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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
Quasicrystals form a class of solid materials characterized by an aperiodic but deterministic structural arrangement of elements that is intermediate between the ordered crystal and the amorphous structure [1, 2]. Light propagation in such systems is now extensively studied [3, 4]. Aperiodic deterministic structures can possess rotational symmetry of higher order than traditional photonic crystals [5, 6] and also allow for localized states of the light [7, 8]. Most of the previous studies of photonic quasicrystals were focused on passive structures.
In our works [9, 10], 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], 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, 12], where the neighboring QWs are coupled by tunneling. When plasmon-polaritons are studied in aperiodic lattices [13], they are coupled through an evanescent electromagnetic field [14].
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]. Due to these polariton features, our systems fundamentally differ from the light-emitting Thue-Morse structures based on non-resonant dielectrics studied before [8]. 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 Fibonacci chain LSLLSLSL… is the textbook example of a one-dimensional quasicrys-talline structure sharing the name with the Fibonacci numbers due to the construction rule that the next complete sequence is the present sequence plus the previous sequence, labeling the first sequence as S and the second as L (corresponding below to short and long separations between quantum wells) [2]. Nevertheless, almost all previous studies were focused on the case when the ratio between the widths of long and short segments in the chain was equal to the golden mean. In this work, we bring into consideration the “noncanonical” Fibonacci structures where such a ratio is arbitrary. We present a general equation for the structure factor of the one-dimensional quasicrystalline chains, including “noncanonical” Fibonacci lattices, and formulate the generalized resonant Bragg condition for these systems.
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.
The incommensurate chains and related structures [15, 16] are studied since the 1960s, even before the term “quasicrystal” was introduced in [1]. The coordinates of the QW centers are written in the form
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
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]. 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+1 − zm take one of the two values,
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
Moreover, the ratio N S/N L of numbers of the spacings L and S in an infinite lattice is related with t by
Under certain conditions imposed upon the values t and φ [17, 18], the QW arrangement can also be obtained by the substitution rules acting on the segments L and S as follows
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]. The correspondence between the two definitions is established by the relation t = 1 + (λ 1 − α)/γ between a value of t and indices α, β, γ, δ, where and w = βγ − αδ. For the quasicrystals, w must be equal to ±1 [20].
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]. 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].
The optical properties of the chain, Eqs. (1) and (2), are described by its structure factor f(q)
In the limit N → ∞, the structure factor of a quasicrystal [23] consists of δ-peaks corresponding to the Bragg diffraction and characterized by two integer numbers h and h′, see [2, 18],
with the diffraction vectors
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
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]. 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],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]
The parameters of this structure are given by
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] and, for L p/S p equal to the golden mean τ, it becomes the canonical Fibonacci chain [9]. In the noncanonical Fibonacci structures, this ratio is different from 1 and τ.
The resonant Bragg condition [9] 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
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]. 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 o − S 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, 26]. 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.
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]. 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.
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, 28] 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]. 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
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, 31] 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.
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
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, 28, 29]. 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
Based on the very good agreement of linear and nonlinear theoretical and experimental results, we now use our theory to investigate the origin of different spectral properties in more detail. Figure 5 shows results obtained from a switch-off analysis where spectra calculated with the same QW susceptibilty as used in the previous investigations (shaded area) are compared to identical simulations except that either the real part of the QW susceptibility (blue line), or its imaginary part (red line), or the whole QW susceptibility χ (black line) is set to zero. It can clearly be seen that the main spectral features, such as the narrow deep dip close to the hh-resonance position (vertical line) as well as the valley between the hh- and the lh- resonance position, result from Re[χ]. Thus, they stem from cavity-like effects. While Re[χ] alone produces almost the correct spectral shape as well as a number of additional sharp peaks and dips, Im[χ] alone simply leads to absorption peaks at the respective 1s-resonance positions. In the full computation, Im[χ] leads to a smearing out of some of the sharp features.
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]. 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 [33,34]. Similar slow light studies have also been made in a waveguide with periodic side coupling to resonators [35, 36].
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
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] 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.
In contrast to the first Bragg resonance, the structure factor of the second Bragg resonance, i.e. j = 2 with (h,h′) = (F 2,F 1), has an average spacing of D j=2 = τλ/2 = 0.8090λ. Thus, the fully periodic situation, ρ = 1, yields equal L o = S o = D j=2 = 0.8090λ, which produces a destructive interference in reflection because the coupled QWs are separated by irrational fractions of λ, unlike for the first Bragg condition. According to Fig. 8, the structure factor f vanishes at ρ = 1, which shows that the destructive interference is complete. When ρ is increased to two, the short intervals have a distance S o = D/τ = λ/2 such that the large interval becomes L o = 2S o = λ. Since the coupled QWs are now separated by integer multiples of λ/2, the corresponding reflection experiences a constructive interference, as seen also in Fig. 8. Figure 7(b) shows the full reflection spectra around the 1s hh resonance for five different values of ρ. We observe that the trends predicted by f and the simple arguments above are still valid even though this computation has D = 0.8115λ that slightly deviates from the perfect Bragg condition.
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.
We next investigate how the refractive index details of the 275 layers of the 54 QW sample influence the reflection. Figure 9(a) presents the spectrum for the sample FIB13 (shaded area) and the theoretical reflectivity of the same sample is shown as the red line when an anti-reflection coating (ARC) is added at the sample-air interface. Clearly, the ARC removes the background reflection, as it should, while the resonance and dip structure remains intact. We then simplify the 275 layers between the QWs giving them a common constant index of refraction, n = nQW, everywhere while the optical lengths of the layers were kept the same as in the sample FIB13. The result is plotted as the blue line in Fig. 9(a). Also this case does not change the Fibonacci structure much. Consequently, the dip must be caused by the QW arrangement, not by interference effects due to the dielectric environment of the QWs. We have additionally studied 54 periodically spaced QWs in Fig. 9(a) (black line). Even this case displays a strong dip below the 1s hh resonance (vertical line). That dip follows from the non-ideal average spacing detuned slightly away from λ/2.
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).
To check the effect of negative detuning, we compute R(ω) using D = 0.4992λ that is tuned slightly below λ/2. In case of periodic spacing (black line), R(ω) does not have any dip in contrast to the Fibonacci spacing (shaded area), as shown in Fig. 9(b). In particular, the difference of the reflectivity in the dip minimum and the reflectivity maximum next to the dip became only slightly smaller due to the different average spacing. Therefore, one may conclude that the dip in Fig. 9(a) is primarily caused by the average spacing differing from the ideal λ/2 value. Nonetheless, the Fibonacci spacing leads to the formation of a dip as well due to the quasi-periodic nature. As a result of the interplay of these two effects, the Fibonacci spacing produces a narrower and deeper dip than the one obtained in the periodic structure, c.f. Figs. 7 and 9.
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.
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