Picosecond beats in coherent optical spectra of semiconductor heterostructures: photonic Bloch and exciton-polariton oscillations This article has been downloaded from IOPscience. Please scroll down to see the full text article. 2001 Semicond. Sci. Technol. 16 R1 (http://iopscience.iop.org/0268-1242/16/3/201) Download details: IP Address: 138.73.1.36 The article was downloaded on 06/05/2013 at 06:21 Please note that terms and conditions apply. View the table of contents for this issue, or go to the journal homepage for more Home Search Collections Journals About Contact us My IOPscience
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Picosecond beats in coherent optical spectra of semiconductor heterostructures: photonic
Bloch and exciton-polariton oscillations
This article has been downloaded from IOPscience. Please scroll down to see the full text article.
2001 Semicond. Sci. Technol. 16 R1
(http://iopscience.iop.org/0268-1242/16/3/201)
Download details:
IP Address: 138.73.1.36
The article was downloaded on 06/05/2013 at 06:21
Please note that terms and conditions apply.
View the table of contents for this issue, or go to the journal homepage for more
Home Search Collections Journals About Contact us My IOPscience
Picosecond beats in coherent opticalspectra of semiconductorheterostructures: photonic Bloch andexciton–polariton oscillationsGuillaume Malpuech and Alexey Kavokin
Laboratoire des Sciences et Materiaux pour l’Electronique, et d’Automatique, UMR 6602 duCNRS, Universite Blaise Pascal–Clermont-Ferrand II, 63177 Aubiere Cedex, France
Received 5 July 2000, accepted for publication 15 January 2001
AbstractPropagation of short pulses of light in multilayer semiconductor structurescontaining excitons is modelled by solving time-dependent Maxwellequations with the use of the scattering state technique. This approachallows this problem to be reduced to finding the eigenstates (scatteringstates) of the system, subject to the stationary Maxwell equations withappropriate boundary conditions. Then, the time-dependent electric field inthe system can be found by Fourier integration of the scattering states.Application of this technique to the problem of light-propagation in laterallyconfined Bragg mirrors reveals the effect of photonic Bloch oscillations, i.e.,oscillations of photons between two inclined mini-gaps of the opticalsuperlattice, analogous to the well-known electronic Bloch oscillations. Thescattering state technique is applied to the problem of propagation ofexciton–polaritons in semiconductor films. The pulsed excitation induces agrating of dielectric polarization in the direction of propagation of light,which arises through interference between two exciton–polariton branches.The grating evolves backwards relative to the light propagation directionbecause of the multiple re-emission and re-absorption of photons byexcitons. Inhomogeneous broadening of excitons exerts a dramatic influenceon the time-resolved coherent optical spectra of semiconductor structuresthrough the vertical motional narrowing effect in bulk crystals and multiplequantum wells (MQWs). The polariton interference governs the resonantRayleigh scattering (RRS) spectra of the MQWs. In particular, a drasticdifference between the RRS spectra of Bragg-arranged andanti-Bragg-arranged MQWs is predicted.
1. Introduction
This paper addresses picosecond and femtosecond scale os-cillations in linear optical spectra of multilayer semiconductorheterostructures. Time-resolved optical spectroscopy becamea widely diffused experimental technique towards the end ofthe 20th century, giving access to a number of intriguing phe-nomena of the light–matter coupling. Picosecond and fem-tosecond quantum beats in the optical spectra of semiconductor
heterostructures are frequently reported, while their interpre-tation often represents a non-trivial problem. Here an attemptis given to provide a theoretical background for the descriptionof a few important classes of short-scale oscillations, namelyto those caused by the optical interference effects in multilayerstructures and those caused by propagation effects of exciton–polaritons.
The physics of light–matter interaction in solids hasbeen strongly advanced by the invention of low-dimensional
semiconductor structures. In the conventional quantumwell (QW) structures, three-dimensional photons may becoupled to two-dimensional (2D) excitons, which givesrise to quasi-2D exciton–polariton excitations [1]. Inquantum wires and quantum dots, photons are coupledto one-dimensional (1D) and zero-dimensional electronicexcitations [2]. In microcavities, 2D photons are coupled withexciton states of different dimensionalities [3]. The variety ofdifferent regimes of light–matter coupling leads to a very richspectrum of effects associated with exciton–polaritons.
Exciton–polaritons are quasi-particles exhibiting proper-ties of both photons and excitons and arising in semiconduc-tor crystals illuminated by light [4]. An exciton–polaritoncan be considered a virtual pair of an electron and a hole,which recombines emitting a photon and then forms an exci-ton by absorbing the same photon. The importance of exciton–polaritons for fundamental issues of light–matter interaction insolids, as well as their strong influence on the optical proper-ties of semiconductor structures, has inspired a huge numberof theoretical and experimental works in the last 40 years (fora review see [5]).
One of the powerful tools for studying the exciton–polariton dynamics in semiconductors is the coherent time-resolved optical spectroscopy including pump-probe andup-conversion experiments allowing the measurement ofthe time-resolved reflection, transmission and resonantRayleigh scattering (RRS) [6–9]. To describe adequatelythese experiments, one should solve the time-dependentMaxwell equations for multilayer structures containing excitonresonances.
The problem is additionally complicated by an essentialinhomogeneous broadening of excitons due to the potentialfluctuations in the plane of thin semiconductor films (seee.g. papers in [10]). In the last decade, a number oftheoretical papers on the inhomogeneous exciton broadeningin QWs have appeared [11–14]. Basically, the analysis hasbeen performed within two principal approaches, namely,the quantum approach based on solving the semiconductorMaxwell–Bloch equations and quantum treatment of theexciton–polaritons [11–13], and the semi-classical approachwhich involves solving the Maxwell equations with accounttaken of the non-local excitonic contribution to the dielectricsusceptibility of the QWs [14]. The quantum approachallows the energy distribution of the laterally confined excitonstates within a model fluctuation potential to be calculated,with the contributions from discrete exciton states to theoptical response of the QW summed. Within the semi-classical approach, the exciton energy distribution functionis introduced phenomenologically, with the exciton–lightcoupling taken into account in a simple quasi-analytical way.This allows the optical spectra to be easily calculated withaccount taken of all kinds of interference effects.
Without going into depth about the details of the differenttheoretical approaches, we would like to point out that one ofthe principal difficulties for all models is associated with thedifferent scales of roughness affecting excitons and photons.Actually, the excitons are sensitive to potential fluctuationshaving the scale of their Bohr radius, while the roughness scaleaffecting the properties of the photons is much larger (aboutthe wavelength of light in the media). The questions as to
which scale is important for the mixed exciton-photon states(exciton–polaritons) and whether exciton–polaritons behave asplane-waves or as quasi-particles remain open.
The idea that potential fluctuations in semiconductorstructures may be averaged by extended exciton–polaritonstates has inspired a series of works concerned with themotional narrowing effect [15–18, 9]. Speaking about themotional narrowing, one should distinguish between thehorizontal effect, i.e., averaging in the plane of the light wave[15–17], and the vertical effect, i.e., averaging of fluctuationsin the direction of light propagation [9, 18]. In both cases, thetheoretical model supposes the existence of extended exciton–polariton modes having macroscopic dimensions.
The recent experiments on the RRS from multiplequantum well (MQW) structures pose a new challenge to thetheory. The essential difference between the RRS [19] andthe photoluminescence (PL) is that the optical coherence ismaintained in the former case and broken in the latter. TheRRS in QWs is due to the elastic scattering of excitons by thein-plane disorder potential that necessarily arises owing to themonolayer QW width fluctuations and alloy fluctuations. It isvery difficult to distinguish experimentally between the RRSand PL.
The recent spectroscopic data on the RRS by excitons insemiconductor quantum wells have been interpreted in termsof exciton scattering by a disorder potential in the plane ofthe QWs [20–22]. Strictly speaking, this interpretation iscontroversial, since the mechanical excitons [4] themselvesare not coherent with the incident light. Actually, they areformed because of the dissociation of exciton–polaritons withthe loss of coherence. That is why the mechanical excitonsgovern the PL spectra, while the RRS spectra should bedescribed in terms of the exciton–polaritons. This difference isextremely important for the case of MQW structures where themechanical excitons are localized in the individual QWs, butthe exciton–polaritons occupy the entire structure. Recently,Rabi-oscillations in the RRS spectra of microcavities have beenreported [23], which is a purely polaritonic effect.
In the present work, we address the problem ofpropagation of exciton–polaritons excited by short pulses oflight in various semiconductor structures. In particular, wefocus our attention on the effect of inhomogeneous excitonbroadening on the time-resolved reflection and transmissionspectra of thin semiconductor layers and MQW structures, andon the RRS in MQWs. We use a semi-classical approach basedon the non-local dielectric response theory [24, 25], extendedto describe the disorder effects [14], and on the scattering statetechnique [26] for simulation of light pulse propagation insemiconductor structures. The optical interference phenomenain the structures under study are accounted for in the frameworkof the transfer matrix technique.
The paper is organized as follows: in section 2, weintroduce the scattering state technique for solving thetime-dependent Maxwell equations in multilayer structures.As an example, section 3 considers an original effect ofphotonic Bloch oscillations in laterally confined Bragg mirrors,simulated using the scattering state technique. In section 4,the dynamics of propagation of exciton–polaritons throughbulk semiconductor films is analysed. The appearance ofa polariton-induced grating of dielectric polarization in a
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semiconductor is discussed. The concept of weak localizationof exciton–polaritons is introduced. Section 5 presentscalculations of time-resolved reflection spectra for single QWand MQW structures, containing inhomogeneously broadenedexciton resonances. The vertical motional narrowing effect isdiscussed. Section 6 is concerned with the RRS in MQWs.The role of exciton–polariton effects in the RRS spectrais emphasized. Particular cases of Bragg- and anti-Bragg-arranged QWs are considered. Finally, section 7 summarizesthe contribution of exciton–polaritons to coherent opticalspectra of semiconductor quantum structures.
2. The formalism: solving time-dependent Maxwellequations
2.1. Maxwell equations without sources
From the viewpoint of the classical electrodynamics, thegeneration and propagation of an electromagnetic wave in asolid film is described by the time- and coordinate-dependentMaxwell equations, with the Maxwell boundary conditions atthe interfaces
divD(r, t) = 4πρ(r, t) (2.1)
rotE(r, t) = −1
c
∂B(r, t)
∂t(2.2)
divB(r, t) = 0 (2.3)
rotH(r, t) = 4π
cj(r, t) +
1
c
∂D(r, t)
∂t. (2.4)
Here r is the coordinate, t is the time, D is the displacementfield, E is the electric field, ρ is the charge density, B is themagnetic induction, H is the magnetic field, j is the currentdensity and c is the speed of light in the vacuum.
In most cases, the problems of generation and propagationof an electromagnetic wave can be separated. The propagationproblem is represented by Maxwell equations without sources,i.e., with ρ = 0 and j = .
The displacement field is related to the electric field andpolarization P (r, t) of the medium by
D(r, t) = εbE(r, t) + 4πP (r, t) (2.5)
where εb is the background dielectric permittivity. Further,we seek the dielectric response of a non-magnetic structure,assumed to be linear and isotropic. In this case, linear scalarrelations between the polarization and the electric field can bewritten
P (k, ω) = χ(k, ω)E(k, ω) (2.6)
where k is the wave vector of light, ω is its frequency, andχ(k, ω) is the dielectric susceptibility of the medium, whichis the key characteristic of the optical response of materials.In general, a quantum microscopic consideration is requiredto obtain this parameter. Given the function χ(k, ω), thepresent theory of propagation of electromagnetic waves insemiconductors is semi-classical.
2.2. Spectral decomposition
In the linear case, we can solve the Maxwell equations in thereciprocal space in a basis of monochromatic waves and obtain
the time-dependent electric field as a linear combination of thefrequency- and wave vector-dependent solutions:
E(r, t) =(
1
2π
)4 ∫E(k, ω) exp [−i(ωt − kr)] dk dω.
(2.7)In a non-magnetic medium, B = H . In k space, three otherMaxwell equations read
ik × E(k, ω) = 1
ciωH(k, ω) (2.8)
ikH(k, ω) = 0 (2.9)
ik × H(k, ω) = − iω
c[E(k, ω) + 4πP (k, ω)] . (2.10)
In this framework, and for a planar medium, the Maxwellboundary conditions for the electric field require a continuityof the in-plane component of E and H .
Using equations (2.8), (2.9), (2.10) and (2.6), we obtain
k2 − ω2
c2ε(k, ω) = 0 (2.11)
for the transverse waves and
ε(k, ω) = 0 (2.12)
for the longitudinal waves, where ε(k, ω) = εb + 4πχ(k, ω)is the dielectric permittivity. The refractive index is then givenby
n(k, ω) =√ε(k, ω) (2.13)
while Re (n(k, ω)) > 0.Equation (2.7) can be modified as follows
E(r, t) =(
1
2π
)2 ∫ +∞
−∞
∫ +∞
−∞E(ω, ky)
× exp [−i(ωt − kr)] dky dω
=(
1
2π
)2 ∫ +∞
−∞
∫ +∞
−∞E(ω, ky, z)
× exp(−iωt) exp(ikyy)dω dky (2.14)
where
E(ω, ky, z) = E(ω, ky) exp[ikz(ω, ky, z)
](2.15)
z is the structure axis, E(ω, ky) are the solutions of equations(2.8), (2.9) and (2.10) with adapted Maxwell boundaryconditions and ky is the in-plane component of the wave-vector.
