## Abstract

The solution of the nondegenerate two-wave-mixing problem in an absorptive Kerr medium including the contribution from nonlinear absorption is obtained for the first time to our knowledge. This solution enables us to analyze the influence of nonlinear absorption on the two-wave-mixing gain and account for previously unexplained results reported for ruby and Cr^{3+}:YAlO_{3}. The effect of nonlinear absorption of a pump beam on the gain of a probe beam is shown to be minimized when incident intensities of the interacting beams are equal. The intensity dependence of the two-wave-mixing gain for dispersive (ruby and Cr^{3+}:YAlO_{3}) media and the gain measurements for absorptive (fluorescein-doped glass) media are also in agreement with the new theory.

© 1996 Optical Society of America

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### Equations (19)

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(1)
$$\begin{array}{ll}\frac{\mathrm{d}{I}_{1}(z)}{\mathrm{d}z}=& -\alpha {I}_{1}+\beta {{n}_{2}}^{\u2033}({I}_{1}+{I}_{2}){I}_{1}+\frac{\beta {{n}_{2}}^{\u2033}}{1+{\Delta}^{2}}{I}_{1}{I}_{2}\\ & +\frac{\beta {{n}_{2}}^{\prime}\Delta}{1+{\Delta}^{2}}{I}_{1}{I}_{2},\end{array}$$
(2)
$$\begin{array}{ll}\frac{\mathrm{d}{I}_{2}(z)}{\mathrm{d}z}=& -\alpha {I}_{2}+\beta {{n}_{2}}^{\u2033}({I}_{1}+{I}_{2}){I}_{2}+\frac{\beta {{n}_{2}}^{\u2033}}{1+{\Delta}^{2}}{I}_{1}{I}_{2}\\ & -\frac{\beta {{n}_{2}}^{\prime}\Delta}{1+{\Delta}^{2}}{I}_{1}{I}_{2},\end{array}$$
(3)
$${I}_{1}(z)=\frac{{I}_{1}(0)}{H(z)[H{(z)}^{-D}(1-{C}_{1})+{C}_{1}]}.$$
(4)
$${I}_{2}(z)=\frac{{I}_{2}(0)}{H(z)[H{(z)}^{D}(1-{C}_{2})+{C}_{2}]},$$
(5)
$$H(z)=kI(0)-\text{exp}(\alpha z)[kI(0)-1].$$
(6)
$${\Gamma}_{\text{ant}}=\frac{1}{2}\text{ln}\frac{{I}_{2}(+\Delta )}{{I}_{2}(-\Delta )}={{n}_{2}}^{\prime}\beta I(0)\frac{1-\text{exp}(-\alpha L)}{\alpha}\frac{\Delta}{1+{\Delta}^{2}}.$$
(7)
$$\frac{{I}_{1}(L)}{{I}_{1}(0)}=\frac{(1+A)\text{exp}(-\alpha L)}{1+A\text{exp}(\gamma )},$$
(8)
$$\frac{{I}_{2}(L)}{{I}_{2}(0)}=\frac{(1+{A}^{-1})\text{exp}(-\alpha L)}{1+{A}^{-1}\text{exp}(-\gamma )},$$
(9)
$$\gamma =\beta (1+A){\Gamma}_{2}{I}_{1}(0)\frac{1-\text{exp}(-\alpha L)}{\alpha},$$
(10)
$${\Gamma}_{1}={{n}_{2}}^{\prime}\frac{\Delta +r({C}^{-1}+2+{C}^{-1}{\Delta}^{2}+{\Delta}^{2})}{1+{\Delta}^{2}},$$
(11)
$${\Gamma}_{2}={{n}_{2}}^{\prime}\frac{\Delta -r(C+2+C{\Delta}^{2}+{\Delta}^{2})}{1+{\Delta}^{2}},$$
(12)
$$\frac{\mathrm{d}I(z)}{\text{dz}}=-\alpha I(z)+\beta {{n}_{2}}^{\u2033}{I}^{2}(z),$$
(13)
$${I}_{1}(z)+{I}_{2}(z)=\frac{I(0)\text{exp}(-\alpha z)}{1+\frac{\beta {{n}_{2}}^{\u2033}}{\alpha}I(0)[\text{exp}(-\alpha z)-1]},$$
(14)
$$\begin{array}{ll}{\Gamma}_{\text{ant}}=\frac{1}{2}\text{ln}\frac{{I}_{2}(+\Delta )}{{I}_{2}(-\Delta )}\approx & {{n}_{2}}^{\prime}\beta I(0)\frac{1-\text{exp}(-\alpha L)}{\alpha}\frac{\Delta}{1+{\Delta}^{2}}\\ & \times \left[1+{{n}_{2}}^{\u2033}\beta I(0)\frac{1-\text{exp}(-\alpha L)}{\alpha}\right].\end{array}$$
(15)
$$\frac{\mathrm{d}{I}_{{\Gamma}_{1}}}{\mathrm{d}z}=-\alpha {I}_{{\Gamma}_{1}}+\gamma {I}_{{\Gamma}_{1}}{I}_{{\Gamma}_{2}},$$
(16)
$$\frac{\mathrm{d}{I}_{{\Gamma}_{2}}}{\mathrm{d}z}=-\alpha {I}_{{\Gamma}_{2}}-\gamma {I}_{{\Gamma}_{1}}{I}_{{\Gamma}_{2}},$$
(17)
$$\frac{{I}_{1}(L)}{{I}_{1}(0)}=\frac{(1+C)\text{exp}(-\alpha L)}{1+C\text{exp}(\gamma )},$$
(18)
$$\frac{{I}_{2}(L)}{{I}_{2}(0)}=\frac{(1+{C}^{-1})\text{exp}(-\alpha L)}{1+{C}^{-1}\text{exp}(-\gamma )}.$$
(19)
$$\gamma =\beta {{n}_{2}}^{\prime}[{I}_{1}(0)+{I}_{2}(0)]\frac{\Delta}{1+{\Delta}^{2}}\frac{1-\text{exp}(-\alpha L)}{\alpha}.$$