## Abstract

For low-symmetry crystals that belong to 2 and *m* point groups, two kinds of nonlinear optical coefficients matrix were derived after the positive directions of the optical coordinate axes were defined with the aid of the piezoelectric coordinate axes. These two kinds of nonlinear optical coefficient matrix are the cause of the nonsymmetry of the effective nonlinear optical coefficients about the *x*, *y*, and *z* optical coordinate planes. For the corresponding wave vector $k(\theta ,\phi )$ in the $N\text{th}$ quadrant, the effective nonlinear optical coefficient expressions are $d_{\mathrm{eff}}{}^{\mathrm{I}}=a_{i}{}^{{e}_{2}}{\left[{d}_{ijk}({\omega}_{3},{\omega}_{2},{\omega}_{1})\right]}_{N+2}a_{j}{}^{{e}_{1}}a_{k}{}^{{e}_{1}}$ (Type I) and $d_{\mathrm{eff}}{}^{\mathrm{II}}=a_{i}{}^{{e}_{2}}{\left[{d}_{ijk}({\omega}_{3},{\omega}_{2},{\omega}_{1})\right]}_{N+1}a_{j}{}^{{e}_{1}}a_{k}{}^{{e}_{2}}$ (Type II), where *N* is the number of the quadrant. Only one kind of nonlinear optical coefficient matrix is the maximum effective nonlinear optical coefficient. It is necessary to find out which quadrant, in which the wave vector $k(\theta ,\phi )$ is located, corresponds to the maximum effective nonlinear optical coefficient.

© 2005 Optical Society of America

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