing

power" in classical optics, it would probably only create confusion to

replace it by a more appropriate term.

§25. Case of Three Media: Lamellar Absorbing Power: We now turn

our attention to the case of three media, separated by the parallel

planes z = 0 and z = - r as shown in Fig. 22, which is, except for minor

modifications, a reproduction of Fig. 18.

In our physical application to the problem of photoelectric emission,

the middle substance will ordinarily be a thin film of alkali metal, the

bottom one the base upon which it is deposited, and the upper one air or

vacuum. It will, therefore, save a great deal of circumlocution if we refer

Ircdent \eam /

eam 2

Beam 1\ / eam 3

Y ~~~Z=-t

Beam 4

FIG. 22.

to them simply as "film," "base" and "air." However, the results which

we shall obtain will be valid no matter what materials are used for

film and base, provided only that the upper medium is a transparent

dielectric. In other words, unless exceptions are explicitly noted, the

only restrictions are that all three materials shall be isotropic and that

the extinction coefficient of the upper one shall be zero.

Moreover, so far as the present section is concerned, the film need

not be "thin." It can have any thickness from zero (which is a case of

two media) to infinity (which is again a case of two media).

A beam of plane-polarized light is incident upon the upper surface of

the film. Part of it is reflected as Beam 2, part is transmitted into the

base as Beam 4 and the rest is absorbed by the film, in which there are a

refracted Beam 1 and an internally reflected Beam 3.

Symbols which refer to these various beams will be supplied with

corruponlding 505bcriptq1 Thii9, if they have no subscripts, they refer

to the Incident Beam, while if they have subscripts 1, 2, 3 or 4 they refer

316


TABLE 3. Reflection and refraction from a film. E normal to the plane of incidence.

1=0 a=r 1'=-cm=1 b=O m'=0n=O c=-(1I-r 2 )1/2 n=a

11 =0 a, = qrlql 11 = =- Cm = 1 b==O act=°n =0 cl= -(1 -q2r'/ql 2

)l2 ni' = a,

12=0 a2=r 12'= -C2m2 = 1 b2 =0 M2 =0fl2=0 C2= +(1-r 2 )1/2 n2' =a2

13=0 a3 = qr/q 1 3=-cfM3= 1 b3=0 m3,=0n3=0 c3= +(l- q

2r2/ql

2)I12 i3 =a3

14 = 0 a4 = qr/q4 14'= -C4m4 = 1 b4 =0 M4 =014=0 C4 = -(1 -q2j/q 4

2 )1I2 = a4

El 2k(k 1 + k4 )X2

E (k + k)(k + k4)X2 + (k - )(k- k4)E2 (k - k)(k 1 + k4)X2 + (k + k)(k - k4 )E (k + k)(k + k4)X2 + (k - k1 )(k - k 4) (77 1)E3 2k(ki - k 4)E (k + k)(k + k4)X' + (k - ki)(k - k4 )

E4' 4kkX

E (k + k)(k + k4)X2 + (k - k)(k - k4)

Notes:1. The symbol k is used for the combinaton iqc/h, the q, c, and h all being understood to

have the same subscripts as the k.2. The symbol X is used for eiq 1,1.3. It is not necessary to write the H's. They are defined by (9), which is valid for any

medium, provided q, h, E and H all have the subscript appropriate to that medium.4. The signs affixed to the radicals denote the signs of the real parts.

to Beams 1, 2, 3 or 4.9 In addition, since we have no occasion to discussthe values of the various electric and magnetic forces at other placesthan the bounding surfaces, we shall understand all such symbols todenote values at the upper surface z =0 unless a prime is attached, inwhich case they denote values at z =-

In order to facilitate reference, we assemble in Tables 3 and 4 the in-formation needed to define the amplitudes, directions of propagation,

9Certain quantities which measure intrinsic properties of the three substances necessarilyhave the same values for the Incident Beam and Beam 2, and also for Beams 1 and 3. The index

June, 1932] 317

TABLE 4. Reflectiois and refraction front a filn. H normal to the plane of incidence.

