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Full Article | PDF Article**Journal of the Optical Society of America**- Vol. 15,
- Issue 3,
- pp. 125-130
- (1927)
- •doi: 10.1364/JOSA.15.000125

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

[CrossRef] - G. Mie, Ann. der Physik, 25, pp. 377–455; 1908.

[CrossRef] - 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.

[CrossRef] - W. Shoulejkin, Phil. Mag. 47, p. 307; 1924.

[CrossRef] - An explicit consideration of distinct light fluxes directed back and forth has already been introduced 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 of equal distribution (i.e. ζ=1/2), which would suit only very small particles. Moreover, he has no term representing the collimated radiation but, in accordance with the nature of his subject, considers only the scattered fluxes in either sense. This leads to two equations only which (apart from the emission terms) differ somewhat from (2) and (3).

[CrossRef] - This differs, even for ζ=1/2, from Schuster’s result (loc. cit.) as regards the coefficients In fact, Schuster’s formula for the total emergent radiation R reduces, in absence of emission, toR=4c/[(1+c)2eβn-(1-c)2e-βn],where, in our symbols c=κκ+A, while β=κ(κ+A), identical with our β for ζ=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 thickness of the layer (cf. infra) is essentially the same.Schuster’s type of formula has also been obtained, by an entirely different reasoning, by Channon, Renwick, and Storr, Proc. Roy. Soc., 94, p. 222; 1918.

[CrossRef]

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

[CrossRef]

As e.g. by R. Gans, Ann. der Physik, 76, p. 29; 1925.

[CrossRef]

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

[CrossRef]

This differs, even for ζ=1/2, from Schuster’s result (loc. cit.) as regards the coefficients In fact, Schuster’s formula for the total emergent radiation R reduces, in absence of emission, toR=4c/[(1+c)2eβn-(1-c)2e-βn],where, in our symbols c=κκ+A, while β=κ(κ+A), identical with our β for ζ=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 thickness of the layer (cf. infra) is essentially the same.Schuster’s type of formula has also been obtained, by an entirely different reasoning, by Channon, Renwick, and Storr, Proc. Roy. Soc., 94, p. 222; 1918.

[CrossRef]

G. Mie, Ann. der Physik, 25, pp. 377–455; 1908.

[CrossRef]

An explicit consideration of distinct light fluxes directed back and forth has already been introduced 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 of equal distribution (i.e. ζ=1/2), which would suit only very small particles. Moreover, he has no term representing the collimated radiation but, in accordance with the nature of his subject, considers only the scattered fluxes in either sense. This leads to two equations only which (apart from the emission terms) differ somewhat from (2) and (3).

[CrossRef]

[CrossRef]

As e.g. by R. Gans, Ann. der Physik, 76, p. 29; 1925.

[CrossRef]

Unless the particle is a perfect conductor, when almost the whole light is scattered backwards. Cf Mie, l.c., p. 430.

G. Mie, Ann. der Physik, 25, pp. 377–455; 1908.

[CrossRef]

[CrossRef]

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

[CrossRef]

[CrossRef]

[CrossRef]

[CrossRef]

G. Mie, Ann. der Physik, 25, pp. 377–455; 1908.

[CrossRef]

As e.g. by R. Gans, Ann. der Physik, 76, p. 29; 1925.

[CrossRef]

[CrossRef]

[CrossRef]

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

[CrossRef]

[CrossRef]

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$$\zeta AI$$

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

$$\frac{dS}{dn}=\zeta AI-\kappa S-(1-\zeta )AS+(1-\zeta )A{S}^{\prime},$$

$$-\frac{d{S}^{\prime}}{dn}=(1-\zeta )AI-\kappa {S}^{\prime}-(1-\zeta )A{S}^{\prime}+(1-\zeta )AS,$$

$$\frac{dS}{dn}=\zeta AI-\alpha S+(1-\zeta )A{S}^{\prime},$$

$$-\frac{d{S}^{\prime}}{dn}=(1-\zeta )AI-\alpha {S}^{\prime}+(1-\zeta )AS,$$

$$\alpha =\kappa +(1-\zeta )A.$$

$$S=0,\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\text{for}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}n=0,$$

