Taking account of the contribution of spin-orbit splitting, the energy band of GaN/InGaN quantum wells (QWs) has been calculated. For the quadratic electro-optic effect (QEOE), the resonant third-order nonlinear optical susceptibility due to the interband transition of the mode, whose polarization is parallel to the [0 0 1] direction of the QWs, has been analysed as a function of the well width and the concentration of In.

The electron states confined in wurtzite In x Ga 1-x N/GaN strained quantum dots (QDs) have been investigated in the effective -mass approximation by solving the Schrödinger equation, in which parabolic confined potential and strong built-in electric field effect (due to the piezoelectricity and spontaneous polarization) have been taken into account. The real part (3) Re (0,0, ) χ ω and the imaginary part (3) Im (0,0, ) χ ω of the third-order susceptibility describe quadratic electro-optic effects and electro-absorption process of the QDs respectively. And both of them have been calculated in directions parallel and vertical to z axis. Furthermore, the study shows (3) Re (0,0, ) χ ω and (3) Im (0,0, ) χ ω increase under resonant conditions with the QDs' radius and height increase, and the same results occur when the content increase. In addition, the resonant position shift to the lower energy region when the parabolic frequencies increase.GaN-based semiconductor materials have been attracting great interest lately due to their wide direct band gaps, high thermal conductivity, high electron saturated drift velocity and small dielectric constants [1][2][3][4] . Studies show that due to the large piezoelectric constants along the [0001] orientation of the group III nitrides [5][6][7] , the presence of the strong built-in piezoelectric field in wurtzite QDs leads to a modification of the confined carrier states [8] . Liming Liu considered the built-in piezoelectric field in wurtzite cylindrical QDs and studied the third-order nonlinear susceptibility's dependence on dots' size as well as distribution [9] . At the same time, because of the quantum size effect, there is a confined potential of carriers in both three dimensions. WANG Guanghui discussed the third-order nonlinear susceptibility in disk-like QDs with parabolic confine potential taken into account [10] and many other studies on built-in electric effect in these materials are also investigated [11][12][13][14][15] . Self-assembled In x Ga 1-x N/GaN QDs combined these two factors provide a good model to investigate the nonlinear properties. Besides, many studies on the transition in the conduction bands of quantum wells have been reported [11,12] .In this paper, we present a calculation of the intraband transition of electrons in the conduction band of the cylindrical In x Ga 1-x N/GaN QDs, which are confined in parabolic potential and strong built-in electric field at various directions. The third-order susceptibility on quadratic electro-optic effects and electro-absorption

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