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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 8 — Apr. 13, 2009
  • pp: 6813–6828
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One dimensional resonant Fibonacci quasicrystals: noncanonical linear and canonical nonlinear effects

M. Werchner, M. Schafer, M. Kira, S. W. Koch, J. Sweet, J. D. Olitzky, J. Hendrickson, B. C. Richards, G. Khitrova, H. M. Gibbs, A. N. Poddubny, E. L. Ivchenko, M. Voronov, and M. Wegener  »View Author Affiliations


Optics Express, Vol. 17, Issue 8, pp. 6813-6828 (2009)
http://dx.doi.org/10.1364/OE.17.006813


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Abstract

A detailed experimental and theoretical study of the linear and nonlinear optical properties of different Fibonacci-spaced multiple-quantum-well structures is presented. Systematic numerical studies are performed for different average spacing and geometrical arrangement of the quantum wells. Measurements of the linear and nonlinear (carrier density dependent) reflectivity are shown to be in good agreement with the computational results. As the pump pulse energy increases, the excitation-induced dephasing broadens the exciton resonances resulting in a disappearance of sharp features and reduction in peak reflectivity.

© 2009 Optical Society of America

1. Introduction

In our works [9

9. A. N. Poddubny, L. Pilozzi, M. M. Voronov, and E. L. Ivchenko, “Resonant Fibonacci quantum well structures in one dimension,” Phys. Rev. B 77, 113306 (2008). [CrossRef]

, 10

10. J. Hendrickson, B. C. Richards, J. Sweet, G. Khitrova, A. N. Poddubny, E. L. Ivchenko, M. Wegener, and H. M. Gibbs, “Excitonic polaritons in Fibonacci quasicrystals,” Opt. Express 16, 15382–15387 (2008). [CrossRef] [PubMed]

], the study of the optical properties of the aperiodic lattices was extended to resonant systems based on multiple quantum well (MQW) structures with two different inter-well distances satisfying the Fibonacci-chain rule with the golden ratio between the long and short inter-well distances. We focused on the linear light propagation in such a medium around the frequency ω 0 corresponding to the excitonic resonance of a quantum well (QW). The widths of the inter-well barriers were determined from the resonant Bragg condition [9

9. A. N. Poddubny, L. Pilozzi, M. M. Voronov, and E. L. Ivchenko, “Resonant Fibonacci quantum well structures in one dimension,” Phys. Rev. B 77, 113306 (2008). [CrossRef]

], specifying the constructive interference of the waves reflected from the MQWs at the excitonic resonance. We chose to study the Fibonacci sequence because it is the most well-known example of 1D quasiperiodic structures; its distinctive features are a dense quasi-continuous pure point Fourier spectrum and a direct connection with the 2D and 3D quasicrystals, the Penrose lattices.

Since the inter-well barriers are of the order of half of the wavelength of the exciting light, the QWs are coupled only via the electromagnetic field. This is a strong qualitative difference between the considered system and short-period semiconductor Fibonacci superlattices [11

11. R. Merlin, K. Bajema, R. Clarke, F. Y. Juang, and P. K. Bhattacharya, “Quasiperiodic GaAs-AlAs Heterostruc-tures,” Phys. Rev. Lett. 55, 1768–1770 (1985). [CrossRef] [PubMed]

, 12

12. M. Kohmoto and J. R. Banavar, “Quasiperiodic lattice: Electronic properties, phonon properties, and diffusion,” Phys. Rev. B 34, 563–566 (1986). [CrossRef]

], where the neighboring QWs are coupled by tunneling. When plasmon-polaritons are studied in aperiodic lattices [13

13. E. L. Albuquerque and M. G. Cottam, “Theory of elementary excitations in quasiperiodic structures,” Phys. Rep. 376, 225–337 (2003). [CrossRef]

], they are coupled through an evanescent electromagnetic field [14

14. E. L. Ivchenko, Optical spectroscopy of semiconductor nanostructures (Alpha Science International, Harrow, UK, 2005).

].

In our investigations, the QWs couple resonantly to the propagating light such that the ex-citonic resonance strongly modifies the optical response yielding the so-called excitonic polaritons. In the MQW Fibonacci structures the quasicrystalline long-range order results in an excitonic polariton stopband similar to that of photonic crystals while the lack of periodicity in the quasicrystal results in efficient photoluminescence emission in the direction normal to the layer planes [10

10. J. Hendrickson, B. C. Richards, J. Sweet, G. Khitrova, A. N. Poddubny, E. L. Ivchenko, M. Wegener, and H. M. Gibbs, “Excitonic polaritons in Fibonacci quasicrystals,” Opt. Express 16, 15382–15387 (2008). [CrossRef] [PubMed]

]. Due to these polariton features, our systems fundamentally differ from the light-emitting Thue-Morse structures based on non-resonant dielectrics studied before [8

8. L. D. Negro, M. Stolfi, Y. Yi, J. Michel, X. Duan, L. C. Kimerling, J. LeBlanc, and J. Haavisto, “Photon band gap properties and omnidirectional reflectance in Si/SiO2 Thue-Morse quasicrystals,” Appl. Phys. Lett. 84, 5186–5188 (2004). [CrossRef]

]. In those systems, the light-matter interaction is enhanced due to the formation of the localized states of light.

In this paper, we present a detailed study of the linear and nonlinear optical properties of different MQW Fibonacci structures. The detailed comparison between the theoretical results and the experiment for the molecular-beam-epitaxy-grown samples based on GaAs/AlGaAs Fibonacci MQWs is performed.

The rest of the paper is organized as follows. In Sec. 2, we briefly discuss the general definitions of different one-dimensional quasicrystals and present the resonant Bragg condition. Section 3 is devoted to the description of the experimental setup and theoretical approaches used to study the Fibonacci MQW structures. Experimental and theoretical nonlinear reflectivity of the canonical Fibonacci structures is discussed in Sec. 4. In Sec. 5, we analyze the linear reflection spectra of different noncanonical Fibonacci structures tuned to and slightly detuned from the Bragg resonance. The brief summary of the main results is given in Conclusions.