For the simplest case of an optically homogeneousmedium, with the spatial dispersion neglected, k isindependent of the index.
Under this assumption, the electromagnetic field in themedium is a transverse plane wave, and equation (2.11) reads
k2 = n2(ω)ω2
c2. (2.16)
There exists only a single solution of equations (2.12) and(2.13) for a given direction ofk. In the case of normal incidence(k ‖ z), equation (2.14) becomes, for a given polarization σ ,
E(z, σ, t) = 1
2π
∫ +∞
−∞E(ω, σ ) exp [−i(ωt)] dω (2.17)
with
E(ω, σ ) = E(ω, σ ) exp
[iωn(ω)
cz
].
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G Malpuech and A Kavokin
2.3. The transfer matrix method
In the case of a planar multilayer (and, consequently,inhomogeneous) structure grown along the z-axis, thefunctions E(ω, ky, σ, z) can be easily obtained by the transfer-matrix method [27]. Let us consider a monochromatic planewave incident on such a structure, with a wave vector k:kx = 0, kz = k0 cos θ0, ky = k0 sin θ0, where θ0 is the angle ofincidence. If the optical index in the medium to the left of thestructure isn0, then k0 = n0(ω/c). For a given polarization anda given incidence angle, the electric field can be decomposedinto two plane waves propagating from the left to the right (E+)and from the right to the left (E−):
E(ω) = E+ + E−.
Let us attribute to each layer and each interface a 2 × 2matrix which connects the vector (E+, E−) (of the beginningof a layer) to the vector (E+, E−) (of the end of the layer)or, for the interface, connects the vector (E+, E−) just to theleft of the interface to the same vector just to the right of theinterface (referred to as the transfer matrix [27]). For a layerof thickness d, the transfer matrix has the form
Ad(ω) =(
exp (−ikz(ω)d) 00 exp (ikz(ω)d)
). (2.18)
For an interface between two layers 1 and 2, we can obtain theangles θ1 and θ2 from the Snell–Descartes laws.
n0 sin θ0 = n1 sin θ1 = n2 sin θ2. (2.19)
Thus, for the TM polarization the transfer matrix looks like
ATM(ω) = n1(ω) cos θ2 + n2(ω) cos θ1
2n1(ω) cos θ1
×(
1 n1(ω) cos θ2−n2(ω) cos θ1n1(ω) cos θ2+n2(ω) cos θ1
n1(ω) cos θ2−n2(ω) cos θ1n1(ω) cos θ2+n2(ω) cos θ1
1
)(2.20)
and for the TE polarization:
ATE(ω) = n1(ω) cos θ1 + n2(ω) cos θ2
2n1(ω) cos θ1
×(
1 n1(ω) cos θ1−n2(ω) cos θ2n2(ω) cos θ2+n1(ω) cos θ1
n1(ω) cos θ1−n2(ω) cos θ2n2(ω) cos θ2+n1(ω) cos θ1
1
). (2.21)
We choose the initial conditions that correspond to theincidence of a δ-pulse of light on the structure from the left(z → −∞). Let us number all the layers and interfaces in thestructure from the left to the right. In this case, the transfermatrix across the entire structure has the form
A(ω, ky, σ ) =N∏i=1
Ai(ω, ky, σ ) (2.22)
where N is the total number of layers and interfaces. Theamplitude reflection and transmission coefficients of thestructure are determined by a linear system:(
1r(ω, ky, σ )
)= A(ω, ky, σ )
(t (ω, ky, σ )
0
)(2.23)
so that
t (ω, ky, σ ) = 1
a11(ω, ky, σ )(2.24)
r(ω, ky, σ ) = a21(ω, ky, σ )
a11(ω, ky, σ ). (2.25)
In the case of pulsed excitation, the transmission and reflectionamplitudes are given by
tg(ω, ky, σ ) = t (ω, ky, σ )g(ω) (2.26)
andrg(ω, k
y, σ ) = r(ω, ky, σ )g(ω) (2.27)
respectively, where g(ω) is the spectral function of the incidentpulse.
Further, we can introduce a matrix Az(ω, ky, σ ) which isa product of the transfer matrices from the coordinate z to theend of the structure. Thus, we obtain:(
Hereafter the amplitudes E(ω, ky, σ, z) will be referred to asthe scattering states of the system [26]. Any solution of theMaxwell equations for a multilayer structure can be written asa linear combination of the scattering states.
2.4. Choice of the initial conditions
In a linear system, the stationary solutions of the Maxwellequations can be written as
where h(ky) is the envelope function of the incident pulse ink space.
Equation (2.17) gives the time- and coordinate-dependentelectric field created by any incident pulse as
Eg,h(y, z, σ, t) =(
1
2π
)2 ∫ +∞
−∞
∫ +∞
−∞Eg,h(ω, k
y, σ, z)
×∑
exp(−iωt) exp(ikyy)dω dky (2.31)
where∑ = 1 in the TE polarization, and
∑=[
1 −(
ky
n0ω/c
)2]1/2
in the TM polarization.In the same way, and with the use of equation (2.6), the
time- and coordinate-dependent polarization field can be foundas
Pg(y, z, σ, t) =(
1
2π
)2 ∫ +∞
−∞
∫ +∞
−∞Pg(ω, k
y, σ, z)
×∑
exp(ikyy) exp(−iωt)dω dky (2.32)
where
Pg(ω, ky, σ, z) = χ(ω)Eg(ω, k
y, σ, z). (2.33)
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Topical Review
80
70
60
50
40
30
20
10
0
Z (nm)0 500 1000 1500 2000 2500 3000
Tim
e (f
s)
n = 2 n = 4 n = 2
Figure 1. Propagation of a 1 fs long pulse of light in a 3-layerstructure surrounded by vacuum. The optical indices of the layersare 2, 4 and 2, respectively.
If the spatial dispersion can be neglected, which is quite a usualcase, equation (2.30) reduces to the form
Eg(ω, ky
0 , σ, z) = E(ω, ky
0 , σ, z) g(ω) (2.34)
where ky
0 is the in-plane component of the wave vectorcorresponding to the angle of incidence. In this case,
Eg(z, ky
0 , σ, t) = 1
2π
∫ +∞
−∞Eg(ω, k
y
0 , σ, z) exp(−iωt) dω.
(2.35)Figure 1 shows the propagation of a 1 fs pulse incident fromz = −∞ which enters a three-layer structure surrounded by thevacuum at t = 0 at normal incidence, calculated according toequation (2.35). The optical indices of the layers are, from theleft to the right: 2, 4 and 2. One can clearly see the reflectionand the transmission of light at each interface. The speed oflight is two times less in the layer having the refractive indexn = 4 than in two other layers with n = 2. The figure provesthe validity of the proposed method.
2.5. Maxwell equations with a source
To model a number of optical experiments like photolumi-nescence, Rayleigh scattering, optical emission of electricallypumped systems etc, we should assume the presence of a lightemitter inside the modelled structure. Here we propose anoriginal method which allows the emission of light (and thedistribution of the electromagnetic field within the structure)to be calculated with full account of the optical interferenceand the eventual acts of absorption re-emission of light.
Let us consider a planar multilayer structure of totalthickness d. The left surface of the structure is at the pointz = 0. We assume the presence of an emitter at the pointz = z0. Thus, we neglect any spatial distribution of the emitter,which is, of course, a strong assumption. In the followingformulas, A is the transfer matrix of the entire structure, Ar isthe matrix of the right part of the structure (z0 < z < d), andAl is the inverse transfer matrix of the left part of the structure(0 < z < z0).
rr , tr , rl , tl are the reflection (r) and transmission (t)coefficients for the light incident from the point z = z0 tothe left (l) or to the right (r).
We adopt the following boundary conditions: no light isincident either from z = −∞ or from z = +∞. At t = 0, aδ-pulse of light is emitted at the point z = z0 in the positivedirection. This choice allows us to obtain the scattering statesof the system Ez0,+(ω, ky, σ, z). If the emitted pulse hasspectral and spatial functions g(ω) and h(ky), respectively,the scattering states can be represented as
Ez0,+g,h (ω, k
y, σ, z) = g(ω)h(ky)Ez0,+(ω, ky, σ, z). (2.36)
It is convenient now to consider our structure as a Fabry–Perot resonator with an infinitely thin central layer, wherein thesource is placed. Taking into account all the multiple reflectionacts in the resonator, we can write the electric field just to theright of the source as
Ez0,++ (ω, ky, σ, z0) = 1 + rrrl + (rrrl)
2 + . . .
= 1
1 − rrrl(2.37)
and the field just to the left of the source as
Ez0,+− (ω, ky, σ, z0) = rσr
[1 + rrrl + (rrrl)
2 + . . .]
= rr
1 − rrrl. (2.38)
In the same way, the electric field at the right and left surfacesof the structure can be found as
Ez0,++ (ω, ky, σ, d) = tr
1 − rrrl(2.39)
Ez0,+− (ω, ky, σ, 0) = rr tl
1 − rrrl(2.40)
respectively.The electric field in the right- and left-hand parts of the
structure can be found now by the transfer-matrix method,taking into account that the incident pulse for the right part isEz0,++ (ω, z0), and that for the left part, Ez0,+− (ω, z0). Let Az
r bea transfer matrix from the coordinate z > z0 to the coordinated and Az
l a transfer matrix from the coordinate z<z0 to thecoordinate 0. Then, using equation (2.26), the generalizedscattering states of the system can be found as(Ez0,++ (ω, ky, σ, z)
Ez0,+− (ω, ky, σ, z)
)= Az
r(ω)
×(Ez0,++ (ω, ky, σ, d)
0
)if z > z0
(2.41)(Ez0,+− (ω, ky, σ, z)
Ez0,++ (ω, ky, σ, z)
)= Az
l (ω)
×(Ez0,+− (ω, ky, σ, 0)
0
)if z<z0.
In the same manner we can find the amplitudes Ez0,−+ (ω, ky,
to a δ-pulse emitted from z = z0 in the negative direction. Tofind all these quantities, we just have to substitute the indicesr by l and vice versa in equations (2.37), (2.38), (2.39), (2.40)and (2.41).
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G Malpuech and A Kavokin
n = 2 n = 4 n = 280
70
60
50
40
30
20
10
0
Z (µm)0 5 10 15 20 25 30
Tim
e (f
s)
Figure 2. Propagation of a pulse emitted from the centre of thesame structure as in figure 1.
Finally, we consider a pulse with spectral distributiong(ω)h(ky) propagating in the positive direction and a pulsewith spectral distribution p(ω)g(ky) propagating in theopposite direction. This is the case in the Rayleigh scatteringproblem addressed in section 6. In this case, the solution ofthe frequency- and coordinate-dependent Maxwell equationsis given by
Ez0g,h,p,q(ω, k
y, σ, z)
= g(ω)h(ky)[Ez0,+
+ (ω, ky, σ, z) + Ez0,+− (ω, ky, σ, z)]
+p(ω)q(ky)[Ez0,−
+ (ω, ky, σ, z) + Ez0,−− (ω, ky, σ, z)].
(2.42)
The y-components of the time-dependent electric field andpolarization field are given by
Ez0g,h,p,q(y, z, σ, t)
=(
1
2π
)2 ∫ +∞
−∞
∫ +∞
−∞Ez0g,h,p,q(ω, k
y, σ, z)
×∑
exp(−iωt) exp(ikyy)dω dky (2.43)
Pz0g,h,p,q(y, z, σ, t)
=(
1
2π
)2 ∫ +∞
−∞
∫ +∞
−∞Pz0g,h,p,q(ω, k
y, σ, z)
×∑
exp(−iωt) exp(ikyy)dω dky. (2.44)
Figure 2 shows the propagation of a 1 fs long pulse in a three-layer structure surrounded by the vacuum. The optical indicesof the layers are, from the left to the right: 2, 4 and 2. Thepulse is generated at t = 0 in the positive direction at the centreof the structure, which is the only difference from the casein figure 1. It can be seen that the emitted pulse propagatesinitially in the positive direction, then experiences multiplereflections, penetrates into the left part of the structure, etc.The figure proves the validity of the present technique.
To summarize this part, we have formally divided ourstructure into two substructures lying to the left and to the rightfrom the source. A proper choice of the boundary conditionsfor these substructures allowed us to calculate the time- andcoordinate-dependent electric field in the structure, taking intoaccount all the interference acts. This approach enabled theapplication of the scattering-state technique to light-emittingsystems.
3. Photonic bloch oscillations in laterally confinedBragg mirrors
3.1. Analogy between electronic and photonic crystals
In this section, we use the scattering-state technique to modelthe propagation of short light pulses in a 1D photonic band gapstructure. We show that photons in laterally confined Braggmirrors may experience Bloch oscillations similar to those inthe case of electrons in semiconductor superlattices subject toan electric field. As will be seen below, an effective electricfield is produced in the photonic case by variation of the lateralsize of the structure as one moves along the structure axis.
