Analytical and numerical modeling of optical second harmonic generation in anisotropic crystals using ♯SHAARP package

Abstract

Electric-dipole optical second harmonic generation (SHG) is a second-order nonlinear process that is widely used as a sensitive probe to detect broken inversion symmetry and local polar order. Analytical modeling of the SHG polarimetry of a nonlinear optical material is essential to extract its point group symmetry and the absolute nonlinear susceptibilities. Current literature on SHG analysis involves numerous approximations and a wide range of (in)accuracies. We have developed an open-source package called the Second Harmonic Analysis of Anisotropic Rotational Polarimetry (♯SHAARP.si) which derives analytical and numerical solutions of reflection SHG polarimetry from a single interface (.si) for bulk homogeneous crystals with arbitrary symmetry group, arbitrary crystal orientation, complex and anisotropic linear dielectric tensor with frequency dispersion, a general SHG tensor and arbitrary light polarization. ♯SHAARP.si enables accurate modeling of polarimetry measurements in reflection geometry from highly absorbing crystals or wedge-shaped transparent crystals. The package is extendable to multiple interfaces.

MgSiP2: An Infrared Nonlinear Optical Crystal with a Large Non‐Resonant Phase‐Matchable Second Harmonic Coefficient and High Laser Damage Threshold

“…The advantage of using a wedge sample is that only the SHG generated from the front surface needs to be considered, which has proven robustness in extracting the SHG tensor. [ 3,36 ] The reflected SHG field was decomposed into p ‐polarized (∥) and s ‐polarized (⊥) by an analyzer and detected by a photo‐multiplier tube. For the point group $$\bar{4}2m$$, the d tensor in Voigt notation is: $$$$\begin{eqnarray}d\ = \left( { \def\eqcellsep{&}\begin{array}{cccccc} 0&0&0&{{d}_{14}}&0&0\\ 0&0&0&0&{{d}_{14}}&0\\ 0&0&0&0&0&{{d}_{36}} \end{array} } \right)\ \end{eqnarray}$$$$…”

“…The theoretical expressions for the SHG intensity in normal reflection geometry were generated by the modeling tool ♯SHAARP: [ 36 ] $$$$\begin{eqnarray} I_X^{2\omega } &=& 1.501 \times {{10}}^{ - 5}{{\left( {1. + \cos 2\psi } \right)}}^2d_{14}^2\nonumber\\ && +\, \big[ - 2.324 \times {{10}}^{ - 5} + 3.123 \times {{10}}^{ - 5}\cos 2\psi\nonumber\\ && +\, 5.447 \times {{10}}^{ - 5}{{\left( {\cos 2\psi } \right)}}^2 \big] {d}_{14}{d}_{36}\nonumber\\ && +\, \big[ 9.001 \times {{10}}^{ - 6} - 4.219 \times {{10}}^{ - 5}\cos 2\psi\nonumber\\ && +\, 4.943 \times {{10}}^{ - 5}{{\left( {\cos 2\psi } \right)}}^2 \big]d_{36}^2\nonumber\\ I_Y^{2\omega } &=& 9.532 \times {{10}}^{ - 5}{{\left( {\sin 2\psi } \right)}}^2d_{14}^2 \end{eqnarray}$$$$…”

“…The numbers are the numerical values of the relevant Fresnel coefficients (for conversion from 1550 to 775 nm) and that of the rotation matrix that transforms the crystal physics coordinates (Z 1 , Z 2 , Z 3 ) to the experimental coordinate system (X, Y, Z). The completely analytical expressions of the SHG electric field can be generated using the ♯SHAARP package; [ 36 ] it is not shown here for brevity. To obtain the d ij coefficients of MgSiP 2 , the polar plots were fitted to Equation () (Figure 3b) and compared with that measured on a wedged x‐cut LiNbO 3 reference under the same experimental conditions (Figure S7, Supporting Information).…”

“…The advantage of using a wedge sample is that only the SHG generated from the front surface needs to be considered, which has proven robustness in extracting the SHG tensor. [3,36] The reflected SHG field was decomposed into p-polarized (∥) and s-polarized (⊥) by an analyzer and detected by a photo-multiplier tube. For the point group 42m, the d tensor in Voigt notation is:…”

“…The theoretical expressions for the SHG intensity in normal reflection geometry were generated by the modeling tool ♯SHAARP: [36] I 2𝜔 X = 1.501 × 10 (2)…”