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arXiv:0704.1697 · Physics of the Solid State

Effect of spatial dispersion on the shape of a light pulse propagating through a quantum well

L. I. Korovin, I. G. Lang, S. T. Pavlov

Published Oct 1, 2007• 0 citations
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Abstract

The reflection, transmission, and absorption of a symmetric electromagnetic pulse are calculated. The carrier frequency of the pulse is close to the frequency of direct interband transitions in a quantum well (QW). The QW energy levels are assumed to be discrete, with two closely spaced excited levels being taken into account. The QW width is assumed to be sufficiently large and comparable to the light wavelength corresponding to the pulse carrier frequency. In this case, the dependence of the momentum matrix element for the interband transition on the light wave vector should be taken into account. The refractive indices of the QW and barriers are assumed to be equal. The problem is solved for an arbitrary relation between the radiative and non-radiative lifetimes of the excited electronic states. It is shown that spatial dispersion considerably affects the shapes of the reflected and transmitted pulses. The greatest changes occur in the case where the inverse radiative lifetime is close to the difference between the frequencies of the interband transitions considered.

DOI
10.1134/s1063783407100290

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The reflection, transmission, and absorption of a symmetric electromagnetic pulse are calculated. The carrier frequency of the pulse is close to the frequency of direct interband transitions in a quantum well (QW). The QW energy levels are assumed to be discrete, with two closely spaced excited levels being taken into account. The QW width is assumed to be sufficiently large and comparable to the light wavelength corresponding to the pulse carrier frequency. In this case, the dependence of the momentum matrix element for the interband transition on the light wave vector should be taken into account. The refractive indices of the QW and barriers are assumed to be equal. The problem is solved for an arbitrary relation between the radiative and non-radiative lifetimes of the excited electronic states. It is shown that spatial dispersion considerably affects the shapes of the reflected and transmitted pulses. The greatest changes occur in the case where the inverse radiative lifetime is close to the difference between the frequencies of the interband transitions considered.
The reflection, transmission, and absorption of a symmetric electromagnetic pulse whose carrier frequency is close to the frequency of the interband transition in a quantum well are calculated. The energy levels in the quantum well are assumed to be discrete, and one excited level is taken into account. Consideration is given to the case of a sufficiently wide quantum well when the pulse wavelength corresponding to the carrier frequency is comparable to the quantum well width and when allowance should be made for the dependence of the matrix element of the interband transition on the photon wave vector. The calculations are performed with due regard for the difference between the refractive indices of the material of the quantum well and the barrier at an arbitrary ratio of the reciprocal radiative to nonradiative lifetimes of the excited level of the electronic system. It is demonstrated that the inclusion of the spatial dispersion and the difference in the refractive indices most strongly affects the reflection of the electromagnetic pulse, because the reflection due to interband transitions in the quantum well is accompanied by an additional reflection from the quantum well boundaries. Compared to the previously considered model, the most radical changes in the reflection are observed in the case when the reciprocal nonradiative lifetime of the excited state is substantially longer than the reciprocal radiative lifetime.
The reflectance, transmittance, and absorbance of a symmetric electromagnetic pulse with a carrier frequency close to the frequency of direct interband transitions in a quantum well are calculated. The energy levels in the quantum well are assumed to be discrete, and two closely spaced excited levels are taken into account. The theory holds true for quantum wells of an arbitrary width at which the quantum confinement is retained. The calculations are performed with due regard for the difference between the refractive indices of the quantum well and the barriers. In this case, there appears an additional reflection from the quantum-well boundaries. The additional reflection results in a substantial change in the shape of the reflected pulse as compared to that characteristic of a homogeneous medium. The reflection from the quantum-well boundaries disappears at specific ratios between the carrier frequency of the exciting pulse and the quantum-well width.
Reflectance, transmittance and absorbance of a symmetric light pulse, the carrying frequency of which is close to the frequency of interband transitions in a quantum well, are calculated. Energy levels of the quantum well are assumed discrete, and two closely located excited levels are taken into account. The theory is applicable for the quantum wells of arbitrary widths when the size quantization is preserved. A distinction of refraction indices of barriers and quantum well is taken into account. In such a case, some additional reflection from the quantum well borders appears which changes essentially a shape of the reflected pulse in comparison to homogeneous medium. The reflection from the borders disappears at some definite ratios of the carrying frequency of the stimulating pulse and quantum well width.
A theory for the response of a 2 D two-level system to irradiation by a symmetric light pulse is developed. Under certain conditions, such an electron system approximates an ideal solitary quantum well in a zero field or a strong magnetic field H perpendicular to the plane of the well. One of the energy levels is the ground state of the system, while the other is a discrete excited state with energy ℏω 0 , which may be an exciton level for H =0 or any level in a strong magnetic field. It is assumed that the effect of other energy levels and the interaction of light with the lattice can be ignored. General formulas are derived for the time dependence of the dimensionless “coefficients” of the reflection ℛ( t ), absorption A ( t ), and transmission ℐ( t ) for a symmetric light pulse. It is shown that the ℛ( t ), A ( t ), and ℐ( t ) time dependences have singular points of three types. At points t 0 of the first type, A ( t 0 )= T ( t 0 )=0 and total reflection takes place. It is shown that for γ r ≫γ, where γ r and γ are the radiative and nonradiative reciprocal lifetimes, respectively, for the upper energy level of the two-level system, the amplitude and shape of the transmitted pulse can change significantly under the resonance ω l =ω 0 . In the case of a long pulse, when γ l <γ r , the pulse is reflected almost completely. (The quantity γ l characterizes the duration of the exciting pulse.) In the case of an intermediate pulse duration γ l ≅γ r , the reflection, absorption, and transmission are comparable in value and the shape of the transmitted pulse differs considerably from the shape of the exciting pulse: the transmitted pulse has two peaks due to the existence of the point t 0 of total reflection, at which the transmission is zero. If the carrier frequency ω l of light differs from the resonance frequency ω 0 , the oscillating ℛ( t ), A ( t ), and ℐ( t ) time dependences are observed at the frequency Δω=ω l −ω 0 . Oscillations can be observed most conveniently for Δω≅γ l . The position of the singular points of total absorption, reflection, and transparency is studied for the case when ω l differs from the resonance frequency.
The reflected and transmitted energy flux corresponding to the linear response of a quantum well (QW) is calculated analytically for the cases of monochromatic and pulsed incident radiation normal to the QW plane. The light–matter interaction has been taken to all orders of perturbation theory, resulting in the appearance of radiative broadening in the expressions for the secondary fields. Three time scales determine the decay of the response after the pulse excitation while the quantum beats are superimposed on the exponentially decaying contributions at the frequency which (i) equals detuning from the electronic transition or (ii) corresponds to a characteristic excitation in the system.
It has been shown that the non-sinusoidal character oscillations appear in the transmitted, reflected and absorbed light fluxes when light pulses irradiate a semiconductor quantum well (QW), containing a set of a large number of the equidistant energy levels of electronic excitations. The oscillation amplitude is comparable to the flux values for the short pulses, duration of which γ_l^{-1} \leq \hbar/ΔE. A damping echo of the exciting pulse appears through the time intervals 2π\hbar/ΔE in the case of the very short light pulses γ_l^{-1}\ll \hbar/ΔE. Symmetrical and asymmetrical pulses with a sharp front have been considered. Our theory is applicable for the narrow QWs in a strong magnetic field, when the equidistant energy levels correspond to the electron-hole pairs with different Landau quantum numbers.