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Influence of Anomalous Dispersion on Optical Characteristics of Quantum Wells

Frequency dependencies of optical characteristics (reflection, transmission and absorption of light) of a quantum well are investigated in a vicinity of interband resonant transitions in a case of two closely located excited energy levels. A wide quantum well in a quantizing magnetic field directed normally to the quantum-well plane, and monochromatic stimulating light are considered. Distinctions between refraction coefficients of barriers and quantum well, and a spatial dispersion of the light wave are taken into account. It is shown that at large radiative lifetimes of excited states in comparison with nonradiative lifetimes, the frequency dependence of the light reflection coefficient in the vicinity of resonant interband transitions is defined basically by a curve, similar to the curve of the anomalous dispersion of the refraction coefficient. The contribution of this curve weakens at alignment of radiative and nonradiative times, it is practically imperceptible at opposite ratio of lifetimes . It is shown also that the frequency dependencies similar to the anomalous dispersion do not arise in transmission and absorption coefficients.

the module of the light wave vector. Actually above mentioned works are true in a zero approximation on the parameter κd.
On the other hand, for wide quantum wells the parameter κd can be ≈ 1. For example, for the GaAs hetero-laser radiation (wave length 0.8µ) and a quantum-well width d ∼ = 500Å, the parameter κd ∼ = 1.5. In this case, it is necessary to take into account the spatial dispersion of the electromagnetic wave, since its amplitude strongly varies inside of the quantum well. Besides, for wide quantum wells, an inequality d ≫ a 0 (a 0 is the lattice constant) is very strong, that allows to use Maxwell equations for continuous matters at determination of the electromagnetic field. Such approach allows to take into account distinction in refraction coefficients of barriers and quantum wells.
In 20,21 , the theory considering spatial dispersion of an electromagnetic wave at its passage through a quantum well is developed. One excited energy level was considered (i.e., one interband transition) and alongside with the spatial dispersion, different refraction coefficients of barriers and of quantum well were introduced for monochromatic 20 and for pulse 21 irradiation. Work 22 is devoted to the account of a spatial dispersion of an electromagnetic wave in the case of two closely located interband resonant transitions. That corresponds to the magnetopolaron state in a quantum well 23 .
At calculation of optical characteristics of a quantum well in the resonance, the contribution of resonant transitions is allocated from dielectric permeability. It is considered separately. These resonant transitions lead to occurrence of a high-frequency current, in which alongside with the contribution, frequency dependence of which corresponds to absorption, there is the contribution, frequency dependence of which is similar to a curve of the anomalous dispersion of the refraction coefficient. Below, the role of these two contributions in formation of frequency dependencies of the light reflection, transmission and absorption by a quantum well is investigated as an example of two closely located resonant transitions. It is generalization of two previous works of authors: Unlike of 20 , two excited energy levels are taken into account, and results of 22 are generalized on a case, when refraction coefficients of barriers and of quantum well are different.

I. BASIC RELATIONSHIPS
The system consisting of a semiconductor quantum well, located in an interval 0 ≤ z ≤ d, and of two semi-infinite barriers is considered. A constant quantizing magnetic field is directed perpendicularly to the quantum-well plane (along the z axis). The external plane electromagnetic wave extends along the z axis from negative z. It is considered that barriers are transparent for the electromagnetic wave, but in the quantum well the wave is absorbed, causing resonant interband transitions. Zero temperatures are assumed, a valence band is filled completely and a conduction band is empty. In linear approximation on a wave amplitude, the excited states are excitons. The frequencies of light close to the energy gap are considered, when a small part of electrons of the valence band takes part in light absorption. These electrons are located near the band gap and the effective mass method is applicable for them. For deep quantum wells in this case, it is possible to neglect by electron tunneling into barriers. Besides, the energy levels close to the quantum-well bottom can be considered in approximation of infinitely high barriers. This restriction is not principal and the theory can be extended on quantum wells of final depths.