Very recently the phenomenon of electron Bloch-oscillations has been experimentally demonstrated insemiconductor superlattices [28]. The problem of theexistence of such oscillations gave rise to long andcontroversial debates since the 1950s when Wannierformulated the theory of electronic states within a crystal in thepresence of a uniform electric field [29–32]. The theory hasmet much difficulties because of the problem of the unboundedpotential operator eFz (where e is the electron charge, F isthe electric field and z is the real-space coordinate) and onlyrecently a rigorous formalism has been developed [33]. Anelectron in a crystal experiences Bloch oscillations if thedephasing time is longer than the oscillation timeh/ef d, whered is the lattice period. Two main mechanisms responsiblefor the damping of Bloch oscillations are the electron-phononscattering and the Zener tunnelling [32, 34–37]. For Blochoscillations to be observed, a large supercell in real space,i.e., a large period d, is needed, which corresponds to a smallBloch-oscillation time. Moreover, there exists a limitationon the strength of the applied electric fields, since theyinduce Zener tunnelling between adjacent crystal bands (orsuperlattice minibands). In fact, Bloch oscillations have onlybeen observed in semiconductor superlattices.
The analogy between an electron in a real crystal and aphoton in a dielectric medium [38, 39] allows us to speculateabout the possibility of photonic Bloch oscillations in speciallydesigned photonic crystals. Since photons have a muchgreater coherence length as compared with electrons, theymight appear to be better candidates for observation of theBloch oscillations. In recent studies, the photonic Stark-ladderhas been addressed [40, 41]. In this section we review theresults obtained in [42] and demonstrate the existence of suchoscillations in photonic crystals, namely, in laterally confinedBragg mirrors.
3.2. Design of a photonic superlattice
The inset of figure 3 shows schematically the model structurechosen for studies of the photonic Bloch oscillations. Itis an optical superlattice (Bragg mirror) made of poroussilicon with two different degrees of porosity. An effectiveelectric field acting on photons is provided by the varyinglateral confinement of the structure in the y direction (unlikein [40, 41] where the effective field was created by variationof the refractive index). The porous silicon seems to be avery convenient material for our purposes because it allows therefractive index to be varied widely. The technology of Braggmirror growth from porous silicon is quite well developed,
R6
Topical Review
E0 = 400 meV
L Z
Z (nm)0 10 20 30
500
450
400
350
300
250
200
150
100
500
0
Vph
(meV
)
Light
Figure 3. The potential profile of a confined optical superlattice inits growth direction for a photon having an energy E0 = 400 meV.The inset shows a schematic of the structure in question. Thestructure is periodical along the growth direction and confined in theplane. The light is incident along the growth direction.
and its chemical etching and metallization have become quiteordinary operations. We have chosen the same refractiveindices for two kinds of porous silicon as in [43] (na = 1.27and nb = 2.25). The corresponding layer thicknesses areLa = 380 nm and Lb = 300 nm, respectively. The structurecomprises 46 periods. The variation of the lateral dimensionL(z) of the structure can be achieved by chemical etching.Then the etched surfaces should be metallized in order toachieve a required degree of the in-plane optical confinement.Further, we suppose that an electric field in the structure haszeros at the metallized surfaces and that the lateral size of thestructure varies slowly as one moves along the structure axis.In this case, the in-plane photonic wave vector componentsand those in the light propagation direction are, respectively
ky
ph(z) = jπ
L(z)(3.1)
and
kzph(z) =√[nαω
c
]2−[ky
ph(z)]2
(3.2)
where hω = E is the photon energy; nα is the refractive indexwith α = a, b for two kinds of layers within the structure;L(z) is the waveguide width; j = 1, 2, 3, . . . is the index ofthe confined photonic mode. Here, we suppose that the photonelectric field in the structure can be factorized as a product ofthe plane wave propagating along the axis and the cosine-likeenvelope function in the direction of the lateral confinement(the adiabatic approximation). In what follows we supposej = 1 for simplicity. Note that ky is constant in our case.
In order to find the electric field in the system, one shoulduse equation (2.35) with the scattering states E(ω, z) obtainedfrom equation (2.29) as follows
Eg(z, t) = 1
2π
∫ +∞
−∞Eg(ω, z) exp(−iωt)dω (3.3)
whereEg(ω, z) = E(ω, z)g(ω) (3.4)
g(ω) is the spectral function of the incident pulse. Further, weconsider the incident pulse having a form
g(ω) = h√π ,
exp
[−(hω − E0
,
)2]
(3.5)
with E0 = 530 meV and , = 45 meV.It is instructive to consider the photonic motion in the
optical superlattice in terms of an approximate analytical‘effective mass’ model. If a light pulse with spectral width, hπnc/L is incident on the structure, one can write kph
in the form
kzph =√
2mph(E − Vph)
h2 (3.6)
where
Vph(z) = E0
2+
h2c2
2n2αE0
[π
L(z)
]2
(3.7)
mph = n2α
c2E0 (3.8)
andE0 is the energy of the pulse centre (E0 ,). To make theoptical system analogous to the electronic superlattice subjectto an external electric field, it is convenient to choose L in theform
L(z) = θ√Q + z
(3.9)
where θ and Q are constants, z = 0 corresponds to thebeginning of the structure. In the model structure consideredhereQ = 680 nm and θ = 1.98×105 nm3/2. For this particulargeometry, we can rewrite
Vph(z) = E0
2+
h2c2
2n2αE0
(πθ
)2Q + eFeffz (3.10)
where
eFeff = h2c2
2n2αE0
(πθ
)2. (3.11)
Here Feff is the effective electric field acting on photons. Thepotential (3.10) is plotted in figure 3 for E0 = 400 meV.In each type (a or b) of material, the potential is a linearfunction of the coordinate near E = E0 as in the case ofa conventional superlattice subject to an electric field. Note,however, that the effective electric field strength is not the samein the two kinds of layers we used. In the material (a) wehave F = 7.6 × 103 V m−1, while in the material (b) F =2.4 × 103 V m−1. Also, the potential and electric field seen byan incident photon depend on its energy, which is a peculiarityof our optical systems. The effect of lateral confinementon the propagation of photons is strongly different in high-and low-index materials. Another difference from the caseof a conventional superlattice is that the optical superlatticepotential does not exceed the photon energy E0 in most of thestructure. This phenomenon is less pronounced at low E0,whereas the approximation E0 , is more adapted to largevalues. This specific feature complicates direct application ofthe existing theory of Bloch-oscillations in a Stark-superlattice
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G Malpuech and A Kavokin
800
700
600
500
400
300
200
100
0
Minigap
Z (µm)0 5 10 15 20 25 30
Ene
rgy
(meV
)
Minigap
Figure 4. Photonic band diagram of the confined opticalsuperlattice in its growth direction. White regions represent theminibands, dark regions, the minigaps.
to our system, but does not affect the existence of the effect,as will be seen below.
Variation of n can also be used to model the electric fieldfor photons [40, 41]; however, one should take into accountthe strong effect of n on the photonic mass and on the slope ofthe potential curve.
The real miniband structure of our model sample along thelight-propagation direction (z-direction) is shown in figure 4.The allowed miniband energies are found from the followingcondition (which is a consequence of the Bloch theorem)
−1 � 12 (a11 + a22) � 1 (3.12)
where a11 and a22 are the diagonal elements of the transfermatrix across the period of the structure
Azd = Ib→aL
zaIa→bL
zb. (3.13)
The matrices I and L are, respectively, the transfer matricesacross an interface and across a layer, defined in section 2.3.They depend on z because of the z-dependence of the wave-vector.
One can see inclined minibands and minigaps in figure 4.Note that the energies of the band edges of photonic gapschange almost linearly with the coordinate, which confirmsthe analogy with a conventional superlattice subject to anelectric field applied along the growth direction. The valueextracted from the inclination angle of the minibands is F =7.6 × 103 V m−1, in perfect agreement with the value given byequation (3.11).
Figure 5 shows the calculated electric field induced by theincident pulse (3.5) in the structure as a function of time andcoordinate in the growth direction. The brightness in the figureis proportional to the electric field intensity. One can see aninitial decrease of the intensity of the pulse when it tunnelsthrough the minigap region 0 < z < 6µm, then oscillationsstart approximately between z = 4µm and z = 30µm.Indeed, these are the Bloch oscillations of light within theinclined miniband. Note that the inclined minigap at thebeginning of the structure filters the low-frequency wing ofthe pulse, which mostly participates in oscillations.
Z (µm)0 5 10 15 20 25 30
2.5
2.0
1.5
1.0
0.5
0
Tim
e (p
s)
Figure 5. Propagation of a Gaussian pulse of light in the confinedoptical superlattice. The brightness is proportional to the electricfield intensity.
102
100
10−2
10−4
10−6
10−80 2 4 6 8 10
(b)
(a)
Ref
lect
ion
(arb
. uni
ts)
Time (ps)
Figure 6. Calculated time-resolved reflection spectra of theconfined Bragg mirror under study. Pulse centred at 530 meV(curve (a)), pulse centred at 470 meV (curve (b)).
As follows from figure 5, the period of the observed Blochoscillations of light is about T = 0.78 ps. The classicalformula
T = h
eFd(3.14)
with an electric field F = 7.6 × 103 V m−1 obtained from theband diagram in figure 4 yields T = 0.8 ps, in agreement withnumerical simulations.
Experimentally, the photonic Bloch oscillations can beobserved in time-resolved reflection spectra, as it can be seenin figure 6. A calculation was performed by equation (3.4) withz = 0 for two different pulse energies. Curve (a) correspondsto a pulse centred at 530 meV. Note quite clear and pronouncedoscillations due to the Bloch oscillations. Their amplitudeis very low since light is mostly reflected by the minigap.The second curve, (b), was calculated with a pulse centredat 470 meV. The oscillations disappear after a few periods butthe amplitude of the 4–5 first maxima is relatively high, so thatthis pulse energy seems to be more suitable for experimentaldetection of the photonic Bloch oscillations.
In conclusion, the scattering state technique allowedus to simulate the propagation of a short light pulse ina specially designed photonic band-gap structure. This
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Topical Review
numerical experiment shows that observation of photonicBloch oscillations is a realistic task. The formalismsdescribing electronic and optical Bloch oscillations are ratherdifferent, while the photonic Bloch oscillations do havethe same physical nature as electronic oscillations andcan be observed in laterally confined periodical structures.Fabrication of such laterally confined porous silicon Braggmirrors seems to be achievable, whereas time-resolvedreflectivity experiments with 0.5 ps resolution are currentlyperformed. Alternative structures suitable for observation ofphotonic Bloch oscillations could be photonic wires with across-section varying along the wire axis.
4. Propagation of exciton–polaritons inhomogeneous and inhomogeneous semiconductorfilms
4.1. Introduction
In this section we address the problem of propagation ofexciton–polaritons excited by short pulses of light in disorderedmedia, adopting a semi-classical model of extended exciton–polariton states whose in-plane size is much larger than allfluctuation scales in the problem. We assume that the photon(polariton) wave-vector is conserved in the plane and thatthere is no Rayleigh scattering. Our model semiconductorfilm is optically homogeneous both in the plane and in thegrowth direction. Moreover, we neglect the spatial dispersionof exciton–polaritons.
Within these approximations, and using the scattering-states method described in section 2, we find the temporaldynamics of propagation of the exciton–polaritons insemiconductor films illuminated with a short light pulse. Bothhomogeneous and inhomogeneous broadening of excitonsis analysed. The inhomogeneous broadening of excitonresonances is taken into account within a local model.Approximate analytical expressions for the temporal dynamicsof the dielectric polarization in the semiconductor films willbe obtained using the steepest-descent method [44, 45] andcompared with the quantum model description accounting forthe mesoscopic nature of the polariton transport [46].
This section is organized as follows: in section 4.2 wepresent the semi-classical theory of propagation of exciton–polaritons in disordered media. Section 4.3 discusses the time-resolved transmission and the time- and frequency-dependentpolarization for the case of a homogeneously broadenedslab. These results are to be compared with those for aninhomogeneously broadened case, presented in section 4.4.In section 4.5, concluding remarks are made and mesoscopiceffects in the propagation of exciton–polaritons in disorderedmedia are addressed.
4.2. Semi-classical approach to the exciton–polaritonproblem
In the vicinity of an excitonic resonance, we can write thedielectric susceptibility in the framework of the local modelas:
χ(ω, ω0) = 1
4π
εbωLT
ω0 − ω − iγ(4.1)
where εb is the background dielectric constant; ω0, ωLT
represent, respectively, the exciton resonance frequencyand the exciton longitudinal-transverse splitting in the bulkmaterial; and γ is the exciton non-radiative homogeneousbroadening, supposed here to be frequency-independent.Throughout the section we assume the background refractiveindices nb = √
εb of the slab and of the surrounding medium tobe the same. This assumption allows us to avoid complicationscaused by any optical interference effects.
