## Abstract

We present a theory of ultrashort-pulse difference-frequency generation (DFG) with quasi-phase-matching (QPM) gratings in the undepleted-pump, unamplified-signal approximation. In the special case of a cw (or quasi-cw) pump, the spectrum of the generated idler is related to the spectrum of the signal through a transfer-function relation that is valid for arbitrary dispersion in the medium. The engineerability of this QPM-DFG transfer function establishes the basis for arbitrary pulse shaping. Experimentally we demonstrate QPM-DFG devices operating in a frequency-degenerate type II configuration and producing pulse-shaped output at 1550 nm from 220-fs pulses at 1550 nm.

© 2001 Optical Society of America

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### Equations (24)

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(1)
$$\frac{{\partial}^{2}}{\partial {z}^{2}}{\stackrel{\u02c6}{E}}_{i}(z,\omega )+{k}^{2}(\omega ){\stackrel{\u02c6}{E}}_{i}(z,\omega )=-{\mu}_{0}{\omega}^{2}{\stackrel{\u02c6}{P}}_{\mathrm{NL}}(z,\omega ),$$
(2)
$$\frac{{\partial}^{2}}{\partial {z}^{2}}{\stackrel{\u02c6}{E}}_{s}(z,\omega )+{k}^{2}(\omega ){\stackrel{\u02c6}{E}}_{s}(z,\omega )=0,$$
(3)
$$\frac{{\partial}^{2}}{\partial {z}^{2}}{\stackrel{\u02c6}{E}}_{p}(z,\omega )+{k}^{2}(\omega ){\stackrel{\u02c6}{E}}_{p}(z,\omega )=0,$$
(4)
$${\stackrel{\u02c6}{P}}_{\mathrm{NL}}(z,\omega )=2{\epsilon}_{0}d(z){\int}_{-\infty}^{+\infty}\stackrel{\u02c6}{{E}_{s}^{*}}(z,\omega -{\omega}^{\prime}){\stackrel{\u02c6}{E}}_{p}(z,{\omega}^{\prime})\mathrm{d}{\omega}^{\prime},$$
(5)
$${\stackrel{\u02c6}{E}}_{m}(z,\omega )={\stackrel{\u02c6}{A}}_{m}(z,{\mathrm{\Omega}}_{m})exp[-\mathit{ik}({\omega}_{m}+{\mathrm{\Omega}}_{m})z],$$
(6)
$${\stackrel{\u02c6}{E}}_{m}(z,\omega )={\stackrel{\u02c6}{B}}_{m}(z,{\mathrm{\Omega}}_{m})exp[-\mathit{ik}({\omega}_{m})z],$$
(7)
$$\frac{\partial}{\partial z}{\stackrel{\u02c6}{A}}_{i}(z,{\mathrm{\Omega}}_{i})=-i\frac{{\mu}_{0}{\omega}_{i}^{2}}{2{k}_{i}}{\stackrel{\u02c6}{P}}_{\mathrm{NL}}(z,{\mathrm{\Omega}}_{i})exp[\mathit{ik}({\omega}_{i}+{\mathrm{\Omega}}_{i})z],$$
(8)
$$\frac{\partial}{\partial z}{\stackrel{\u02c6}{A}}_{s}(z,{\mathrm{\Omega}}_{s})=0,$$
(9)
$$\frac{\partial}{\partial z}{\stackrel{\u02c6}{A}}_{p}(z,{\mathrm{\Omega}}_{p})=0,$$
(10)
$${\stackrel{\u02c6}{P}}_{\mathrm{NL}}(z,\mathrm{\Omega})=2{\epsilon}_{0}d(z){\int}_{-\infty}^{+\infty}\stackrel{\u02c6}{{A}_{s}^{*}}(z,-\mathrm{\Omega}+{\mathrm{\Omega}}^{\prime}){\stackrel{\u02c6}{A}}_{p}(z,{\mathrm{\Omega}}^{\prime})\times exp\{i[k({\omega}_{s}-\mathrm{\Omega}+{\mathrm{\Omega}}^{\prime})-k({\omega}_{p}+{\mathrm{\Omega}}^{\prime})]z\}\mathrm{d}{\mathrm{\Omega}}^{\prime},$$