2. One-dimensional quasicrystals and the structure factor

In this section, we present three equivalent definitions of quasicrystalline chains and find their general diffractive properties. The structure under consideration consists of N semiconductor QWs with their centers positioned at the points z = zm (m = 1…N) arranged in an aperiodic lattice. Different approaches to introduce the concept of a one-dimensional quasicrystal go back to (i) the incommensurate chains, (ii) the substitution rules and (iii) the cut-and-project method.

zm=z0+md¯+r(m),
(1)

where d¯ is the mean period of the lattice, z 0 is an arbitrary shift of the lattice as a whole, and the modulation r(m) is the periodic function

r(m)=Δ{mt+φ},
(2)

where {x} stands for the fractional part of x. Here Δ, t, and φ are the structure parameters, with t being irrational and φ being noninteger. At vanishing Δ, Eqs. (1) and (2) specify a simple periodic lattice with the period d¯ (without loss of generality we assume hereafter that t > 1). In the case of rational t, the structure is still periodic but has a compound supercell, whereas for irrational values of t Eq. (1) leads to a deterministic aperiodic chain termed also as “modulated crystal” [2

2. C. Janot, Quasicrystals. A Primer (Clarendon Press, Oxford, UK, 1994).

]. The parameter Δ describes the modulation strength and the value of φ specifies the initial phase of the function r(m). For zm defined according to Eqs. (1) and (2), the physical spacings z m+1zm take one of the two values,

Sp=d¯+Δ/tandLp=d¯+Δ(1/t1).
(3)

In the following, we generally denote the large (small) QW spacing by L (S). In particular, we introduce the subscript “p” for the physical lengths of the spacings, L p and S p, while the optical pathlengths of these spacers are denoted as L o and S o. The value of Δ should not be too large so that the spacings L p and S p remain positive. Excluding Δ in Eqs. (3), one can find the relation

d¯=Sp+(LpSp)/t.
(4)

NSNL=t1.
(5)

Under certain conditions imposed upon the values t and φ [17

17. Z. Lin, M. Goda, and H. Kubo, “A family of generalized Fibonacci lattices: self-similarity and scaling of the wavefunction,” J. Phys. A 28, 853–866 (1995). [CrossRef]

, 18

18. J. M. Luck, C. Godreche, A. Janner, and T. Janssen, “The nature of the atomic surfaces of quasiperiodic self-similar structures,” J. Phys. A 26, 1951–1999 (1993). [CrossRef]

], the QW arrangement can also be obtained by the substitution rules acting on the segments L and S as follows

Lσ(L)=M1M2Mα+β,
Sσ(S)=N1N2Nγ+δ.
(6)

Each of the symbols Mk and Nk in the right-hand side of Eq. (6) stands for L or S, α and β denote the numbers of letters L and S in the sequence σ(L), and γ and δ are the numbers of L and S in σ(S), respectively [19

19. X. Fu, Y. Liu, P. Zhou, and W. Sritrakool, “Perfect self-similarity of energy spectra and gap-labeling properties in one-dimensional Fibonacci-class quasilattices,” Phys. Rev. B 55, 2882–2889 (1997). [CrossRef]

]. The correspondence between the two definitions is established by the relation t = 1 + (λ 1α)/γ between a value of t and indices α, β, γ, δ, where λ1=(v+v2+4w)/2,v=α+δ and w = βγαδ. For the quasicrystals, w must be equal to ±1 [20

20. M. Kolář, “New class of one-dimensional quasicrystals,” Phys. Rev. B 47, 5489–5492 (1993). [CrossRef]

].

The structure described by Eqs. (1) and (2) can be equivalently defined by the cut-and-project method based on a projection from the two-dimensional space upon a straight line [21

21. M. C. Valsakumar and V. Kumar, “Diffraction from a quasi-crystalline chain,” Pramana 26, 215–221 (1986). [CrossRef]

]. Sizes of the unit cell of the two-dimensional lattice (rectangular or oblique) are determined by the ratio L p/S p of spacings Eq. (3). However, the order of the segments L and S is determined only by t and φ and can be obtained by the projection of the square lattice [22

22. Z. Lin, H. Kubo, and M. Goda, “Self-similarity and scaling of wave function for binary quasiperiodic chains associated with quadratic irrationals,” Z. Phys. B: Condensed Matter 98, 111–118 (1995). [CrossRef]

].

The optical properties of the chain, Eqs. (1) and (2), are described by its structure factor f(q)

f(q)=limNf(q,N),f(q,N)=1Nm=1Ne2iqzm.
(7)

In the limit N → ∞, the structure factor of a quasicrystal [23

23. D. Levine and P. J. Steinhardt, “Quasicrystals. I. Definition and structure,” Phys. Rev. B 34, 596–616 (1986). [CrossRef]

] consists of δ-peaks corresponding to the Bragg diffraction and characterized by two integer numbers h and h′, see [2

2. C. Janot, Quasicrystals. A Primer (Clarendon Press, Oxford, UK, 1994).

, 18

18. J. M. Luck, C. Godreche, A. Janner, and T. Janssen, “The nature of the atomic surfaces of quasiperiodic self-similar structures,” J. Phys. A 26, 1951–1999 (1993). [CrossRef]

],

f(q)=h,h=δ2q,Ghh,fhh,
(8)

with the diffraction vectors

Ghh′=2πd¯(h+h′t),
(9)

filling the wavevector axis in a dense quasicontinuous way. We point out that in the periodic lattice (Δ = 0), the structure factor has non-zero peaks at the single-integer diffraction vectors Gh = 2πh/d¯ with ∣fh∣ = 1. One can show that, for irrational values of t and Δ ≠ 0, the structure-factor coefficients are given by

fhh′=sinShh′Shh′eiθhh′,θhh′=(z0+Δ{φ})Ghh′+Shh′,
Shh′=πΔhd¯+πh′(1+Δtd¯)=πh′+Δ2Ghh′.
(10)

The above equations for the structure factor are valid for arbitrary values of the phase φ in Eq. (2) and are original to the best of our knowledge. For the specific case when φ = 0, they can be obtained by a straightforward transformation of the result presented in [21

21. M. C. Valsakumar and V. Kumar, “Diffraction from a quasi-crystalline chain,” Pramana 26, 215–221 (1986). [CrossRef]

]. Although the absolute value of the structure factor is independent of φ, the order of segments L and S between the QWs determined by Eq. (1) does depend on this phase. On the other hand, Eq. (10) generalizes the results of [23

23. D. Levine and P. J. Steinhardt, “Quasicrystals. I. Definition and structure,” Phys. Rev. B 34, 596–616 (1986). [CrossRef]

],derived for the Fibonacci lattice shifted by the phase φ < 1, to the quasicrystal with arbitrary d¯, Δ and t.

The one-dimensional Fibonacci lattice, being one of the most studied quasicrystals, is determined by the substitution rule [2

2. C. Janot, Quasicrystals. A Primer (Clarendon Press, Oxford, UK, 1994).

]

LLS,SL.
(11)

The parameters of this structure are given by

t=τ(5+1)/21.618,
Δ=SpLp,φ=0,d=Sp+(LpSp)/τ.
(12)

If the center of the first QW is chosen at the plane z = 0 then z 0 = −S p. Here, the ratio L p/S p is arbitrary, so that for L p = S p the structure becomes periodic [24

24. E. L. Ivchenko, A. I. Nesvizhskii, and S. Jorda, “Bragg reflection of light from quantum-well structures,” Phys. Solid State 36, 1156–1161 (1994).