Since the size of heterogeneity (what is the quantum well) is comparable with the light wavelength, optical characteristics of such systems must be determined from the corresponding Maxwell equation, where a current density is obtained on the basis of microscopic consideration.
In the theory, interband matrix elements p c v of quasimomentum operator are essential. These matrix elements correspond to the direct interband transition, i.e., to creation of an electron-hole pair with coinciding coordinates of an electron and hole. As well as in previous works 18,19,24 , the model is used, in which the vector p c v for two types of excitations I and II looks like: where e x (y) is the unite vector along the x(y) axis, p c v is some real constant. This model corresponds to heavy holes in semiconductors with the zinc blend structure, if the z axis is directed along the 4th order symmetry axis 25,26 . If the vectors of circular polarization of stimulating light are introduced, the property of preservation of a polarization vector is carried out. Thus, wave functions of the electron-hole pair and energy levels do not depend on indices I and II.
As is known, the electron-hole pair in a magnetic field is free, if two conditions are satisfied. First, the Lorentz's force must be large in comparison with Coulomb and exchange forces of electron -hole interactions in pairs. Then, the wave function of an electron-hole pair can be represented in the form of a product of two functions: Φ(z) and the function depending on r ⊥ in the xy plane of the quantum well. Estimations show 27,28 that in GaAs for a magnetic field corresponding to formation of magnetopolarons, this condition is carried out. The second condition is some excess of a size-quantization energy over the Coulomb and exchange energy of interaction in the electron-hole pair. Then, it is possible to consider the electron-hole pair as free and to use the approximation of infinitely high barriers, which is used below. The wave function describing dependence on z is as follows: and Φ λ (z) = 0 in barriers. The index λ = (m c , m v ) depends on quantum numbers of size-quantization of electrons m c and holes m v . Let us note that in GaAs at κd ≥ 1, the second condition of existence of free electron-hole pairs is not carried out and the function Φ(z) will be another there. However, the approximation (4), essentially simplifying calculations, does not affect qualitatively (as it is shown in Section 4) on frequency dependence of optical characteristics of a quantum well.
In the present work, monochromatic light of frequency ω ℓ and normal incidence on a quantum-well plane is considered. According to these assumptions, an electric field of stimulating light wave looks like: where E 0 is the complex amplitude, ν 1 is the real material refraction coefficient in barriers. It is supposed also that there are two closely located excited energy levels in a quantum well, and the others excited energy levels in the quantum well are located far enough. Such situation is possible, for example, for a magnetopolaron state 19,23 .
In case of a direct interband transition, the average induced charge density in a quantum well ρ(z, t) = 0, that allows to introduce the gauge ϕ(z, t) = 0, where ϕ(z, t) is the scalar potential. Then, the amplitude of electric field in barriers is determined by the equation and in a quantum well (0 ≤ z ≤ d), it is determined by the equation c is the light velocity in vacuum; ν 1 , ν are the refraction coefficient in barriers and in the quantum well, respectively.J(z) is the the Fourier-component of the current density (averaged on the system ground state), which is induced in the quantum well by a monochromatic plane wave. In case of two excited energy levels,J(z) for a quantum well with infinitely high barriers looks like (The general formula is represented, for example, in 20,24 .): andJ(z) = 0 in barriers. Here, γ rj is the radiative broadening of the excited states of an energy doublet in the case of narrow quantum wells.
If the doublet is formed by a magnetopolaron A to which the Landau quantum number of a hole n = 1 (according to the magnetopolaron classification 23 ), then, where (hω g is the energy gap, m 0 is the free electron mass, H is the magnetic field, e is the electron charge). The factor determines a change of radiative lifetime at deviation of the magnetic fields from the resonant value defined by equality Ω c = ω LO . ∆E pol is the polaron splitting 24 , Ω c , ω LO are the cyclotron frequency and longitudinal optical phonon frequency, respectively. In the resonance, λ = 0, Q 0j = 1/2 and γ r1 = γ r2 . Resonant denominators in (8) are: where ω j are the frequencies of resonant transitions of the doublet energy levels, γ j are the radiative broadenings of these levels. In (8), resonant denominators are taken into account only. Subscripts j = 1 and j = 2 of the function Φ j (z) correspond to pairs of quantum numbers of the size-quantization, between which there is a transition. m c (1), m v (1) correspond to subscript j = 1; m c (2), m v (2) correspond to subscript j = 2. The Landau quantum number n is kept at the direct transition. The total field E enters on the right-hand side of the equation (8), that is connected with refusal from the perturbation theory on the coupling constant e 2 /hc.