The local description is appropriate for thick semiconduc-tor films. In order to describe the inhomogeneous broaden-ing we assume that the excitons have a frequency distributioncentered at a frequency ω0 given by the function f (ω, ω0).As discussed in detail in [14], the coherent contribution fromall excitons of the distribution is correctly taken into accountwithin the constant in-plane wave-vector approximation by re-placing the dielectric susceptibility χe (equation (4.1)) by thataveraged with the exciton distribution function:
χ(ω) =∫ +∞
−∞χe(ω, ν)f (ν, ω0) dν. (4.2)
We emphasize here that solving the Maxwell equationswith a weighted dielectric susceptibility is not equivalent tosolving these equations for a single exciton resonance withfurther averaging of the electric field. Such a difference isquite essential because the two procedures describe differentschemes of interaction between excitons and light. This pointis analysed in detail in section 5.
In this work, the exciton distribution function is assumedto be Gaussian
f (ω, ω0) = h√π ,
exp
[−(ω0 − ω
,
)2]. (4.3)
, will be referred to as the parameter of inhomogeneousbroadening. The advantage of the Gaussian distribution isthat it allows analytical calculation of the susceptibility (4.2),which becomes
χ(ω) = ihεb
4√π
ωLT
,w(z) (4.4)
where z = (ω − ω0 + iγ )/, and w(z) = e−z2erfc(−iz),
erfc(z) being the complementary error function [47]. It can beseen that once the susceptibility of a medium is calculated, forexample, in terms of a microscopic quantum model, it becomespossible to directly apply the results of the scattering-statestechnique to calculation of the optical properties of structures.
4.3. Propagation of exciton–polaritons in homogeneoussemiconductor films
4.3.1. Time-resolved transmission. Figure 7 shows the time-resolved transmission induced by a 300 fs long pulse of lightin GaAs films of thickness varied in the range between 500 nmand 150µm. The excitonic parameters used in this calculationare the following: hωLT = 0.08 meV, hω0 = 1515 meV,hγ = 0.05 meV and nb = 3.54. We consider a normalincidence case, so that the in-plane wave vector ky = 0. Thus,the scattering states of the system (2.29) can be written in the
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G Malpuech and A Kavokin
1010
105
100
10−5
0 20 40 60 80
Tra
nsm
issi
on (
arb.
uni
ts)
Time (ps)
150 µm
10 µm
0.5 µm
Figure 7. Time-resolved transmission through GaAs slabs of variedthickness: 150 µm, 10 µm and 0.5 µm. Excitonic parameters usedin this calculation: hωLT = 0.08 meV, hω0 = 1515 meV andhγ = 0.05 meV. The background refractive index is 3.54. The300 fs long incident pulse is centred at the exciton resonancefrequency.
form E(ω, 0, z). The central energy of the pulse g(ω) incidenton the structures is chosen equal to the exciton resonanceenergy. The time-resolved transmission can then be foundfrom equation (2.35)
G(t) = Eg(d, 0, t) = 1
2π
∫ +∞
−∞Eg(ω, 0, d) exp(−iωt) dω.
(4.5)
In the limit d λ, whereλ is the wavelength of light in themedium, multiple reflections at each surface of the structurecan be disregarded, so that G(t) becomes
G(t) =∫ +∞
−∞
dω
2π
4nbn(ω)
[nb + n(ω)]2 exp[i8(ω)] (4.6)
with 8(ω) = ω [n(ω)d/c − t] being a rapidly oscillatingphase. G(t) can now be approximately evaluated using thesteepest-descent method [44, 45] (which coincides with thestationary-phase method for Im8(ω) = 0). This approachis based on the observation that the integral in equation (4.6)is dominated by frequencies corresponding to large exp(i8),i.e., by the saddle points for Im8(ω) (in fact, the imaginarypart of an analytical function has no maxima or minima).Thus, in order to calculate the integral, it is convenient tochoose integration paths passing through the saddle pointsand concentrating large exp(−Im8) values on the shortestinterval. These paths are readily determined from the Riemanncondition for analytical functions, stating that the lines ofmaximum variation for Im8(ω) are equipotential lines forRe8(ω).
In [44], G(t) was evaluated by the steepest-descentmethod for a homogeneous medium described by the localsusceptibility χ (equation (4.1)), in the time window d/υ t (d/2υ)(ω0/ωLT) (υ = c/nb). It was shown that theresponse function is given by
G(t) = 1√2π
(4d
ν
ω2c
t3
)1/4
exp[i8(hom)
0
]exp(−γ t)
× cos(∣∣∣,8(hom)
S
∣∣∣ +π
4
)(4.7)
where ∣∣∣,8(hom)S
∣∣∣ = 2ωc
(d
ν
)1/2
t1/2 (4.8)
and
8(hom)0 = −ω0 t − d
ν
ωLT
2. (4.9)
ωc is the polariton splitting: ωc = (ω0ωLT/2)1/2, and t =t − d/ν.
As can be seen from figure 7 and equation (4.7), in thecase of pure homogeneous broadening the decay rate of thesignal is independent of the slab thickness. Moreover, with theparameters of figure 7, the exponential decay exp(−γ t ) largelysurpasses the t−3/4 decay entailed by the pre-exponentialfactor. The spectra exhibit pronounced oscillations; the timeinterval τ (hom) between two successive minima, depending ontime and slab thickness, is given by
τ (hom) = π
ωc
(νt
d
)1/2
(4.10)
as can be deduced from equation (4.7). The oscillations are atypical feature of the interference between exciton–polaritonbranches, which was already observed experimentally [48]a few years ago. The nature of the beats can be simplyunderstood if we consider the exciton–polaritons as wavepackets composed by plane waves having different frequenciesand different group velocities. The intensity of the transmittedsignal at a time t is governed by the polariton waves with groupvelocity υ = ∂ω/∂k = d/t .
For each group velocity one can find two differentpolariton-waves coming from the two different branches.The interference between these two waves gives rise to theoscillations in time-resolved transmission spectra. Note thepicosecond time scale in the spectra, the time scale typical ofthe exciton–polariton transport processes (to be distinguishedfrom the time of flight of a free photon across the structure,which is about 100 fs for a 10 µm thick slab). Longertimes correspond to interference between polaritons withlarger group velocity, i.e., between polaritons that are closerto resonance. The decrease in the splitting between twointerfering modes, which eventually reaches the longitudinal-transverse splitting ωLT, corresponds to an increase in theperiod of oscillations with time, given by the reciprocal ofthe frequency splitting.
4.3.2. Time-dependent dielectric polarization. Figure 8shows the temporal evolution of the excitonic polarizationin a 10 µm thick GaAs slab excited by a short light pulseas a function of coordinate and time, calculated within thescattering-state approach (section 2, equations (2.32) and(2.33)) to be
P excg (z, 0, t) = 1
2π
∫ +∞
−∞P excg (ω, 0, z) exp(−iωt) dω (4.11)
withP excg (ω, 0, z) = g(ω)χE(ω, 0, z). (4.12)
All the other parameters are the same as in figure 7; thebrightness of colour is proportional to the absolute value of thedielectric polarization.
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Topical Review
Z (µm)
16
14
12
10
8
6
4
2
00 2000 4000 6000 8000 10000
Tim
e (p
s)
Figure 8. Coordinate- and time-dependent dielectric polarizationinduced by excitons excited by a 300 fs light pulse in a 10 µm thickGaAs slab with the same excitonic parameters as in figure 7. Thebrightness is proportional to the absolute value of the dielectricpolarization.
It can be seen that the dielectric polarization in figure 8is a strongly non-monotonic function of the coordinate andtime. This means that a pronounced grating of dielectricpolarization appears. The period of oscillation increases withtime and decreases with coordinate, which is consistent withequation (4.10) (where the layer thickness d should be replacedby the coordinate z). Interestingly, the grating moves backwardwith respect to the light propagation direction, which seems,at first glance, to be in contradiction with the wave-vectorconservation law. Note, however, that the excitons in ourmodel are infinitely heavy, so that the motion of the dielectricgrating cannot be associated with the motion of excitons, butonly with absorption and coherent re-emission of light bydifferent excitons in the structure. Thus, the wave-vectorconservation law is not violated in this calculation.
Let us interpret the peculiar behaviour of the excitonicpolarization in terms of the re-absorption–re-emission oflight by excitons. At very short times the polarization ishomogeneous, which is a natural consequence of the filmhomogeneity. Then coherent re-emission of light starts:photons re-emitted at different points z1 and z2 interfere atz3, and a z-dependent intensity pattern develops, which inturn influences the re-absorption processes responsible forfurther changes in the dielectric polarization, thus producingan intensity pattern which is also time-dependent. The timeindependence of the polarization at z = 0 follows from thefact that, owing to wave-vector conservation, no interferencepattern produced by photons re-emitted at different pointsarises, and the polarization is only determined by the photonsemitted at z = 0. The appearance of the grating can also beanalysed in terms of the scattering-state model. Figure 9 showsthe excitonic polarization of a 10-µm GaAs slab, determined asa function of frequency and coordinate (equation (4.12)). Onceagain, the brightness of colour is proportional to the absolutevalue of the dielectric polarization. It can be seen that the initialsingle-resonance distribution of the dielectric polarizationat the exciton resonance frequency changes drastically ingoing deeper into the semiconductor medium. A pronouncedcamelback structure appears, with two maxima spaced by
1518
1517
1516
1515
1514
1513
1512
Z (µm)0 2000 4000 6000 8000 10000
Ene
rgy
(meV
)
Figure 9. Coordinate- and frequency-dependent dielectricpolarization induced by excitons excited by a 300 fs light pulse in a10 µm thick GaAs slab with the same parameters as in figure 1. Thebrightness is proportional to the absolute value of the dielectricpolarization.
about 2 meV. Apparently, the beats in the temporal dynamics ofthe polarization are due to the double-resonance structure of itsfrequency-resolved spectra. Such a structure is characteristicof the regime with strong coupling between excitons andphotons. The spacing between the polarization maxima isroughly proportional to the number of acts of absorption and re-emission of a photon by an exciton during the propagation of apolariton mode, thus growing with the coordinate. The periodof the beats in time is inversely proportional to the spacingbetween the two eigenstates of the system and, therefore,decreases with the coordinate. Thus, the behaviour of theexcitonic polarization in the frequency domain well fits thequantum absorption–re-emission model.
The classical model explains the appearance of thecamelback structure by the interplay between the transmissionand absorption of light in a semiconductor slab. If we neglectthe reflection by the surface, the intensity of the transmittedlight at the point z can be represented as
T (ω, z) = 1 − ξ(z)α(ω) (4.13)
where α(ω) is the absorption coefficient, and ξ(z) is amonotonically increasing function of z. Since
P excg (ω, z) ∝ α(ω)T (ω, z) (4.14)
the excitonic polarization peak experiences a bifurcation at acertain value of z, which is seen in figure 9.
4.4. Exciton–polaritons in the case of inhomogeneousbroadening
Figure 10(a) shows the time-resolved transmission of GaAsfilms of varied thickness (from 500 nm to 150 µm), inducedby a 300 fs pulse of light. The pulse is centred at the excitonresonance frequency while its spectral width much exceeds theexciton broadening. The excitonic parameters used for thiscalculation are the same as in figure 7; the only differenceis that in figure 10(a) the exciton is assumed to be alsoinhomogeneously broadened with h, = 1 meV. A remarkable
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G Malpuech and A Kavokin
10−5
100
105
1010T
rans
mis
sion
(ar
b. u
nits
)
150 µm
10 µm
0.5 µm
10−5
100
105
1010
Tra
nsm
issi
on (
arb.
uni
ts)
0 20 40 60 80Time (ps)
150 µm
10 µm
0.5 µm
(a)
(b)
Figure 10. Time-resolved transmission through GaAs slabs ofvaried thickness: 150, 10 and 0.5 µm: (a) numerical calculation bythe scattering state method, (b) results of the analytical model.Excitonic parameters used in this calculation: hωLT = 0.08 meV,hω0 = 1515 meV, hγ = 0.05 meV, h, = 1 meV. The 300 fsincident pulse is centred at the exciton resonance frequency.
feature of these spectra is that, in contrast to the homogeneouscase, the transmission decay rate depends strongly on the slabthickness. It decreases with increasing thickness, approachingfinally (for a 150 µm thick slab) a value close to the decay ratein the case of only homogeneous broadening, as indicated bycomparison of figures 10(a) and 7. Note also that the period ofoscillations of the time-resolved transmission is independentof time and slab thickness in figure 10, while in figure 7 itdepends on both the parameters. Apparently, the time-resolvedtransmission is highly sensitive to the mechanism of excitonbroadening.
