JOURNAL OF THE OPTICAL SOCIETY OF AMERICA VOLUME 34. NUMBER 10 OCTOBER, 1944

Note on Specular Densities and Forward Scattering FRANZ URBACH

Institute of Optics, University of Rochester, Rochester, New York (Received July 7, 1944)

THE measurement of specular photographic density should be an absorption measure­

ment in which no light scattered by the photo­graphic layer interferes with the measurement of the transmitted light. In fact, a certain con­tribution of light scattered in the direction of the transmitted beam is never completely avoidable.

F. H. G. Pitt1 has shown how this contribution can be determined, using the fact that the trans­mitted and the scattered beam depend differently on the geometrical conditions of the measure­ment. This makes possible the determination on the one hand of the "true" specular density of a given sample, and, on the other hand, of the scattering of the sample in the direction of the transmitted beam. The latter is usually difficult to determine just because of the interference of the directly transmitted light. This "forward scattering" of the sample can be characterized by an empirical magnitude, the "scattering con­stant" S. If the sample be regarded as a second­ary light source, S represents its brilliancy in the direction of the transmitted beam, per unit illumination of the sample by the primary light source. It has been shown by Pitt that .S is in fact independent of the geometrical configura­tion. It is, however, naturally strongly dependent on the density of the sample. Pitt gives some empirical data concerning this dependence with­out further discussion although at least their qualitative interpretation will certainly have occurred to him.

It will be shown here that a crude quantitative treatment leads to a more complete understand­ing of Pitt 's empirical data and indicates that one of his results is an example for a rather general rule.

An absorbing and scattering sample of area a is placed between a point-like source and any light measuring instrument. Let the light source be at a distance d from the receiver and d1 from the sample. The specularly transmitted light will be, with a given light source, proportional to

10_D/d2 where D represents the specular density of the sample. The light scattered by the sample onto the receiver will be proportional to (1/d1)2×aS/(d — d1)2. The apparent transmission of the sample measured in this arrangement will be:

This formula has been derived and tested by Pitt, and the "scatter-constant" S has been found to be actually independent of the position of the sample.

Concerning the dependence of S on the density, Pitt summarizes the result of his measurements as follows: " I t can be seen that S increases rapidly to a density of approximately 0.4 and then decreases asymptotically towards 0. . . . The ratio of the illumination due to the scattered light to the specular illumination, when d1 = d/2 increases almost linearly with density." Table I gives Pitt 's numerical results.

TABLE I.

1 F. H. G. Pitt, Phot. J. 78, 486 (1938).

The behavior of S as a function of D is ob­viously owing to the fact that at low densities the scattering increases with the amount of the silver deposit, while at higher densities the absorption of the scattered light in the layer overcom-pensates this increase. The mathematical formu­lation can be based on the following assumptions. For an infinitesimally thin layer, taken as a secondary light source, the brilliancy of the layer is proportional to its density dD and to the flux density in the layer. The latter assumption will hold if the directional distribution of the flux is

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S P E C U L A R D E N S I T I E S AND F O R W A R D S C A T T E R I N G 593

sufficiently constant. The first assumption cer­tainly holds if the particle size distribution is constant, independent of the density, since the density is then simply a measure for the number of scattering centers. But also with varying grain sizes the assumption will be good within reasonable limits because for not too small grains both the density and the scattering power should increase with increasing grain size.

Consider now an infinitesimally thin layer within the sample parallel to the surface of the sample and at such a depth that the density between the layer and the surface which faces the light source is D'. If the incident flux density is unity at the surface of the layer it is 10-D' in the infinitesimal layer considered. If we disregard the illumination produced within the layer by its own scattering, which is certainly justified for low densities, we may put as a fair approximation

where the last factor represents the absorption of the scattered light in the layer on its way towards the measuring instrument. D' is variable while D is constant; the integration from D' = 0 to D' = D yields

c should be a constant characterizing the material

F I G . 1.

and independent of D. In fact the function (2) has the general shape described by Pitt, and the quantitative agreement (see Fig. 1) can be re­garded as fair if we take into account the uncer­

tainty of S shown by the irregular behavior of the empirical points in Pitt 's Fig. 4. Note that the relatively complicated curve is represented with a single adjustable parameter (the value c=10 has been used in Fig. 1).

F I G . 2.

If we calculate now the ratio R of the scattered light to the specular illumination, that is, the ratio between the two terms in formula (1), using our expression (2) for S we obtain

In agreement with Mr. Pitt's statement quoted above, this ratio is a linear function of D. Beyond this agreement our derivation suggests that the slope of the straight line is not an independent empirical magnitude but is determined on the one hand by the geometrical data of the arrangement and on the other hand by the empirical constant c of formula (2). A check of this relation with the data given by Pitt (d=l50 cm, d1=75 cm, a = 6.6 cm2) and with the constant from Fig. 1 we obtain

Figure 2 shows again satisfactory agreement of our deductions with Pitt 's data.

The most striking feature of our interpretation of Pitt 's experiment is, however, obtained by analyzing the position of the maximum in Fig. 1. By differentiation of (1) we find for the density at which the maximum occurs:

In Pitt 's paper the position of the maximum occurs as a purely empirical result and the specific value given by Pitt as quite incidental.

594 B O O K R E V I E W

According to our consideration, however, this value is by no means incidental. It is independent not only of the experimental arrangement but also of the material as long as it fulfills our basic assumptions. It represents the intrinsic rela­tionship of the definitions of scattering and ab­

sorption. It should hold, thus, quite generally also in other fields of physics (like x-ray or neutron scattering) that the forward scattering of an absorbing and slightly scattering layer reaches a maximum if the fraction of the incident beam, which is transmitted specularly, is just equal to 1/e.