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The excitonic effects on the nonlinear optical properties of small quantum dots with a semiparabolic confining potential are studied under the density matrix formalism. First, within the framework of the strong confinement approximation, we present the excitonic states and then calculate the excitonic effects on the nonlinear optical properties, such as second harmonic generation, third harmonic generation, nonlinear absorption coefficient and refractive index changes. We find the explicit analytical expressions between the corresponding nonlinear optical properties with and without considering the excitonic effects. It is seen that these analytical expressions are related only to ratios of the effective masses of electron and hole. These explicit expressions indicate that the excitonic effects on the corresponding nonlinear optical properties become more important with increasing orders of the optical susceptibilities. In addition, we suggest a scaling rule for the nth-order susceptibility as γ(n+1)/2. The effect of the confining potential frequency on the corresponding nonlinear optical properties is also studied. Our results show a remarkable dependence of nonlinear optical properties on both the excitonic effects and the confining potential frequency.

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