] and, for L p/S p equal to the golden mean τ, it becomes the canonical Fibonacci chain [9

9. A. N. Poddubny, L. Pilozzi, M. M. Voronov, and E. L. Ivchenko, “Resonant Fibonacci quantum well structures in one dimension,” Phys. Rev. B 77, 113306 (2008). [CrossRef]

]. In the noncanonical Fibonacci structures, this ratio is different from 1 and τ.

The resonant Bragg condition [9

9. A. N. Poddubny, L. Pilozzi, M. M. Voronov, and E. L. Ivchenko, “Resonant Fibonacci quantum well structures in one dimension,” Phys. Rev. B 77, 113306 (2008). [CrossRef]

] is obtained by equalizing the double light wavevector at the exciton resonance frequency ω 0 to one of the diffraction vectors Ghh′ with a large value of the coefficient ∣fhh′∣. This condition is equivalent to

q(ω0)d¯=π(h+h′τ),
(13)

where q(ω 0) is the light wavevector at the excitonic resonance frequency.

It is important to notice that the Bragg condition Eq. (13) is written for the structure without the nontrivial QW dispersion and without the structured dielectric QW environment that will modify the Bragg condition, in analogy with the case of the periodic QWs [25

25. E. L. Ivchenko, M. M. Voronov, M. V. Erementchouk, L. I. Deych, and A. A. Lisyansky, “Multiple-quantum-well-based photonic crystals with simple and compound elementary supercells,” Phys. Rev. B 70, 195106 (2004). [CrossRef]

]. In realistic samples, several different layers are grown in between the active QW material layers such as, e.g., barrier layers, adjuster layers, and spacer layers. According to the different refractive indices of the respective materials, it is impossible to present a rigorous analytical generalization of Eq. (13), taking into account the effects of various layers and the QW dispersion. Instead, we use the concept of optical pathlength of layers defined as the products of the physical layer widths and their indices of refraction. Hence, we will always refer to the vacuum wavelength λ and treat the quantity q(ω 0)d¯ in Eq. (13) as the product of the vacuum light wavevector at the exction resonance frequency q(ω 0) = ω 0/c ≡ 2π/λ and the average optical path D. The value of D is determined similar to Eq. (12), so that D = S o + (L oS o)/τ. Now, S o and L o are determined as the optical path lengths from the center of one QW to the center of the next-neighboring one. In Sec. 5, we will demonstrate that the exact Bragg condition can be slightly different from that given by Eq. (13) due to the effects of the QW dispersion.

We stress that the set of the diffraction vectors is independent of the ratio ρ = L o/S o, see Eq. (9), and is the same both for canonical and noncanonical Fibonacci MQWs with the equal mean period D. In the case when ρ = 1, Eq. (13) reproduces the well-known Bragg condition for periodic structures. For the periodic resonant Bragg spacings, there is a series of spacer thicknesses which fulfill the Bragg condition, namely integer multiples of half the resonance wavelength [24

24. E. L. Ivchenko, A. I. Nesvizhskii, and S. Jorda, “Bragg reflection of light from quantum-well structures,” Phys. Solid State 36, 1156–1161 (1994).

, 26

26. M. Hübner, J. Kuhl, T. Stroucken, A. Knorr, S. W. Koch, R. Hey, and K. Ploog, “Collective Effects of Excitons in Multiple-Quantum-Well Bragg and Anti-Bragg Structures,” Phys. Rev. Lett. 76, 4199–4202 (1996). [CrossRef] [PubMed]

]. For the canonical Fibonacci structures, the f(q) resonances, see Eq. (8), have largest values of ∣fhh′∣ ≈ 1, which corresponds to h and h′ equal to the subsequent Fibonacci numbers: (h,h′) = (Fj,Fj-1). For the noncanonical structures, the coefficients ∣fhh′∣ are maximal when the ratio of h′/h is close to L o/S o = 1, as one can see from the analysis of Eqs. (10) and (12). For comparison, the first three sets of spacers are shown in Table 1 for the resonant periodic Bragg spacing (ρ = 1) and for the resonant canonical Fibonacci spacing (ρ = τ).

Table 1. Comparison of spacers for periodic Bragg spacing (ρ = 1) and canonical Fibonacci spacing (ρ = τ) for the first three Bragg resonances, j = 1,2,3.

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3. Experimental and theoretical approaches

The samples FIB10, containing a single GaAs/AlGaAs QW, and FIB13, containing 54 Fibonacci-spaced GaAs/AlGaAs QWs, were grown by MBE on a (001) GaAs substrate [10

10. J. Hendrickson, B. C. Richards, J. Sweet, G. Khitrova, A. N. Poddubny, E. L. Ivchenko, M. Wegener, and H. M. Gibbs, “Excitonic polaritons in Fibonacci quasicrystals,” Opt. Express 16, 15382–15387 (2008). [CrossRef] [PubMed]

]. The sample FIB13 corresponds to the canonical Fibonacci spacing with ρ = τ and j = 1, see Table 1. A schematic of the experimental apparatus is shown in Fig. 1. What is referred to as Fibonacci-spaced sequence of QWs can have any number N of QWs. Computations show that nothing special occurs for N equal to a Fibonacci number; i.e., all measurable quantities vary slowly as N passes through a Fibonacci number. FIB13 with N = 54 is, of course, one short of a Fibonacci number of QWs. The data are taken in a single-beam reflection geometry using the 100fs output pulse from an 80-MHz modelocked Ti:sapphire laser. The sample is mounted in a liquid-helium cryostat and maintained close to 4K. The use of a polarization-independent beam splitter sends the entire reflected signal to a spectrometer and CCD camera. The illuminated spot on the sample is approximately 7 microns in diameter. Typical integration times for the data presented below are 0.1 second. Even though a single beam is employed, when the pulse spectrum is centered above the continuum band-edge the experiment is equivalent to an above-band pump and resonant-probe experiment, as shown below.

Fig. 1. Experimental setup

To explain the experimental observations with a microscopic theory, we solve the self-consistent coupling between the macroscopic QW polarization P and the wave equation. More specifically, we evaluate the linear optical response from the semiconductor Bloch equations [27

27. M. Lindberg and S.W. Koch, “Effective Bloch Equations for Semiconductors,” Phys. Rev. B 38, 3342 (1988). [CrossRef]

, 28

28. H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors (fifth ed., World Scientific Publishing, Singapore, 2009).

] in steady state with constant carrier densities. The approach fully describes the microscopic polarization P k∣∣, for all relevant carrier momenta k ∣∣, and its dynamics influenced by the phase-space filling, the Coulomb renormalizations to the Rabi as well as to the single-particle energies, and the Coulomb-induced two-particle correlations. These correlations are systematically treated with the so-called cluster-expansion approach up to the two-particle scattering level [29

29. M. Kira and S. W. Koch, ”Many-body correlations and excitonic effects in semiconductor spectroscopy,” Prog. Quantum Elec. 30, 155–296 (2006). [CrossRef]

]. As a result, our analysis also includes a microscopic description of screening effects as well as the Coulomb-induced scattering of polarization that yields excitation-induced dephasing and energy renormalizations to the excitonic resonances.

