Journalof the

Optical Society of Americaand

Review of Scientific InstrumentsVol. 15 SEPTEMBER, 1927 Number 3

THE TRANSPARENCY OF TURBID MEDIA*BY LUDWIK SILBERSTEIN

In a recent paper on the transparency of a developed photographicplate as observed under various conditions' it was assumed that thetotal light scattered by each particle (grain) is sent forward, i.e. inthe direction of and under acute angles with the incident light beam.It is well known that if the scattering particle, supposed to be spherical,is very small compared with the wave length, then, according to Ray-leigh's elementary theory and Mie's developed form of the electromag-netic theory,2 one-half of the scattered light is sent forward and one-halfbackward,3 the distribution of intensity being perfectly symmetricalwith respect to the plane passing through the particle and normal tothe incident beam. For particles, however, whose size is comparablewith the wave length there is a tendency of the scattered light to con-centrate in the forward direction. This tendency, derived theoreticallybut tested also experimentally,4 is for gold particles, for instance,already very pronounced at a diameter of 160mbt (with X550), while forsomewhat larger particles (180m,u) practically the whole scattered lightis sent forward, as a glance on Mie's figures (l.c., p. 429) will show.These and similar examples computed by Shoulejkin,5 who pushed the

* Communication No. 318 from the Research Laboratory of the Eastman Kodak Com-pany.

1 L. Silberstein and C. Tuttle. The Relation between the, Specular and Diffuse Photo-graphic Densities. J.O.S.A. & R.S.I., 14; , p. 365; 1927.

2 G. Mie, Ann. der Physik, 25, pp. 377-455; 1908.3 Unless the particle is a perfect conductor, when almost the whole light is scattered

backwards. Cf Mie, l.c., p. 430.As e.g. by R. Gans, Ann. der Physik, 76, p. 29; 1925.

6 W. Shoulejkin, Phil. Mag. 47, p. 307; 1924.

125

[J.O.S.A. & R.S.I., 15

ratio of particle size to wave length up to the value of two or threeunits, have suggested the adoption of the said assumption in the caseof the black silver grains of photographic emulsions, the size of thesegrains being in general of the order of the wave length of visible light.(Commonly, their diameters range from a few tenths to several wholemicrons.) The assumption of a unilateral spreading of the scatteredlight recommended itself also by the obvious simplicity of its conse-quences. Moreover, the object of that paper was only to correlate witheach other the two kinds of photographic "density" (diffuse and spec-ular) without studying each of them in detail as a function of thesize and the number of particles, and for this purpose the assumptionturned out to be accurate enough. It has, therefore, been adopted inthat connection, although implying the rather uncertain extrapolationfrom Mie's spherical metallic particles to the generally larger, shape-less, and spongelike silver grains of the photographic plate.

In the present paper the problem of light transmission through turbidmedia will be treated without this special, simplifying hypothesis. Itwill be assumed that of the light scattered (or rescattered) by each par-ticle a certain fraction r is sent forward and the remainder backward,the numerical value of this fraction being left free. The equationscorresponding to this general case are somewhat more complicatedbut can be solved without trouble.

Consider a plane-parallel layer of absorbing and scattering particleswhich will be supposed to be all equal. Let lo be the intensity of theincident light collimated normally to the layer at its front surface(x =0) and I the intensity of the directly transmitted light which haspenetrated to any depth x. The total energy scattered (per unittime) by a particle which is struck by this light can again be writtenAl, where A, the scattering coefficient, will be a function of wave

length, etc. Of this energy let the fraction

PAI

be sent forward and the remainder (1- I) Al backward (along rays,that is, which make with the incident beam angles 0 to 7r/2 and 7r/2to 7r respectively). Let KI be the energy absorbed by a particle, Nthe number of particles per unit volume, and dn =Ndx. Then, in thefirst place, we will have for the directly transmitted light

dl-- (K+), (1)

LUDWIK SILBERSTEIN126

TRANSPARENCY OF TURBID MEDIA

which'is independent of the partition of the scattered light. Next, if,at any depth x, S be the total flux of scattered light energy directedforward and S' that directed backward, a first contribution to dS/dnwill be PAI and (1 -A) Al will be one to -dS'/dn, the role of x beingfor the retrograde flux S' replaced by -x. Further, of the scatteredradiation S the amount KS will be absorbed and the amount AS willbe rescattered, per particle; of the latter, however, the fraction PASis again sent forward so that only (1 -P) AS is lost to the S-flux.Finally, the amount of S' rescattered is AS' and of this PAS' is sentin the negative and (1 -A) AS' in the positive direction of the x-axis.Thus the equation for S becomes

dS- = AI-,S-(1-)AS+(1-)AS',dn

and similarly, the equation for S',

dS1- = (1- )AI- KS'- (1- )AS'+ (1- -)AS,

dn

or, collecting the terms,

dS- =PA I - S+ (1- )AS', (2)dn

dS'--= (- )A I - aS'+ (1 - )AS, (3)dn

where

a=K+(-)A.(1), (2), (3) are the required differential equations for the three lightfluxes.' In addition to these we have the boundary conditions, viz.at the front surface (x = 0) of the layer, I = Io = 1, say, and

S=0, for n=O, (4)

and at the back surface,

S'=O, for n=fz, (5)

6 An explicit consideration of distinct light fluxes directed back and forth has already beenintroduced in A. Schuster's treatment of the problem of an incandescent "foggy atmosphere,"Astrophysical Journal 21, p. 1; 1905. Schuster limits himself, however, to the special case ofequal distribution (i.e. = 1/2), which would suit only very small particles. Moreover, hehas no term representing the collimated radiation but, in accordance with the nature of hissubject, considers only the scattered fluxes in either sense. This leads to two equations onlywhich (apart from the emission terms) differ somewhat from (2) and (3).