II. ELECTRIC FIELD OF AN ELECTROMAGNETIC WAVE
The further calculation is spent in the assumption that that, in particular, corresponds to the magnetopolaron A. In this case, the equality takes place and the formula (8) becomes The solution of the equation (6), determining the amplitude of the field E(z) in barriers, is C R determines the amplitude of the reflected wave, C T determines the amplitude of the wave which passed through the quantum well. It is convenient to represent the integro-differential equation (7) for the field amplitude in a quantum well in the form of the Fredholm integral equation of the second type 20 where For arbitrary m c and m v , F (z) is equal Φ(z) is defined in (4), It follows from (19) and (20) that Multiplying the equation (18) on Φ(z) and integrating on z from 0 up to d, we obtain where designations are introduced: As a result, complex amplitude of the electric field in a quantum well becomes Parameter ε, as well as in the case of one excited energy level, determines renormalization of the radiative broadening ε ′ and shift ε" for each of two excited levels. It follows from (24) and (20) that ). Thus, the real radiative broadening of energy levels of the doublet is determined by values That fact that ε ′ and ε " are identical for both energy levels of the doublet is connected with the assumption of equality of the size-quantization quantum numbers m  c(v) . Arbitrary constants C 1 and C 2 enter, according to (25), in the function h. Constants C 1 , C 2 , C R and C T were determined from continuity conditions of the magnetic field and the tangential projection of the electric field on borders z = 0 and z = d. Normal projections of an electric field are equal to zero. Arbitrary constants are equal: In (29) -(32), the designation is introduced and the function N looks like: At obtaining of this formula, the equality is used. Uncertain coefficients (29) -(30) on their form coincide with obtained in 20 for one excited energy level. Distinction consists in the function N , which in case of γ r1 (or γ r2 ) → 0 passes in N (the formula (41) in 20 ).
The curve N (ω ℓ ) is a function with two extrema, etch of which corresponds to the resonant transition. It may be represented more evidently, namely, Here, the renormalized resonant frequencies Ω 1 and Ω 2 and total broadening of each energy level Γ 1 and Γ 2 are equal: Values β ′ 1(2) and β ′′ 1(2) make complex parameters They determine mutual influence of energy levels. In the function (36), two peaks are presented in the explicit form, but they are not of the Lorentz-type, since parameters β 1 and β 2 ( bringing the contribution in broadening, and in shift of peaks) depend on frequency. It is naturally that representation of the function N in the form of the sum Lorentzian's (as it is made in 19,22 ) and non-Lorentzian's curves leads to identical results. Let us represent also an obvious form of parameters β ′ j and β " j : In a quantum well in case of plane monochromatic exciting wave in time representation, the vector of the electric field looks like where E(z) = [e iκz + (F (z)/F (0))N ]C 1 + [e −iκz + (−1) mc+mv e −iκd (F (z)/F (0))N ]C 2 (40) is the sum of plane waves exp(±iκz), which are connected with reflection from quantum-well borders. Besides, (as it follows from a kind of F (z)) the field in a quantum well contains oscillating terms, describing coordinate dependence of wave functions of electrons and holes. In time representation, field vectors to the left (E ℓ (z, t)) and the right (E r (z, t)) of the quantum well are: In a limiting case of a homogeneous environment (ν = ν 1 , ζ = 1), the electric field (after substitution (29) and (30)) becomes (κd = 0): If, on the contrary, to neglect by the spatial dispersion of the light wave (κd = 0), but to consider environment as not uniform (ζ = 1), for allowed transitions m c = m v , It is visible that γ ri and ζ enter only in the form of product γ ri ζ. It means that at ν = ν 1 in case of narrow quantum wells, the refraction coefficient of barriers ν 1 enters in radiative damping, and a refraction coefficient of quantum well ν does not appear anywhere. Physical sense of this result is clear: At κd ≪ 1, it is possible to pass to the limit d → 0, when the substance of a quantum well is absent, but the induced current, corresponding to transitions with an exciton creation, is kept. Thus, it is proved that obtained earlier results for narrow quantum wells are true and at ν = ν 1 , since formulas include only refraction coefficients of barriers.