In the case of a distributed resonance frequency, therefractive index has a nonvanishing imaginary part even atγ → 0, which in turn implies that the phase8 in equation (4.6)has both a real and an imaginary part: 8(ω) = Re8(ω) +i Im8(ω) (with Im8(ω) > 0). In order to calculateG(t) (equation (4.6)) using the steepest-descent method, thestationary points should be evaluated at each instant of time t .These points are solutions of the following equation
ζw(ζ ) = i√π
[1 +
1
2
(,
t=
)2]
(4.15)
with = = =(t) = ωc[(d/ν)(1/t)
]1/2. The stationary points
ωS = ωS(t) = ω0 ±=(t)− iγ (4.16)
(with the plus and minus signs referring to the upper and lowerpolariton branches, respectively) are correctly obtained in thelimiting case , → 0 from equation (4.15), as can be easily
verified using the asymptotic expansion for ζ → ∞ up to thirdorder in (1/ζ ) [45].
Once the points ζS = ζR + iζI are calculated (where ζS
are the complex solutions of equation (4.15), and ζR and ζI arethe real and imaginary parts of ζS, respectively) the responsefunction at a time t can be evaluated
G(t) =√
2
π
sin(2β)
|Re8′′| ei80 e−Im8S cos (|,8S| + β) . (4.17)
In equation (4.17), all the quantities are to be calculated at thestationary points. Here
80 = 8(hom)0 − ωLT
4
d
ν
(,
t=
)2
(4.18)
,8S = −ζR,
{t +(ωc
,
)2 d
ν
[1 +
1
2
(,
t=
)2]
|ζ |−2
}(4.19)
Im8S = Im8(hom)S + Im8
(inh)S = γ t − ζI,
×{t −
(ωc
,
)2 d
ν
[1 +
1
2
(,
t=
)2]
|ζ |−2
}(4.20)
8′′S = ∂28
∂ω2= Re8′′
S + i Im8′′S
= 2d
ν
ω2c
,3
{1
ζ
[1 +
1
2
(,
t=
)2]
−(,
t=
)2
ζ
}(4.21)
β = 1
2arctan
{−ζR
ζI
[1 + 1
2
(,t=
)2
|ζ |2 −(,
t=
)2]
×[
1 + 12
(,t=
)2
|ζ |2 +
(,
t=
)2]−1
. (4.22)
Both the homogeneous and inhomogeneous broadeningscontribute to the decay of the time-resolved spectra(equation (4.17)) through the term exp(−Im8S). As far as γis supposed to be energy-independent, the former broadeningproduces a thickness-independent exponential decay. Onthe other hand, the latter broadening is responsible for anon-exponential time-dependent decay rate, since exciton–polaritons have an energy-dependent damping in the case ofinhomogeneous broadening. The polaritons that are closer tothe resonance exhibit a larger damping. At the same time,for reasonable values of ,, polaritons with larger splittinghave a damping that is close to the homogeneous one. Forthis reason, the decay is also expected to be thickness-dependent: polaritons which determine the transmitted signalat a fixed time t become closer to resonance with decreasingslab thickness, which in turn implies a faster decay for thinnerslabs.
Let us first examine the case (,/t=) 1: we namethis situation the ‘minor disorder case’. Using the asymptoticexpansion of w(ζ ) for ζ → ∞ and retaining only correctionsto equation (4.15) up to the lowest order in (,/=), thestationary points ωS are given by
ωS = ω0 ±=
[1 +
3
4
(,
t=
)2]
− iγ (4.23)
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(with the plus and minus signs referring to the upper and lowerpolariton branches, respectively). The resulting transmittedsignal is given by
G(t) = 1√2π
(4d
ν
ω2c
t3
)1/4[
1 − 3
8
(,
t=
)2]
× exp(−γ t)exp(i8hom
0
)cos
(|,8S| +
π
4
)(4.24)
with
|,8S| =∣∣∣,8(hom)
S
∣∣∣[
1 +1
4
(,
=
)2]. (4.25)
Equation (4.23) shows that in this case, , produces onlya small correction to the real part of the energy eigenvaluesof the polariton modes. This, in turn, implies that the decayof the transmitted signal is still quasi-exponential, as in thehomogeneous case, with, however, the period of the beatsmodified and shortened by the inhomogeneous broadening tobecome
τ (inh) = τ (hom)
1 + 34
(,t=
)2 . (4.26)
Figure 10(b) shows the time-resolved transmissionthrough the same slabs of GaAs as in figure 10(a), calculated byequation (4.24). A very good agreement is observed betweenthe analytical model and numerical calculations.
The dependence of the time-resolved transmission on theslab thickness can also be qualitatively explained in terms of aquantum model interpreting the exciton–polariton propagationas a chain of virtual emission–absorption acts.
In the case of only homogeneous broadening, any photonis always absorbed and re-emitted by the same excitoncharacterized by the homogeneous broadening γ whichgoverns the decay time of the time-resolved transmission. Onthe other hand, a photon can be absorbed and re-emitted byany exciton from the frequency distribution in the case ofinhomogeneous broadening. Suppose that during its motionacross the slab the photon experiences N absorption–re-emission acts (apparently, on average, N increases with slabthickness). In the case of N = 1 (which might be named ‘theQW limit’), the probability ρ for a photon having frequencyω to be absorbed by an exciton with frequency ω′ depends ontwo factors. The first is the imaginary part of the variation ofthe dielectric susceptibility, associated with this exciton, andthe second, the probability of meeting this exciton. Thus, ρcan be estimated to be
ρ(ω, ω′) ∝ εbωLTγ
(ω′ − ω)2 + γ 2f (ω′, ω0). (4.27)
Equation (4.27) shows that a photon can interact coherentlyonly with excitons situated in the γ -vicinity of its frequencyω.Moreover, photons with frequencies close to ω0 have betterchances to be absorbed by an exciton than those havingfrequencies far from the centre of the excitonic distribution.For the same reason, excitons at the distribution’s centre havebetter chances to be excited by photons than the excitons inthe distribution’s wings. In this ‘QW limit’, the transmissiondecays with a Gaussian envelope and a decay parameter givenby 1/,, as discussed in detail in [14].
Now suppose that , hγ . The probability that thesame photon is virtually absorbed-emitted by an exciton fromits γ -vicinity N times is proportional to f N(ω′, ω0). Withincreasing N , this function becomes progressively sharper,finally approaching the Dirac delta function δ(ω′ − ω0). Inturn, N must increase with increasing slab thickness, since inour model the exciton–polaritons are supposed to maintain thespatial coherence over the whole slab thickness. (Note that thisis still true when dephasing processes, neglected by our model,are taken into account, provided the slab thickness does notexceed the coherence length of polaritons.) That is why in thecase of very thick films only the exciton–polariton mode withfrequency ω = ω0 survives. This mode has a homogeneousbroadening γ which governs its decay rate. So, in this limit,the influence of the inhomogeneous broadening on the decaytime of the transmitted signal is completely suppressed. Thisis what we see in figure 10 (the case of a 150 µm thick slab).
The variation of the transmission decay time with samplethickness is a manifestation of the vertical motional narrowingeffect. Actually, the origin of the effective narrowing of thefrequency distribution of exciton–polaritons with increasingslab thickness is the averaging of the disorder in the slabby extended polariton modes filtering out the exciton statesfrom the wings of the Gaussian distribution. Recently, thevertical motional narrowing effect has been observed in MQWstructures where the decay time of the time-resolved reflectionincreased with growing number of QW [9].
We call here for a similar experimental study of themotional narrowing effect in thick semiconductor films, whichseems to have a substantial fundamental importance andpossible application in ultrafast optoelectronics.
Note the second scale of modulation of the transmittedsignal in figure 10 (weak minima at 20 ps for the 10 µm thickslab and at 62 ps for the 150 µm thick slab). This modulationhas the same nature as the oscillations in the time-resolvedreflection of the single QW (see section 5, figure 12), namely,the interference within a single polariton branch caused byinhomogeneous broadening.
4.5. Mesoscopic nature of the exciton–polariton transport
In this section, we analyse the exciton–polariton propagationin semiconductor slabs in the framework of a simplifiedsemiclassical model of dielectric susceptibility, using also thescattering-states technique (section 2). This model assumesthe slabs to be optically homogeneous even though the excitonresonance is inhomogeneously broadened. The model neglectsall scattering processes in the semiconductor, but takes intoaccount coherent multiple acts of re-absorption–re-emission oflight by excitons with different resonance energies, the effectexerting a profound influence on the time-resolved opticalresponse of the slabs.
It seems rather instructive to consider the propagation ofexciton–polaritons as a chain of virtual emission-absorptionacts. A photon entering the structure can excite an excitonstate, which can coherently re-emit the photon, which isthen absorbed again, etc. Because of the random potentialfluctuations, exciton states in semiconductors are characterizedby slightly different energies (this is what we call theinhomogeneous broadening case). In this case, a photon
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G Malpuech and A Kavokin
passing through the film may interact with several differentexcitons from this distribution. Since different photonsinteract with different exciton resonances, the transmissionof light through an optically homogeneous medium with aninhomogeneously broadened exciton resonance is to someextent analogous with the problem of the electronic transportin mesoscopic structures containing randomly arranged elasticscatters.
Consideration of the exciton–polariton propagation asa problem of mesoscopics assumes a quantum approachemphasizing the quasi-particle nature of polaritons. Wespeculate on this issue using the results of a semiclassicalmodel. This is possible since the quantum and classicalapproaches to the light–matter interaction problems areformally equivalent in the linear regime.
In this section, we calculate numerically and describe an-alytically the appearance of peculiar interference phenomenarelated to the propagation of exciton–polaritons in semicon-ductor slabs.
For both homogeneously and inhomogeneously broad-ened excitons, we predict the formation of a dielectric polariza-tion grating induced by a short light pulse propagating througha semiconductor slab. Interestingly, the grating moves back-ward with respect to the light-propagation direction.
Striking differences are predicted between the time-resolved transmission spectra of semiconductor films forinhomogeneously and homogeneously broadened excitons.In particular, in the former case the rate of transmissiondecay depends strongly on the slab thickness. The signaldecays with a non-exponential envelope governed mostly bythe inhomogeneous broadening parameter , in the limit ofvery thin films. On increasing the slab thickness, the decaybecomes progressively slower, eventually approaching thevalue characteristic of the decay in homogeneously broadenedmedia. This behaviour, which is a manifestation of thevertical motional narrowing effect, can be explained in a simpleintuitive way if the propagation of the exciton–polaritons isinterpreted in terms of virtual re-absorption re-emission of lightby excitons. The explanation can be summarized as follows:statistically, photons have a larger probability to interact withexcitons at the centre of the energy distribution. In the limitof an infinite number of emission-absorption acts, all excitonresonances but one are filtered out, so that the total decayrate equals the (non-radiative) decay rate of this single excitonresonance.
Using the steepest-descent method, we derived an analyticformula for the time-resolved transmission and interpretedthe preceding results in terms of the classical optics. Inthis framework, the inhomogeneous broadening is predictedto affect the dispersion and group velocity of exciton–polaritons, as well as their damping rates. The interferenceof polariton modes from the upper and lower branches, havingthe same group velocities, governs the polarization at a givenpoint and time. The frequency dependence of the dampingrate is responsible for the thickness-dependent decay of thetransmitted signal.
5. Time-resolved reflection of light from QWstructures
5.1. Specifics of multiple QWs
Here we address the effect of exciton inhomogeneousbroadening on the time-resolved reflection spectra of QWs.As it can be readily verified, the inhomogeneous broadening isresponsible for any deviations of the time-resolved reflectiondecay from an exponential function. This follows from thefact that, in the absence of inhomogeneous broadening, theexciton resonance yields a Lorentzian contour in the frequency-resolved reflection r(ω). Being an inverse Fourier transformof r(ω), the time-resolved reflection r(t) must be nothing butexponential in this case.
Experimentally, the time-resolved reflection spectra fromQW structures exhibit a strongly non-exponential decay and,in some cases, oscillations [9]. An adequate theory able todescribe these peculiarities has been difficult to build, becauseboth the polaritonic effect and the disorder potential effecton the exciton states are to be taken into account. Thesemiclassical non-local dielectric response theory formulatedfor the case of light coupling with free excitons in ideal QWsis to be extended to the general case of light coupling witha quasi-infinite number of localized, quasi-localized or quasi-free exciton states in an inhomogeneous QW.
One of the key questions to be answered in formulatinga theoretical approach to the problem is whether the non-conservation (scattering) of exciton (and light) momentumin the QW plane is essential in the reflection experiments.Apparently, consideration of excitons localized in thefluctuation potential as individual scatterers yields the effectof the Rayleigh scattering of light, recently discussed andobserved experimentally [6–8]. On the other hand, theprobability of scattering appears to be much lower (by about3 orders of magnitude) than the probability of reflection ortransmission of light through the QW at the exciton resonanceregion [14]. That is why the model of an inhomogeneouslybroadened exciton resonance, assuming the conservation of thein-plane wave vector [14, 17, 18] (the ‘plane wave’ approach),seems reasonable.
The plane-wave and individual-scatterer approaches tothe exciton–polariton problem in disordered media are similarto some extent to the Bragg and von Laue models of X-raydiffraction in crystals (see e.g. [50]). The difference is that inthe Laue model all atoms are identical and the crystal has aregular lattice (which yields a formal equivalence between theLaue and Bragg models). In contrast, in our case the excitonsfrom the inhomogeneous distribution have different energiesand are randomly distributed over the fluctuation potential.That is why the conclusions of the ‘Laue-like’ models [11, 12]are somewhat different from the conclusions of the ‘Bragg-like’ models [13, 14] (the difference is mostly related to theeffect of Rayleigh scattering of light by localized excitons).