The semiconductor Bloch equations produce the linear QW response, i.e., the QW susceptibility

χ(ω)=P(ω)ε0E(ω),P(ω)=dcvҮkPk(ω),
(14)

where E(ω) is the Fourier transform of the field that excites the QW. The macroscopic polarization is a sum over P k∣∣, scaled by the dipole-matrix element dcv and the quantization area Ү. In general, χ(ω) is defined completely by the internal properties of the QW, i.e., χ(ω) is the same for each QW and it is not influenced by other QWs or by the radiative environment.

The self-consistent coupling between the QWs and the light follows after we solve the semiconductor Bloch equations together with the Maxwell’s equations. Thus, one needs to propagate light through the experimental structure where each dielectric layer and each QW is accounted for. For the linear response, this can be performed using the so-called transfer-matrix approach [30

30. E. Merzbacher, Quantum Mechanics (first ed., Wiley, New York, 1961).

, 31

31. M. Kira, F. Jahnke, W. Hoyer, and S. W. Koch, ”Quantum theory of spontaneous emission and coherent effects in semiconductor microstructures”, Prog. Quantum Elec. 23, 1891961).279 (1999). [CrossRef]

] where the optical response of each layer is included via its refractive index while each QW is described through χ(ω). As a result of the self-consistent coupling, also the radiative dephasing of the QW polarization is described microscopically. Altogether, the SQW sample contains ten and the 54 QW sample contains 275 dielectric layers which are all included in the analysis; see Table 2 for the layer thicknesses and refractive indices used in the transfer matrix computations.

Table 2. Layer widths and refractive indices used in the analysis of the samples. The sample FIB10 contains only “cap” and “buffer and substrate” layers while the sample FIB13 contains additional “large spacer” and “small spacer” layers, producing a Fibonacci-spaced chain of QWs. The layers in both samples can be categorized into seven types. The refractive indices of only four layer types (indicated by a *) out of these seven were changed to match theory with experiment. The barriers are Al0.3Ga0.7As, the spacers are Al0.04Ga0.96As, the QWs and adjuster are GaAs, and the AlAs/GaAs SL is a six times repetition of 2 nm AlAs followed by 2 nm GaAs.

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Besides the excitation induced dephasing effects, the excitonic resonances in the experimen-tally realized samples are additionally broadened via excitation-level-independent disorder as well as phonon scattering. These are modeled by adding a frequency-dependent dephasing

γbg(ω)=γbgexp((h¯ωEx+ΔEcut)/C)+1
(15)

that enters as the background dephasing constant to the semiconductor Bloch equations. The appearing constants are matched by comparing the full computation with the single-QW experiment. Figure 2(a) shows that the experimental (shaded area) and the theory (solid line) reflection spectra agree when we use γ bg = 0.163meV (0.110meV) for the heavy-hole (light-hole) 1s resonance, positioned at Ex. The cut-off energy is set to ΔE cut = 0.25meV and the constant is chosen to be C = 0. 1meV. In addition, the heavy-hole and the light-hole dipole matrix elements are defined from a separate bandstructure calculation giving dvc = 0.78e nm and dvc = 0.5e nm, respectively. The refractive indices are fitted using a single small set of parameters such that the calculations match the measured linear SQW and 54QW reflectance, respectively.

To check the quality of the parameter choices, we use exactly the same material inputs to compute the optical properties of the complicated sample. Figure 2(b) presents the calculated (solid line) and the measured (shaded area) reflection for the 54 Fibonacci-spaced QWs. Due to the Fibonacci spacing, the spectrum displays multiple features which all are very well reproduced by the theory. The best match between the experiment and theory is found using an average QW spacing of D 0 = 0.5016λ and a ratio of large spacing to small spacing of ρ 0 = 1.643. In our further analysis, we are particularly interested in the deep dip below the hh 1s resonance (vertical dashed line).

Fig. 2. Linear fits (blue line) to measured data (shaded area) for (a) a single QW (FIB10) and (b) for 54 Fibonacci-spaced QWs (FIB13) using a frequency-dependent dephasing γ(ω) and identical fit parameters. For both samples, theory and experiment agree excellently.

4. Nonlinear reflectivity of canonical Fibonacci quantum wells

Based on the parameter assignment of the linear evaluations, we proceed to analyze nonlinear experiments. Assuming 40K carrier distributions in thermodynamic quasi-equilibrium, we obtain the experiment-theory comparison shown in Fig. 3. For densities n = 1 × 109cm-2 (shaded area), n = 5 × 109 cm-2 (red line), n = 2 × 1010 cm-2 (blue line), and n = 5 × 1010 cm-2 (black line), the computed nonlinear reflectance R(ω) of the 54QW sample is shown (a) which is in very good agreement with the nonlinear experimental results (b) obtained with pump powers of 76.6μW (shaded area), 871μW (red line), 3.7mW (blue line), and 11.1mW (black line). The real part (c) and the imaginary part (d) of the computed QW susceptibility are shown for the different densities. The excitation-induced dephasing leads to a bleaching and broadening of the susceptibility with increasing carrier density.

Fig. 3. Very good agreement between (a) computed and (b) experimental nonlinear reflectance is obtained for FIB13. Theory used densities n = 1 × 109 cm-2 (shaded area), n = 5 × 109 cm-2 (red line), n = 2 × 1010 cm-2 (blue line), and n = 5 × 1010 cm-2 (black line) while in the experiment the pump power was P = 76.6μW (shaded area),P = 871μW (red line), P = 3.7mW (blue line), P = 11.1mW (black line). An average spacing of D 0 = 0.5016λ is deduced from the fit parameters. The real part (c) and the imaginary part of the corresponding computed susceptibilities (d) show broadening and bleaching with elevated carrier densities.

In particular, the pronounced dip below the hh 1s resonance (vertical line) gradually disappears from R(ω) (frames (a) and (b)) for elevated excitations. We also see that the reflection stopband becomes smeared out for the largest excitations. The corresponding computed nonlinear true absorption probabilities A (ω) are plotted in Fig. 4(a) (solid lines) together with the experimentally applied pulses (shaded areas, scaled). The actual true absorption, Fig. 4(b), is gained by multiplication of the pulse spectrum with the respective absorption probability. The resonant excitation as well as the above-band excitation shows considerable absorption in a wide spectral range. Accordingly, the nonlinear reflectance obtained with on-resonance or with above-resonance excitation look very much the same, compare Fig. 3(b) with Fig. 4(c), which explains the match of experimental resonant-pump data and pump-probe-like calculated spectra, c.f. Fig. 3(a) and (b).

