Superradiant modes in resonant quasi-periodic double-period quantum wells C. H. Chang, C. H. Chen, C. W. Tsao, and W. J. Hsueh* Photonics Group, Department of Engineering Science, National Taiwan University, 1, Sec. 4, Roosevelt Road, Taipei, 10660, Taiwan * [email protected]
Abstract: This paper firstly proposes the existence of superradiant modes in resonant quasi-periodic double-period quantum wells (QWs), which has not been observed from analyzing the structure factor by traditional methods. Using the gap map method, the reflection spectra under the relevant conditions show that there are dips in the middle and the linewidth grows linearly, despite the dips, as the number of QWs increases, which is a direct demonstration of superradiance. It is also found that the relevant conditions are divided into three regions, each of which has a different width of bandgaps. ©2015 Optical Society of America OCIS codes: (140.6630) Superradiance, superfluorescence; (240.5420) Polaritons; (230.5590) Quantum-well, -wire and -dot devices; (160.5293) Photonic bandgap materials.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
Z. V. Vardeny, A. Nahata, and A. Agrawal, “Optics of photonic quasicrystals,” Nat. Photonics 7(3), 177–187 (2013). Y. H. Cheng, C. H. Chang, C. H. Chen, and W. J. Hsueh, “Bragg-like interference in one-dimensional doubleperiod quasicrystals,” Phys. Rev. A 90(2), 023830 (2014). M. C. Rechtsman, H.-C. Jeong, P. M. Chaikin, S. Torquato, and P. J. Steinhardt, “Optimized structures for photonic quasicrystals,” Phys. Rev. Lett. 101(7), 073902 (2008). M. E. Zoorob, M. D. B. Charlton, G. J. Parker, J. J. Baumberg, and M. C. Netti, “Complete photonic bandgaps in 12-fold symmetric quasicrystals,” Nature 404(6779), 740–743 (2000). C. H. Chen and W. J. Hsueh, “Enhancement of tunnel magnetoresistance in magnetic tunnel junction by a superlattice barrier,” Appl. Phys. Lett. 104(4), 042405 (2014). 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(20), 15382–15387 (2008). C. H. Chang, C. H. Chen, and W. J. Hsueh, “Strong photoluminescence emission from resonant Fibonacci quantum wells,” Opt. Express 21(12), 14656–14661 (2013). M. Gross and S. Haroche, “Superradiance: An essay on the theory of collective spontaneous emission,” Phys. Rep. 93(5), 301–396 (1982). M. O. Scully and A. A. Svidzinsky, “Physics. The super of superradiance,” Science 325(5947), 1510–1511 (2009). L. Ostermann, H. Zoubi, and H. Ritsch, “Cascaded collective decay in regular arrays of cold trapped atoms,” Opt. Express 20(28), 29634–29645 (2012). T. V. Teperik and A. Degiron, “Superradiant optical emitters coupled to an array of nanosize metallic antennas,” Phys. Rev. Lett. 108(14), 147401 (2012). A. N. Poddubny, L. Pilozzi, M. M. Voronov, and E. L. Ivchenko, “Resonant Fibonacci quantum well structures in one dimension,” Phys. Rev. B 77(11), 113306 (2008). L. Pilozzi, A. D’Andrea, and K. Cho, “Spatial dispersion effects on the optical properties of a resonant Bragg reflector,” Phys. Rev. B 69(20), 205311 (2004). M. Hübner, J. P. Prineas, C. Ell, P. Brick, E. S. Lee, G. Khitrova, H. M. Gibbs, and S. W. Koch, “Optical lattices achieved by excitons in periodic quantum well structures,” Phys. Rev. Lett. 83(14), 2841–2844 (1999). D. Goldberg, L. I. Deych, A. A. Lisyansky, Z. Shi, V. M. Menon, V. Tokranov, M. Yakimov, and S. Oktyabrsky, “Exciton-lattice polaritons in multiple-quantum-well-based photonic crystals,” Nat. Photonics 3(11), 662–666 (2009). T. Ikawa and K. Cho, “Fate of the superradiant mode in a resonant Bragg reflector,” Phys. Rev. B 66(8), 085338 (2002).