In the reflection or coherent-transmission geometry, thedifference between the two approaches does not seem to becrucial. A theoretical study of the reflection of light bya regular grating of identical quantum dots [2] yielded thesame form of the reflection and transmission coefficients asin a homogeneous QW model (which confirms the analogywith the Bragg–Laue case). Estimations [14] have shown
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that even for an irregular grating of non-equivalent quantumdots with real parameters, only a small fraction (less than1%) of light is scattered, with most of the photons keepingtheir in-plane wave-vector constant. The ‘Bragg-like’ modelprovided explanations for both the horizontal [17] andvertical [18] motional narrowing effects in microcavities andMQW structures, respectively.
An important advantage of the ‘Bragg-like’ model is itssimplicity. Actually, it allows one to find the linear opticalresponse of complex semiconductor structures with the use ofthe Maxwell equations, fully taking into account the acts ofabsorption-emission of light by excitons (i.e., the polaritoniceffect).
Here we present an analysis of the time-resolved reflectionspectra of SQW and MQW structures in terms of the ‘Bragg-like’ model (described below). In the next section, devotedto the resonant Rayleigh scattering of light in MQWs, theapproach is extended in order to describe the RRS.
5.2. Theoretical model of the linear optical response ofexcitons in a disordered QW
In the framework of the linear non-local response theory, theMaxwell equations for a light wave normally incident on asingle QW in the vicinity of the exciton resonance frequencycan be written in the form
∇ × ∇ × E = ω2
c2D (5.1)
whereD(z) = ε∞E(z) + 4πPexc(z) (5.2)
Pexc(z) =∫χ(z, z′)E(z′) dz′ (5.3)
and the non-local susceptibility is expressed as
χ(ω, z, z′) = χ(ω)8(z)8(z′) (5.4)
where 8(z) is the exciton envelope function taken with equalelectron and hole coordinates, and z is normal to the QW plane.Taking into consideration only the exciton ground state, we canwrite, in the absence of inhomogeneous broadening,
χ(ω − ω0) = ε∞ωLTπa3Bω
20/c
2
ω0 − ω − iγ(5.5)
whereω0 is the exciton resonance frequency, γ is non-radiativehomogeneous broadening, ωLT is the exciton longitudinal-transverse splitting in the bulk material, and aB is the bulkexciton Bohr radius.
Solving equation (5.1) with the susceptibility givenby (5.5) and supposing for simplicity the background dielectricconstants in the well and barrier to be equal, we can obtain theamplitude reflection and transmission coefficients for a singleQW in the form [14]
rQW = iαχ
1 − iαχ, tQW = 1 + rQW (5.6)
where α = ?0c2/(ε∞ωLTπω
20a
3B). Here the slight
renormalization of the exciton resonance frequency duethe long-range exchange interaction is omitted. Note that
formula (5.6) is quite general. It is independent of aconcrete form of the susceptibility function, being valid forany symmetric planar layer.
According to the model [14], the exciton inhomogeneousbroadening can be introduced in the formalism by averagingthe dielectric susceptibility (5.5) with a continuous distributionfunction of the exciton resonance frequency (chosen to be aGaussian function for simplicity)
χ(ω, γ ) = h√π,
∫χ(ω, ω1, γ )
× exp
[−h2
(ω1 − ω0
,
)2]
dω1. (5.7)
This modified susceptibility should be substituted intoequation (5.6), which yields for the reflection coefficient
rQW = −√π?0w(z)
,h +√π?0w(z)
(5.8)
where w(z) = e−z2erfc(−iz), z = h(ω − ω0 + iγ )/,, and
erfc(z) is the complementary error function.In the framework of the ‘planar wave’ model one should
use equation (5.6) for the reflection coefficient of the QW,since it follows from the Maxwell equations for a plane waveincident on a planar layer. The inhomogeneous broadeningeffect can only be accounted for by a variation of the dielectricsusceptibility (as, e.g., in equation (5.7)). Averaging of thereflection coefficient for a single exciton resonance (calculatedwith the susceptibility (5.5)) with some frequency distributionfunction is inconsistent with the Maxwell equations and has nophysical relevance. In the next part of this section we commenton this issue in detail.
The optical response of a MQW structure can be foundby the transfer matrix method. In the basis of amplitudes oflight waves propagating towards positive and negative z, thetransfer matrix across a QW has a form:
TQW = 1
tQW
(1 −rQW
rQW t2QW − r2QW
). (5.9)
The transfer matrix across the entire structure is a product oftransfer matrices across all layers and interfaces. The reflectioncoefficient of the entire structure can be found as described insection 2.3.
The time-dependent reflection amplitude for an incidentpulse g(ω) is given by
rg(t) =∫ +∞
−∞
dω
2πrQW(ω)g(ω)e
−iωt . (5.10)
5.3. Weighted susceptibility or weighted optical response?
It was pointed out in section 4 and the preceding subsectionthat the correct way to describe the inhomogeneous broadeningwithin the constant in-plane wave-vector approximation is toconsider the weighted dielectric susceptibility (5.7), ratherthan the weighted optical response (reflection or transmissioncoefficient) of a structure. Owing to its relevance to theentire following discussion of the inhomogeneous broadeningeffects, we address this question here in detail.
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G Malpuech and A Kavokin
all excitons E
any photon
(b)
nth exciton Enth photon
(a) E1st exciton
1st photon
Figure 11. Illustration of the two schemes of interaction between aphoton and an inhomogeneously broadened exciton resonance (seethe text). (a) First the optical response of each single exciton statebelonging to the inhomogeneous distribution is found, then the totalresponse of the structure is calculated as the sum of individualresponses. (b) First the dielectric susceptibility containingcontributions of all excitons is calculated, then the light interactionwith a layer characterized by this susceptibility is considered.
The physical difference between the solution of theMaxwell equations obtained considering first a single excitonresonance and then averaging the electric field relatedto different resonances and the solution with weightedsusceptibility can be easily understood with the aid of figure 11.In the former case (figure 11(a)) each given photon can interactwith only a single exciton resonance from the frequencydistribution; thus, the photon is always emitted and re-absorbedby the same exciton. The averaging of all single-photon–single-exciton interaction schemes yields the optical responseof the system within this model.
In contrast, if the Maxwell equations are solved withweighted dielectric susceptibility (5.7), a photon emitted bya given exciton is allowed by the model to be absorbed byanother exciton from the distribution (figure 11(b)). Thus,the averaging of the dielectric susceptibility allows us to takeinto account all the possible chains of coherent absorption-emission acts. The difference between the two approaches isalmost negligible for cw optical spectra, but it appears to behuge in time-resolved spectra where fine interference effectsplay an important role.
This is illustrated by figure 12 which shows the cw(inset) and time-resolved reflection spectra for a single QW.The following set of excitonic parameters h?0 = 0.2 meV,hγ = 0.1 meV and, = 1.9 meV was used. These parametersyield the best fit to the experimental reflection and absorptionspectra of a 172 Å thick In0.06Ga0.94As/GaAs QW [51]. Thetime-resolved reflection spectrum (figure 12, curve (a)) and thecw reflectivity spectrum (R(ω) = |rQW(ω)|2, inset of figure 12,full curve) were calculated in terms of the non-local model byequation (5.6) with the averaged dielectric susceptibility (5.7).
These spectra are to be compared with time-resolved andcw reflection spectra (figure 12, curve (b)) and the broken curvein the inset of figure 12) calculated by averaging the reflectioncoefficient of the quantum well
rQW(ω, ω0) =∫ +∞
−∞rQW(ω, ω
′0)f (ω
′0, ω0) dω′
0 (5.11)
Pulse
Time (ps)
100
105
1010
1015
Ref
lect
ion
(arb
. uni
ts) Ref
lect
ivity
(ar
b. u
nits
)
(a)
(b)
1450 1460Energy (meV)
0 1 2 3 4 5
Figure 12. Time-resolved reflectivities calculated for a single QWcontaining an inhomogeneously broadened exciton resonance. Theinhomogeneous broadening is taken into account in the dielectricsusceptibility of the QW (curve (a)); frequency-resolved reflectionspectra calculated for individual exciton resonances from theinhomogeneous distribution are averaged with the exciton frequencydistribution function (curve (b)). Time-resolved spectra wereobtained by an inverse Fourier transform of the frequency-resolvedspectra (shown in the inset by full (for case (a)) and broken (forcase (b)) curves with the spectral function of the incident pulse(shown in the figure).
where rQW(ω, ω0) is given by equation (5.6) with the dielectricsusceptibility (5.5). In this case, the time-resolved reflectionwas also found from equation (5.10) with a 300 fs long pulse.
It can be seen that the time-resolved reflection spectrumcalculated with the averaged dielectric susceptibility exhibitspronounced oscillations absent in the spectrum calculated withthe averaged reflection coefficient. Quantum beats in time-resolved spectra of a QW with a single inhomogeneouslybroadened exciton resonance were predicted in [14]. Nodirect experimental evidence of the beats has been reportedyet to the best of our knowledge, while an indirect proof hasbeen produced recently [51]. The appearance of the beats inthe time-resolved response of a single optical resonance is astriking and unexpected effect. (Note that usually the beats takeplace between two discrete, closely spaced quantum states.)
In [14], the beats were interpreted in the framework ofthe semiclassical approach as resulting from the interferenceof photons emitted at frequencies slightly lower and higherthan ω0. The beats can also be interpreted in terms ofthe quantum absorption–re-emission model. Considering thepropagation of exciton–polaritons in a semiconductor as achain of emission-absorption acts (as illustrated by figure 11),we attribute the beats to the interference between those photonswhich interact with only a single exciton resonance from theenergy distribution (they induce a monotonic decay of R(t)shown in figure 10(b)) and those which interact with differentexcitons during their lifetime in a QW. Passing through
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different chains of emission-absorption acts, the photons gaindifferent phases which gives rise to interference.
This makes propagation of exciton–polaritons in quantumstructures containing inhomogeneously broadened excitonresonances to some extent analogous to the transport ofelectrons in mesoscopically disordered media. (See also thediscussion of this issue for the case of bulk exciton–polaritonsin section 4). It should be noted that the emission-absorptionacts we are speaking about are virtual. Moreover, in the caseof a QW there is no real propagation of exciton–polaritons,since the excitons are confined to the z-direction. The analogyrelies on the fact that, in the presence of inhomogeneousbroadening, the photon in a QW is coupled with all excitonsfrom the inhomogeneous contour. This means that during apolariton’s lifetime it may be virtually emitted-absorbed bydifferent excitons from the distribution, in the same way asan electron may be scattered by any scattering centre in themedium.
5.4. Absorption of light by inhomogeneously broadenedexcitons in QWs
One of the conclusions of the semiclassical model ofinhomogeneous broadening, proposed in [14], was seeminglya paradox and could have created doubts in the validity ofthe model. Namely, the model predicted that the integratedabsorption of the QW in the presence of inhomogeneousbroadening is not a continuous function of the excitonhomogeneous broadening γ . At γ = 0 the integratedabsorption is zero (since there is no energy dissipation inthe system [52]), but at γ → 0 it retains some finite valuedependent on the inhomogeneous broadening parameter ,.
In this section, we address the problem of light absorptionat an inhomogeneously broadened exciton resonance in aQW, which is of particular importance for understandingthe effect of disorder. We show that the model presentedin [14] is basically correct, while in the limit γ → 0 thecontinuous distribution of the exciton resonance energy shouldbe substituted by a discrete function containing a finite numberof resonances. We have recently published a short paper onthis matter [53].
Let us consider a QW containing a single excitonresonance. Taking into consideration only the groundexciton state, we can represent the excitonic contributionto the dielectric susceptibility of the QW in the absence ofinhomogeneous broadening by formula (5.5).
In real structures, the exciton resonance is broadenedinhomogeneously, which means that instead of the onlyresonance frequency ω0 we should take into account a hugenumber of resonances corresponding to exciton eigenstatesconfined by a lateral fluctuation potential. The model [14]introduces the inhomogeneous broadening by averaging thedielectric susceptibility (5.5) with a continuous distributionfunction of the exciton resonance frequency (a Gaussian), asdescribed by formula (5.7). This model assumes a continuousdistribution of the exciton resonance frequency. In the presentsection, we are going to take into account the discrete energyspectrum of an exciton in a QW.
Actually, for any kind of spatially limited potential thedistribution of energy levels is discrete, but can be supposed
continuous if the maximum difference β between these levelsis too small to be detected experimentally. In our case, thiscriterion can be formulated as
d2 λ2 (5.12)
where d is the average distance between two excitons withthe energy difference β smaller than the exciton homogeneousbroadening: β < hγ . The value of β depends on the profileof the fluctuation potential. If the criterion (5.12) is satisfied,the QW plane is optically homogeneous.
