Experimental Study of Light Scattering in Turbid Media*)

The problem of the propagation of light in turbid media, involving allowance for multiple scattering, is of extremely great importance in a whole range of branches of science and technology. At the same time, despite the abundance of work in this field, it is still very far from being solved. The reason for this state of affairs should be seen in the extreme complexity of the phenomena accompanying the propagation of light in a turbid medium, as a consequence of which both the mathematical treatment and the difficulties of experimental investigation are complex. As a result, both the theoretical premises and the experimental methods usually have to be subjected to such a strong simplification that the results obtained with their aid prove to be of comparatively little significance. The way out, evidently, consists in carrying out a series of extensive, comprehensive investigations that would make it possible to embrace the problem not one-sidedly, but sufficiently broadly. Therefore the appearance of sufficiently extensive and comprehensive investigations in this field, such as the paper under review, should be regarded as a significant event.

The author set himself the task of experimentally investigating the distribution of brightness in a turbid medium filling an infinite half-space (i.e., a sufficiently large volume), under various illumination conditions. As the object, media were taken with a very large scattering coefficient and a very small absorption coefficient (diluted milk, rosin sol, soap solutions). The dependence of the brightness \(B\) on the direction of observation and on the depth of immersion in the medium was studied. The measurements were carried out by means of photoelectric photometers (with a selenium photocell). Photometers of two types were used: tubular and lens photometers—the input plano-convex lens directed the light flux onto the photocell; the angle of vision was limited by a diaphragm placed in the focal plane of the lens. Readings were taken with a galvanometer (without amplification), and it was found that the galvanometer readings at small illuminances were proportional to the brightness. The field angle of the photometers \(\omega\) varied from \(0^\circ .39\) to \(10^\circ .1\). The attenuation coefficient of the turbid medium \(K\) was measured on a König-Martens spectrophotometer by

) V. A. Timofeeva, Proceedings of the Marine Hydrophysical Institute of the Academy of Sciences of the USSR 3*, 35 (1953).

attenuation of light as it passes through a cuvette of definite thickness. The measurements showed that, for an optical thickness of the layer less than 2 relays (i.e. \(K l_1 < 2\), where \(l_1\) is the thickness of the layer), Bouguer’s law is well satisfied;

Fig. 1. Dependence of the brightness \(B\) in a milky medium \((K = 1.90\ \text{cm}^{-1},\ \omega = 0^\circ.39)\) on depth for various observation angles \(\varphi\).

at greater optical thicknesses of the layer, noticeable deviations occur, evidently due to multiple scattering.

The medium being measured filled a tank of dimensions \(50 \times 50 \times 20\ \text{cm}\) (i.e., not too large) and was illuminated through a hatch in the tank lid by direct vertical rays of the Sun (with the aid of a system of mirrors). The brightness was measured at various angles \(\varphi\) relative to the direction toward the zenith.

as a function of the depth of immersion of the photometer. The general picture of the distribution of brightness in a turbid medium \((K = 1.90\ \mathrm{cm}^{-1},\ \omega = 0^\circ39)\) is shown in Fig. 1. The following circumstances attract attention: a) for direct rays \((\varphi = 0)\) the brightness decreases with depth \(l\) according to the exponential law \(e^{-K_1 l}\) down to a certain depth corresponding to an optical thickness \(K_1 l \sim 5\) relays (the measurements showed that \(K_1 \approx 1.15 K\), where \(K\) was measured on a spectrophotometer). Then the rate of decrease of brightness with depth diminishes. b) At a depth of the order of 12 relays an exponential law of decrease of brightness is again established, of the form \(e^{-K'(l-l_0)}\), where (for \(0.3\ \mathrm{cm}^{-1} < K < 12\ \mathrm{cm}^{-1}\)) \(K' \sim \sqrt{AK}\) (Fig. 2), \(A\) is a parameter of the medium depending, in the author’s opinion, on the size of the scattering particles and on the absorption coefficient of the medium, and is, in order of magnitude, equal to \(0.003\)—\(0.01\) for various media; \(l_0\) is a certain parameter of the medium (according to the measurements \(K l_0 \sim 10 \div 12\) relays); c) for \(0^\circ < \varphi \lesssim 120^\circ\) the brightness first increases (from some initial value) and, after reaching a maximum, begins to decrease. For \(120^\circ \lesssim \varphi < 180^\circ\) no increase in brightness is observed—the brightness decreases monotonically with depth; d) beginning at a depth of the order of 10—12 relays, an exponential decrease of brightness with depth is established according to the law \(e^{-K'(l-l_0)}\), and \(K'\) does not depend on \(\varphi\) and has the same value as for \(\varphi = 0\), while \(K l_0\), remaining as before of the order of 12 relays, depends somewhat on \(\varphi\).