I=c a=r l'=0m1=0 b=0 n' =1n= -a c= -(1r2)1/ 2 n'=0

11=Cl a, = qr/q, 1,'=0O

mIn,=O bi=O ml'= 1

11= -a, c =-(1 -q2r2/q 2) 12 ' 0

12 = C2 a2=r 12'=5

M2 =0 b2 =0 mt' =1

112= -a2 C2= +(1 -r2)12 n2

13 = c3 a3 = qr/gq 13 =0

z3 =O b3=O M3 = 13 = -a3 C3 = + (1- q2r21qy 2)12 ns'= 0

14=c4 a4 =qr/q4 14'=O

1114=0 b4 =O in 4 ' 1

n4= -a4 C4= -(1-q2r2/q42)ll2 n4 '=O

HI 2K(K, + 4 )X2

Hl (K + KI)(KI + K 4 )X2 + (K - KI)(KI - 14)

H2 (K - Ki)(Ki + K4 )X2 + (K + KI)(Ki - 4)

1 (x + K)(Ki + 4)X2 + (x-x)(xi-K4) (77 )

H3 2K(Ki - K4)

H (K + K)(Ki ± K4 )X2 + (K - KI)(KI - K4)

H4' 4KK1X

H (K + K,)(Ki + Q 4 )X2 + (K - K,)(KI - 14)

Notes:1. The symbol K is used for the combination iqc/g, the q, c, and g being understood to have

the same subscripts as the K.

2. The symbol X is used for e"l"i.

3. It is not necessary to write the E's. They are defined by (9) which is valid for any

medium, provided g, It, E and H all have the subscript appropriate to that medium.

4. The signs affixed to the radicals denote the signs of the real parts.

damping, and states of polarization of the various beams. This material

is derived from §§14-17, and its interpretation should be clear in the

light of the extended comment of §§2-10. The form of presentation is

identical with that used in Tables 1 and 2.

of refraction N, the extinction coefficient Ko and the propagation constant q belong to this

class. In the case of such quantities the subscript 1 is used consistently for the film, even when

we are talking about the internally reflected Beam 3, and no subscript at all for air, even when

dealing with Beam 2.

[J.O.S.A., 22THORNTON C. FRY318


With this information before us, we are in a position to compute thez-components of the Poynting vectors (or rather the time-average ofthese components) for the Beams 1, 2 and 4. As explained in §11, theserepresent, respectively, the amount of incident energy which falls persecond upon a square centimeter of the upper surface, the amount car-ried away from this square centimeter by the reflected beam, and.theamount carried through a square centimeter of the lower surface into thebase. If we call these S, S2z and S4z, respectively, it is obvious that eachsquare centimeter of illuminated film must absorb ISz I -S2z - IS units of energy per second. The proportion absorbed, or the lamellarabsorbing power of the film, is

AL =

However, since the energy-flow in the incident and transmitted beamsis downward (that is, in the negative z-direction), while that in the re-flected beam is upward, the signs of S. and S4z will be negative, and thatof S positive. Hence, when algebraic signs are fully accounted for,

S 2z S 42AL = 1 + -- * (78)sz So

We now proceed to the evaluation of S, S2, and S4z, taking the Icase first.

As the upper medium is a dielectric, the electric and magnetic vectorsare in phase and hence

S = (pAo/167 2) [EH] 0,

where the brackets denote the vector product. But, in the I case,[EH]Z = [Emg] EH Ic, and therefore

S = (pAo/167r2) EH c.

We may remark in passing that, by the use of (8), (9) and (10), it ispossible to throw this into the alternative forms

S = (pA 0 /167r2) E2 k = - (pAo/167 2 ) H2 K, (79)

which we shall have occasion to use later on. It also follows that, if thebeam is of unit intensity, (that is, if E 12=2) the amount of energywhich falls upon one square centimeter of the plane z = 0 is by (67)10

10 The negative sign is explained as follows: Since the beam is directed downward, c, k, andtherefore S also, are negative. But by "amount of energy" we evidently mean an essentiallypositive quantity. Hence we must replace the negative numbers c and k by positive numbers-c and -k.