Sept., 1927] 127

LUDWIK SILBERSTEIN

where hi is the total number of particles per unit area of the'layer.(For =1 this system of equations reduces at once to our previousformulae, loc. cit.)

By (1),

where p = K+A. Thus the directly transmitted light would still obeyBeer's simple law.

To satisfy the equations (2) and (3) put

S= re-Pn+aefn+be-InSI = r'e-pn+ a'e In+ b''

where r, r', etc. are constants to be determined. Substituting into (2),(3) and comparing the coefficients of the three exponentials we find,first of all,

1-D.r+-r'=-1,a+Pr- r=1

so that r'= and r -1.Next,

a' a+3 (1- )A- = -X, say,

a (1- )A a-fl

bI CZ-O' (1 - )Aand - ( = y, say.

The first pair of these equations gives = [ 2- (1 -) 2A2]1/2 and thesecond pair gives exactly the same expression for /3'. Thus, eitherfl'=Bor P'=-/3. But in the latter case we would have X=,u and, bythe boundary condition (5), a+b=O. This, however, would clash withthe condition (4) which calls for a+b = 1. We are left, therefore, with

A =.8= [2_ (I _r)2A2]1/2= [K2+ 2 (I -)A ]1/2. (6)

(Taking the negative square root we would only interchange the rolesof a and b.)

The two fluxes now become

S=aeon+ bean_epn (7)

[J.O.S.A. & R.S.I., 15128

S'=XaePn+pbe-ftn, (8)

TRANSPARENCY OF TURBID MEDIA

whereca+ a-3

A= -X A=- * ~~~~~~~~(9)(1 - OA (-

The coefficients a, b will be determined by the boundary conditions(4) and (5), i.e.

a+b= 1, Xaen+ tbe-# =O,

whence, writing henceforth n instead of ii,

=-On __ .e b n__e_. (10)

Thus, by (7),

S= _e-pn,Xe On-. e-# n

where X, y are as in (9).Ultimately, therefore, the emergent scattered light flux, proceeding

forward, isS= __e-~20(K+A)n (1

(a+,) el n- (a- ) e-0 where

a= (I - )A+K, ,=[C2 +2 (I -)AK ]1/2,

and the intensity of the directly transmitted, collimated light,

I=e-+ n

Thus the total transmitted light flux S+I or (since o= ) the totaltransparency of the layer, as recorded, say, by means of an integratingsphere, will be

213S+I= +l)e2sn (a _ j)e-in (12)

which contains the original three constants only through their twocombinations, a and 13.7 It is this total transparency which is un-

I This differs, even for r= 1/2, from Schuster's result (loc. cit.) as regards the coefficientsIn fact, Schuster's formula for the total emergent radiation R reduces, in absence of emission, to

R=4c/ [(1+c)2eR-(1 C)2 e-],

where, in our symbols c =4 ,"+A while = IU (K+A), identical with our ,3 for = 1/2.

The difference in the structure of the coefficients is due to the fact that no collimated light (flux

I) has been taken into account. The dependence on n, however, or equivalently on the thick-ness of the layer (cf. infra) is essentially the same. Schuster's type of formula has also been ob-tained, by an entirely different reasoning, by Channon, Renwick, and Storr, Proc. Roy. Soc.,94, p. 222; 1918.

Sept., 1927] 129

LUDWIK SILBERSTEIN

ambiguously observable, while it would be hard to isolate I from thesuperposition of some unknown fraction of S. In photographic nomen-clature -log (S+I) would be the "diffuse density." Notice that ingeneral only I itself satisfies Beer's law, while S and S+1 do not obeythis law. (In the particular case = 1 we have a= =K=f3, so thatS=e-Kn-epn, as in the first paper on this subject. In this case both Iand I+S obey Beer's law.) For small values of fn the total trans-parency (12) reduces to 1/ (1 +can), and for large fin to 2fie-On/ (a+:).

It may still be interesting to determine the total scattered light S'thrown backward and emerging at the front surface (x = 0) of the layer.This will be obtained by putting in (8) n = 0. Thus

S' = a+,ub,i.e., by (10) and (9),

-~~~~~~~~~~~~~l -o e-flnS'= (1 - )A (a+)l-(-~-t 13)

or, compared with the total forward flux,

S/ (1- )AS (l-¢)A(ean- efn) (14)

S+I 2fi

For indefinitely increasing n the emerging backward flux (13) tends to

S., (1 - )A

a finite value, as might have been expected.In all these formulae n or fNdx is the number of particles over unit

area, whether their distribution in depth is uniform or not. If it isuniform (N=const.), the same formulae hold good when n stands forthe thickness of the layer, only that K and A are then the original con-stants multiplied by N, i.e. the absorption and the scattering coeffi-cients not per particle but per unit volume.

EASTMAN KODAK RESEARCH LABORATORY,ROCHESTER, N. Y.,

MAY 12, 1927.

[J.O.S.A. & R.S.I. is130