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Received 11 Feb 2015; revised 13 Apr 2015; accepted 18 Apr 2015; published 28 Apr 2015 4 May 2015 | Vol. 23, No. 9 | DOI:10.1364/OE.23.011946 | OPTICS EXPRESS 11946
17. 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(19), 195106 (2004). 18. H. M. Gibbs, G. Khitrova, and S. W. Koch, “Exciton-polariton light-semiconductor coupling effects,” Nat. Photonics 5(5), 273–282 (2011). 19. S. M. Sadeghi, W. Li, X. Li, and W.-P. Huang, “Photonic electromagnetically induced transparency and collapse of superradiant modes in Bragg multiple quantum wells,” Phys. Rev. B 74(16), 161304 (2006). 20. 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(24), 241106 (2006). 21. Z. S. Yang, N. H. Kwong, R. Binder, and A. L. Smirl, “Distortionless light pulse delay in quantum-well Bragg structures,” Opt. Lett. 30(20), 2790–2792 (2005). 22. A. N. Poddubny, L. Pilozzi, M. M. Voronov, and E. L. Ivchenko, “Exciton-polaritonic quasicrystalline and aperiodic structures,” Phys. Rev. B 80(11), 115314 (2009). 23. 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(8), 6813–6828 (2009). 24. W. J. Hsueh, C. H. Chang, and C. T. Lin, “Exciton photoluminescence in resonant quasi-periodic Thue-Morse quantum wells,” Opt. Lett. 39(3), 489–492 (2014). 25. M. S. Vasconcelos, P. W. Mauriz, F. F. de Medeiros, and E. L. Albuquerque, “Photonic band gaps in quasiperiodic photonic crystals with negative refractive index,” Phys. Rev. B 76(16), 165117 (2007). 26. E. Liviotti, “A study of the structure factor of Thue-Morse and period-doubling chains by wavelet analysis,” J. Phys. Condens. Matter 8(27), 5007–5015 (1996). 27. W. J. Hsueh, C. T. Chen, and C. H. Chen, “Omnidirectional band gap in Fibonacci photonic crystals with metamaterials using a band-edge formalism,” Phys. Rev. A 78(1), 013836 (2008). 28. C. H. Chang, C. W. Tsao, and W. J. Hsueh, “Superradiant modes in Fibonacci quantum wells under resonant conditions,” New J. Phys. 16(11), 113069 (2014).
Since the concept of quasicrystals was first proposed, quasi-periodic materials have been the subject of many studies, because of their extensive experimental and theoretical applications [1–5]. Recently, studies have concentrated on an investigation of the propagation of exciton polaritons in multiple quantum well structures that are arranged in different quasi-periodic sequences. Previous studies have concentrated on the differences between the polaritonic properties in quasi-periodic and periodic structures [6,7]. A phenomenon that is analogous to the Dicke superradiance, which is a cooperative radiation phenomenon that was first proposed by Dicke, has been of particular interest [8–11]. For periodic quantum wells (PQWs), the superradiant mode occurs when the structures satisfy the Bragg condition, q(ω0) = mπ, where q(ω0) is the light wave vector at the exciton resonance frequency, d is the periodic thickness and m is a non-zero positive integer. The evolution of the reflection spectrum for the superradiant mode has two stages, depending on the increase in the number of QWs, N: superradiant and photonic crystal regimes [12–16]. When N is small, the reflection spectrum has a Lorentzian shape and the linewidth grows linearly as the number of QWs increases. This also results in a decay rate that is proportional to the number of QWs in the time domain, which is a direct demonstration of Dicke superradiance. If N is sufficiently large, there is a transition from the superradiant to the photonic crystal regime occurs. The frequency range for a high reflectance that is close to unity is also consistent with the bandgap in an infinite periodic structure. The width of the bandgap is given by 2Δ / m , where Δ = 2Γ 0ω0 / π and Γ0 is the exciton radiative damping rate in a single QW [12,17]. Because of these unique