June, 1932] 319

- pAok/87r2 = - pAoNc/87r'L.

By an exactly similar argument we find the z-component of the Poyn-

ting vector in the reflected beam to be

S22 = - (pAo/167 2 ) E2 I k = (pAo/16r 2) I H22 K. (80)

Dividing this equation by (79), and replacing E2 /E 12 by its value as

given in Table 3, we have"

S2z I (k - k)(k, + k4)X2 + (k + k)(ki - k4) 12

Sz I (k + kl)(kl + k4)X2 + (k - ki)(k, - k4)

This result has been obtained for the I case. It is physically obvious,

however, that (79) and (80) must be true for the other case also, since the

relationship of the intensity of the light to the magnitude of its electric

force-vector does not depend upon the plane of polarization; and this

conclusion is easily verified by direct computation of S, and S22 from the

data of Table 4. If, then, we again divide (80) by (79) and replace

|H2 /H 12 by its value as given in Table 4, we get the lamellar reflecting

power of the film for the I| case. It is

S22 (K - K) (K, + K4) X2

+ (K + K) (K, - K4) 2

= - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ (81 1)Sz (K + K) (K, + K4) X

2 + (K - K,) (K, - K4)

We are now ready to turn our attention to the light which passes into

the base. Here the situation is a bit more complicated owing to the fact

that this beam is "hybrid." Again, however, we may follow the proce-

dure laid down in §11.

We take first the I case, and note that, since E4 ' lies in the y-direc-

tion, we need only use the x-component of the magnetic force in our

computation of S4z. However, by Table 3, the x-component of magnetic

force is readily found to be H4'14', which, by (9) and Table 3 is easily

seen to be given byH4'14V/E4'= -iq4C4/

114 = -k 4.

If the base is a metal, k4 is a complex quantity. 2 Calling it

k4 = Re

11 This is the lantellar reflecting power of the filint in the presence of tile base. It reduces to the

bulk reflecting power of the base when the thickness of the film is made zero (this corresponds

to X= 1), and to the bulk reflecting power of the film material alone, provided the thickness

of the film is made infinite (this corresponds to X=0). As we have said, the negative sign

merely reflects the fact that the energy flows downward in the incident beam and upward in 4

the reflected one.12 This is also true even when the base is a dielectric, if total reflection takes place at the

[J.O.S.A., 22THORNTON C. FRY320


the argument of §1 shows that

S4 = (pAa/167w2) I E4'H4'14' I cos = (pAo/1672)| E4 ' 2R cos

= (PAa/16,r2) E4' 12 Re k4.

Dividing this by (79), and taking the value of E4'2/E2 from Table3, we get

S4Z, l4kk 1X 2

S2 (k + k1)(k + k4)X2 + (k - k1 )(k - k4) Re (k4 /k)* (821)

This formula applies to the I case. For the || case we find withoutgreat difficulty that

542 4KKIX 2S42 4KK 1X 2~~~~~~~~~~Re (K4/K) . (82 fl)SI (K + K) (Ki + K4) X2 + (K - K1) (K1 - 4)

By substituting (81) and (82) in (78) we find the absorbing power ofthe film in the presence of its backing plate. The formula thus obtained,however, may be considerably simplified by means of certain tediousthough not especially difficult algebraic transformations. As they arepurely formal, and involve no physical concepts whatever, we shallomit them and state our result at once in the final form which appearsto be most convenient for purposes of calculation:

ALI = 2 [| (k + k4)X + (k - k4) 21 k(X + 1) 12 Re k1(X-1)k(X + 1)

+ (k + k4)X - (k1-k4 ) 2 | k(X- 1) Re k(X + 1]

| (k + k)(k1 + k4)X2 + (k - k)(k1 - k4) 2 (831)

The formula for AL11 is identical, except that all k's are replaced byK's. We shall refer to it as (83 ||).

