## One dimensional resonant Fibonacci quasicrystals: noncanonical linear and canonical nonlinear effects

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

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

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/SiO_{2} 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]

*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]

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]

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]

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/SiO_{2} Thue-Morse quasicrystals,” Appl. Phys. Lett. **84**, 5186–5188 (2004). [CrossRef]

*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.

## 2. One-dimensional quasicrystals and the structure factor

*N*semiconductor QWs with their centers positioned at the points

*z*=

*z*(

_{m}*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.

*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

*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

*z*defined according to Eqs. (1) and (2), the physical spacings

_{m}*z*

_{m+1}−

*z*take one of the two values,

_{m}*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

*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. 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]

*L*and

*S*as follows

*M*and

_{k}*N*in the right-hand side of Eq. (6) stands for

_{k}*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]

*t*= 1 + (

*λ*

_{1}−

*α*)/

*γ*between a value of

*t*and indices

*α*,

*β*,

*γ*,

*δ*, where

*w*=

*βγ*−

*αδ*. 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]

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

*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]

*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]

*δ*-peaks corresponding to the Bragg diffraction and characterized by two integer numbers

*h*and

*h*′, see [2, 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]

*G*= 2

_{h}*πh*/

*d*¯ with ∣

*f*∣ = 1. One can show that, for irrational values of

_{h}*t*and Δ ≠ 0, the structure-factor coefficients are given by

*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]

*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]

*φ*< 1, to the quasicrystal with arbitrary

*d*¯, Δ and

*t*.

*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

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]

*τ*.

**77**, 113306 (2008). [CrossRef]

*ω*

_{0}to one of the diffraction vectors

*G*with a large value of the coefficient ∣

_{hh′}*f*∣. This condition is equivalent to

_{hh′}*q*(

*ω*

_{0}) is the light wavevector at the excitonic resonance frequency.

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]

*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*≡ 2

*π*/

*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.

*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

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]

*f*(

*q*) resonances, see Eq. (8), have largest values of ∣

*f*∣ ≈ 1, which corresponds to

_{hh′}*h*and

*h*′ equal to the subsequent Fibonacci numbers: (

*h*,

*h*′) = (

*F*,

_{j}*F*). For the noncanonical structures, the coefficients ∣

_{j-1}*f*∣ are maximal when the ratio of

_{hh′}*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*).

## 3. Experimental and theoretical approaches

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]

*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.

*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]

*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]

*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.

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]

*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.

*γ*

_{bg}= 0.163meV (0.110meV) for the heavy-hole (light-hole) 1s resonance, positioned at

*E*. The cut-off energy is set to Δ

_{x}*E*

_{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

*d*= 0.78e

_{vc}*nm*and

*d*= 0.5e

_{vc}*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.

*D*

_{0}= 0.5016

*λ*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).

## 4. Nonlinear reflectivity of canonical Fibonacci quantum wells

*n*= 1 × 10

^{9}cm

^{-2}(shaded area),

*n*= 5 × 10

^{9}cm

^{-2}(red line),

*n*= 2 × 10

^{10}cm

^{-2}(blue line), and

*n*= 5 × 10

^{10}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 76.6

*μ*W (shaded area), 871

*μ*W (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.

*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).

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

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

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

*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.

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]

*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 × 10

^{9}cm

^{-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

*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*= 0.500

*λ*(

*D*= 0.501

*λ*) up to

*D*= 0.503

*λ*(

*D*= 0.503

*λ*) 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}= 0.5016

*λ*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*.

*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}= 0.5016

*λ*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*.

*j*= 2 with (

*h*,

*h*′) = (

*F*

_{2},

*F*

_{1}), has an average spacing of

*D*

_{j=2}=

*τλ*/2 = 0.8090

*λ*. Thus, the fully periodic situation,

*ρ*= 1, yields equal

*L*

_{o}=

*S*

_{o}=

*D*

_{j=2}= 0.8090

*λ*, 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}= 2

*S*

_{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*= 0.8115

*λ*that slightly deviates from the perfect Bragg condition.

*n*=

*n*, 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

_{QW}*λ*/2.

*R*(

*ω*) using

*D*= 0.4992

*λ*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.

*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.

## 6. Conclusion

*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

## References and links

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2. | C. Janot, |

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. |

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14. | E. L. Ivchenko, |

15. | M. Y. Azbel, “Energy spectrum of a conduction electron in a magnetic field,” Sov. Phys. JETP |

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28. | H. Haug and S. W. Koch, |

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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. |

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. |

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 |

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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. |

36. | P. Chak, S. Pereira, and J. E. Sipe, “Coupled-mode theory for periodic side-coupled microcavity and photonic crystal structures,” Phys. Rev. B |

**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

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