Why is there such good agreement between experimental data and the theoretical computations? The analysis of many pulsed nonlinear experiments of quantum wells in the past have led to the conclusion that near band edge absorption of a 100 fs pulse results in carriers that can be quite adequately described by an equilibrium carrier distribution with temperature of 40-50K with negligible cooling or redistribution occurring within 100 fs. The nonresonant excitation can thus be characterized by a single parameter (the carrier density); it does not introduce any coherent polarizations in the quantum well. The density-dependent quantum-well nonlinearities are computed fully microscopically [27

27. M. Lindberg and S.W. Koch, “Effective Bloch Equations for Semiconductors,” Phys. Rev. B 38, 3342 (1988). [CrossRef]

, 28

28. H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors (fifth ed., World Scientific Publishing, Singapore, 2009).

, 29

29. M. Kira and S. W. Koch, ”Many-body correlations and excitonic effects in semiconductor spectroscopy,” Prog. Quantum Elec. 30, 155–296 (2006). [CrossRef]

]. Using these individual quantum-well results, the reflection or transmission of a resonant probe incident normal to the multiple-quantum-well structure is then computed by the well known transfer matrix technique giving the effects of propagation through the complete structure. As the carrier density is increased the dominant carrier dependent nonlinearity is the so called excitation dependent dephasing, basically the increased relaxation of the probe-induced polarization due to carrier collisions. As a result any narrow spectral features, such as the sharp dip here, are broadened and disappear as the carrier density is increased. In the experiment just described with the peak of the pulse centered in the quantum well continuum, carriers are generated incoherently; their effect on narrow spectral features is monitored by the weak resonant part of the pulse, corresponding to the probe of the theory. Since the entire reflected beam is detected, there is an averaging over carrier densities; this is not much of a problem either since the broadening changes relatively slowly with density. Clearly the qualitative behavior with increased excitation power is reproduced by the theory, and a detailed comparison of exact carrier densities seems unwarranted.

Fig. 4. The spectra of the experimentally applied pulses (shaded areas) are shown together with the computed absorption probability in frame (a) for low (blue line) and high (red line) excitation. The corresponding true absorption (b) is plotted for on-resonance and above-resonance excitation, showing that in both cases most of the absorption generates free carriers. Nonlinear reflectances obtained by above-band pumps are shown in frame (c). The spectra look very similar to those obtained by pumping resonantly.

5. Linear reflectivity: origin of sharp dip and sensitivity to D and ρ

Fig. 5. Calculated reflectance spectra using the full QW susceptibility (shaded area) and with the real part (blue line), imaginary part (red line), or total susceptibility (black line) set to zero. Fine structures are introduced by the real part of the susceptibility while the imaginary part makes them partially vanish again.

A sharp spectral feature is a possible candidate for a high-speed optical switch because the reflectivity could be changed from a low to a high value in a very short time by shifting the entire spectrum, for example by the optical Stark effect [32

32. D. Hulin, A. Mysyrowicz, A. Antonetti, A. Migus, W. T. Masselink, H. Morkoc, H. M. Gibbs, and N. Peygham-barian, “An ultrafast all optical gate with subpicosecond on and off response time,” Appl. Phys. Lett. 49, 7491961).751 (1986). [CrossRef]

]. It also has potential applications to slow light as explored for the interference fringes that occur within the spectral stopband of a very large number of slightly detuned excitonic Bragg periodic quantum wells [33

33. Z. S. Yang, N. H. Kwong, R. Binder, and A. L. Smirl, “Stopping, storing, and releasing light in quantum-well Bragg structures,” J. Opt. Soc. Am. B 22, 21441961).2156 (2005). [CrossRef]

,34

34. J. P. Prineas, W. J. Johnston, M. Yildirim, J. Zhao, and A. L. Smirl, “Tunable slow light in Bragg-spaced quantum wells,” Appl. Phys. Lett. 89, 241106 (2006). [CrossRef]

]. Similar slow light studies have also been made in a waveguide with periodic side coupling to resonators [35

35. M. F. Yanik, W. Suh, Z. Wang, and S. Fan, “Stopping light in a waveguide with an all-optical analog of electro-magnetically induced transparency,” Phys. Rev. Lett. 93, 233903 (2004). [CrossRef] [PubMed]

, 36

36. P. Chak, S. Pereira, and J. E. Sipe, “Coupled-mode theory for periodic side-coupled microcavity and photonic crystal structures,” Phys. Rev. B 73, 035105 (2006). [CrossRef]

].

To address the influence of the Fibonacci spacing on the spectral features, we vary either the average spacing D while the ratio of the two spacers, ρ = L o/S o, is kept constant, or we vary ρ while D is kept constant. The dependence of the spectrum on a variation of D or ρ is investigated in Fig. 6 for 54 QWs using the lowest density n = 1 × 109cm-2. To quantify the deviation between the computed reflection spectrum R(ω) and the original Fibonacci R 0((ω) in Fig. 3(a), we evaluate

ε=ω1,s,hhΔω1,s,hh+ΔR(ω)R0(ω)ω1,s,hhΔω1,s,hh+ΔR0(ω).
(16)

where h¯ω 1s,hh = 1523.4meV and h¯Δ = 1.5meV. The computed ε is shown in Fig. 6(a) as a function of D when ρ is fixed to be ρ 0 = 1.643. We observe that the spectrum is very sensitive to the average spacing because varying D within ± 1% already results in 25% changes. The strong dependence of the spectral shape on the average spacing can easily be understood based on the generalized Bragg condition, Eq. (13). Since the resonant Bragg condition defines a corresponding average spacing, it is obvious that the spectrum should have this strong dependence on D. In particular, the deep dip is present in the spectra within a narrow range from D = 0.500λ (D = 0.501λ) up to D = 0.503λ (D = 0.503λ) for the Fibonacci (periodic) spacing. We also have investigated the e deviation when the ratio of the spacer widths is changed while the average spacing D 0 = 0.5016λ is kept constant. Figure 6(b) shows that ε changes only very little as a function of ρ. In particular, we have changed ρ/ρ 0 almost an order of magnitude more than D/D 0 in Fig. 6(a) and get an ε deviation of only few percents, in contrast to the strong D dependence. Thus the Fibonacci features present a certain robustness of the spectrum to variations in ρ.

The effect of stronger variations of ρ can be seen in Fig. 7(a) which shows the reflectance of a sweep of ρ for the first Bragg resonance, i.e. j = 1 with (h,h′) = (F 1,F 0) = (1,0). The ratio ρ = L o/S o is tuned from the well-known [14

14. E. L. Ivchenko, Optical spectroscopy of semiconductor nanostructures (Alpha Science International, Harrow, UK, 2005).

