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Chinese Journal of Luminescence

Third-order Nonlinear Optical Susceptibility in Quantum Disks

Cui Liu

Published Jan 1, 2001• 1 citation
OpenAlex

Abstract

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′=ezE(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.

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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′=ezE(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.
I recent years, significant attention has been given to nonlinear optical phenomena, such as nonlinear optical absorption and dispersion, derived from a driven quantum transition of semiconductor nanostructures, such as semiconductor quantum wells, quantum disks and mainly quantum dots. An important issue that is widely studied is the effect of probe electromagnetic field intensity on the total (nonlinear) absorption coefficient and on the total (nonlinear) index of refraction. Some of the phenomena that are commonly found for large electromagnetic field intensities are the creation of a strong dip in the absorption spectrum near or at resonance (the so-called bleaching effect), negative absorption near or at resonance that leads to optical gain and the change of the slope of the total index of refraction near or at resonance from negative to positive that changes the behavior of the system from fast to slow light. Unfortunately, these phenomena do not exist and are artifacts of a common abuse of the theoretical methodology for the study of the total absorption coefficient and the total index of refraction. The main error is that the total absorption coefficient and the total index of refraction was initially derived from density matrix equations using perturbation theory and includes first-order (linear) as well as third-order (nonlinear) terms. This method gives correct results for small electromagnetic field intensities and it leads to saturation of absorption and dispersion in this regime. However, for larger intensities it gives the misleading results mentioned above due to either failure of the order of perturbation theory considered in the derivation of the total absorption coefficient and the total index of refraction or even failure of perturbation theory at all in the treatment of the problem. In this work, we revisit the problem of the total absorption coefficient and the total index of refraction in a symmetric semiconductor quantum dot structure (we choose the example of a core-shell quantum dot) under a strong probe field excitation. We use the two-level model, solve the relevant density matrix equations under the rotating wave approximation and under steady state conditions, and obtain the correct form of the nonlinear optical susceptibility that is then used for the derivation of the formulae of the total absorption coefficient and the total index of refraction under the interaction with a strong probe field. Actually, the commonly used perturbation theory formulae can be obtained by our formulae using a series expansion and keeping terms up to second order in electric field amplitude. We also derive the electronic structure of the quantum dot system by solving numerically the time-independent Schrodinger equation, under the effective mass approximation, using the potential morphing method and obtain the energy difference and the electric dipole matrix element for the relevant transitions of the quantum dot system. Then, for the specific quantum dot system we compare the results of the total absorption coefficient and the total index of refraction for the two methodologies (the perturbation result and our result) for different electromagnetic field intensities.
To increase the optical nonlinearity effects in hybrid structures comprising a quantum dot and a nanoparticle, a quantum dot is taken as a three-level system. Under the dipole–dipole interactions between the quantum dot and the nanoparticle, the density matrix elements are derived within the rotating approximation. Under the quasi-static approximation by taking the nanoparticle polarization as a function of its geometry, the induced fields around the quantum dot in the presence of one/two nanoparticle(s) are obtained and the increasing factors of the local fields are calculated. The third-order nonlinear optical susceptibility of quantum dot is achieved by using the perturbation expansion theory of density matrix for different geometries of the nanoparticle, and the effects of the structural parameters are investigated to optimize the nonlinear optical response of the system. It can be seen that the presence of two nanorods results in great optical nonlinearities in the hybrid structure, being suitable for on/off switches, quantum communication, and quantum computing applications.
The effects of quantum dot (QD) radius, incident light intensity, temperature and other factors on the optical absorption coefficients (OACs) of cylindrical QDs are investigated numerically for typical GaAs/Ga[Formula: see text]Al[Formula: see text]As. We theoretically studied the cylindrical QDs with like-inversely quadratic Hellmann potential and parabolic potential. The expressions for the energy levels and wave functions were obtained by the Nikiforov–Uvarov method, and then the expressions for the OACs were derived by using the density matrix theory and the iterative method. Finally, the results show that the structural and external parameters affect the sensitivity of the optical absorption properties by influencing the matrix elements and energy levels, and the quantum confinement effects affect their applicable frequencies, which makes the QD system easier to tune.
Linear and nonlinear optical absorptions in one-dimensional semi-parabolic quantum dot is investigated. The analytical expression of linear and third-order nonlinear optical absorption coefficients are obtained by using compact density-matrix approach and iterative method. Numerical results are presented for an GaAs/AlGaAs quantum dot and the dependence on the quantum confinement of the semi-parabolic quantum dot, the incident optical intensity, and on the relaxation rate are also investigated. The results indicate that the optical absorption coefficients are strongly influenced by the confining frequency, relaxation time, and the incident optical intensity, respectively. However, their influences for the linear optical absorption, third-order nonlinear optical absorption, and the total optical absorption are different. We think our theoretical study may make a great contribution to experimental studies and may open up new opportunities for practical exploitation of the quantum-size effect in optical devices.
In the past few years advance in nanofabrication techniques have made it possible to make well-like compound semiconductor structures of low nanometer size.In these structures,the state of the electron in the direction perpendicular to the length of the well is quantized with sufficiently large separation in energy levels and the motion of the electron parallel to the length is that of a quasi-free electron.One of the most remarkable properties of these one-dimensional electron system is that the optical transitions between the size-quantized subbands are feasible.The result shows there is a large optical nonlinearities in asymmetric quantum wells.The problem of optical saturation of intersubband absorption has recently attracted considerable attention.The problem is very important to the development of superlattice devices such as fast infrared detectors and(modulators) because one needs to know the range of input optical intensities for which linear operation is(possible.) The analytical expression of the linear and the third-order nonlinear optical intersubband absorption coefficients are obtained in Pschl-Teller wells using the density matrix formalism taking into account the(intrasubband) relaxation.Since there are two adjustable parameters κ and λ in Pschl-Teller wells,the shapes of the wells will change with the change of the parameters κ and λ,accordingly the linear and the third-order nonlinear optical absorption coefficients will change with the parameters κ and λ.It is shown that the optical absorption coefficient depends on the parameter,and the optical intensity strongly. The absorption saturation will occur with I increase.Numerical results are presented for a typical AlGaAs/GaAs Pschl-Teller well.
Abstract The Kerr nonlinearity and the nonlinear absorption coefficient in a four-level M-model of a GaAs cylindrical quantum dot (QD) with parabolic potential under electromagnetically induced transparency are investigated. By solving the density matrix equations in the steady-state, the third order susceptibility is obtained. Then, by using the real and imaginary parts of third order susceptibility, the Kerr nonlinearity and the nonlinear absorption coefficient, respectively, for this system are computed. The effects of the radius and height of the cylindrical QD are then investigated. In addition, the effects of the control laser fields on the Kerr nonlinearity and the nonlinear absorption coefficient are investigated.