## Abstract

We investigate the effect of beam temporal coherence on four-wave mixing via transmission gratings in photorefractive media. We take into account the propagation of mutual coherence and self-coherence in space and the evolution of mutual coherence and self-coherence with time. Analytical solutions for nondepleted-pump cases and numerical results for general cases are obtained. We find that the mutual coherence of the signal beam and the pump beam can be enhanced or reduced, depending on the coupling constant and the beam ratio of the signal beam and the pump beam. A decrease of the initial mutual coherence can lead to a lower phase-conjugate reflectivity in the nondepleted-pump case. Even with low initial mutual coherence, high phase-conjugate reflectivity can still be obtained in the case with pump depletion. As a result of the shift of the grating profile, partial coherence can lead to an enhancement of the phase-conjugate reflectivity when the signal–pump-beam ratio is small. We also found that when the initial value of the phase-conjugate beam is zero, the phase-conjugate beam and the pump beam are in phase during propagation. Results for beam intensity and mutual coherence are presented and discussed to provide an insight into four-wave mixing with partially temporally coherent waves.

© 1999 Optical Society of America

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### Equations (51)

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(1)
$${E}_{j}(z,\omega )={A}_{j}(z,\omega -{\omega}_{0})exp(-{\mathit{ik}}_{j}z),$$
(2)
$${A}_{j}(z,\omega -{\omega}_{0})={\int}_{-\infty}^{+\infty}{A}_{j}(z,t)exp[i(\omega -{\omega}_{0})t]\mathrm{d}t,$$
(3)
$$\frac{\partial {A}_{1}(z,t)}{\partial z}+\frac{1}{\nu}\frac{\partial {A}_{1}(z,t)}{\partial t}=-\frac{\gamma}{2}\frac{Q(z,t){A}_{2}(z,t)}{{I}_{0}},$$
(4)
$$\frac{\partial {A}_{2}(z,t)}{\partial z}+\frac{1}{\nu}\frac{\partial {A}_{2}(z,t)}{\partial t}=\frac{\gamma}{2}\frac{{Q}^{*}(z,t){A}_{1}(z,t)}{{I}_{0}},$$
(5)
$$\frac{\partial {A}_{3}(z,t)}{\partial z}-\frac{1}{\nu}\frac{\partial {A}_{3}(z,t)}{\partial t}=\frac{\gamma}{2}\frac{Q(z,t){A}_{4}(z,t)}{{I}_{0}},$$
(6)
$$\frac{\partial {A}_{4}(z,t)}{\partial z}-\frac{1}{\nu}\frac{\partial {A}_{4}(z,t)}{\partial t}=-\frac{\gamma}{2}\frac{{Q}^{*}(z,t){A}_{3}(z,t)}{{I}_{0}},$$
(7)
$${\tau}_{p}\frac{\partial Q(z,t)}{\partial t}+Q(z,t)={A}_{1}(z,t){{A}_{2}}^{*}(z,t)+{A}_{3}(z,t){{A}_{4}}^{*}(z,t),$$
(8)