$${S}^{\prime}=0,\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\text{for}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}n=\overline{n},$$

$$I={e}^{-pn}$$

$$\begin{array}{lll}\hfill S& =& r{e}^{-pn}+a{e}^{\beta n}+b{e}^{-{\beta}^{\prime}n}\\ \hfill {S}^{\prime}& =& {r}^{\prime}{e}^{-pn}+{a}^{\prime}{e}^{\beta n}+{b}^{\prime}{e}^{-{\beta}^{\prime}n}\end{array}$$

$$\begin{array}{r}r+\frac{1-\zeta}{\zeta}{r}^{\prime}=-1,\\ r-\frac{\alpha +p}{(1-\zeta )A}{r}^{\prime}=-1,\end{array}$$

$$\begin{array}{r}\frac{{a}^{\prime}}{a}=\frac{\alpha +\beta}{(1-\zeta )A}=\frac{(1-\zeta )A}{\alpha -\beta}=\mathrm{\lambda},\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\text{say},\\ \text{and}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\frac{{b}^{\prime}}{b}=\frac{\alpha -{\beta}^{\prime}}{(1-\zeta )A}=\frac{(1-\zeta )A}{\alpha +{\beta}^{\prime}}=\mu ,\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\text{say}.\end{array}$$

$${\beta}^{\prime}=\beta =[{\alpha}^{2}-{(1-\zeta )}^{2}{A}^{2}{]}^{1/2}=[{\kappa}^{2}+2(1-\zeta )\kappa A{]}^{1/2}.$$

$$S=a{e}^{\beta n}+b{e}^{-\beta n}-{e}^{-pn},$$

$${S}^{\prime}=\mathrm{\lambda}a{e}^{\beta n}+\mu b{e}^{-\beta n},$$

$$\mathrm{\lambda}=\frac{\alpha +\beta}{(1-\zeta )A},\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mu =\frac{\alpha -\beta}{(1-\zeta )A}.$$

$$a+b=1,\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\lambda}a{e}^{\beta \overline{n}}+\mu b{e}^{-\beta \overline{n}}=0,$$

$$a=\frac{\mu {e}^{-\beta n}}{\mu {e}^{-\beta n}-\mathrm{\lambda}{e}^{\beta n}},\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}b=\frac{-\mathrm{\lambda}{e}^{\beta n}}{\mu {e}^{-\beta n}-\mathrm{\lambda}{e}^{\beta n}}.$$

$$S=\frac{\mathrm{\lambda}-\mu}{\mathrm{\lambda}{e}^{\beta n}-\mu {e}^{-\beta n}}-{e}^{-pn},$$

$$S=\frac{2\beta}{(\alpha +\beta ){e}^{\beta n}-(\alpha -\beta ){e}^{-\beta n}}-{e}^{-(\kappa +A)n},$$

$$\alpha =(1-\zeta )A+\kappa ,\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\beta =[{\kappa}^{2}+2(1-\zeta )A\kappa {]}^{1/2},$$

$$I={e}^{-(\kappa +A)n}.$$

$$S+I=\frac{2\beta}{(\alpha +\beta ){e}^{\beta n}-(\alpha -\beta ){e}^{-\beta n}}$$

$${S}^{\prime}=\mathrm{\lambda}a+\mu b,$$

$${S}^{\prime}=(1-\zeta )A\frac{{e}^{\beta n}-{e}^{-\beta n}}{(\alpha +\beta ){e}^{\beta n}-(\alpha -\beta ){e}^{-\beta n}}$$

$$\frac{{S}^{\prime}}{S+I}=\frac{(1-\zeta )A}{2\beta}({e}^{\beta n}-{e}^{-\beta n}).$$

$${{S}_{\infty}}^{\prime}=\frac{(1-\zeta )A}{\alpha +\beta},$$

$$R=4c/[{(1+c)}^{2}{e}^{\beta n}-{(1-c)}^{2}{e}^{-\beta n}],$$

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