## Abstract

The Beer–Lambert–Bouguer absorption law, known as Beer’s law for absorption in an optical medium, is precise only at power densities lower than a few kW. At higher power densities this law fails because it neglects the processes of stimulated emission and spontaneous emission. In previous models that considered those processes, an analytical expression for the absorption law could not be obtained. We show here that by utilizing the Lambert *W*-function, the two-level energy rate equation model is solved analytically, and this leads into a general absorption law that is exact because it accounts for absorption as well as stimulated and spontaneous emission. The general absorption law reduces to Beer’s law at low power densities. A criterion for its application is given along with experimental examples.

© 2008 Optical Society of America

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### Equations (15)

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(1)
$$\frac{\text{d}\varphi}{\text{d}z}=-\alpha \varphi \mathrm{.}$$
(2)
$$\varphi (z)=\varphi (0){e}^{-\alpha z}\mathrm{.}$$
(3)
$$\frac{\text{d}{N}_{0}}{\text{d}t}=-\sigma \varphi {N}_{0}+\sigma \varphi {N}_{1}+{\tau}^{-1}{N}_{1},$$
(4)
$$\frac{\text{d}{N}_{1}}{\text{d}t}=\sigma \varphi {N}_{0}-\sigma \varphi {N}_{1}-{\tau}^{-1}{N}_{1}\mathrm{.}$$
(5)
$${N}_{0}+{N}_{1}={N}_{t}\mathrm{.}$$
(6)
$$-\sigma \varphi {N}_{0}+(\sigma \varphi +{\tau}^{-1}){N}_{1}=0.$$
(7)
$$\left(\begin{array}{cc}-\sigma \varphi & \sigma \varphi +{\tau}^{-1}\\ 1& 1\end{array}\right)\left(\begin{array}{c}{N}_{0}\\ {N}_{1}\end{array}\right)=\left(\begin{array}{c}0\\ {N}_{t}\end{array}\right)\mathrm{.}$$
(8)
$$\frac{\text{d}\varphi}{\text{d}z}=\sigma \varphi ({N}_{1}-{N}_{0})\mathrm{.}$$
(9)
$$\frac{\text{d}\varphi}{\text{d}z}=-\frac{{\tau}^{-1}{N}_{t}\sigma \varphi}{{\tau}^{-1}+2\sigma \varphi}\mathrm{.}$$
(10)
$$2\sigma \tau \varphi +\frac{\text{d}\varphi}{\varphi}=-{N}_{t}\sigma \text{d}z\Rightarrow 2\sigma \tau \varphi (z)-2\sigma \tau \varphi (0)+\mathrm{ln}\left[\frac{\varphi (z)}{\varphi (0)}\right]\phantom{\rule{0ex}{0ex}}=-{N}_{t}\sigma z\Rightarrow \mathrm{exp}[2\sigma \tau \varphi (z)]2\sigma \tau \varphi (z)\phantom{\rule{0ex}{0ex}}=\mathrm{exp}[2\sigma \tau \varphi (0)]2\sigma \tau \varphi (0)\mathrm{exp}[-{N}_{t}\sigma z]\mathrm{.}$$
(11)
$$\varphi (z)=\frac{1}{2\sigma \tau}W(2\sigma \tau \varphi (0)\mathrm{exp}[2\sigma \tau \varphi (0)-{N}_{t}\sigma z])\mathrm{.}$$
(12)
$$\text{d}\varphi =-{N}_{t}\sigma \varphi \frac{1}{1+2\sigma \tau \varphi}\text{d}z\mathrm{.}$$
(13)
$$\frac{1}{1+2\sigma \tau \varphi}\approx 1-2\sigma \tau \varphi +(2\sigma \tau \varphi {)}^{2}-\dots \text{higher orders in \hspace{0.17em}}2\sigma \tau \varphi \mathrm{.}$$
(14)
$$\text{d}\varphi =-{N}_{t}\mathrm{\sigma \varphi}\text{d}z+{N}_{t}\sigma \varphi (2\sigma \tau \varphi )\text{d}z-\dots \text{higher orders in \hspace{0.17em}}2\sigma \tau \varphi \mathrm{.}$$
(15)
$$2\sigma \tau \varphi \ll 1.$$