-This equation, then, gives the lamellar absorbing power of the filmfor any angle of incidzence, for any thickness of film and (with the k'sreplaced by KS) for either plane of polarization of the incident light. It isvalid whether the film and base are conductors or dielectrics. It is evenpossible for the film and base to be of identical material, in which case

lower boundary of the film. Our argument, though nominally carried out for a metallic base, isgood for dielectric bases also, whether totally reflecting or not, since real values of k4 merelyimply thatq =O.

June, 1932] 321

THORNTON C. FRY

(83) gives the proportion of the incident radiation which is absorbed by

a layer of this material adjacent to the bounding surface.§26. Approximate Formulae: Lamellar Absorbing Power of Very Thin

Films: We have already indicated that the purposes of this study have

to do primarily with very thin films, so thin in fact that they are virtu-

ally transparent. For such films we can derive approximate formulaewhich are both simpler than the ones given in §25, and also easier tointerpret physically.

To this end, it is convenient to introduce the notation.

e = iqici = klIzhP, (84)

and assume that this e is a small quantity.'3 Then we have approximately

X 1, X- 1 E, X + 1-2; (85)and therefore (83 1) becomes

2k 2I kie k 4 e 2 k,1ALl I Re + - Re- (861)

k+k4 L k ki ke.

By an identical argument we also get from (83 1)

2K 2 K1E K4,E 2 K1lAL[ K R +Le- Re-J (8611)

g + K4 K KI Ke

The formal analogy between the I and || cases ends at this point,and we shall therefore find it most convenient to drop the || case for atime and concentrate upon the study of (86 1).

To begin with, we must recall that the upper medium is a dielectric.Hence by (67)

' ~~~k = NcIA

is real. Therefore, upon using (66), (67) and (84), we find without diffi-

culty that

13 The physical significance of this assumption is easily seen. As explained in §§4 and 6, the

real part of iqlcit measures the amount by which light is attepuated in its travel from the

upper to the lower surface of the film. The imaginary part measures the phase change that

takes place in the same vertical distance (which is less than the phase change which takes

place in the same distance when measured in the direction of propagation). To say that e is

small, therefore, means that the film is thin enough to be nearly transparent, and also that it

is thin when compared with the wave length of the light in the film (not its ether wave length).

In order to meet the latter restriction r ought not exceed some such value as A/20.

In the experimental study with which the present paper is mainly concerned, the film is

usually a monomolecular layer of an alkali metal. For such films the conditions are probably

met for ether wave lengths greater than 1OA.

[J.O.S.A., '22322


Re(kil/kE) = 0,

Re(kie/k) = -47rgNiKio/culAoN.

Finally, noting that -c is the cosine of the angle of incidence, (86 1)takes the form

(NK1OP) (K 2k4>

ALl_ 27rl lk + 1 (87(NAop cos I)

For purposes of physical interpretation (87 1) has been divided intofour factors. We see from (60) that the first in the numerator is the in-trinsic absorbing power of the material of which the film is composed.That is, it is the rate at which heat energy is produced in unit volumeof film material by a wave of unit electric intensity, this intensity beingmeasured inside the film.

The interpretation of the second factor is clear from (69). It is theratio of the total intensity just above the surface of thefilm to the inten-sity of the incident beam, computed as if the film were absent and thereflection took place directly from the base. It need hardly be notedthat by Boundary Condition II, §14, the intensity just below the surfaceis the same as that just above.

The factor in the denominator is the rate at which energy is carried tounit area of film by an incident beam of unit intensity.

The product of these three factors, then, gives the ratio of the energyabsorbed per unit volume of film to the amount falling upon unit areaof film.

Finally the fourth factor, , is the volume of film material per squarecentimeter of exposed area.

Turning now to the || case and applying the transformation (64) tothe last term, we readily throw (86 11) into the form

2K 2 K1E

2K4 2 KE

A L -I- .~ Re + ~Re~- (88)K + K4 K K + K4I Ki

But by using (84); (65), (66), (67) and (68); and then finally (8), wereadily find that

Re L1 = r2 : g 2 (NlKlop)( 87r~ ) K gi 27 N/l NAp 8 I

and

June, 1932] 323

THORNTON C. FRY

* / NiKiop\ 87r 21Re-= (1_ 2) - - I

K1 2ir/ l NAop cos I

Inserting these in (88), and arranging the terms somewhat after thefashion of those in (87 1), we get finally:

NlK1.oP( 2K4 2 g 2

2 K2(--)( K (1 - 2) + - ± K 2

A (NAop cos I)

87r2A

We have again four factors, of which three are identical with three in(87 1). The meaning of the other is easily seen by reference to (70),

(71) and (72). It contains the components of electric intensity just above

thefilm, computed as if the film were absent, with the normal componentcorrected for the discontinuity that must take place on passing theboundary.

