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Full Article | PDF Article**Journal of the Optical Society of America**- Vol. 14,
- Issue 5,
- pp. 365-373
- (1927)
- •doi: 10.1364/JOSA.14.000365

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- An alternative treatment (applied, e.g., by Callier) consists in recording in both cases only the normally emergent light flux but using diffused incident light (opal glass) in one and collimated incident light in the other case. The former is also the method usually employed in densitometry.

- Replacing the usual but less convenient symbols d|| and d††.

- The fraction ϵ might be a function of Is itself. But it will appear that the observations are well represented by attributing to ϵ, for each given material, a constant value.

- One might think of applying Mie’s scattering formulas (Annalen der Physik, 25, p. 377, 1908). These, of course, are derived for spherical particles only. But even if one wished to disregard the departure from sphericity, there would still remain the difficulty that the two optical constants appearing in Mie’s formulas are not known for the spongelike state of silver as it occurs in the developed grains and can scarcely be identified with those found for metallic silver.

- It can be shown that the treatment based on this expression is ultimately equivalent to one which takes a full account of the effect of overlapping of grains piled above each other haphazardly in different strata dx.

- Thus it comes that in Schuster’s treatment of the problem of a foggy atmosphere (Astrophys. Journal, 21, p. 1; 1905) the scattered light is, at each stage, assumed to be sent in equal amounts backward and forward.

[Crossref] - Cf. a paper by W. Shoulejkin, Phil. Mag., 47, p. 307; 1924, based on G. Mie’s general form of the solution for a spherical scattering particle. This forward tendency of scattered light with increasing size of scatterer is already very marked in the case of silver particles of diameter 153 μμ for λ=525 μμ, as found by R. Gans (Ann. der Physik, 76, p. 29; 1925) theoretically, by Mie’s formula, and also verified by him experimentally. A much more marked concentration of the scattered radiation in the anterior hemisphere is found for gold particles of 160 and 180 μμ, with λ=550, as investigated in Mie’s own paper, Ann. der Physik, 25, p. 429; 1908. In the latter case almost the whole scattered light is sent forward. Although the optical constants of the spongelike silver grains of the photographic plate will, no doubt, differ from those of metallic silver and gold, the said tendency can be supposed to hold essentially also for these grains. This, moreover, is borne out by the fact, that in our case a direct experimental estimate gave for the total light thrown back, including that reflected at the surface, only 4 per cent of the incident light.

[Crossref] - Thus, e.g., in the case of log (1−z)=−1.401z, corresponding to the emulsion Eastman 40, the successive approximations, starting from z=0.9, were .945, .9525, .9536, .9539, .9539, converging very rapidly, indeed.

Cf. a paper by W. Shoulejkin, Phil. Mag., 47, p. 307; 1924, based on G. Mie’s general form of the solution for a spherical scattering particle. This forward tendency of scattered light with increasing size of scatterer is already very marked in the case of silver particles of diameter 153 μμ for λ=525 μμ, as found by R. Gans (Ann. der Physik, 76, p. 29; 1925) theoretically, by Mie’s formula, and also verified by him experimentally. A much more marked concentration of the scattered radiation in the anterior hemisphere is found for gold particles of 160 and 180 μμ, with λ=550, as investigated in Mie’s own paper, Ann. der Physik, 25, p. 429; 1908. In the latter case almost the whole scattered light is sent forward. Although the optical constants of the spongelike silver grains of the photographic plate will, no doubt, differ from those of metallic silver and gold, the said tendency can be supposed to hold essentially also for these grains. This, moreover, is borne out by the fact, that in our case a direct experimental estimate gave for the total light thrown back, including that reflected at the surface, only 4 per cent of the incident light.

[Crossref]

One might think of applying Mie’s scattering formulas (Annalen der Physik, 25, p. 377, 1908). These, of course, are derived for spherical particles only. But even if one wished to disregard the departure from sphericity, there would still remain the difficulty that the two optical constants appearing in Mie’s formulas are not known for the spongelike state of silver as it occurs in the developed grains and can scarcely be identified with those found for metallic silver.

