## Abstract

A model of energy transfer by nearly degenerate two-beam coupling in media exhibiting two-photon and excited state absorption is presented and discussed. The two beams include an incident laser, which has a nonlinear time dependent phase, and an elastically backscattered wave derived from the incident laser. Superposition of these two waves creates an index modulation, arising from a spatially modulated, two-photon-created excited-state population, which couples the forward and backward propagating waves. Reflectance of the stimulated wave and transmittance of the incident laser are computed, including the effects of two-photon absorption and excited state absorption as well as energy transfer between the two beams, and the relevance of the results to experimental measurements is discussed. The backscattered wave has a frequency that is unshifted with respect to that of the incident laser, and the small signal gain is proportional to the square of the incident laser intensity. The different effects due to multimode and chirped waves are also discussed.

© 2005 Optical Society of America

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### Equations (15)

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(1)
$$E\left(z,t\right)={A}_{L}\left(z,t+\tau \right)\mathrm{exp}\left\{i\left[\mathit{kz}-\mathit{\omega t}-b{\left(t+\tau \right)}^{2}\right]\right\}$$
(2)
$$+{A}_{S}\left(z,t-\tau \right)\mathrm{exp}\left\{i[-\mathit{kz}-\mathit{\omega t}-b{\left(t-\tau \right)}^{2}]\right\}+c.c.$$
(3)
$$P={\epsilon}_{0}\left({\chi}_{g}^{\left(1\right)}+\left(\frac{{N}_{e}}{N}\right)\Delta {\chi}^{\left(1\right)}+3{\chi}^{\left(3\right)}\u3008{E}^{2}\u3009\right)E$$
(4)
$$\frac{\partial {N}_{e}}{\partial t}=\frac{{\sigma}_{2}N{I}^{2}}{2\mathit{\u0127}\mathit{\omega}}-\frac{{N}_{e}}{{T}_{e}}$$
(5)
$$\frac{{N}_{e}}{N}={C}_{0}+\left\{{C}_{1}\mathrm{exp}\left[i\left(2\mathit{kz}-4\mathit{b\tau t}\right)\right]+{C}_{2}\mathrm{exp}\left[i\left(4\mathit{kz}-8\mathit{b\tau t}\right)\right]+c.c.\right\},$$
(6)
$${C}_{0}=\frac{{\sigma}_{2}{T}_{e}}{2\mathit{\u0127}\mathit{\omega}}\left({I}_{L}^{2}+{I}_{S}^{2}+4{I}_{L}{I}_{S}\right),$$
(7)
$${C}_{1}=\frac{{\sigma}_{2}{T}_{e}}{2\mathit{\u0127}\mathit{\omega}}\left({I}_{L}+{I}_{S}\right)\left(\frac{4{\epsilon}_{0}\mathit{nc}{A}_{L}{A}_{S}^{*}}{1-i4\mathit{b\tau}{T}_{e}}\right),$$
(8)
$${C}_{2}=\frac{{\sigma}_{2}{T}_{e}}{2\mathit{\u0127\omega}}\frac{{\left(2{\epsilon}_{0}\mathit{nc}{A}_{L}{A}_{S}^{*}\right)}^{2}}{1-i8\mathit{b\tau}{T}_{e}},$$
(9)
$$\frac{d{I}_{L}}{\mathit{dz}}=-g\left(1-\frac{z}{d}\right)\left({I}_{L}+{I}_{S}\right){I}_{L}{I}_{S}-{\gamma}_{\mathit{eff}}\left({I}_{L}^{2}+3{I}_{S}^{2}+6{I}_{L}{I}_{S}\right){I}_{L}-\beta \left({I}_{L}+2{I}_{S}\right){I}_{L}$$
(10)
$$\frac{d{I}_{S}}{\mathit{dz}}=-g\left(1-\frac{z}{d}\right)\left({I}_{L}+{I}_{S}\right){I}_{L}{I}_{S}+{\gamma}_{\mathit{eff}}\left({3I}_{L}^{2}+{I}_{S}^{2}+6{I}_{L}{I}_{S}\right){I}_{S}+\beta \left({2I}_{L}+{I}_{S}\right){I}_{S}$$
(11)
$$g=\frac{8b{T}_{e}\mathit{\omega}\Delta {\chi}_{R}^{\left(1\right)}d}{{c}^{2}{I}_{\mathit{sat}}^{2}},$$
(12)
$${\gamma}_{\mathit{eff}}=\frac{N\Delta \sigma}{{I}_{\mathit{sat}}^{2}},$$
(13)
$${\eta}_{0}=\frac{R{\left[\left(1-R\right)\left(1-R+{\eta}_{0}\right)+2{\eta}_{0}^{2}\right]}^{\frac{1}{2}}}{{\left(1+R\right)}^{2}\mathrm{exp}\left(\Gamma \right)}{\left(\frac{1-R+2{\eta}_{0}}{1-R+{\eta}_{0}}\right)}^{\frac{3}{2}},$$
(14)
$$\Gamma =\frac{1}{2}{\left(1-R\right)}^{2}g{I}_{L}^{2}\left(0\right)d.$$
(15)
$${\eta}_{0}\approx \frac{R\left(1-R\right)}{{\left(1+R\right)}^{2}\mathrm{exp}\left(\Gamma \right)}.$$