The net result of the argument then is this:In the case of thin 4 films, the lamellar absorbing power can be found to a

high degree of accuracy by

(a) ignoring the presence of the film and finding the components of totalelectric intensity that would exist just above the surface if reflection took

place directly between air and BASE [by using (69), (70) and (71) ];

(b) correcting the normal component for the discontinuity which musttake place in crossing the surface between air and FILM [by using (72) ];

(c) finding the rate at which energy is absorbed by the film material from

a wave of this amplitude [by use of the intrinsic absorbing power (60) and

the thickness r ];

(d) and finally dividing this by the rate at which energy is conveyed to thefilm by the incident beam.

In other words, both (87 1) and (87 I) can be written in a single

form:NiVIKIOP (E 2 E 2 i 2 Q e 2 )(9

\7 .t E E gi ElAL ~~~~~~~~~~~~~~~(89)(NAop cos I

87r2y A

which applies to any plane of polarization; it being understood that thetotal electric intensities fS, e, E, are those which would exist ifreflection took place directlyfrom the base.

14 That is, thin enough to justify the use of the approximations (hS). (See Footnote 13.)

[J.O.S.A., 22324

June, 19321 ABSORPTION OF LIGHT BY METALS 325

.It may be noted in passing that the only factors of (89) that dependupon the character of the film material are its intrinsic absorbing powerand the discontinuity ratio g/gi, neither of which is in any way depend-ent upon the angle of incidence of the light, or upon its state of polari-zation.

§27. Some Numerical Illustrations: As a first illustration of the use towhich the formulae of §§23-26 may be put, we may compute the bulkabsorbing power of rubidium, using the optical constants N = 0.134,

0

>,

DL

C3

L4-

0.Q

0'

L

0,61n-L2

0 30 60 90Angle of Incidence

(a)


(b)

FIG. 23. ABSORBING POWER VS. ANGLE OF INCIDENCE

The electric vector in (a) is normal to, and in (b) parallel to, the plane of incidence.The optical constants used are those of rubidium at A =4893 A: N=0.134, Ko =0.868.The dotted curve is bulk absorbing power; the heavy curve lantellar absorbing power of rubidium

on rubidium; and the lighter curves the lamellar absorbing powers of thin layers at depths below thesurface (in cm X 10-6) indicated by the numbers on the curves.

Ko = 0.868 corresponding to wave length Ao=4893A. For this purpose,of course, we make use of equations (62 1) and (62 ||) of §24.

The result is shown by the dotted curves of Fig. 23 for the two planesof polarization.

As a second example, we may consider the lamellar absorbing powerof rubidium on rubidium: that is, the absorbing power of a very thinlayer near the surface of a rubidium plate. It is given by equation (89)

InoC

L

or

L.

0

L

00

.

L0

n

THORNTON C. FRY

of §26, provided we assign the constants of rubidium to both the filmand the base.

The only factors in this expression which vary with the angle of inci-

dence are 1/E 12+ I Q/E 12+ Ig/gi 12 z /E12 and cos I. The other

factors are unimportant so long as we are willing to plot our results inarbitrary units. But the factor I /E 12+ I Q/E 12+ Ig/g 2 I z,/E 12 is

the electric intensity just below the surface, which we have alreadyplotted as the curves marked 0 in Fig. 21. To get the lamellar absorb-ing power of rubidium on rubidium, therefore, we need only multiplythese upper curves of Fig. 21 by the secant of the angle of incidence,thus getting the heavy curves of Fig. 23. The deviation from the bulkabsorbing power is striking.