$$\frac{1}{{\tau}_{p}}=\frac{1}{{\tau}_{1}}+\frac{1}{{\tau}_{2}},$$
(9)
$$\frac{1}{{\tau}_{1}}\sim \u3008|{A}_{1}{|}^{2}\u3009+\u3008|{A}_{2}{|}^{2}\u3009,$$
(10)
$$\frac{1}{{\tau}_{2}}\sim \u3008|{A}_{3}{|}^{2}\u3009+\u3008|{A}_{4}{|}^{2}\u3009.$$
(11)
$$\frac{\partial {\mathrm{\Gamma}}_{12}(z,\tau )}{\partial z}=\frac{\gamma}{2{I}_{0}}Q(z)[{\mathrm{\Gamma}}_{11}(z,\tau )-{\mathrm{\Gamma}}_{22}(z,\tau )],$$
(12)
$$\frac{\partial {\mathrm{\Gamma}}_{11}(z,\tau )}{\partial z}=-\frac{\gamma}{2{I}_{0}}Q(z){{\mathrm{\Gamma}}_{12}}^{*}(z,-\tau )-\frac{\gamma}{2{I}_{0}}{Q}^{*}(z){\mathrm{\Gamma}}_{12}(z,\tau ),$$
(13)
$$\frac{\partial {\mathrm{\Gamma}}_{22}(z,\tau )}{\partial z}=\frac{\gamma}{2{I}_{0}}Q(z){{\mathrm{\Gamma}}_{12}}^{*}(z,-\tau )+\frac{\gamma}{2{I}_{0}}{Q}^{*}(z){\mathrm{\Gamma}}_{12}(z,\tau ),$$
(14)
$$\frac{\partial {\mathrm{\Gamma}}_{34}(z,\tau )}{\partial z}=\frac{\gamma}{2{I}_{0}}Q(z)[{\mathrm{\Gamma}}_{44}(z,\tau )-{\mathrm{\Gamma}}_{33}(z,\tau )],$$
(15)
$$\frac{\partial {\mathrm{\Gamma}}_{33}(z,\tau )}{\partial z}=\frac{\gamma}{2{I}_{0}}Q(z){{\mathrm{\Gamma}}_{34}}^{*}(z,-\tau )+\frac{\gamma}{2{I}_{0}}{Q}^{*}(z){\mathrm{\Gamma}}_{34}(z,\tau ),$$
(16)
$$\frac{\partial {\mathrm{\Gamma}}_{44}(z,\tau )}{\partial z}=-\frac{\gamma}{2{I}_{0}}Q(z){{\mathrm{\Gamma}}_{34}}^{*}(z,-\tau )-\frac{\gamma}{2{I}_{0}}{Q}^{*}(z){\mathrm{\Gamma}}_{34}(z,\tau ),$$
(17)
$$\frac{\partial {\mathrm{\Gamma}}_{13}(z,\tau )}{\partial z}=-\frac{2}{\nu}\frac{\partial {\mathrm{\Gamma}}_{13}(z,\tau )}{\partial \tau}-\frac{\gamma}{2{I}_{0}}Q(z){\mathrm{\Gamma}}_{23}(z,\tau )+\frac{\gamma}{2{I}_{0}}{Q}^{*}(z){\mathrm{\Gamma}}_{14}(z,\tau ),$$
(18)
$$\frac{\partial {\mathrm{\Gamma}}_{14}(z,\tau )}{\partial z}=-\frac{2}{\nu}\frac{\partial {\mathrm{\Gamma}}_{14}(z,\tau )}{\partial \tau}-\frac{\gamma}{2{I}_{0}}Q{\mathrm{\Gamma}}_{24}(z,\tau )-\frac{\gamma}{2{I}_{0}}Q{\mathrm{\Gamma}}_{13}(z,\tau ),$$
(19)
$$\frac{\partial {\mathrm{\Gamma}}_{23}(z,\tau )}{\partial z}=-\frac{2}{\nu}\frac{\partial {\mathrm{\Gamma}}_{23}(z,\tau )}{\partial \tau}+\frac{\gamma}{2{I}_{0}}{Q}^{*}(z){\mathrm{\Gamma}}_{13}(z,\tau )+\frac{\gamma}{2{I}_{0}}{Q}^{*}(z){\mathrm{\Gamma}}_{24}(z,\tau ),$$
(20)
$$\frac{\partial {\mathrm{\Gamma}}_{24}(z,\tau )}{\partial z}=-\frac{2}{\nu}\frac{\partial {\mathrm{\Gamma}}_{24}(z,\tau )}{\partial \tau}+\frac{\gamma}{2{I}_{0}}{Q}^{*}(z){\mathrm{\Gamma}}_{14}(z,\tau )-\frac{\gamma}{2{I}_{0}}Q(z){\mathrm{\Gamma}}_{23}(z,\tau ).$$
(21)
