## Abstract

A geometrical algorithm for optical ray tracing is used to trace the trajectories of probe laser beam rays in pulsed photothermal deflection spectroscopy (PDS) of a static gas. Rays are traced as a function of various parameters such as time, absorbed pump beam energy, radius, and pulse duration. The exit angle is also computed and compared with the deflection angle calculated analytically with very good agreement for a pump pulse duration shorter than the thermal diffusion time. For a long pump pulse duration, the analytical expression is inaccurate due to the neglect of thermal diffusion during the pump pulse. Probe beam ray crossing is examined and found to be unimportant for most common PDS situations. The saturation of the PD signal is found for high absorbed pump pulse energy; this is due to excessive heating of the absorber. Finally, the maximum probe beam deflection occurs when the deflection is measured at a time corresponding to the end of the pump pulse duration.

© 1987 Optical Society of America

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### Equations (23)

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(1)
$$\nabla n=\frac{d}{ds}(n\tau ),$$
(2)
$$\tau =\frac{d\mathbf{r}}{ds},$$
(3)
$$\nabla n=\frac{d}{ds}\left(n\frac{d\mathbf{r}}{ds}\right).$$
(4)
$$\mathrm{\Delta}\mathbf{r}={\mathbf{r}}_{2}-{\mathbf{r}}_{1}={\tau}_{1}\mathrm{\Delta}s.$$
(5)
$${\mathbf{r}}_{2}={\mathbf{r}}_{1}+{\tau}_{1}\mathrm{\Delta}s.$$
(6)
$$\nabla nds=d(n\tau ),$$
(7)
$$\nabla n\mathrm{\Delta}s=\mathrm{\Delta}(n\tau )=n({\mathbf{r}}_{2}){\tau}_{2}-n({\mathbf{r}}_{1}){\tau}_{1},$$
(8)
$${\tau}_{2}=\frac{n({\mathbf{r}}_{1})}{n({\mathbf{r}}_{2})}{\tau}_{1}+\frac{\mathrm{\Delta}s}{n({\mathbf{r}}_{2})}\nabla n.$$
(9)
$$n=1+\delta =1+{\delta}_{0}\hspace{0.17em}\left(\frac{{T}_{0}}{T}\right).$$
(10)
$$T={T}_{0}+\frac{2\alpha cD{E}_{0}}{\pi \mathrm{\lambda}({w}_{0}^{2}+8Dt)}\text{exp}\hspace{0.17em}\left(\frac{-2{r}^{2}}{{w}_{0}^{2}+8Dt}\right)$$
(11)
$$\frac{\partial n}{\partial x}=\frac{\partial n}{\partial T}\frac{\partial T}{\partial x}=\frac{\partial \delta}{\partial T}\frac{\partial T}{\partial r}\frac{\partial r}{\partial x},$$
(12)
$$\frac{\partial n}{\partial x}=-{\delta}_{0}\left(\frac{{T}_{0}}{{T}^{2}}\right)\frac{\partial T}{\partial r}\frac{x}{r},$$
(13)
$$\varphi =-\frac{1}{n}\frac{dn}{dT}\frac{4\alpha cD{E}_{0}(vt-a)}{\sqrt{2\pi}\mathrm{\lambda}{({w}_{0}^{2}+8Dt)}^{3/2}}\text{exp}\hspace{0.17em}\left[-\frac{2{(vt-a)}^{2}}{({w}_{0}^{2}+8Dt)}\right]$$
(14)
$$\frac{\partial T}{\partial r}=\frac{\alpha c{E}_{0}}{\pi \mathrm{\lambda}\mathrm{\Delta}t2r}\left[\text{exp}\hspace{0.17em}\left(\frac{-2{r}^{2}}{{w}_{0}^{2}+8Dt}\right)-\text{exp}\hspace{0.17em}\left(\frac{-2{r}^{2}}{{w}^{2}}\right)\right],$$
(15)
$${w}^{2}=\{\begin{array}{ll}{w}_{0}^{2}\hfill & \text{if}\hspace{0.17em}0\le t\le \mathrm{\Delta}t;\hfill \\ {w}_{0}^{2}+8D(t-\mathrm{\Delta}t),\hfill & t>\mathrm{\Delta}t.\hfill \end{array}$$
(16)
$$T={T}_{0}+\frac{\alpha c{E}_{0}}{\pi \mathrm{\lambda}\mathrm{\Delta}t2}{\int}_{{r}_{0}}^{\infty}[\text{exp}(-\beta {r}^{2})-\text{exp}(-\gamma {r}^{2})]\frac{dr}{r},$$
(17)
$$\beta =\frac{2}{{w}_{0}^{2}+8Dt},$$
(18)
$$\gamma =\frac{2}{{w}^{2}}.$$
(19)
$$\eta =\beta {r}^{2},$$
(20)
$${\int}_{{x}_{0}}^{\infty}\text{exp}(-\eta )\frac{d\eta}{\eta}.$$
(21)
$$l=-{\left(\frac{d\varphi}{dy}\right)}^{-1}.$$
(22)
$$\rho C\left[\frac{\partial T(r,t)}{\partial t}\right]=\mathrm{\lambda}{\nabla}^{2}T(r,t)+Q(r,t),$$
(23)
$$\frac{\partial \rho}{\partial t}=-\hspace{0.17em}(\nabla \xb7\rho \mathbf{v}).$$