Apparently, in the limit γ → 0, the discrete character ofthe excitonic energy distribution should be taken into account.Thus, we propose to rewrite formula (5.7) as follows
χ(ω, γ ) =p=+∞∑p=−∞
χ(ω, ω0 + pβ/h, γ )
× exp
[−(hω0 + pβ
,
)2]
×{p=+∞∑p=−∞
exp
[−(hω0 + pβ
,
)2]}−1
(5.13)
where β has some finite value. Let us compare the integratedexcitonic absorption of light
∫A(ω)h dω, where
A(ω) = 1 − |r|2 − |t |2 (5.14)
is calculated in terms of the present model and the modeldescribed in [14].
Figure 13 shows the evolution of the integrated absorptionas a function of γ in the case of continuous (full curve) anddiscrete (broken curve) energy distributions. We used thefollowing parameters: hω0 = 1515 meV, h?0 = 0.02 meV,, = 1 meV, β = 0.5 meV. It can be seen that two methodsyield the same result for hγ > β, which confirms thevalidity of the continuous approximation in a wide range ofconditions. On the other hand, in the opposite limit, our modelprovides a continuous decrease of the integrated absorptionto zero in agreement with [52], while the model [14] yields adiscontinuity of the integrated absorption at γ → 0.
Physically, this means that, if hγ > β, the exciton canbe easily (quasi-resonantly) scattered from each discrete stateto a neighbouring state. Being scattered, the exciton does notcontribute to the coherent emission of photons and, thus, makesno contribution to reflection or transmission. According toequation (5.14), it contributes to the absorption of light.
It should be noted at this point that the quantity definedas absorption by equation (5.14) includes, in fact, the realabsorption (i.e., transmission of the energy of photons to thecrystal lattice) and all incoherent processes (inelastic scatteringof photons). At small γ , the heating of the crystal by opticalabsorption decreases proportionally to γ , so that ourAmostlyrepresents a contribution from inelastic scattering.
Figure 14 illustrates this statement. It shows theabsorption spectra of our QW calculated with hγ = 1 meV (a),hγ = 0.1 meV (b) and hγ = 0.001 meV (c), in terms of themodels of continuous and discrete distributions of the excitonresonance energy (broken and full curves, respectively). Thecases (a), (b) and (c) correspond to the regimes hγ > β,
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G Malpuech and A Kavokin
0.12
0.10
0.08
0.06
0.04
0.02
00 0.002 0.004 0.006
Inte
grat
ed a
bsor
ptio
n (m
eV)
Inte
grat
ed a
bsor
ptio
n (m
eV)
γ (meV)
γ (meV)0 0.5 1.0 1.5 2.0
0.14
0.12
0.10
0.08
0.06
0.04
0.02
0
Figure 13. Integrated excitonic absorption in a QW as a function ofγ , calculated in the cases of continuous and discrete distributions ofthe exciton resonance energy (broken and full curves, respectively).Parameters of the calculation: hω0 = 1515 meV, h?0 = 0.02 meV,, = 1 meV, β = 0.5 meV. The inset shows a zoom on the initialpart of the curves.
hγ < β and hγ β, respectively. In the case (a), thefull and broken curves coincide completely, while in the cases(b) and (c) the broken curves show pronounced multiple-peakstructures appearing because of the discrete character of theexciton resonance distribution. In the case (b), the discretepeaks still overlap, which enhances the integrated absorption.In the case (c), the neighbouring excitons are not coupled,so that the total integrated absorption is nothing but a sumof integrated absorptions of individual resonances, decreasinglinearly with γ in this regime.
Interestingly, the central peaks in the case (c) have largerwidths and smaller amplitudes than the peaks spaced by 1 meVfrom hω0. This can be understood from equations (5.13),(5.14). Actually, in this limit the absorption at each individualresonance ωi is given by
Ai(ω) = 2γ ?0
(ωi − ω)2 +(γ + ?0
)2 (5.15)
where
?0 = ?0 exp
{−[h(ωi − ω0)
,
]2}
×{p=+∞∑p=−∞
exp
[−(hω0 + pβ
,
)2]}−1
. (5.16)
?0 decreases with increasing |ωi − ω0|. The amplitude ofAi has maxima Ai(ωi) = 1/2 at frequencies ωi given by thecondition ?0 = γ , while the halfwidth of the peak is the largestat ωi = ω0. Taking into account that in our case ?0 = 20γ ,
1510 1512 1514 1516 1518 1520Energy (meV)
0.25
0.50
0
0.05
0.10
0
0.02
0.04(a)
(b)
(c)
Abs
orpt
ion
Abs
orpt
ion
Abs
orpt
ion
Figure 14. Frequency-resolved excitonic absorption of light in aQW, calculated in the cases of continuous and discrete distributionsof the exciton resonance energy (broken and full curves,respectively) with the same parameters ω0, ?0, ,, β as in figure 13for (a) hγ = 1 meV, (b) hγ = 0.1 meV and (c) hγ = 0.001 meV.
we readily obtain Ai(ωi) ≈ 1/2 at hωi ≈ 1.5135 eV andhωi ≈ 1.5165.
In conclusion of this sub-section, we have shown thatthe model [14] assuming a continuous distribution of theexciton resonance energy through inhomogeneous broadeningin a QW is valid in the case hγ > β. Here β is themaximum difference between the exciton energy levels in thelateral fluctuation potential, and γ is the exciton homogeneousbroadening. This approximation is fulfilled in most realsystems and formula (5.8) is used throughout the rest of thisreview. This approximation fails when hγ becomes smallerthan β. In this limit, the discrete distribution of the excitonresonance energy should be used, which allows one to obtainthe physically correct tendency of a continuous decrease ofthe integrated absorption with γ . Fine peculiarities of the lineshape of the excitonic absorption spectrum have been revealedin this case.
5.5. Vertical motional narrowing of exciton–polaritons inmultiple QWs
Recently, it has been shown experimentally [9] andtheoretically [18] that the radiative coupling between excitonsconfined in different QWs in a MQW structure may reduce theeffective disorder acting on the exciton. The signature of thiseffect, known as the vertical motional narrowing (VMN), isan increase in the decay time of the time-resolved reflectionsignal resulting from the averaging of the disorder potential ina MQW structure by extended exciton–polariton modes thatoccupy the entire structure. It should be distinguished from
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Topical Review
the horizontal motional narrowing (HMR) which consists inthe averaging of the potential fluctuations in the QW plane byexciton–polaritons [15].
Here we examine theoretically the time-resolved responseof GaN/AlGaN MQWs, a modern semiconductor systempromising for various applications in optoelectronics. Weshow that, except in Bragg-arranged MQWs, the VMN effectgoverns the excitonic decay rate in these structures. Fourparameters determine the VMN. These are the exciton radiativerecombination rate ?0, exciton inhomogeneous broadening,,spacing between wells d and number of wells N . Theparameter , is in the range 5–10 meV for the best qualityMQW structures based on the presently available nitrides[54, 55]. The exciton oscillator strength in nitrides is an orderof magnitude larger than in GaAs, so that h?0 is as low as0.4 meV in GaN/AlGaN QWs [55]. The coincidence of thestrong disorder and huge exciton oscillator strength makes usexpect an extremely pronounced VMN effect in nitride-basedMQWs.
We apply here the semiclassical model of the non-local dielectric response of QWs containing inhomogeneouslybroadened exciton resonances, developed in [14] and describedin section 5.2. In the numerical calculations, hγ = 0.1 meV,d = 70 nm and hω0 = 3.6 eV were taken for all the QWsunder consideration. We consider either ‘disordered’ QWstructures with, = 5 meV, typical of GaN/AlGaN QWs [54],or ‘homogeneous’ QWs with , = 0 (for reference).
Figure 15 shows time-resolved reflection spectra ofa disordered SQW (a), disordered MQWs containing10 wells (b), 50 wells (c), 100 wells (d) and a homogeneousSQW (e). Comparing the curves (a) and (e) in figure 15,we see that the exciton inhomogeneous broadening leads toa dramatic decrease of the decay time of the time-resolvedreflection. This reflects the appearance of additional channelsof energy relaxation for excitons as the disorder increases inthe structure [17, 18]. Note the strong decrease of the excitondecay time with increasing number of QWs in the structure.This is a manifestation of the VMN effect, which is visiblystronger than that observed experimentally in GaAs/AlGaAsMQWs [9]. Note also pronounced oscillations in the spectraof MQWs and a disordered SQW (figure 15, curves (a)–(d)). These oscillations have been widely discussed in recentyears ([9, 14, 18, 51]). Basically, they arise from boththe interference of different exciton–polariton modes and theinterference within each single exciton–polariton mode due tothe inhomogeneous broadening.
In the rest of this section, we present an analyticaltheory aimed to express the decay time of the coherentoptical response of MQWs and the period of oscillationsvia structural parameters and excitonic characteristics. Weadopt the effective dielectric media (EDM) approximation[25] describing the MQW structure as a homogeneouslayer containing a single inhomogeneously broadened excitonresonance. This approximation is valid if ?0 γ and we canfind an integer n such that
kd − nπ 1 (5.17)
where k is the wave vector of light in the medium at theexciton resonance frequency. The formalism allowing one tocalculate the time-resolved response of a semiconductor film
Time (ps)
102
100
10−2
10−4
10−6
Ref
lect
ion
(arb
. uni
ts)
0 1 2 3 4 5
(a)(b)
(c)
(d)
(e)
Figure 15. Time-resolved reflection from GaN/AlGaN QWstructures: SQW with , = 5 meV (curve (a)); MQW, N = 10,, = 5 meV (curve (b)); MQW, N = 50, , = 5 meV (curve (c));MQW, N = 100, , = 5 meV (curve (d)) and SQW with, = 0 meV (curve (e)).
containing an inhomogeneously broadened exciton resonancewas developed in section 4. To describe the MQWs, we shouldsubstitute 2?0/kd for the exciton longitudinal-transversesplittingωLT in (4.4) andNd for the film thickness d in (4.18)–(4.21) and (4.24) [25].
Figure 16 shows the time-resolved transmission of thestructure (figure 15, curve (c)) calculated within the EDMapproximation, in comparison with the results of an exactcalculation. Note that the criterion (5.17) is satisfied for thisstructure at n = 1. The excellent agreement between the exactand approximate calculations allows us to apply the steepestdescent method presented in section 4.4 to find the decay time τand the period of oscillations T in the time-resolved reflectionof the MQWs. Using (4.26), we can obtain
T (t) = π√t√
N?0[1 + (3,2t)/(4Nh?0)
] . (5.18)
The inset of figure 16 shows the distances between minima inthe time-resolved transmission plotted as a function of time incomparison with the dependence T (t) from formula (5.18) forthe structure (figure 15, curve (c)). A very good agreementbetween the analytical theory and numerical calculation canbe seen.
In order to find a compact analytical formula for thedecay time τ we have to extend substantially the formalismof section 4.4 and the EDM approximation. It will be recalledthat the spectrum of exciton–polariton eigenmodes in a MQWstructure consists of several super-radiant modes having hugeoscillator strength and many dark modes having very smalloscillator strengths. If the condition (5.17) is satisfied forshort-period MQWs or quasi-Bragg-arranged MQWs, thereis only a single super-radiant mode with an extremely shortdecay time. On the picosecond time scale the exciton radiativedecay is governed by the dark modes forming two bands similarto the bulk exciton–polariton bands. That is why the EDMapproximation works so well for this kind of MQWs.
In the limit , Nh?0, the sum of the oscillatorstrengths of the exciton–polariton modes can be assumed tobe independent of ,. Thus, the oscillator strength of the
R19
G Malpuech and A KavokinT
rans
mis
sion
(ar
b. u
nits
)
Time (ps)0 1.50.5 1.0
10−4
10−3
10−2
10−1
100
101
Time (ps)
�
�
��
� �
0 0.4 0.8 1.2
0.3
0.2
0.1
0
Peri
od o
fos
cilla
tions
(ps
)
Figure 16. Time-resolved transmission of light through aGaN/AlGaN MQW structure containing 50 QWs (full curve) andthrough a thin film of bulk semiconductor characterized by aneffective dielectric susceptibility according to the EDMapproximation (broken curve). The inset shows the distancesbetween minima in the time-resolved transmission spectrum of thisstructure (points) compared to the function T (t) given byequation (5).
dark modes can be estimated as a total oscillator strength(proportional to N?0) minus the oscillator strength of thesuper-radiant mode, which decreases with increasing ,.Within this hypothesis and with the use of equation (4.19), wecan find the decay rate of the time-resolved optical responseof a MQW structure as follows
1
2τ= γ +
,2
2[N?0 − (,2/2Nh?0)
]h2 . (5.19)
Figure 17 shows the decay rate for a GaN/AlGaN MQWstructure with the same period and excitonic parameters asbefore versus N (full curve). The broken curve representsthe calculation by equation (5.19). It can be seen that thetendency for the decay rate to decrease with increasing N ,which is a signature of the VMN effect, is reproduced byformula (5.19). The significant deviations at small N arecaused by the contributions to the decay rate from the super-radiant modes that become important on the picosecond timescale in this regime.
