## Abstract

We present the results of degenerate four-wave mixing (DFWM) studies in three samples of *p*-type Ge with a wide range of absorptive properties. Three mechanisms for DFWM are observed in this material. A model, which includes the mechanisms of the inhomogeneously broadened saturable absorption of *p*-type Ge and the optical Kerr effect that is due to bound electrons in Ge but not the mechanism of bulk plasma formation that occurs only at extremely high pump intensities, is developed and compared with the experimental results. Pump-beam attenuation by the medium is shown to be an important effect in modeling DFWM. The model, which has no free parameters, fits the experimental data well for samples with small-signal transmission of greater than 20%.

© 1983 Optical Society of America

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### Equations (26)

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(1)
$$P=\frac{in{\alpha}_{r}}{2\pi k}E+\frac{in{\alpha}_{0}}{2\pi k}\frac{E}{{(1+\mid E{\mid}^{2}/{{E}_{s}}^{2})}^{1/2}}+3{\chi}_{3}\mid E{\mid}^{2}E.$$
(2)
$$P={P}_{0}+\mathrm{\Delta}P,$$
(3)
$${P}_{0}=\frac{in{\alpha}_{r}{E}_{s}}{2\pi k}{E}_{0}+\frac{i{\alpha}_{0}{E}_{s}}{2\pi k}\frac{{E}_{0}}{{(1+\mid {E}_{0}{\mid}^{2})}^{1/2}}+3{\chi}_{3}{{E}_{s}}^{3}\mid {E}_{0}{\mid}^{2}{E}_{0}$$
(4)
$$\mathrm{\Delta}P=\frac{in{\alpha}_{r}{E}_{s}}{2\pi k}\mathrm{\Delta}E+\frac{i{\alpha}_{0}{E}_{s}}{2\pi k}\frac{(2+\mid {E}_{0}{\mid}^{2})\mathrm{\Delta}E-{{E}_{0}}^{2}\mathrm{\Delta}E*}{{(1+\mid {E}_{0}{\mid}^{2})}^{3/2}}+6{\chi}_{3}{{E}_{s}}^{3}\mid {E}_{0}{\mid}^{2}\mathrm{\Delta}E+3{\chi}_{3}{{E}_{s}}^{3}{{E}_{0}}^{2}\mathrm{\Delta}E*.$$
(5)
$${\nabla}^{2}{E}_{0}+{n}^{2}{k}^{2}{E}_{0}=-\frac{4\pi {k}^{2}}{{E}_{s}}{P}_{0}$$
(6)
$${\nabla}^{2}\mathrm{\Delta}E+{n}^{2}{k}^{2}\mathrm{\Delta}E=-\frac{4\pi {k}^{2}}{{E}_{s}}\mathrm{\Delta}P.$$
(7)
$$\frac{\text{d}{E}_{f}}{\text{d}z}=[-{\alpha}_{p}+i\beta ({{\rho}_{f}}^{2}+2{{\rho}_{b}}^{2})]{E}_{f},$$
(8)
$$\frac{\text{d}{E}_{b}}{\text{d}z}=[{\alpha}_{p}-i\beta ({{\rho}_{b}}^{2}+2{{\rho}_{f}}^{2})]{E}_{b},$$
(9)
$$\frac{\text{d}{E}_{1}}{\text{d}z}=\alpha {E}_{1}+i\kappa *{E}_{2}*\hspace{0.17em}\text{exp}(i{\varphi}_{+}),$$
(10)
$$\frac{\text{d}{E}_{2}}{\text{d}z}=-\alpha {E}_{2}-i\kappa *{E}_{1}*\hspace{0.17em}\text{exp}(i{\varphi}_{+}),$$
(11)
