Table of symbols

Variables	Explanation
$(E_{L_1}^{T,e,\omega},E_{L_2}^{T,e,\omega},E_{L_3}^{T,e,\omega})$	Components of the unit vector of the extraordinary electric field at $\omega$ frequency in the $(L_1,L_2,L_3)$ coordinate system.
$(E_{L_1}^{T,o,\omega},E_{L_2}^{T,o,\omega},E_{L_3}^{T,o,\omega})$	Components of the unit vector of the ordinary electric field at $\omega$ frequency in the $(L_1,L_2,L_3)$ coordinate system.
$(E_{L_1}^{T,e,2\omega},E_{L_2}^{T,e,2\omega},E_{L_3}^{T,e,2\omega})$	Components of the unit vector of the extraordinary electric field at $2\omega$ frequency in the $(L_1,L_2,L_3)$ coordinate system.
$(E_{L_1}^{T,o,2\omega},E_{L_2}^{T,o,2\omega},E_{L_3}^{T,o,2\omega})$	Components of the unit vector of the extraordinary electric field at $2\omega$ frequency in the $(L_1,L_2,L_3)$ coordinate system.
$(\tilde{n}_1^{\omega},\tilde{n}_2^{\omega},\tilde{n}_3^{\omega})$	Complex refractive indices in principal coordinate system at $\omega$.
$(\tilde{n}_1^{2\omega},\tilde{n}_2^{2\omega},\tilde{n}_3^{2\omega})$	Complex refractive indices in principal coordinate system at $2\omega$.
$(P_{L_1}^{T,ee,2\omega},P_{L_2}^{T,ee,2\omega},P_{L_3}^{T,ee,2\omega})$	Nonlinear polarization generated by two extraordinary waves.
$(P_{L_1}^{T,oo,2\omega},P_{L_2}^{T,oo,2\omega},P_{L_3}^{T,oo,2\omega})$	Nonlinear polarization generated by two ordinary waves.
$(P_{L_1}^{T,eo,2\omega},P_{L_2}^{T,eo,2\omega},P_{L_3}^{T,eo,2\omega})$	Nonlinear polarization generated by extraordinary and ordinary waves.
$(\theta^i,\theta^R,\theta^{T,e,\omega},\theta^{T,o,\omega},\theta^{T,e,2\omega},\theta^{T,o,2\omega})$	Incident angle; reflective angle; refractive angle for extraordinary, ordinary wave at $2\omega$; refractive angle for extraordinary, ordinary wave at $2\omega$.

Varaibles	Explanation
$(L_1,L_2,L_3)$	Orthogonal lab coordinate system (LCS) where experiments are performed.
$(Z_1,Z_2,Z_3)$	Orthogonal crystal physics coordinate system (ZCS) where materials properties are described.
$(a,b,c)$	Non-orthogonal crystallographic coordinate system (CCS) where lattice parameters are defined.
$(Z_1^{Principal},Z_2^{Principal},Z_3^{Principal})$	Orthogonal principal coordinate system (PCS) where dielectric/refractive index tensors are diagonalized.
$(\varepsilon_{L_1L_1}^{\omega},\varepsilon_{L_1L_2}^{\omega},\varepsilon_{L_1L_3}^{\omega},\varepsilon_{L_2L_2}^{\omega},\varepsilon_{L_2L_3}^{\omega},\varepsilon_{L_3L_3}^{\omega})$	Dielectric tensor components at $\omega$ frequency described in the lab coordinate system.
$(\varepsilon_{L_1L_1}^{2\omega},\varepsilon_{L_1L_2}^{2\omega},\varepsilon_{L_1L_3}^{2\omega},\varepsilon_{L_2L_2}^{2\omega},\varepsilon_{L_2L_3}^{2\omega},\varepsilon_{L_3L_3}^{2\omega})$	Dielectric tensor components at $2\omega$ frequency described in the lab coordinate system.
$(C_{L_1}^{T,ee,2\omega},C_{L_2}^{T,ee,2\omega},C_{L_3}^{T,ee,2\omega})$	Electric fields radiated by nonlinear polarization $(P_{L_1}^{T,ee,2\omega},P_{L_2}^{T,ee,2\omega},P_{L_3}^{T,ee,2\omega})$
$(C_{L_1}^{T,oo,2\omega},C_{L_2}^{T,oo,2\omega},C_{L_3}^{T,oo,2\omega})$	Electric fields radiated by nonlinear polarization $(P_{L_1}^{T,oo,2\omega},P_{L_2}^{T,oo,2\omega},P_{L_3}^{T,oo,2\omega})$
$(C_{L_1}^{T,eo,2\omega},C_{L_2}^{T,eo,2\omega},C_{L_3}^{T,eo,2\omega})$	Electric fields radiated by nonlinear polarization $(P_{L_1}^{T,eo,2\omega},P_{L_2}^{T,eo,2\omega},P_{L_3}^{T,eo,2\omega})$
$(k^i,k^R,k^{T,e,\omega},k^{T,o,\omega})$	Wavevectors of incident wave, reflective wave; refractive waves for extraordinary, ordinary wave at $\omega$
$(k^{T,e,2\omega},k^{T,o,2\omega},)$	Refractive waves for extraordinary, ordinary wave at $2\omega$
$(k^{T,ee,2\omega},k^{T,oo,2\omega},k^{T,eo,2\omega})$	Refractive waves for three inhomogeneous wave at $2\omega$
$(d_{ij}, i=1-3, j=1-6)$	SHG Tensor