In this limiting case, the field in a quantum well does not depend on coordinate, since in dipole approximation, the phase of a light wave does not vary in a quantum well.

III. OPTICAL CHARACTERISTICS OF A QUANTUM WELL
In this Section, formulas for the reflection, transmission and absorption of a plane monochromatic electromagnetic wave by a quantum well are represented in the general case, when ν = ν 1 and κd = 0, and in limiting cases of a homogeneous environment and κd = 0.
The reflection coefficient R is defined by standard way as the ratio of the module of the reflected energy flux to the module of the incident energy flux: Similarly, the transmission coefficient T is defined as The dimensionless absorption coefficient A (a share of energy absorbed by a quantum well), according to (46) and (47), looks like Using (31), (32), and (30), it is possible to represent the reflection R in the form The denominator ∆, defined by (21), is been transformed to Functions L and G determine the frequency dependence of reflection in the region of interband transitions into two excited energy levels: Coefficients v, v 1 , X, X 1 , Y, Z and Z 1 depend on parameters ζ = ν/ν 1 and κd: If, in a quantum well, there is one excited energy level, it is necessary to pass in the formula (49) to the limit γ r2 → 0, γ r1 = γ r , γ 1 → γ, ω 1 = ω 0 . In this case, the reflection coefficient looks like: Since refraction coefficients of barriers and quantum well, as a rule, do not differ enough (i.e., ζ ≃ 1), in the denominator |∆| 2 , v ≃ 4 plays a dominating role. Other multipliers contain factors ζ 2 −1 or (ζ −1) 2 and are small in comparison with v even in the resonance, when L ≃ 1 and G ≃ 1. In numerator R, the value v 1 ≪ 1 and the contribution of other terms is significant. In particular, a sign-variable term ∼ L, whose frequency dependence is similar to a curve of an anomalous dispersion, plays the essential role. Such dependence takes place in the refraction coefficient in the region of frequencies corresponding to absorption peaks.
The transition coefficient T is equal and does not contain sign-variable terms in the numerator. The same takes place and for absorption For one excited energy level, we obtain In approximation of a homogeneous medium (ζ = 1), There is no the term ∼ L in the reflection, i.e., this approximation does not take into account the contribution into the reflection, originating from sign-variable terms in the current (8).

IV. LIGHT REFLECTION, ABSORPTION AND TRANSMISSION FOR ARBITRARY Φ(z)
In Sections 2 and 3, coefficients R, T and A were obtained for Φ(z) in the form of (4), that is true for free electron-hole pairs. In this Section, Φ(z) is considered as arbitrary real function, in particular, it may be an excitonic function. Calculation, similar to Section 2, leads to following expressions for constants C R and C T : where In (64) -(66), function N is defined by the formula (34) (or (36)), and constants w (real) and w R (complex) are: At obtaining of these formulas, relationships F (d) = e iκd F * (0), Reε = ε = |F (0)| 2 , which follow from (19), were used (Φ(z) is considered real since represents onedimensional movement). If (4) is used for Φ(z), then w → 1, w R → ζ 2 + 1 and formulas (64) -(66)) coincide with (30) and (31). It is visible from comparison of expressions (64) -(66) with coefficients C R , C T and ∆, obtained in Section 2, that the structure of the function N , which basically determines the frequency dependence, does not depend on type of Φ(z). Only values Re ε = ε ′ and Im ε = ε" depend on type of Φ(z), i.e., the factor at γ r and the shift of resonant frequencies. Factors w and w R , entering in ∆ 1 and C R , can change only a ratio between contributions in optical characteristics of values L and G. Therefore, changes of optical characteristics will be hardly radical due to changes of Φ(z).