$${\alpha}_{p}=\frac{{\alpha}_{r}}{\text{cos}(\theta )}+\frac{{\alpha}_{0}}{\pi \hspace{0.17em}\text{cos}(\theta ){[1+{({\rho}_{f}+{\rho}_{b})}^{2}]}^{1/2}}K(b),$$
(12)
$$\beta =\frac{6\pi k{\chi}_{3}{{E}_{s}}^{2}}{n\hspace{0.17em}\text{cos}(\varphi )},$$
(13)
$$\alpha =-\frac{{\alpha}_{r}}{\text{cos}(\theta )}-\frac{{\alpha}_{0}}{\pi \hspace{0.17em}\text{cos}(\theta ){[1+{({\rho}_{f}+{\rho}_{b})}^{2}]}^{1/2}}\times \left[K(b)+\frac{E(b)}{1+{({\rho}_{f}-{\rho}_{b})}^{2}}\right]+2i\beta ({{\rho}_{f}}^{2}+{{\rho}_{b}}^{2}),$$
(14)
$$\kappa =\frac{i{\alpha}_{0}}{\text{cos}(\theta ){[1+{({\rho}_{f}+{\rho}_{b})}^{2}]}^{1/2}}\left\{\frac{{{\rho}_{f}}^{2}+{{\rho}_{b}}^{2}}{2{\rho}_{f}{\rho}_{b}}[K(b)-E(b)]-\frac{{({\rho}_{f}-{\rho}_{b})}^{2}}{1+{({\rho}_{f}-{\rho}_{b})}^{2}}E(b)\right\}+2i\beta {\rho}_{f}{\rho}_{b}.$$
(15)
$$\frac{\text{d}{\rho}_{f}}{\text{d}z}=-{\alpha}_{p}{\rho}_{f},$$
(16)
$$\frac{\text{d}{\rho}_{b}}{\text{d}z}={\alpha}_{p}{\rho}_{b},$$
(17)
$$\frac{\text{d}{\varphi}_{+}}{\text{d}z}=\beta ({{\rho}_{b}}^{2}-{{\rho}_{f}}^{2}).$$
(18)
$${E}_{\begin{array}{l}1\\ 2\end{array}}={A}_{\begin{array}{l}1\\ 2\end{array}}\hspace{0.17em}\text{exp}\pm i{\int}_{0}^{z}{\alpha}_{I}(x)\text{d}x,$$
(19)
$$\frac{\text{d}{A}_{1}}{\text{d}z}={\alpha}_{R}{A}_{1}+i\kappa *{A}_{2}*\hspace{0.17em}\text{exp}(i{\varphi}_{+})$$
(20)
$$\frac{\text{d}{A}_{2}}{\text{d}z}=-{\alpha}_{R}{A}_{2}-i\kappa *{A}_{1}*\hspace{0.17em}\text{exp}(i{\varphi}_{+}).$$
(21)
$$\frac{\text{d}{\rho}_{1}}{\text{d}z}={\alpha}_{R}{\rho}_{1}-[{\kappa}_{R}\hspace{0.17em}\text{sin}(\varphi )-{\kappa}_{I}\hspace{0.17em}\text{cos}(\varphi )]{\rho}_{2},$$
(22)
$$\frac{\text{d}{\rho}_{2}}{\text{d}z}=-{\alpha}_{R}{\rho}_{2}+[{\kappa}_{R}\hspace{0.17em}\text{sin}(\varphi )-{\kappa}_{I}\hspace{0.17em}\text{cos}(\varphi )]{\rho}_{1},$$
(23)
$${\rho}_{1}\frac{\text{d}{\varphi}_{1}}{\text{d}z}=[{\kappa}_{R}\hspace{0.17em}\text{cos}(\varphi )+{\kappa}_{I}\hspace{0.17em}\text{sin}(\varphi )]{\rho}_{2},$$
(24)
$${\rho}_{2}\frac{\text{d}{\varphi}_{2}}{\text{d}z}=-[{\kappa}_{R}\hspace{0.17em}\text{cos}(\varphi )+{\kappa}_{I}\hspace{0.17em}\text{sin}(\varphi )]{\rho}_{1}.$$
(25)
$$\text{tan}[\varphi (0)]=\frac{-{\kappa}_{R}(0)}{{\kappa}_{I}(0)}.$$
(26)
$$\frac{\text{d}\varphi}{\text{d}z}=\frac{\text{d}{\varphi}_{+}}{\text{d}z}-[{\kappa}_{R}\hspace{0.17em}\text{cos}(\varphi )+{\kappa}_{I}\hspace{0.17em}\text{sin}(\varphi )]\hspace{0.17em}\left(\frac{{\rho}_{2}}{{\rho}_{1}}-\frac{{\rho}_{1}}{{\rho}_{2}}\right).$$