## Abstract

The importance of Lambertian transmission, rather than
total transmission, is argued and expressions for both are given. Exact
expressions for output radiance are given in terms of both total sphere
area and port area. A formula for choosing sphere size to give the
maximum Lambertian transmission is developed. Port fractions in the
range of 0.1–0.2 are recommended.

Full Article |

PDF Article
### Equations (16)

Equations on this page are rendered with MathJax. Learn more.

(1)
$${a}_{1}\prime =\frac{\pi {D}^{2}}{2}\left(1-\sqrt{1-{{d}_{1}}^{2}/{D}^{2}}\right).$$
(2)
$${a}_{1}=A{f}_{1}\left(1-{f}_{1}\right)={a}_{1}\prime \left(1-{f}_{1}\right).$$
(3)
$${L}_{L}=\frac{\Phi {\rho}^{2}}{\pi A}\left[1+\rho \left(1-f\right)+{\rho}^{2}{\left(1-f\right)}^{2}+\cdots \right]=\frac{\Phi}{\pi A}\text{\hspace{0.17em}}\frac{{\rho}^{2}}{1-\rho \left(1-f\right)}\mathrm{.}$$
(4)
$$T=\rho f+\rho \left(1-f\right)\rho f+{\rho}^{2}{\left(1-f\right)}^{2}\rho f+\cdots =\frac{\rho f}{1-\rho \left(1-f\right)}\mathrm{.}$$
(5)
$${L}_{ave}=\frac{\Phi T}{\pi a}=\frac{\Phi}{\pi a}\text{\hspace{0.17em}}\frac{\rho f}{1-\rho \left(1-f\right)}\mathrm{.}$$
(6)
$${T}_{L}=T-\rho f={\rho}^{2}f\left(1-f\right)\left[1+\rho \left(1-f\right)+{\rho}^{2}{\left(1-f\right)}^{2}+\cdots \right]=\rho \left(1-f\right)T=\frac{{\rho}^{2}f\left(1-f\right)}{1-\rho \left(1-f\right)},$$
(7)
$${L}_{L}=\frac{\Phi {T}_{L}}{\pi a}=\frac{\Phi}{\pi a}\text{\hspace{0.17em}}\frac{{\rho}^{2}f\left(1-f\right)}{1-\rho \left(1-f\right)}.$$
(8)
$${f}_{\mathrm{max}}=\frac{\sqrt{1-\rho}\left(1-\sqrt{1-\rho}\right)}{\rho}\text{,}$$
(9)
$$a={a}_{1}+{a}_{2}=A{f}_{1}\left(1-{f}_{1}\right)+A{f}_{2}\left(1-{f}_{2}\right)=A(f-{{f}_{1}}^{2}-{{f}_{2}}^{2})$$
(10)
$$\text{\hspace{0.17em}}\Rightarrow \frac{1}{A}=\frac{f-{{f}_{1}}^{2}-{{f}_{2}}^{2}}{a}\mathrm{.}$$
(11)
$${L}_{L}=\frac{\Phi}{\pi a}\text{\hspace{0.17em}}\frac{{\rho}^{2}}{1-\rho \left(1-f\right)}(f-{{f}_{1}}^{2}-{{f}_{2}}^{2})=\frac{\Phi {T}_{L}}{\pi a}\text{\hspace{0.17em}}\frac{\left(1-\frac{{{f}_{1}}^{2}+{{f}_{2}}^{2}}{f}\right)}{\left(1-f\right)}\text{,}$$
(12)
$$\frac{{{d}_{1}}^{2}}{{D}^{2}}=\frac{\pi {{d}_{1}}^{2}}{\pi {D}^{2}}=\frac{4{a}_{1}}{A}\mathrm{.}$$
(13)
$${f}_{1}=\frac{{a}_{1}\prime}{A}=\frac{1}{2}\left(1-\sqrt{1-{{d}_{1}}^{2}/{D}^{2}}\right)\Rightarrow 1-{{d}_{1}}^{2}/{D}^{2}={\left(1-2{f}_{1}\right)}^{2}=1-4{f}_{1}+4{{f}_{1}}^{2}\Rightarrow \frac{{{d}_{1}}^{2}}{{D}^{2}}=\frac{4{a}_{1}}{A}=4{f}_{1}\left(1-{f}_{1}\right)\Rightarrow {a}_{1}=A{f}_{1}\left(1-{f}_{1}\right)={a}_{1}\prime \left(1-{f}_{1}\right).$$
(14)
$${f}_{1}=\frac{1}{2}\left(1-\sqrt{1-{{d}_{1}}^{2}/{D}^{2}}\right)\approx \frac{{{d}_{1}}^{2}}{4{D}^{2}}\left(1+\frac{{{d}_{1}}^{2}}{4{D}^{2}}\right)\approx \frac{1}{4}{\left(\frac{{d}_{1}}{D}\right)}^{2}\text{,}$$
(15)
$$\text{\hspace{1em} \hspace{1em}}\frac{\text{d}{T}_{L}}{\text{d}f}=\frac{{\rho}^{2}\left(1-2f\right)\left[1-\rho \left(1-f\right)\right]-{\rho}^{3}f\left(1-f\right)}{{\left[1-\rho \left(1-f\right)\right]}^{2}}.$$
(16)
$$\rho {{f}_{\mathrm{max}}}^{2}+2\left(1-\rho \right){f}_{\mathrm{max}}-\left(1-\rho \right)=0.$$