In conclusion, we have shown that the vertical motionalnarrowing effect is of great importance for the coherentoptical spectra of the GaN/AlGaN MQWs. The approximateanalytical expressions for the decay rate and period ofoscillations of the time-resolved reflection and transmissionspectra of MQWs have been derived.
6. Resonant Rayleigh scattering ofexciton–polaritons in multiple quantum wells
In this section we present the concept of resonant Rayleighscattering (RRS) of exciton–polaritons in MQWs. The opticalcoupling between excitons in different QWs is shown to governthe RRS spectra, giving rise to the characteristic picosecondscale oscillations. We give an explanation of the existingexperimental data on the RRS from MQWs, which have been
�
�
�
�
�
�
�
�
�
� ��
0 50 100 150 200Number of QWs
Dec
ay r
ate
(ps
)−1
5
4
3
2
1
0
Figure 17. The decay rate of the time-resolved reflection fromGaN/AlGaN MQW structures as a function of the number of QWs,obtained numerically (•) and compared with the result obtainedusing the analytical theory (broken curve).
puzzling so far. The Bragg- and anti-Bragg-arranged QWstructures with the same excitonic parameters are predictedto have drastically different RRS spectra.
Previous studies of the RRS from the MQWs can bebriefly summarized as follows. The spectra reported in [6–8]typically exhibit an RRS signal that initially rises quadraticallywith time. Then it decays rapidly and non-exponentially onthe time-scale of the inverse inhomogeneous broadening ofthe exciton peak. This fast decay is followed by a slowerexponential decay. The rise and the fast initial decay havebeen explained in terms of a theoretical model [20] thatassumes an ensemble of classical oscillators moving within arandom potential with a finite correlation length. The coherentcontrol technique [7, 8] eliminated the light-hole–heavy-holebeats in the GaAs QWs and revealed that the time-resolvedRRS signal exhibits a behaviour more complicated than thatpredicted by the model [20]. Recent experiments on differentMQW structures demonstrated a complex non-monotonicdecay sometimes showing oscillations with a period of severalpicoseconds [8, 56]. These oscillations were related in [57]to the interference between light-waves scattered by differentlocalized exciton states in the QW plane. Experiments onsingle QWs [57] show no features of this kind and are wellexplained using a correlation length shorter than the excitonradius. It has been shown [58] that secondary emission (SE)from QWs is made possible by quantum fluctuations of thelight field. This theory, written for an ideal single QW, bearsno direct relation to the discussed experiments performed inthe linear regime on the a priori disordered MQW samples.Finally, an attempt to take into account the polariton effectsin the RRS has been made by Citrin [22]. However, allthe presented numerical calculations ignored the effect ofpolaritons, so that their role, especially in the RRS of MQWs,remained unclear.
The necessity for extension of the exciton–polaritontheory in order to explain the RRS data is quite clear now. Inour opinion, this requires a revision of the physical concept ofthe RRS in QWs, namely, the concept of scattering of coherentexciton–polariton states is to be introduced.
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Topical Review
QW
π − Θ
− Θπ + ΘΘ
Figure 18. Four scattering directions to be taken into account whenmodelling the RRS from QWs.
The algorithm we use in the RRS calculations is as follows:
• The scattering state method is used to calculate thecoordinate- and time-dependent electric field in the systemilluminated by an initial pulse of light, ignoring the RRS.
• We consider the scattering of light waves incident on eachindividual QW from both sides. A phenomenologicalfunction is introduced, giving the amplitude of the wavescattered at the angle A as a function f (τ) of the delaybetween the incident and scattered signals. Here we use animportant consequence of the assumed mirror symmetryof our QWs: the scattering probabilities at the angles A,−A, π − A and π + A are equal (see figure 18). Notethat the directions A and π − A are equivalent since theemission in these directions is due to the same QW excitonstate.
• Therefore, the amplitudes of the signal scattered at thesefour angles are equal and can be found as a convolutionof the time-dependent amplitude of the incident electricfield at the QW with the scattering function f (τ).
• Then, we are looking for the scattering spectrum ofthe structure at the angle A. The scattered amplitudeis a sum of the amplitudes scattered by all QWs asdescribed above. Each QW is considered to be an effectiveemitter. The propagation of pulses emitted by eachQW at four scattering angles is considered in terms ofthe scattering state method, taking into account all thereflection-transmission acts during its propagation in thestructure but ignoring the secondary scattering.
From the technical point of view, this algorithm requiressolving the Maxwell equations for a planar structure containinga light emitter. The formalism we used to solve this problemis the generalized scattering state technique described insection 2.4.
In our model, the dielectric response of the QWscontaining inhomogeneously broadened exciton resonancesis described in the framework of the non-local model [14],section 5.2. The parameters used are the exciton radiativedamping rate ?0, the homogeneous broadening γ , and theinhomogeneous broadening ,. For a MQW structure, weindex with zj the coordinate of the j th QW and calculate theelectric fieldsEg(zj , ω) = g(ω)E(zj , ω) (with j varying from1 toN ). HereE(z, ω) is the stationary solution of the Maxwellequations corresponding to a plane light-wave incident on the
structure from the left and g(ω) is the spectral function ofthe incident pulse. In all further calculations we assume theincident pulse of light to be 500 fs long.
The probability of creating an exciton having a frequencyω and a wave-vector k‖ = 0 in j th QW is proportional to theelectric field intensity in this well. The amplitude of the electricfield of the scattered exciton–polariton state with k‖ �= 0 canthen be found using equation (28) of [56]:
AS(zj , ω) = PS(ω)Eg(zj , ω) (6.1)
where
PS(ω) =∫ +∞
−∞
(∫ {exp
[(t
t0
)2
fR
]− 1
}dR
)1/2
× exp
[−(
t√2t0
)2]
exp(iωt) dt (6.2)
with the rise time t0 = h√
2/, and fR being a potentialcorrelation function chosen in the form:
fR = exp
(− R2
2ξ 2
)(6.3)
where ξ is the potential correlation length further assumed to be10 nm, i.e., approximately the Bohr radius of the QW exciton.Note that equation (6.2) is written in the classical limit. Thetime-dependent amplitude of the scattered light in the right-hand part of equation (6.2) is real, while its Fourier transformPS(ω) is a complex function. No random phase fluctuationsare assumed.
The scattered amplitudes can be thought of as light-pulses emitted by each well in the positive and the negativedirections. Using the generalized scattering-states method(section 2.5), we calculate their propagation in the scatteringdirection, namely, we apply the formula (2.42), replacing thepulses g(ω)h(ky) and p(ω)q(ky) by AS(zj , ω)δ(k
yscat), where
kyscat is the absolute value of the y-component of the wave
vector of light in all the considered scattering directions (A,−A, π − A, and π + A). Let us consider an emitter of lightsituated at the centre of the j th QW. We neglect the spatialdistribution of the emitter, which is a usual assumption in thenon-local model. We take into account the polariton effectwithin each single QW, i.e., the possibility of emission andabsorption of a photon by the same exciton. We consider theTM polarization of light as in [57]. Formula (2.42) directlygives the amplitude E
zj
AS,δ(kyscat),AS,δ(k
yscat)(ω, k
yscat,TM, 0) of light
scattered by j th QW at z = 0 in theA, −A, π −A, and π +Adirections.
The amplitude of light scattered in a given direction by asystem containing N QWs can be found as:
rS(ω, kyscat,TM) =
N∑j=1
Ezj
AS,δ(kyscat),AS,δ(k
yscat)(ω, k
yscat,TM, 0).
(6.4)The time-resolved RRS signal is given by:
RS(t) =∣∣∣∣ 1
2π
∫ +∞
−∞rS(ω) exp(−iωt) dω
∣∣∣∣2
. (6.5)
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G Malpuech and A Kavokin
0 5 10 15 20Time (ps)
Seco
ndar
y em
issi
on (
arb.
uni
ts)
(a)
(b)
(c)
(d)
Figure 19. Time-resolved RRS spectra calculated for the structurescontaining 1 (curve (a)), 10 (curve (b)), 25 (curve (c)) and40 (curve (d)) identical QWs spaced by 30 nm, with the followingexcitonic parameters: h?0 = 0.026 meV, hγ = 0.08 meV,, = 0.7 meV. The cap layer is 20 nm thick.
In summary, we took into account all the coherentreflection-absorption processes for this scattered light in thestructure. We ignored the secondary scattering, however.
Figure 19 shows the RRS spectra for the structurescontaining 1, 10, 25 and 40 identical QWs spaced by 30 nm,with the following excitonic parameters: h?0 = 0.026 meV,hγ = 0.08 meV, , = 0.7 meV. It can be seen that theRRS spectrum is quite sensitive to the number of QWs in thestructure. It is monotonic in the case of a single QW, and showsminima and oscillations in MQWs. This indicates that theoptical interference of light scattered by different QWs shouldbe taken into account when modelling the RRS. The previouslyreported theories that ignored this effect [20, 21, 56, 58] arevery far from reality in this regard.
Figure 20 shows the RRS calculated for 25 Bragg-arranged and 25 anti-Bragg-arranged MQWs. We assumeda GaAs/Al0.3Ga0.7As structure embedded in Al0.3Ga0.7Asbarriers and the following QW exciton parameters: ?0 =0.028 meV, γ = 0.006 meV, , = 0.5 meV. The cap layerthickness is 15 nm. It can be seen that there are no oscillationsin the RRS spectra for the Bragg-arranged MQW structure.In contrast, the RRS from the anti-Bragg-arranged MQWsexhibit pronounced short-period oscillations (with a period ofabout 5 ps). These oscillations through interference betweendifferent bright polaritonic modes in the MQWs. Thus, theperiod of a MQW structure appears to have a strong influenceon the RRS spectra.
In conclusion, the RRS from MQW structures is governedby the dynamics of the exciton–polaritons, which are extendedmixed exciton-photon states occupying the entire system. Thepresent semi-classical model takes into account the exciton–polariton effect in MQWs. Drastic differences are predictedbetween the time-resolved RRS from Bragg- and anti-Bragg-arranged MQWs.
7. Conclusion
Time-resolved coherent optical spectroscopy is a verypowerful tool to study the kinetics of exciton–polaritonsin semiconductor structures. It allows one to distinguishbetween homogeneous and inhomogeneous broadening of
0 5 10 15 20Time (ps)
(a)
(b)
10−6
10−4
10−2
100
SE in
tens
ity (
arb.
uni
ts)
Figure 20. Time-resolved RRS spectra calculated for 25Bragg-arranged (curve (a)) and 25 anti-Bragg-arranged (curve (b))MQWs. QW exciton parameters: ?0 = 0.028 meV,γ = 0.006 meV, , = 0.5 meV. The cap layer thickness is 15 nm.
excitons, optical dephasing and scattering. Both purelyoptical phenomena, like interference of light, and quantum-mechanical effects, such as quantum confinement of excitons,contribute to the physics of the exciton–polaritons. That iswhy, for its theoretical description one always has a choicebetween approaches based on the classical electrodynamics orquantum mechanics.
In this review we have analysed the time-resolved opticalresponse of different semiconductor structures, using thesemi-classical method. In order to solve the time-dependentMaxwell equations for short light pulses propagating in thestructures, we used the scattering-state technique. Thelight–exciton coupling was described in terms of the non-local dielectric response theory extended to account forthe inhomogeneous broadening of excitons. The resonantRayleigh scattering of light on the QWs was described asemission of light by coherent sources (i.e., QW excitons)excited by the incident pulse. As a result, the following neweffects were found:
• In laterally confined periodical structures (Braggmirrors) with a gradually changing lateral size, thephotons experience oscillations within an inclined one-dimensional photonic miniband. These oscillationsare analogous to the well-known Bloch oscillations ofelectrons in a periodical potential subjected to an electricfield.
• Propagation of exciton polaritons excited by a short pulseof light in a bulk semiconductor results in the appearanceof a grating of the dielectric polarization along thedirection of light propagation. This grating appears owingto the interference between two polaritonic branches. Itmoves backwards with respect to the light propagationdirection owing to coherent re-emission–re-absorption oflight by excitons.
• Interference of light reflected by different QWs within aMQW structure results in the increase of the decay time ofthe MQW structure with increasing number of the wells.This can be interpreted as a vertical motional narrowingeffect arising because of the averaging of the disorderpotential in QWs by extended exciton–polariton modes.
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Topical Review
• The resonant Rayleigh scattering of light in MQWs isstrongly dependent on the number of wells and the periodof the structure. In fact, this is the scattering of exciton–polaritons rather than of excitons. The coherence betweendifferent QWs in the system plays a major role in time-resolved response both in reflection and in scatteringgeometry.
Finally, we believe that new experiments will soonconfirm the theoretical predictions given here, namely, thephotonic Bloch oscillations, the formation of a dielectricgrating along the propagation direction of exciton polaritonsin bulk semiconductors and the drastic dependence of the RRSsignal on the spacing between wells in a MQW structure.
AcknowledgmentsWe are grateful to J Leymarie, B Gil, J J Baumberg,L C Andreani, A Di Carlo, F Rossi and G Panzarini for fruitfulcollaboration and many enlightening discussions. The workwas supported by the European Commission in the frameworkof a contract HPRN-CT-1999-00132.
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