Fig. 2. Dependence of the attenuation coefficient \(K'\) under stationary conditions on the attenuation coefficient \(K\) for the direct ray.

In Fig. 3 are shown polar diagrams of brightness at various depths for the case of Fig. 1. At the top are indicated the corresponding values of the depth \(l\) in cm; the figures below show the values of the brightness at \(\varphi = 0\), the brightness at \(\varphi = 180^\circ\) at the surface \((l = 0)\) being taken as unity. The polar diagrams for \(l = 20\ \mathrm{cm}\) and \(l = 40\ \mathrm{cm}\) are enlarged by a factor of 10. In considering Fig. 3 the following features attract attention:

a) at small \(l\) the polar diagram has a mushroom-like form, indicating a strong elongation of the scattering indicatrix;

\(l \to 0.5\)  \(1.0\)  \(1.5\)  \(2.0\)  \(3.0\)  \(5.0\)  \(7.0\)  \(10\)

\(48000 \to\)  \(1350 \to\)  \([[unclear: vertical numeric label]] \to\)  \(116 \to\)  \(12.2 \to\)

Enlarged 10 times

\(l \to 20\)  \(40\)

\(K = 1.9\ \mathrm{cm}^{-1}\)

Fig. 3. Polar diagrams of the brightness of a turbid medium at various depths \(l\)
(milk medium; \(K = 1.90\ \mathrm{cm}^{-1}\)).

b) the weakening of the direct beam as it penetrates deeper into the medium is accompanied by an increase in the brightness of the light scattered in directions \(\varphi < 120^\circ\)—the lower part of the “head” of the polar diagram increases, whereas the upper part of the “head” \((\varphi > 120^\circ)\) remains at first practically unchanged and then begins to decrease;

c) beginning at a depth \(l \simeq 6\ \text{cm}\) \((Kl \simeq 12)\), the direct beam practically disappears, but there still remains some elongation of the brightness diagram forward, disappearing at a depth of about \(14\ \text{cm}\) \((Kl \simeq 25)\). At this depth

Fig. 4. Depth of occurrence of the brightness maximum as a function of the observation angle \(\varphi\) for various values of the attenuation coefficient \(K\). The measurements were made at different angles of view of the photometer \(\omega\).

a stationary distribution of brightness over angles is established; with further increase of depth the form of the polar brightness diagram remains practically unchanged;

d) the establishment of a stationary distribution of brightness over angles corresponds to the establishment of an exponential law of decrease of brightness with depth, with an attenuation coefficient \(K'\) that is independent of \(\varphi\).

As already indicated, for \(0 < \varphi < 120^\circ\) the brightness first increases and then decreases. The depth at which the maximum occurs depends on \(\varphi\) (Fig. 4), and also on the attenuation coefficient \(K\) of the medium (Fig. 5), with the relation \(l_{\max} K^\sigma = \mathrm{const}\) holding (what determines the value of \(\sigma\), the author did not ascertain).

It is curious that at \(\varphi=0\) the maximum is located not at the surface, but at a depth approximately determined by the relation \(l_{\max}=\dfrac{1}{K}\).

Fig. 5. Dependence of the depth of occurrence of the brightness maximum on the attenuation coefficient \(K\) for \(\varphi=30^\circ\) (rosin gel).

Fig. 6. Dependence of the relative value of the brightness maximum on the observation angle \(\varphi\).

The relative value of the maximum brightness

\[ \frac{B(\varphi)_{\max}}{B(\varphi)_{l=0}} \]

depends on \(\varphi\) (Fig. 6), but apparently does not depend on \(K\) (i.e., on the concentration).

On the basis of the data from measurements of the brightness \(B(\varphi)\), it was possible to find the light fluxes arriving at a given point from all directions \((\Phi)\), and also from above \(\Phi_{\downarrow}\) and from below \(\Phi_{\uparrow}\). The dependence of \(\Phi\) on depth is shown in Fig. 7.

Fig. 7. Variation with depth of the light flux \(\Phi\) arriving at a given point from all directions (\(\Phi\) is proportional to the volume density of light energy at the given point). Milk medium; \(K=2.3\ \text{cm}^{-1}\); \(K'=0.093\ \text{cm}^{-1}\).

Figure 8 shows the dependence on depth of the illumination of a horizontal surface from above \((E_{\downarrow})\), measured by immersing to the corresponding depths a photoelement of known area. (The quantity \(E_{\downarrow}\) does not coincide

with \(\Phi\); the relation between these quantities depends on the form of the polar brightness diagram.) The coefficient of attenuation of illumination with depth decreases somewhat as the depth increases.

