In recent years

the subject of optical nonlinearity in semiconductor quantum confined structures such as quantum wires and quantum dots has attracted a considerable attention because of their relevance for studying practical applications and as a probe for the electronic structure of media.It has been shown that the low-dimensional quantum systems present very large nonlinear optical susceptibilities.Theoretically

most of the studies on quantum dots are focused on spherical quantum dots for simplicity.The shape of quantum dot depends on growth conditions.In fact

quantum dots are better described as thin disk or cylinder.In the present work third-order nonlinear optical susceptibilities in disk-shaped quantum dots due to resonant intersubband transition are analyzed by a compact density matrix approach.It has rarely been discussed in literatures.Firstly

we consider an electron moving in a quantum disk whose radius

R

is very large compared to its thickness d.For simplicity

we assume that the isotropic effective mass approximation is valid within the disk and that the inner potential is null while the outer confining potential is approximately infinite.Using the adiabatic approximation we solve the three dimensional Schr-dinger equation and obtain the analytical expressions of wave functions and energy levels in cylindrical coordinates.Secondly

we consider the system is excited by an electromagnetic field with frequency ω. The electromagnetic field is incident with a polarization vector normal to the disk.We treat

H

'=ez

E

(

t

) as a perturbation term.By using the compact density matrix method

we derive a general formula for third order nonlinear optical susceptibilites employing a four-level model in quantum disk.Finally

numerical results are illustrated for a typical GaAs quantum disk.We calculate the third-order nonlinear optical susceptibilities as a function of the disk thickness.It is shown that the third order nonlinear optical susceptibilities in quantum disk are 10～100 times larger than that in quantum well for various disk thickness.Furthermore

if we choose an optimized thickness of quantum disk

we could obtain a quite large refractive index with a comparatively small absorption coefficient.Therefore

we suggest that the quantum dots would be an excellent system in the field of optical communications and optical switching.