## Abstract

We theoretically studied temporal properties of the second-harmonic generation of a chirped short pulse. Transient coupling-wave equations are used in describing the time-dependent nonlinear process. The pulse distortion of a fundamental wave occurs at the leading or trailing edge, which is determined by the difference of the group velocity between fundamental and harmonic waves. Pulse chirp and phase mismatch disturb the pulse shape and cause phase distortion. Even in exact phase matching, phase distortion is expected because of pulse chirp. Both distortion of pulse shape and phase shift are dependent on the sign and the quantity of the chirp.

© 1998 Optical Society of America

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### Equations (18)

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(1)
$${\mathcal{E}}_{i}(z,t)={E}_{i}(z,t)exp[j({k}_{i}z-{\omega}_{i}t)]\hspace{1em}(i=1,2,3),$$
(2)
$$\left(\frac{\partial}{\partial z}+\frac{1}{{v}_{1}}\frac{\partial}{\partial t}\right){E}_{1}=\frac{j\omega {d}_{\mathrm{eff}}}{2{n}_{1}c}{{E}_{2}}^{*}{E}_{3}exp(i\mathrm{\Delta}\mathit{kz}),$$
(3)
$$\left(\frac{\partial}{\partial z}+\frac{1}{{v}_{2}}\frac{\partial}{\partial t}\right){E}_{2}=\frac{j\omega {d}_{\mathrm{eff}}}{2{n}_{2}c}{{E}_{1}}^{*}{E}_{3}exp(i\mathrm{\Delta}\mathit{kz}),$$
(4)
$$\left(\frac{\partial}{\partial z}+\frac{1}{{v}_{3}}\frac{\partial}{\partial t}\right){E}_{3}=\frac{j2\omega {d}_{\mathrm{eff}}}{2{n}_{3}c}{E}_{1}{E}_{2}exp(-i\mathrm{\Delta}\mathit{kz}),$$
(5)
$${d}_{\mathrm{eff}}=\frac{1}{2}|{\chi}^{(2)}(2\omega ;\omega ,\omega )|.$$
(6)
$${v}_{i}({\omega}_{i})=\frac{d{\omega}_{i}}{{\mathit{dk}}_{i}}=\frac{c}{{n}_{i}+{\omega}_{i}\left(\frac{d{\omega}_{i}}{{\mathit{dn}}_{i}}\right)}.$$
(7)
$${I}_{i}(z,t)=\frac{1}{2}{\left(\frac{{\u220a}_{0}}{{\mu}_{0}}\right)}^{1/2}{n}_{i}|{E}_{i}(z,t){|}^{2}={I}_{0}|{A}_{i}(z,t){|}^{2}\hspace{1em}(i=1,2,3),$$
(8)
$${A}_{i}={E}_{i}{\left[\frac{{n}_{i}}{2{I}_{0}}{\left(\frac{{\u220a}_{0}}{{\mu}_{0}}\right)}^{1/2}\right]}^{1/2}\hspace{1em}(i=1,2,3),$$
(9)
$$\left(\frac{\partial}{\partial z}+\frac{1}{{v}_{1}}\frac{\partial}{\partial t}\right){A}_{1}=j\mathrm{\Gamma}{{A}_{2}}^{*}{A}_{3}exp(j\mathrm{\Delta}\mathit{kz}),$$
(10)
$$\left(\frac{\partial}{\partial z}+\frac{1}{{v}_{2}}\frac{\partial}{\partial t}\right){A}_{2}=j\mathrm{\Gamma}{{A}_{1}}^{*}{A}_{3}exp(j\mathrm{\Delta}\mathit{kz}),$$
(11)
$$\left(\frac{\partial}{\partial z}+\frac{1}{{v}_{3}}\frac{\partial}{\partial t}\right){A}_{3}=j2\mathrm{\Gamma}{A}_{1}{A}_{2}exp(-j\mathrm{\Delta}\mathit{kz}),$$
(12)
$$\mathrm{\Gamma}=\frac{\omega}{c}\frac{{d}_{\mathrm{eff}}}{({n}_{1}{n}_{2}{n}_{3}{)}^{1/2}}{\left[2{I}_{0}{\left(\frac{{\mu}_{0}}{{\u220a}_{0}}\right)}^{1/2}\right]}^{1/2},$$
(13)
$${A}_{i}(0,t)={A}_{i0}exp\left[-(1+{\mathit{jb}}_{i}){\left(\frac{t}{\mathrm{\Delta}t}\right)}^{2}\right]={\left(\frac{{{I}_{i}}^{0}}{{I}_{0}}\right)}^{1/2}exp\left[-(1+{\mathit{jb}}_{i}){\left(\frac{t}{\mathrm{\Delta}t}\right)}^{2}\right]\hspace{1em}(i=1,2),$$
(15)
$${n}^{2}=A+\frac{B}{1-C{\mathrm{\lambda}}^{2}}-D{\mathrm{\lambda}}^{2},$$
(16)
$${k}_{2\omega}(\varphi )={k}_{\omega}\varphi +{k}_{\omega ,z},$$
(17)
$$2{n}_{2\omega}(\varphi )={n}_{\omega}(\varphi )+{n}_{\omega ,z},$$
(18)
$$n(\varphi )={\left[\frac{{cos}^{2}(\varphi )}{{{n}_{y}}^{2}}+\frac{{sin}^{2}(\varphi )}{{{n}_{x}}^{2}}\right]}^{1/2}.$$