We may go a step further in this connection. We know that the rate

at which energy is absorbed by a thin layer at any depth within the rubi-dium plate is proportional to the electric intensity at that depth: that is,

to the various cur zes of Fig. 21. If, then, we divide each of these curvesby cos I, (which enters (87) in computing the rate at which energy

falls on the surface) we will obtain a set of curves showing what propor-

tion of the incident energy is absorbed at various depths. Plotted inarbitrary units, so as more clearly to show their different behavior as theangle of incidence changes, they are given by the light curves of Fig.23. Calling these, for the moment, "lamellar absorbing powers at vari-ous depths," we may state the rather obvious fact that the curve for bulkabsorbing power versus angle of incidence is a sort of average among thelamellar curves at various depths.'5

In saying that this is obvious, however, I cannot refrain from addingthat the dissimilarity of the lamellar curves themselves is a consequenceof the fact that light is damped out more rapidly when incident obliquelythan when incident normally; and I do not believe this fact is eitherobvious or well known.

Digressing for a moment to the subject of photoemission, we mayremark that if a correlation is to be expected at all between the photoemissivity of bulk material and its optical absorbing power, we would

expect it to befound in connection with the dark lamellar curve rather than

the dotted bulk curve; since the electrons emitted at appreciable depthsbelow the surface will probably never emerge.

As a third example, we may consider the variation of absorbing power

15 If we bad kept the scale factors in mind, instead of plotting everything in arbitrary units

as we have done in Fig. 23, the bulk curve would be just the integral of the lamellar curves

will pist to dth.

[J.().S.A., 22326


with wave length. Silver is one of the few metals for which the opticalconstants have been reliably determined over a broad spectral region,and fortunately also presents an interesting case from our point of view.In Fig. 24 we give the values of N and Ko as determined by Minor be-

4

x

x

2

0I1

x

x

gO X * @0% xxx x

I I I I I "I ?

2200 3000 4000

FIG. 24. Optical constants of silver

2

1

I 11 It 1.15000 6000

Wavelength

(Minor's values).

o L2200 3000 4000 5000 6000

Wavelength

FIG. 25. Total electric intensity vs. wave length. ( is te total electric force just above the re-flecting surface due to the superposition of incident and reflected beams. Minor's optical constantsfor silver were used.

tween Ao=2263A and Ao=5893A. Using these values in (69), (70) and(71), and confining our attention to a 60° angle of incidence, we readilyobtain the curves of Fig. 25 for the electric intensities just above thesilver surface (that is, /E 12= (Y/E 12 for the. I case, and I ii/E 12= I (/E 12+ I (z/E' 12 for the 11 case).

June, 1932] 327

THORNTON C. FRY2

Next we obtain the lamellar absorbing powers as given by (87 1)and (87 f1). We can no longer neglect the denominator and the first factorin the numerator, as we did in computing Fig. 23, since N1, Kjo and A0

are all variable. The curves marked ALi and ALII on Fig. 26, therefore,

2

O L2200

AL

3000 4000 5000 6000Wavelength

FIG. 26. Lamnellar absorbing power of silver on silver. Computed frot Minor's constants.

For a flm of thickness r the vertical scale readings are to be multiplied by 105 a.

1L

4,0

-oIn0a,

'.0

0 L2200 3000 4000 5000 6000

Wavelength

FIG. 27. Bulk absorbing power of silver. Computedfrom Minor's constants.

give quantitatively the proportion of incident light which is absorbedby a thin surface layer of the silver mirror from which it is reflected, solong as the thickness of the layer does not exceed the limits defined inFootnote 13. The scale readings are to be multiplied by 10v.

Finally, we substitute the optical constants of Fig. 24 in (73 ) and

6

4

S.W

0

U,

:-0~0-o4

A

[J.O.S.A., 22328

.


(73 1) and obtain the bulk absorbing power of silver over the same spec-tral region. These are shown in Fig. 27. The great dissimilarity betweenlamellar and bulk absorption is again apparent.