$${\mathrm{\Gamma}}_{11}(z,t)+{\mathrm{\Gamma}}_{22}(z,t)=\mathrm{const}.,$$
(22)
$${\mathrm{\Gamma}}_{33}(z,t)+{\mathrm{\Gamma}}_{44}(z,t)=\mathrm{const}.$$
(23)
$$\u3008{A}_{1}{{A}_{1}}^{*}\u3009,\u3008{A}_{4}{{A}_{4}}^{*}\u3009\ll \u3008{A}_{2}{{A}_{2}}^{*}\u3009,\u3008{A}_{3}{{A}_{3}}^{*}\u3009,$$
(24)
$$\u3008{A}_{3}{{A}_{4}}^{*}\u3009\ll \u3008{A}_{1}{{A}_{2}}^{*}\u3009.$$
(25)
$$\frac{\partial {\mathrm{\Gamma}}_{22}(z,\tau )}{\partial z}=\frac{\partial {\mathrm{\Gamma}}_{33}(z,\tau )}{\partial z}\cong 0,$$
(26)
$$\frac{\partial {\mathrm{\Gamma}}_{12}(z,\tau )}{\partial z}=-\frac{\gamma}{2{I}_{0}}{\mathrm{\Gamma}}_{12}(z,0){\mathrm{\Gamma}}_{22}(z,\tau ),$$
(27)
$$\frac{\partial {\mathrm{\Gamma}}_{11}(z,\tau )}{\partial z}=-\frac{\gamma}{2{I}_{0}}{\mathrm{\Gamma}}_{12}(z,0){{\mathrm{\Gamma}}_{12}}^{*}(z,-\tau )-\frac{\gamma}{2{I}_{0}}{{\mathrm{\Gamma}}_{12}}^{*}(z,0){\mathrm{\Gamma}}_{12}(z,\tau ),$$
(28)
$$\frac{\partial {\mathrm{\Gamma}}_{34}(z,\tau )}{\partial z}=-\frac{\gamma}{2{I}_{0}}{\mathrm{\Gamma}}_{12}(z,0){\mathrm{\Gamma}}_{33}(z,\tau ),$$
(29)
$$\frac{\partial {\mathrm{\Gamma}}_{44}(z,\tau )}{\partial z}=-\frac{\gamma}{2{I}_{0}}{{\mathrm{\Gamma}}_{12}}^{*}(z,0){\mathrm{\Gamma}}_{34}(z,\tau )-\frac{\gamma}{2{I}_{0}}{\mathrm{\Gamma}}_{12}(z,0){{\mathrm{\Gamma}}_{34}}^{*}(z,-\tau ).$$
(30)
$${\mathrm{\Gamma}}_{s}(\tau )=exp\left[-{\left(\frac{\pi \mathrm{\Delta}\nu \tau}{2\sqrt{ln2}}\right)}^{2}\right],$$
(31)
$${\mathrm{\Gamma}}_{11}(0,\tau )={\mathrm{\Gamma}}_{s}(\tau ),$$
(32)
$${\mathrm{\Gamma}}_{22}(0,\tau )={\beta}_{21}{\mathrm{\Gamma}}_{s}(\tau ),$$
(33)
$${\mathrm{\Gamma}}_{12}(0,\tau )=\sqrt{{\beta}_{21}}{\mathrm{\Gamma}}_{s}(\tau +\mathrm{\Delta}t),$$
(34)
$${\mathrm{\Gamma}}_{33}(L,\tau )={\beta}_{31}{\mathrm{\Gamma}}_{s}(\tau ),$$
(35)
$${\mathrm{\Gamma}}_{44}(L,\tau )={\beta}_{41}{\mathrm{\Gamma}}_{s}(\tau ),$$
(36)
$${\mathrm{\Gamma}}_{34}(L,\tau )=\sqrt{{\beta}_{31}{\beta}_{41}}{\mathrm{\Gamma}}_{s}(\tau +\mathrm{\Delta}{t}^{\prime}),$$
(37)
$${\mathrm{\Gamma}}_{11}(z,\tau )=\frac{1}{2I_{0}{}^{2}}{\mathrm{\Gamma}}_{12}(0,0){{\mathrm{\Gamma}}_{12}}^{*}(0,0)[{\mathrm{\Gamma}}_{22}(0,\tau )+{{\mathrm{\Gamma}}_{22}}^{*}(0,-\tau )]{\left[exp\left(-\frac{\gamma}{2}z\right)-1\right]}^{2}+\frac{1}{{I}_{0}}[{\mathrm{\Gamma}}_{12}(0,0){{\mathrm{\Gamma}}_{12}}^{*}(0,-\tau )+{{\mathrm{\Gamma}}_{12}}^{*}(0,0){\mathrm{\Gamma}}_{12}(0,\tau )]\left[exp\left(-\frac{\gamma}{2}z\right)-1\right]+{\mathrm{\Gamma}}_{11}(0,\tau ),$$
