Cross-polarized wave generation

Cross polarized wave (XPW) generation is a nonlinear optical process that can be classified in the group of frequency degenerate processes. It can take place only in media with anisotropy of third order nonlinearity. As a result of such nonlinear optical interaction at the output of the nonlinear crystal it is generated a new linearly polarized wave at the same frequency, but with polarization oriented perpendicularly to the polarization of input wave Cross polarized wave (XPW) generation is a nonlinear optical process that can be classified in the group of frequency degenerate processes. It can take place only in media with anisotropy of third order nonlinearity. As a result of such nonlinear optical interaction at the output of the nonlinear crystal it is generated a new linearly polarized wave at the same frequency, but with polarization oriented perpendicularly to the polarization of input wave ω ( ⊥ )   =   ω ( ‖ )   +   ω ( ‖ )   −   ω ( ‖ ) {displaystyle omega ^{(perp )}~=~omega ^{(|)}~+~omega ^{(|)}~-~omega ^{(|)}} . Simplified optical scheme for XPW generation is shown on Fig. 1. It consists of a nonlinear crystal plate (thick 1-2 mm) sandwiched between two crossed polarizers. The intensity of generated XPW has cubic dependence with respect to the intensity of the input wave. In fact this is the main reason this effect is so successful for improvement of the contrast of the temporal and spatial profiles of femtosecond pulses. Since cubic crystals are used as nonlinear media they are isotropic with respect to linear properties (there is no birefringence) and because of that the phase and group velocities of both waves XPW and the fundamental wave(FW) are equal:VXPW=VFW and Vgr,XPW=Vgr,FW. Consequence of that is ideal phase and group velocity matching for the two waves propagating along the crystal. This property allows obtaining very good efficiency of the XPW generation process with minimum distortions of the pulse shape and the spectrum. Consider the case of interaction of two perpendicularly polarized waves in nonlinear media with cubic nonlinearity . The equations describing the self phase modulation of the fundamental wave A and the generation of new wave perpendicularly polarized wave B in condition that |B| << |A| (i.e. when the depletion of the fundamental wave is neglected, self and cross phase modulation of wave B) can be written in following form: where γ ‖ {displaystyle gamma _{|}} and γ ⊥ {displaystyle gamma _{perp }} are coefficients that depend on (i) the orientation of the sample with respect to the crystal axes (see for the expressions for two popular orientations: Z-cut and for holographic cut); (ii) the component χ x x x x ( 3 ) {displaystyle chi _{xxxx}^{(3)}} and (iii) anisotropy of χ ( 3 ) {displaystyle chi _{}^{(3)}} tensor. The solution of this simplified system with initial conditions А(0)=А0 and B (0) = 0 is: where L is the length of the nonlinear media. In case of CW pump, the efficiency η {displaystyle eta } , that is defined as ratio of XPW intensity Iout at the output of the nonlinear media to the intensity of the input wave Iin can be described by sin2 function of input intensity × length product: (1)         η = | B ( L ) | 2 | A 0 | 2 = I o u t I i n = 4 ( γ ⊥ γ | | ) 2 sin 2 ⁡ ( γ | | | A | 2 L / 2 ) {displaystyle eta ={frac {|B(L)|^{2}}{|A_{0}|^{2}}}={frac {I_{out}}{I_{in}}}=4({frac {gamma _{perp }}{gamma _{||}}})^{2}sin ^{2}(gamma _{||}|A|^{2}L/2)} . If self phase modulation is relatively small γ ‖ | A | 2 L ≤ 1 {displaystyle gamma _{|}|A|^{2}Lleq 1} then: