## Abstract

A detected laser signal backscattered from a tilted target is modeled with a laser-pulse shape as a response of a high-pass filter to an exponential input that describes the gain buildup within the laser cavity before a laser pulse is emitted and a single-pole low-pass RC filter for the electronic amplifier. The model is used to maximize the signal-to-noise ratio of the detected peak signal with a proper choice of the integration time constant τ as a function of the laser-pulse shape and the tilt angle of the backscattering target.

© 1996 Optical Society of America

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### Equations (7)

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(1)
$${V}_{L}(t)=a\text{\hspace{0.17em}\hspace{0.17em}exp}(-t/{\text{\tau}}_{2})+(1-a)\text{exp}(-t/{\text{\tau}}_{3})-\text{exp}(-t/{\text{\tau}}_{1}),$$
(2)
$$\begin{array}{l}{V}_{i}(t)=\frac{\text{\rho}}{\text{\Delta}t}{\displaystyle {\int}_{0}^{t}{V}_{L}}({t}^{\prime})\text{d}{t}^{\prime}\text{\hspace{1em}}t<\text{\Delta}t,\\ {V}_{i}(t)=\frac{\text{\rho}}{\text{\Delta}t}{\displaystyle {\int}_{t-\text{\Delta}t}^{t}{V}_{L}({t}^{\prime})\text{d}{t}^{\prime}\text{\hspace{1em}}t\ge \text{\Delta}t.}\end{array}$$
(3)
$${V}_{0}(t)={\text{\tau}}^{-1}{\displaystyle {\int}_{0}^{t}\text{\rho}{V}_{i}}({t}^{\prime})\text{exp}[-(t-{t}^{\prime})/\text{\tau}]\text{d}{t}^{\prime}.$$
(4)
$${\text{SNR}}_{E}=\frac{{\displaystyle {\int}_{0}^{T}{V}_{0}(t)\text{d}t}}{{V}_{n}T},$$
(5)
$${\int}_{0}^{\infty}{V}_{L}}(t)\text{d}t={\displaystyle {\int}_{0}^{T}\frac{{V}_{0}(t)}{\text{\rho}}\text{d}t={\text{\tau}}_{3}-{\text{\tau}}_{1}-a\text{\hspace{0.17em}}({\text{\tau}}_{3}-{\text{\tau}}_{2}),$$
(6)
$$\begin{array}{c}\frac{{\text{SNR}}_{E}}{{\text{SNE}}_{p}}=\frac{{\displaystyle {\int}_{0}^{T}{V}_{0}(t)\text{d}t}}{T{V}_{0}({t}_{0})}=\frac{\text{\rho}{\displaystyle {\int}_{0}^{\infty}{V}_{L}(t)\text{d}t}}{T{V}_{0}({t}_{0})}\\ =\text{\rho}\frac{{\text{\tau}}_{3}-{\text{\tau}}_{1}-a({\text{\tau}}_{3}-{\text{\tau}}_{2})}{T{V}_{0}({t}_{0})}.\end{array}$$
(7)
$$\frac{\text{\rho}(\text{laser-pulse\hspace{0.17em}energy})}{T(\text{laser-pulse\hspace{0.17em}peak\hspace{0.17em}pwoer})}\le \frac{{\text{SNR}}_{E}}{{\text{SNR}}_{p}}\le 1,$$