] periodic case, ρ = 1, towards the canonical Fibonacci spacing, ρ = τ, as well as to even larger values of ρ while the average spacing D 0 = 0.5016λ is kept constant. The reflection maximum is observed for the periodic case, ρ = 1, which is due to the best constructive interference, D 0 = S o = L o. Accordingly, the reflection gets lower and lower the further the large and small spacers differ from each other. This is in agreement with the behavior of the corresponding structure factor f(q), presented in Fig. 8. The strongest (narrowest and deepest) dip is found for ρ = τ.

Fig. 6. Deviations of 54QW reflectance spectra from the fit spectrum (D 0 = 0.5016λ and ρ 0 = 1.643) are shown in dependence of (a) the average distance D and (b) the ratio of layer thicknesses ρ= L o/S o. A strong dependence on D is found while a certain robustnest against changes in ρ is observed.
Fig. 7. For (a) the first Bragg resonance with (h,h′) = (F 1 ,F 0) = (1,0) as well as an average spacing D 0 =0.5016λ, and (b) the second Bragg resonance with (h,h′) = (F 2,F 1) = (1,1) as well as D = 0.8115λ, the reflectance is shown for different ratios L/S, while the respective average spacing is kept constant.
Fig. 8. Structure factors for the Bragg resonances, (h,h′) = (F 1 ,F 0) = (1,0) plotted in red and (h,h′) = (F 2,F 1) = (1,1) plotted in blue.

Fig. 9. Reflectance for 54QWs: (a) computations with parameters according to FIB13 (shaded area), FIB13 with ARC (red line), constant refractive index everywhere but same optical lengths as FIB13 (blue line), and optical-length periodic spacing (black line) with same average spacing D = 0.5016λ as FIB13. Obtaining the dip in all spectra suggests the dip to be caused by the uniform average spacing. (b) For D = 0.4992λ, there is no dip in the spectrum of periodically spaced 54QWs (black line) while there is one for Fibonacci-spaced QWs (area).

As a last point, we investigate how the QW number influences the dip and the resonance structures. Otherwise, we use the parameters corresponding to the sample FIB13 in our computations and present R(ω) as a function of the QW number in Fig. 10. We observe that the spectrum yields only a peak for small QW numbers. With increasing QW number, the dip emerges at an energetic position slightly below the 1s heavy-hole resonance (vertical line). With further increasing QW number, this peak shifts to lower energies and gets broader while additional dips emerge from the wiggles which one can observe near the hh resonance. These additional dips behave analogously to the first dip. This behavior is found for all Fibonacci-spaced sample types treated in Fig. 9(a) such that it has to be attributed to the QW spacing as well.

Fig. 10. Reflectance close to the 1s-hh-resonance position as a function of QW number and energy for parameters according to FIB13.

6. Conclusion

In conclusion, we have presented reflectance measurements on one-dimensional quasicrystals realized in the form of Fibonacci-spaced QWs and applied our microscopic theory to reproduce and understand these spectra. The linear spectra exhibit a pronounced dip in the center of the Bragg-resonance reflectivity stopband. The analysis shows this dip to be a consequence of the real part of the excitonic resonance susceptibility (index effects); it is not caused by interference effects due to the dielectric environment of the QWs. For elevated carrier densities, the dip bleaches due to excitation-induced dephasing. Moreover, the dip is very sensitive to the average QW spacing, D, and shows a certain robustness for variations of the ratio of the spacer widths around the golden mean while it disappears if that ratio is changed too much. With increasing QW number, the dip gets broader and shifts to lower energies. At the same time, additional dips with similar behavior emerge close to the Bragg resonance position.

Acknowledgments

The Marburg group acknowledges support from the Deutsche Forschungsgemeinschaft and AFOSR grant FA9550-07-1-0010 sponsoring the visits of SWK in Tucson/AZ. The Tucson group thanks AFOSR, NSF AMOP and EPDT, NSF ERC CIAN, and JOSP for support. The St. Petersburg work was supported by RFBR and the “Dynasty” Foundation – ICFPM. M. Wegener acknowledges financial support provided by the Deutsche Forschungsgemeinschaft (DFG) and the State of Baden-Württemberg through the DFG-Center for Functional Nanostructures (CFN) within subproject A1.4.

References and links

1.

D. Levine and P. J. Steinhardt, “Quasicrystals: A New Class of Ordered Structures,” Phys. Rev. Lett. 53, 2477–2480 (1984). [CrossRef]

2.

C. Janot, Quasicrystals. A Primer (Clarendon Press, Oxford, UK, 1994).

3.

A. Ledermann, L. Cademartiri, M. Hermatschweiler, C. Toninelli, G. A. Ozin, D. S. Wiersma, M. Wegener, and G. von Freymann, “Three-dimensional silicon inverse photonic quasicrystals for infrared wavelengths,” Nature Mater. 5, 942–945 (2006). [CrossRef]

4.

T. Matsui, A. Agrawal, A. Nahata, and Z. V. Vardeny, “Transmission resonances through aperiodic arrays of subwavelength apertures,” Nature 446, 517–521 (2007). [CrossRef] [PubMed]

5.

E. Yablonovitch, “Inhibited Spontaneous Emission in Solid-State Physics and Electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987). [CrossRef] [PubMed]

6.

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987). [CrossRef] [PubMed]

7.

M. Kohmoto, B. Sutherland, and K. Iguchi, “Localization of optics: Quasiperiodic media,” Phys. Rev. Lett. 58, 2436–2438 (1987). [CrossRef] [PubMed]

8.

L. D. Negro, M. Stolfi, Y. Yi, J. Michel, X. Duan, L. C. Kimerling, J. LeBlanc, and J. Haavisto, “Photon band gap properties and omnidirectional reflectance in Si/SiO2 Thue-Morse quasicrystals,” Appl. Phys. Lett. 84, 5186–5188 (2004). [CrossRef]

9.

A. N. Poddubny, L. Pilozzi, M. M. Voronov, and E. L. Ivchenko, “Resonant Fibonacci quantum well structures in one dimension,” Phys. Rev. B 77, 113306 (2008). [CrossRef]

10.

J. Hendrickson, B. C. Richards, J. Sweet, G. Khitrova, A. N. Poddubny, E. L. Ivchenko, M. Wegener, and H. M. Gibbs, “Excitonic polaritons in Fibonacci quasicrystals,” Opt. Express 16, 15382–15387 (2008). [CrossRef] [PubMed]

11.

R. Merlin, K. Bajema, R. Clarke, F. Y. Juang, and P. K. Bhattacharya, “Quasiperiodic GaAs-AlAs Heterostruc-tures,” Phys. Rev. Lett. 55, 1768–1770 (1985). [CrossRef] [PubMed]

12.

M. Kohmoto and J. R. Banavar, “Quasiperiodic lattice: Electronic properties, phonon properties, and diffusion,” Phys. Rev. B 34, 563–566 (1986). [CrossRef]

13.

E. L. Albuquerque and M. G. Cottam, “Theory of elementary excitations in quasiperiodic structures,” Phys. Rep. 376, 225–337 (2003). [CrossRef]

14.