(11)
$${\stackrel{\u02c6}{A}}_{s}(z,\mathrm{\Omega})={\stackrel{\u02c6}{A}}_{s}(\mathrm{\Omega}),$$
(12)
$${\stackrel{\u02c6}{A}}_{p}(z,\mathrm{\Omega})={\stackrel{\u02c6}{A}}_{p}(\mathrm{\Omega}),$$
(13)
$${\stackrel{\u02c6}{A}}_{i}(L,\mathrm{\Omega})=-i\gamma {\int}_{0}^{L}d(z)\mathrm{d}z{\int}_{-\infty}^{+\infty}d{\mathrm{\Omega}}^{\prime}{\stackrel{\u02c6}{A}}_{s}^{*}({\mathrm{\Omega}}^{\prime}-\mathrm{\Omega})\times {\stackrel{\u02c6}{A}}_{p}({\mathrm{\Omega}}^{\prime})exp[-i\mathrm{\Delta}k(\mathrm{\Omega},{\mathrm{\Omega}}^{\prime})z],$$
(14)
$$\mathrm{\Delta}k(\mathrm{\Omega},{\mathrm{\Omega}}^{\prime})=k({\omega}_{p}+{\mathrm{\Omega}}^{\prime})-k({\omega}_{i}+\mathrm{\Omega})-k({\omega}_{s}+{\mathrm{\Omega}}^{\prime}-\mathrm{\Omega}).$$
(15)
$${\stackrel{\u02c6}{A}}_{p}(\mathrm{\Omega})={E}_{p}\delta (\mathrm{\Omega}=0),$$
(16)
$${\stackrel{\u02c6}{A}}_{i}(L,\mathrm{\Omega})=\stackrel{\u02c6}{d}(\mathrm{\Omega}){\stackrel{\u02c6}{A}}_{s}^{*}(-\mathrm{\Omega}){E}_{p},$$
(17)
$$\stackrel{\u02c6}{d}(\mathrm{\Omega})=-i\gamma {\int}_{-\infty}^{+\infty}d(z)exp[-i\mathrm{\Delta}k(\mathrm{\Omega})z]\mathrm{d}z,$$
(18)
$$\mathrm{\Delta}k(\mathrm{\Omega})=k({\omega}_{p})-k({\omega}_{i}+\mathrm{\Omega})-k({\omega}_{s}-\mathrm{\Omega}).$$
(19)
$${\stackrel{\u02c6}{B}}_{i}(L,\mathrm{\Omega})=\stackrel{\u02c6}{d}(\mathrm{\Omega}){\stackrel{\u02c6}{B}}_{s}^{*}(-\mathrm{\Omega}){E}_{p}exp\{-i[k({\omega}_{i}+\mathrm{\Omega})-k({\omega}_{i})]L\}.$$
(20)
$$\mathrm{\Delta}k(\mathrm{\Omega})=\mathrm{\Delta}{k}_{0}+\delta {\nu}_{\mathit{si}}\mathrm{\Omega}-\sum _{n=2}^{\infty}\frac{1}{n!}[(-1{)}^{n}{\beta}_{\mathit{sn}}+{\beta}_{\mathit{in}}]{\mathrm{\Omega}}^{n},$$
(21)
$$\stackrel{\u02c6}{d}(\mathrm{\Omega})=-i\gamma {\int}_{-\infty}^{+\infty}d(z)exp[-i(\mathrm{\Delta}{k}_{0}+\delta {\nu}_{\mathit{si}}\mathrm{\Omega})z]\mathrm{d}z,$$
(22)
$${\stackrel{\u02c6}{A}}_{2}(L,\mathrm{\Omega})=\stackrel{\u02c6}{D}(\mathrm{\Omega})\stackrel{\u02c6}{{A}_{1}^{2}}(\mathrm{\Omega}),$$
(23)
$$\stackrel{\u02c6}{D}(\mathrm{\Omega})=-i\gamma {\int}_{-\infty}^{+\infty}d(z)exp[-i\mathrm{\Delta}k(\mathrm{\Omega})z]\mathrm{d}z,$$
(24)
$${U}_{i}=\frac{538}{c{\epsilon}_{0}}\frac{|d|}{{n}^{2}\delta {\nu}_{\mathit{si}}}\frac{({\mathrm{\lambda}}_{s}-{\mathrm{\lambda}}_{p}{)}^{3}}{({\mathrm{\lambda}}_{s}+{\mathrm{\lambda}}_{p}{)}^{2}{\mathrm{\lambda}}_{s}^{2}{\mathrm{\lambda}}_{p}^{2}}\frac{{\tau}_{s}}{{\tau}_{p}}{\mathit{fU}}_{s}{U}_{p},$$