V. DISCUSSION OF RESULTS
Frequency dependencies of the light reflection R and transmission T are calculated for a case m c = m v = 1 and γ r1 = γ r2 = γ r , γ 1 = γ 2 = γ, that corresponds to the system of the polaron A and a hole with the Landau quantum number n = 1 in conditions of the magnetophonon resonance 23 . Calculations were spent with the help of formulas (46) and (47). Functions C R and C T from (30) and the denominator ∆ from (31) are used, as well as function N (34).
In Fig.1 and Fig.2, the reflection R is shown for small values of the ratio γ r /γ. It is visible that curves 3 and 4 for the case ζ = ν/ν 1 = 1 and κd = 0 differ considerably from the case of ζ = 1 (a curve 2) at the same value κd. As it was mentioned above, the formula for the current (8) contains sign-variable terms ∼ (ω ℓ − ω i ) and terms ∼ γ i [(ω ℓ − ω i ) 2 + γ 2 i ], corresponding to absorption curve. Curves 3 and 4 concern to a case, when signvariable terms dominate. Therefore, R(ω ℓ ) is similar to the curve of anomalous dispersion in this case. Alongside with changes of the form of the reflection curve, the increase of R (in 15 times approximately in Fig.1 and twice in Fig.2 ) rakes place in comparison with cases κd = 0, ζ = 1 and κd = 0, ζ = 1.
At γ r /γ = 1, influence of the absorption dominates, as it is visible in Fig.3. However, influence of sign-variable terms is still noticeable: Curves 3 and 4 (ζ = 1.1 and ζ = 0.9, respectively) differ from curve 2 (ζ = 1). At the further increase of γ r /γ, the influence of the anomalous dispersion becomes practically imperceptible and functions R, T and A coincide with obtained in 22 . Curves of the transmission T are shown in Figs.4-6 for the same magnitudes of parameters κd, ζ and γ r /γ, as curves in Figs.1-3. It is visible that transmission is poorly sensitive to changes of ζ. It happens due to absence in T (ω ℓ ) terms proportional to ω ℓ − ω 1 and ω ℓ − ω 2 (see (58)). A small distinction is determined by the denominator |∆| 2 which contains terms linear on ω ℓ − ω j . The absorption A, in general, is poorly sensitive to changes of parameter ζ in all area of changes of γ r /γ. R and T for a case ζ = 1, κd = 1.5 (curve 2) are shown also in figures. Some asymmetry of the curve 2 in comparison with the case κd = 0, ζ = 1 (curves 1, 5 and 6) attracts our attention. This asymmetry is a consequence of the account in the theory of supreme orders on interaction of an electromagnetic wave with the electronic system that leads to occurrence of shifts of resonant frequencies. These shifts combine with resonant frequencies ω 1 and ω 2 . At transition to the limit κd → 0, shifts dis- appear and symmetry of curves is restored. Finally, curves 5 and 6 in figures correspond to the case κd = 0, ζ = 1. Curves are symmetric here, and influence of parameter ζ is shown in the form of a parallel shift of curves (if γ r = γ), or changes in the vicinity of extrema (γ r ≤ γ). The main conclusion of our calculations consists that at greater radiative lifetimes of excited states in comparison with nonradiative lifetimes (which are determined, in particular, by electron and hole scattering on impurities and phonons) frequency dependence of reflection is determined basically by sign-variable terms in the expression for the current density. In this case, it is pos- sible to neglect by γ r in resonant denominators, i.e., to solve the problem in linear approximation on interaction of an electromagnetic wave with the electronic system.