E. L. Ivchenko, Optical spectroscopy of semiconductor nanostructures (Alpha Science International, Harrow, UK, 2005).

15.

M. Y. Azbel, “Energy spectrum of a conduction electron in a magnetic field,” Sov. Phys. JETP 19, 634–645 (1964).

16.

M. Y. Azbel, “Quantum Particle in One-Dimensional Potentials with Incommensurate Periods,” Phys. Rev. Lett. 43, 1954–1957 (1979). [CrossRef]

17.

Z. Lin, M. Goda, and H. Kubo, “A family of generalized Fibonacci lattices: self-similarity and scaling of the wavefunction,” J. Phys. A 28, 853–866 (1995). [CrossRef]

18.

J. M. Luck, C. Godreche, A. Janner, and T. Janssen, “The nature of the atomic surfaces of quasiperiodic self-similar structures,” J. Phys. A 26, 1951–1999 (1993). [CrossRef]

19.

X. Fu, Y. Liu, P. Zhou, and W. Sritrakool, “Perfect self-similarity of energy spectra and gap-labeling properties in one-dimensional Fibonacci-class quasilattices,” Phys. Rev. B 55, 2882–2889 (1997). [CrossRef]

20.

M. Kolář, “New class of one-dimensional quasicrystals,” Phys. Rev. B 47, 5489–5492 (1993). [CrossRef]

21.

M. C. Valsakumar and V. Kumar, “Diffraction from a quasi-crystalline chain,” Pramana 26, 215–221 (1986). [CrossRef]

22.

Z. Lin, H. Kubo, and M. Goda, “Self-similarity and scaling of wave function for binary quasiperiodic chains associated with quadratic irrationals,” Z. Phys. B: Condensed Matter 98, 111–118 (1995). [CrossRef]

23.

D. Levine and P. J. Steinhardt, “Quasicrystals. I. Definition and structure,” Phys. Rev. B 34, 596–616 (1986). [CrossRef]

24.

E. L. Ivchenko, A. I. Nesvizhskii, and S. Jorda, “Bragg reflection of light from quantum-well structures,” Phys. Solid State 36, 1156–1161 (1994).

25.

E. L. Ivchenko, M. M. Voronov, M. V. Erementchouk, L. I. Deych, and A. A. Lisyansky, “Multiple-quantum-well-based photonic crystals with simple and compound elementary supercells,” Phys. Rev. B 70, 195106 (2004). [CrossRef]

26.

M. Hübner, J. Kuhl, T. Stroucken, A. Knorr, S. W. Koch, R. Hey, and K. Ploog, “Collective Effects of Excitons in Multiple-Quantum-Well Bragg and Anti-Bragg Structures,” Phys. Rev. Lett. 76, 4199–4202 (1996). [CrossRef] [PubMed]

27.

M. Lindberg and S.W. Koch, “Effective Bloch Equations for Semiconductors,” Phys. Rev. B 38, 3342 (1988). [CrossRef]

28.

H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors (fifth ed., World Scientific Publishing, Singapore, 2009).

29.

M. Kira and S. W. Koch, ”Many-body correlations and excitonic effects in semiconductor spectroscopy,” Prog. Quantum Elec. 30, 155–296 (2006). [CrossRef]

30.

E. Merzbacher, Quantum Mechanics (first ed., Wiley, New York, 1961).

31.

M. Kira, F. Jahnke, W. Hoyer, and S. W. Koch, ”Quantum theory of spontaneous emission and coherent effects in semiconductor microstructures”, Prog. Quantum Elec. 23, 1891961).279 (1999). [CrossRef]

32.

D. Hulin, A. Mysyrowicz, A. Antonetti, A. Migus, W. T. Masselink, H. Morkoc, H. M. Gibbs, and N. Peygham-barian, “An ultrafast all optical gate with subpicosecond on and off response time,” Appl. Phys. Lett. 49, 7491961).751 (1986). [CrossRef]

33.

Z. S. Yang, N. H. Kwong, R. Binder, and A. L. Smirl, “Stopping, storing, and releasing light in quantum-well Bragg structures,” J. Opt. Soc. Am. B 22, 21441961).2156 (2005). [CrossRef]

34.

J. P. Prineas, W. J. Johnston, M. Yildirim, J. Zhao, and A. L. Smirl, “Tunable slow light in Bragg-spaced quantum wells,” Appl. Phys. Lett. 89, 241106 (2006). [CrossRef]

35.

M. F. Yanik, W. Suh, Z. Wang, and S. Fan, “Stopping light in a waveguide with an all-optical analog of electro-magnetically induced transparency,” Phys. Rev. Lett. 93, 233903 (2004). [CrossRef] [PubMed]

36.

P. Chak, S. Pereira, and J. E. Sipe, “Coupled-mode theory for periodic side-coupled microcavity and photonic crystal structures,” Phys. Rev. B 73, 035105 (2006). [CrossRef]

OCIS Codes
(190.5970) Nonlinear optics : Semiconductor nonlinear optics including MQW
(230.5590) Optical devices : Quantum-well, -wire and -dot devices
(160.5293) Materials : Photonic bandgap materials

ToC Category:
Nonlinear Optics

History
Original Manuscript: February 23, 2009
Revised Manuscript: March 30, 2009
Manuscript Accepted: April 1, 2009
Published: April 9, 2009

Citation
M. Werchner, M. Schafer, M. Kira, S. W. Koch, J. Sweet, J. D. Olitzky, J. Hendrickson, B. C. Richards, G. Khitrova, H. M. Gibbs, A. N. Poddubny, E. L. Ivchenko, M. Voronov, and M. Wegener, "One dimensional resonant Fibonacci quasicrystals: noncanonical linear and canonical nonlinear effects," Opt. Express 17, 6813-6828 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-8-6813