As a fourth illustration we may take an example in which the filmand base are of different substances. For this case we again discuss thevariation of absorbing power with angle of incidence, a platinum basewith the constants N4 =1.928, K40 = 3.175, and a wave length A0 = 4359A.-Ignoring the presence of the film, and using (69), (70) and (71), we com-

4

3

2

0


FIG. 28.

The x, y and z components of I /E 2 sec I vs. angle of incidence.The y curve isfor light; the x and z curvesfor 11 light.The optical constants N =1.928, Ko=3.175 correspond to platinum at A0 =4359 A.

pute the quantities 2 (v/E | I sec I, 2 QixlE I I sec I and Ez/E |2 sec I, thusgetting the results shown in, Fig. 28. Except for the factor sec I, theserepresent electric intensities just above the surface, the first one referringto the case of I polarization and the other two to the I| case. The fac-tor sec I has been introduced at this point purely for convenience: itoccurs in every subsequent formula to which we refer, and we save com-putation by introducing it as early as possible.

In accordance with (87 ||) we may get the lamellar absorbing powerALII of any thin film deposited on this platinum base by multiplying

June, 1932] 329

THORNTON C. FRY

I ( 2 /E 12 sec I by the Igig' 12 appropriate to thefilm material, and addingit to |,/E 12 sec I. There is in addition, of course, a scale factor com-

0 30 60 90Angle or Incidence

FIG. 29.

(1) Lamellar absorbing power of potassium on platinum- (II case).

(2) Lainellar absorbing power of platinum on platinum (|| case).

(3) | Cl/EI 2 sec I.(4) Lamellar absorbing power of potassiicm on platinum, or of platinum on platinum (I case);

or l±/l2 sedI.The vertical scale is in arbitrary units.

I

UJ.O.S.A., 22330


posed of various N's, Ko's and Ao's; but this does not vary with the angleof incidence and is of no importance so long as we are interested only inthe functional relationship of AL to I. Similarly, by (87 1), the lamellarabsorbing power ALI is (except for a scale factor) independent of thematerial of the film.

Using these ideas, we plot on Fig. 29 the lamellar absorbing power A LIwhich applies to any substance, and the ALII for films of potassium 6

5

z4 x

3

2

00 3 0 60 90

Angle of IncidenceFIG. 30.

The x, y and z components of /E 2 sec I vs. angle of incidence.The y curve isfor I light; the x and z curvesfor 11 light.The optical constants N=2.438, Ko=3.451 correspond to platinum at A=5461 A.

and of platinum. At this wave length the ratio of selective to normal lamel-lar absorption is nearly twice as high for a thin film of potassium on plati-num, as for the surface layer of platinum itself.

For comparison we also plot the curves

wI /pE12 sec I t= I tE sec I, bivE 1 tsec Is (urfa c m + L /E 12) sec I

which represent the intensity just above the surface multiplied by sec I.16 The constants (adopted from Duncan's data) are N, =0.0690; Kio=0.815.

331June, 1932]

[J.O.S.A., 22332[THORNTON C. FRY

As an illustration of the great changes that may be produced in

lamellar absorption by relatively small changes in the optical constants

of the film we include also Figs. 30 and 31, which are identical with

Figs. 28 and 29 except that they refer to a wave length Ao=5461A.

Though the platinum constants N=2.438, Ko=3.451, are now some-

what different than before, they produce little change in Fig. 30, in the

AL! curve, (Fig. 31) or in the ALl! curve for a platinum film (Fig. 31).

6 30 60 ' 90Anqle of Incidence

FIG. 31.

(1) Lamellar absorbing power of potassium on platinum ( case).

(2) Lamellar absorbing power of platinum on platinum (1l case).

(3) | (Yii/Ej 2 sec I.(4) Lamellar absorbing power of potassium on platinum, or of platinum on1 platinum (I case)

or CL 1/E| 2 sec I.The vertical scale is in arbitrary units.

But the AL II curve for a potassium film rises only half as high as the

(211 curve in Fig. 31, whereas in Fig. 29 it was twice as high. The con-

stants for potassium, N1 =0.0691; Kjo= 1.325, were again taken from

Duncan's data.

I

332

plane waves of light iii

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