Thus it comes that in Schuster’s treatment of the problem of a foggy atmosphere (Astrophys. Journal, 21, p. 1; 1905) the scattered light is, at each stage, assumed to be sent in equal amounts backward and forward.

[Crossref]

[Crossref]

[Crossref]

[Crossref]

Thus, e.g., in the case of log (1−z)=−1.401z, corresponding to the emulsion Eastman 40, the successive approximations, starting from z=0.9, were .945, .9525, .9536, .9539, .9539, converging very rapidly, indeed.

It can be shown that the treatment based on this expression is ultimately equivalent to one which takes a full account of the effect of overlapping of grains piled above each other haphazardly in different strata dx.

An alternative treatment (applied, e.g., by Callier) consists in recording in both cases only the normally emergent light flux but using diffused incident light (opal glass) in one and collimated incident light in the other case. The former is also the method usually employed in densitometry.

Replacing the usual but less convenient symbols d|| and d††.

The fraction ϵ might be a function of Is itself. But it will appear that the observations are well represented by attributing to ϵ, for each given material, a constant value.

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**Table 1** Motion Picture Positive

**Table 2** Par Speed Motion Picture Negative

**Table 3** Eastman 40

**Table 4** Velox Emulsion coated on glass

**Table 5** Superspeed Portrait Film

**Table 6** Motion Picture Positive (New measurements, by thermoelectric method.)

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$${D}^{\prime}=\text{log}\hspace{0.17em}[{I}_{0}/(I+{I}_{s})]$$

$$D=\text{log}\hspace{0.17em}[{I}_{0}/(I+\u03f5{I}_{s})],$$

$$AINdx.$$

$$\kappa \omega INdx.$$

$$dI=-(A+\kappa \omega )INdx.$$

$$\kappa \omega {I}_{s}Ndx,$$

$$d{I}_{s}=(AI-\kappa \omega {I}_{s})Ndx.$$

$$\frac{dI}{dn}=-(A+\kappa \omega )I,$$

$$\frac{d{I}_{s}}{dn}=AI-\kappa \omega {I}_{s},$$

$$\frac{d}{dn}(I+{I}_{s})=-\kappa \omega (I+{I}_{s}),\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\text{ln}\frac{{I}_{0}}{I+{I}_{s}}=\kappa \omega n,$$

$${D}^{\prime}=M\kappa \omega n,$$

$${D}^{\prime}=\text{log}\frac{(a+b){e}^{bn}-(a-b){e}^{-bn}}{2b},$$

$$I={I}_{0}{e}^{-pn},\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}p=A+\kappa \omega ,$$

$${I}_{s}={I}_{0}({e}^{-\kappa \omega n}-{e}^{-pn}).$$

$$D=-\text{log}\hspace{0.17em}[\u03f5{e}^{-\kappa \omega n}+(1-\u03f5){e}^{-pn}].$$

$${10}^{-D}=\u03f5\xb7{10}^{-{D}^{\prime}}+(1-\u03f5)\xb7{10}^{-\beta {D}^{\prime}}$$

$$T=\u03f5{T}^{\prime}+(1-\u03f5){{T}^{\prime}}^{\beta}.$$

$$(\beta -1)\zeta =\frac{D}{{D}^{\prime}}-1,$$

$$\text{log}\hspace{0.17em}\left[1-\frac{1-{10}^{{D}^{\prime}-D}}{\zeta}\right]=-(\beta -1){D}^{\prime}=-\frac{a{D}^{\prime}}{\zeta}$$

$$\text{log}\hspace{0.17em}(1-z)=-bz,\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}z=c/\zeta ,$$

$$\begin{array}{cccc}0.210,& 0.250,& 0.350,& 0.490,\end{array}$$

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