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References

  1. D. Levine and P. J. Steinhardt, "Quasicrystals: A New Class of Ordered Structures," Phys. Rev. Lett. 53, 2477-2480 (1984). [CrossRef]
  2. C. Janot, Quasicrystals, A Primer (Clarendon Press, Oxford, UK, 1994).
  3. A. Ledermann, L. Cademartiri, M. Hermatschweiler, C. Toninelli, G. A. Ozin, D. S. Wiersma, M. Wegener, and G. von Freymann, "Three-dimensional silicon inverse photonic quasicrystals for infrared wavelengths," Nature Mater. 5, 942-945 (2006). [CrossRef]
  4. T. Matsui, A. Agrawal, A. Nahata, and Z. V. Vardeny, "Transmission resonances through aperiodic arrays of subwavelength apertures," Nature 446, 517-521 (2007). [CrossRef] [PubMed]
  5. E. Yablonovitch, "Inhibited Spontaneous Emission in Solid-State Physics and Electronics," Phys. Rev. Lett. 58, 2059-2062 (1987). [CrossRef] [PubMed]
  6. S. John, "Strong localization of photons in certain disordered dielectric superlattices," Phys. Rev. Lett. 58, 2486-2489 (1987). [CrossRef] [PubMed]
  7. M. Kohmoto, B. Sutherland, and K. Iguchi, "Localization of optics: Quasiperiodic media," Phys. Rev. Lett. 58, 2436-2438 (1987). [CrossRef] [PubMed]
  8. L. D. Negro, M. Stolfi, Y. Yi, J. Michel, X. Duan, L. C. Kimerling, J. LeBlanc, and J. Haavisto, "Photon band gap properties and omnidirectional reflectance in Si/SiO2 Thue-Morse quasicrystals," Appl. Phys. Lett. 84, 5186-5188 (2004). [CrossRef]
  9. A. N. Poddubny, L. Pilozzi, M. M. Voronov, and E. L. Ivchenko, "Resonant Fibonacci quantum well structures in one dimension," Phys. Rev. B 77, 113306 (2008). [CrossRef]
  10. J. Hendrickson, B. C. Richards, J. Sweet, G. Khitrova, A. N. Poddubny, E. L. Ivchenko, M. Wegener, and H. M. Gibbs, "Excitonic polaritons in Fibonacci quasicrystals," Opt. Express 16, 15382-15387 (2008). [CrossRef] [PubMed]
  11. R. Merlin, K. Bajema, R. Clarke, F. Y. Juang, and P. K. Bhattacharya, "Quasiperiodic GaAs-AlAs Heterostructures," Phys. Rev. Lett. 55, 1768-1770 (1985). [CrossRef] [PubMed]
  12. M. Kohmoto and J. R. Banavar, "Quasiperiodic lattice: Electronic properties, phonon properties, and diffusion," Phys. Rev. B 34, 563-566 (1986). [CrossRef]
  13. E. L. Albuquerque and M. G. Cottam, "Theory of elementary excitations in quasiperiodic structures," Phys. Rep. 376, 225-337 (2003). [CrossRef]
  14. E. L. Ivchenko, Optical spectroscopy of semiconductor nanostructures (Alpha Science International, Harrow, UK, 2005).
  15. M. Y. Azbel, "Energy spectrum of a conduction electron in a magnetic field," Sov. Phys. JETP 19, 634-645 (1964).
  16. M. Y. Azbel, "Quantum Particle in One-Dimensional Potentials with Incommensurate Periods," Phys. Rev. Lett. 43, 1954-1957 (1979). [CrossRef]
  17. Z. Lin, M. Goda, and H. Kubo, "A family of generalized Fibonacci lattices: self-similarity and scaling of the wavefunction," J. Phys. A 28, 853-866 (1995). [CrossRef]
  18. J. M. Luck, C. Godreche, A. Janner, and T. Janssen, "The nature of the atomic surfaces of quasiperiodic selfsimilar structures," J. Phys. A 26, 1951-1999 (1993). [CrossRef]
  19. X. Fu, Y. Liu, P. Zhou, and W. Sritrakool, "Perfect self-similarity of energy spectra and gap-labeling properties in one-dimensional Fibonacci-class quasilattices," Phys. Rev. B 55, 2882-2889 (1997). [CrossRef]
  20. M. Kol’a?r, "New class of one-dimensional quasicrystals," Phys. Rev. B 47, 5489-5492 (1993). [CrossRef]
  21. M. C. Valsakumar and V. Kumar, "Diffraction from a quasi-crystalline chain," Pramana 26, 215-221 (1986). [CrossRef]
  22. Z. Lin, H. Kubo, and M. Goda, "Self-similarity and scaling of wave function for binary quasiperiodic chains associated with quadratic irrationals," Z. Phys. B: Condens Matter 98, 111-118 (1995). [CrossRef]
  23. D. Levine and P. J. Steinhardt, "Quasicrystals. I. Definition and structure," Phys. Rev. B 34, 596-616 (1986). [CrossRef]
  24. E. L. Ivchenko, A. I. Nesvizhskii, and S. Jorda, "Bragg reflection of light from quantum-well structures," Phys. Solid State 36, 1156-1161 (1994).
  25. E. L. Ivchenko, M. M. Voronov, M. V. Erementchouk, L. I. Deych, and A. A. Lisyansky, "Multiple-quantumwell-based photonic crystals with simple and compound elementary supercells," Phys. Rev. B 70, 195106 (2004). [CrossRef]
  26. M. H¨ubner, J. Kuhl, T. Stroucken, A. Knorr, S. W. Koch, R. Hey, and K. Ploog, "Collective Effects of Excitons in Multiple-Quantum-Well Bragg and Anti-Bragg Structures," Phys. Rev. Lett. 76, 4199-4202 (1996). [CrossRef] [PubMed]
  27. M. Lindberg and S.W. Koch, "Effective Bloch Equations for Semiconductors," Phys. Rev. B 38, 3342 (1988). [CrossRef]
  28. H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors (fifth ed., World Scientific Publishing, Singapore, 2009).
  29. M. Kira and S. W. Koch, "Many-body correlations and excitonic effects in semiconductor spectroscopy," Prog. Quantum Elec. 30, 155-196 (2006). [CrossRef]
  30. E. Merzbacher, Quantum Mechanics, First ed., (Wiley, New York, 1961).
  31. M. Kira, F. Jahnke, W. Hoyer, and S. W. Koch, "Quantum theory of spontaneous emission and coherent effects in semiconductor microstructures," Prog. Quantum Elec. 23, 189-279 (1999). [CrossRef]
  32. D. Hulin, A. Mysyrowicz, A. Antonetti, A. Migus, W. T. Masselink, H. Morkoc, H. M. Gibbs, and N. Peyghambarian, "An ultrafast all optical gate with subpicosecond on and off response time," Appl. Phys. Lett. 49, 749-751 (1986). [CrossRef]
  33. Z. S. Yang, N. H. Kwong, R. Binder, and A. L. Smirl, "Stopping, storing, and releasing light in quantum-well Bragg structures," J. Opt. Soc. Am. B 22, 2144-2156 (2005). [CrossRef]
  34. J. P. Prineas,W. J. Johnston, M. Yildirim, J. Zhao, and A. L. Smirl, "Tunable slow light in Bragg-spaced quantum wells," Appl. Phys. Lett. 89, 241106 (2006). [CrossRef]
  35. M. F. Yanik, W. Suh, Z. Wang, and S. Fan, "Stopping light in a waveguide with an all-optical analog of electromagnetically induced transparency," Phys. Rev. Lett. 93, 233903 (2004). [CrossRef] [PubMed]
  36. P. Chak, S. Pereira, and J. E. Sipe, "Coupled-mode theory for periodic side-coupled microcavity and photonic crystal structures," Phys. Rev. B 73, 035105 (2006). [CrossRef]

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