## Abstract

Improvements to a fast and accurate transmittance-calculation procedure, Optical Path TRANsmittance (optran), are described. The previous version computed a transmittance ratio for an absorbing layer. It required special attention to the interpolation methodology. The new approach reported here computes the absorption coefficient for an absorbing layer. This modified approach is not only simpler but also runs in one twentieth the time of the original optran approach with the same accuracy.

© 1995 Optical Society of America

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### Equations (17)

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(1)
$$\frac{\mathrm{\tau}(\mathrm{\nu},{A}_{i})}{\mathrm{\tau}(\mathrm{\nu},{A}_{i-1})}=\frac{{\mathrm{\tau}}_{\text{ref}}(\mathrm{\nu},{A}_{i})}{{\mathrm{\tau}}_{\text{ref}}(\mathrm{\nu},{A}_{i-1})}+\sum _{j=0}^{N}{C}_{ij}{Z}_{ji},$$
(2)
$$\widehat{k}({A}_{i})={C}_{i0}+\sum _{j=1}^{N}{C}_{ij}{Z}_{ji},$$
(3)
$$k(A)=-\frac{\text{d}}{\text{d}A}(\text{ln}\mathrm{\tau}).$$
(4)
$$k({A}^{*})=\frac{\text{ln}\mathrm{\tau}({A}_{1})-\text{ln}\mathrm{\tau}({A}_{2})}{{A}_{2}-{A}_{1}}.$$
(5)
$$\frac{\mathrm{\tau}({A}_{2})}{\mathrm{\tau}({A}_{1})}=\text{exp}[-k({A}^{*})({A}_{2}-{A}_{1})].$$
(6)
$$[\text{exp}(200\mathrm{\alpha})-1]/[\text{exp}(\mathrm{\alpha})-1]={A}_{200}/{A}_{1}.$$
(7)
$${P}_{Pi}^{*}=\left[\sum _{m=1}^{i}({P}_{m}+{P}_{m-1})({A}_{Pm}-{A}_{P,m-1})\right]/(2{A}_{Pi}),$$
(8)
$${T}_{Pi}^{*}=\left[\sum _{m=1}^{i}({T}_{Pm}+{T}_{P,m-1})({A}_{Pm}-{A}_{P,m-1})\right]/(2{A}_{Pi}),$$
(9)
$${T}_{Pi}^{**}=\left[\sum _{m=1}^{i}({T}_{Pm}{A}_{Pm}+{T}_{P,m-1}{A}_{P,m-1})({A}_{Pm}-{A}_{P,m-1})\right]/{{A}_{Pi}}^{2}.$$
(10)
$${k}_{Pi}^{*}=-\text{ln}({\mathrm{\tau}}_{Pi}/{\mathrm{\tau}}_{P,i-1})/({A}_{Pi}-{A}_{P,i-1}).$$
(11)
$${\mathrm{\tau}}_{Pi}={\mathrm{\tau}}_{P,i-1}\hspace{0.17em}\text{exp}[-{k}_{Pi}^{*}({A}_{Pi}-{A}_{P,i-1})],$$
(12)
$${\mathrm{\tau}}_{Pi}={\mathrm{\tau}}_{P,i-1}\hspace{0.17em}\text{exp}\{-[\widehat{k}({W}_{i})({W}_{Pi}-{W}_{P,i-1})+\widehat{k}({O}_{i})({O}_{Pi}-{O}_{P,i-1})]\}.$$
(13)
$${\mathrm{\tau}}_{o}^{*}=\frac{{\mathrm{\tau}}_{d+o}}{{\mathrm{\tau}}_{d}}.$$
(14)
$${\mathrm{\tau}}_{W}^{*}=\frac{{\mathrm{\tau}}_{d+o+W}}{{\mathrm{\tau}}_{d+o}}.$$
(15)
$${\widehat{\mathrm{\tau}}}_{d+o+w}={\widehat{\mathrm{\tau}}}_{d}{\widehat{\mathrm{\tau}}}_{o}^{*}{\widehat{\mathrm{\tau}}}_{w}^{*}$$
(16)
$$\widehat{k}({W}_{Ai})\approx {C}_{i0}+{C}_{i1}{P}_{Ai}+{C}_{i2}{{P}_{Ai}}^{2}+{C}_{i3}{P}_{Ai}^{*}+{C}_{i4}{T}_{Ai}.$$
(17)
$$\widehat{k}({O}_{Ai})\approx {D}_{i0}+{D}_{i1}{P}_{Ai}+{D}_{i2}{T}_{Ai}+{D}_{i3}{{T}_{Ai}}^{2}+{D}_{i4}{T}_{Ai}^{**}+{D}_{i5}{T}_{Ai}/{\text{sec}}^{2}\hspace{0.17em}\mathrm{\theta}.$$