(38)
$${\mathrm{\Gamma}}_{22}(z,\tau )={\mathrm{\Gamma}}_{22}(0,\tau ),$$
(39)
$${\mathrm{\Gamma}}_{33}(z,\tau )={\mathrm{\Gamma}}_{33}(L,\tau ),$$
(40)
$${\mathrm{\Gamma}}_{44}(z,\tau )=\frac{1}{2I_{0}{}^{2}}{\mathrm{\Gamma}}_{12}(0,0){{\mathrm{\Gamma}}_{12}}^{*}(0,0)[{\mathrm{\Gamma}}_{33}(L,\tau )+{{\mathrm{\Gamma}}_{33}}^{*}(L,-\tau )]{\left[exp\left(-\frac{\gamma}{2}z\right)-exp\left(-\frac{\gamma}{2}L\right)\right]}^{2},$$
(41)
$${\mathrm{\Gamma}}_{12}(z,\tau )=\frac{1}{{I}_{0}}{\mathrm{\Gamma}}_{12}(0,0){\mathrm{\Gamma}}_{22}(0,\tau )\left[exp\left(-\frac{\gamma}{2}z\right)-1\right]+{\mathrm{\Gamma}}_{12}(0,\tau ),$$
(42)
$${\mathrm{\Gamma}}_{34}(z,\tau )=\frac{1}{{I}_{0}}{\mathrm{\Gamma}}_{12}(0,0){\mathrm{\Gamma}}_{33}(L,\tau )\times \left[exp\left(-\frac{\gamma}{2}z\right)-exp\left(-\frac{\gamma}{2}L\right)\right],$$
(43)
$${\mathrm{\Gamma}}_{11}(z,0)=\frac{1}{{I}_{0}}{\mathrm{\Gamma}}_{12}(0,0){{\mathrm{\Gamma}}_{12}}^{*}(0,0)\times [exp(-\gamma z)-1]+{\mathrm{\Gamma}}_{11}(0,0),$$
(44)
$${\mathrm{\Gamma}}_{22}(z,0)={\mathrm{\Gamma}}_{22}(0,0),$$
(45)
$${\mathrm{\Gamma}}_{33}(z,0)={\mathrm{\Gamma}}_{33}(L,0),$$
(46)
$${\mathrm{\Gamma}}_{33}(z,0)=\frac{1}{I_{0}{}^{2}}{\mathrm{\Gamma}}_{12}(0,0){{\mathrm{\Gamma}}_{12}}^{*}(0,0){\mathrm{\Gamma}}_{33}(L,0)\times {\left[exp\left(-\frac{\gamma}{2}z\right)-exp\left(-\frac{\gamma}{2}L\right)\right]}^{2},$$
(47)
$${\mathrm{\Gamma}}_{12}(z,0)={\mathrm{\Gamma}}_{12}(0,0)exp\left(-\frac{\gamma}{2}z\right),$$
(48)
$${\mathrm{\Gamma}}_{34}(z,0)=\frac{1}{{I}_{0}}{\mathrm{\Gamma}}_{12}(0,0){\mathrm{\Gamma}}_{33}(L,0)\times \left[exp\left(-\frac{\gamma}{2}z\right)-exp\left(-\frac{\gamma}{2}L\right)\right].$$
(49)
$${\mathrm{\Gamma}}_{12n}(z,0)=\frac{|{\mathrm{\Gamma}}_{12}(z,0)|}{[{\mathrm{\Gamma}}_{11}(z,0){\mathrm{\Gamma}}_{22}(z,0){]}^{1/2}}=\frac{1}{{\left[1+\left(\frac{1}{\mathrm{\Gamma}_{12n}{}^{2}(0,0)}-1\right)exp(\gamma z)\right]}^{1/2}},$$
(50)
$${\mathrm{\Gamma}}_{34n}(z,0)=\frac{|{\mathrm{\Gamma}}_{34}(z,0)|}{[{\mathrm{\Gamma}}_{33}(z,0){\mathrm{\Gamma}}_{44}(z,0){]}^{1/2}}=1.$$
(51)
$$|\rho {|}^{2}=\frac{{\mathrm{\Gamma}}_{44}(0,0)}{{\mathrm{\Gamma}}_{11}(0,0)}=\frac{1}{I_{0}{}^{2}}\frac{{\mathrm{\Gamma}}_{33}(L,0)}{{\mathrm{\Gamma}}_{11}(0,0)}{\mathrm{\Gamma}}_{12}(0,0){{\mathrm{\Gamma}}_{12}}^{*}(0,0)\times {\left[1-exp\left(-\frac{\gamma}{2}L\right)\right]}^{2}\propto \mathrm{\Gamma}_{12n}{}^{2}(0,0){\left[1-exp\left(-\frac{\gamma}{2}L\right)\right]}^{2}.$$