From Fig. 3 it is seen that the stationary polar brightness diagram has a form close to spherical, while the pole is shifted rather considerably toward the surface. This corresponds to the fact that the luminous flux directed upward is considerably smaller than the luminous flux directed downward. The difference between these fluxes is due to the absorption of radiation in the lower-lying layers.

Fig. 8. Variation with depth of the illumination of a horizontal surface from above. Milky medium; \(K = 1.54\ \text{cm}^{-1}\).

Fig. 9. Dependence of the brightness of a turbid medium on depth for various angles of observation under simultaneous illumination by oblique rays of the Sun and by scattered light of the sky. Camphor soot; \(K = 1.3\ \text{cm}^{-1}\).

Even with a very small coefficient of absorption, the fraction of energy absorbed by the lower-lying layers is large. In the case of milk it reaches, for example, \(\sim 60\%\) and practically does not change with depth. Increasing the absorption coefficient of the medium (by adding a dye), apparently, has little effect on the almost spherical form of the stationary brightness diagram, but, according to the author’s measurements, noticeably

shifts the pole of the diagram toward the surface, i.e., increases the fraction of energy absorbed in the lower-lying layers.

Special experiments showed that, since the brightness diagram deviates from spherical symmetry, the results of the measurements depend on the viewing angle of the photometer \(\omega\). As a consequence, in most cases it is necessary to take measurements at very small viewing angles.

In addition to measurements with vertical illumination by direct rays of the Sun, measurements were carried out with illumination of the turbid medium by oblique rays of the Sun and by the diffuse light of the sky.

Fig. 9 presents the results for a rosin sol with \(K = 1.3\ \mathrm{cm}^{-1}\) (the different curves correspond to different angles \(\varphi\)). Fig. 10 shows a section of the brightness diagram at a depth \(l = 0.5\ \mathrm{cm}\) by the plane of the vertical of the Sun for a milky medium with \(K = 2\ \mathrm{cm}^{-1}\). First of all

Fig. 10. Polar brightness diagram (in the plane of the vertical of the Sun) under oblique illumination. Depth \(l = 5.0\ \mathrm{cm}\). Milky medium; \(K = 2\ \mathrm{cm}^{-1}\).

Fig. 11. Change with depth of the polar diagram and of the direction of maximum brightness in the case of illumination by oblique rays of the Sun. Milky medium; \(K \approx 9\ \mathrm{cm}^{-1}\).

what attracts attention is that the general character of the curves in Fig. 9 remains the same as in Fig. 3. At a certain depth (of the order of 12 relays) a stationary form of the polar brightness diagram is established, and upon further immersion into the medium the brightness changes according to an exponential law with an attenuation coefficient \(K'\) independent of \(\varphi\). However, the polar diagrams in the case of oblique illumination are asymmetric with respect to the direction of the illuminating ray. Moreover, as one penetrates deeper into the medium, the direction of greatest brightness deviates more and more from the direction of the illuminating ray and approaches the vertical (Fig. 11). As a result, the stationary diagram of brightness turns out to be still symmetric with respect to the vertical and does not depend on the direction of the illuminating ray. Under diffuse illumination, the stationary distribution of brightness is established at smaller depths.

It should be noted that a number of the author’s conclusions require further investigation. In particular, it appears that the value found by the author

the dependence of \(K'\) on \(K\) may be due to the influence of the walls of the tank in which the measurements were made (absorption of light by the walls). The influence of the walls on the magnitude of \(K'\) was established by the author himself: as the distance from the walls is increased, the magnitude of \(K'\) decreases. Dimensional considerations show that the coefficient \(A\) must have the dimension \(\text{cm}^{-1}\). From consideration of the transfer equation one should expect its proportionality to the absorption coefficient (and consequently also to the concentration) of the medium, i.e., ultimately to \(K\), which is approximately equal to the scattering coefficient. In this case the dependence of \(K'\) on \(K\) should be linear. The dependences found by the author—of \(K'\) on \(K\) and of \(K'\) on the distance to the wall—do not, generally speaking, rule out such a possibility.

Nevertheless, the work reviewed makes it possible to form a clear picture of the distribution of brightness in turbid media at small absorption coefficients and large scattering coefficients, and also reveals the principal dependences that occur here. This opens up significant possibilities both for a rational simplification of the theoretical treatment of problems of multiple scattering and for the semi-quantitative estimates needed in solving practical problems.

G. R.