properties, PQWs allow the applications in the design of various optical devices [18–21]. In terms of quasi-periodic QWs, the relevant studies demonstrate that there are the resonant Bragg conditions in Fibonacci and Thue-More QWs (FQWs and TMQWs) that are analogous to the Bragg condition in PQWs [22,23]. The resonant Bragg conditions are determined by the structure factors in one-dimensional quasicrystals, where the diffraction vectors depend on the different quasi-periodic structures. Under the resonant Bragg conditions, there are also the similar superradiant and photonic crystal behaviors in the
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Received 11 Feb 2015; revised 13 Apr 2015; accepted 18 Apr 2015; published 28 Apr 2015 4 May 2015 | Vol. 23, No. 9 | DOI:10.1364/OE.23.011946 | OPTICS EXPRESS 11947
reflection spectra for a FQW and a TMQW. The only qualitative distinction between the reflection spectra in quasi-periodic and periodic QWs is that significant structural dips near the exciton resonance frequency occur only in the former [12,24]. For quasi-crystalline schemes, the double-period sequence is another typical structure, as well as Fibonacci and Thue-Morse sequences [25,26]. Previous studies do not give detailed evidence of the existence of the superradiant mode in a double-period QW (DPQW), because the decrease in the structure factor of the system, as N increases, leads to the lack of Bragg peaks at arbitrary wave vectors, except for q = 0 [22,26]. However, this study further determines whether the superradiant mode also occurs in a DPQW using the gap map diagram, which allows a more comprehensive study of this phenomenon. If the superradiant mode in a DPQW was verified, it would give more flexibility and selectivity in the design of optical devices.
Fig. 1. The reflection spectra for DPQWs with (a) D = 0.98λ(ω0) and F = 0.53, (b) D = 1.02λ(ω0) and F = 0.47, (c) D = 0.98λ(ω0) and F = 0.02 and (d) D = 1.02λ(ω0) and F = 0.96, as the generation order increases. The shaded areas show the bandgap regions for v = 9 for the respective cases, which are later discussed in detail. The parameters are as follows: ħω0 = 1.533 eV, ħΓ0 = 50μeV, ħΓ = 100 μeV, and nb = 3.55, where Γ is the exciton nonradiative damping rate in a single QW. The normalized frequency, Ω, is defined as Ω = (ω-ω0)D/(2πc), where D = dA + dB.
This paper studies a DPQW that has two different spacings, A and B, and which follows a double-period scheme: A→AB and B→AA [2,25]. The generation orders of the system, v = 1, 2, 3 and 4, correspond to the different structures, A, AB, ABAA and ABAAABAB. Since there is a QW in each of the spacings, the number of QWs in a DPQW equals N = 2(v-1). Each QW is assumed to be the same. The DPQWs are sandwiched between two semi-infinite barriers. The dielectric contrast between the QW and the barrier is assumed to be negligible. It is also assumed that the barrier is sufficiently thick to avoid interaction between excitons in the QWs, so the excitons only couple with the electromagnetic field. First of all, the reflection spectra in the DPQWs for increasing generation orders demonstrate the superradiance phenomenon, as shown in Fig. 1. The different spacing thicknesses are given by the specific values of the thickness filling factor, defined as F = dA/(dA + dB). These values are determined from an analysis of the gap map diagrams, which is later explained in detail. Maxwell’s equations give that the coupling between the electric field and the excitons for normal incidence on a single layer, where a QW is situated at z = 0 and within the left and right boundaries, z− and z+, is governed by: d 2 E( z) 4πω 2 + q 2 (ω ) E ( z ) = − 2 Pexc , 2 dz c
(1)
where the coordinate, z, is set to the growth direction, q = ωnb/c is the light wave vector, nb is the background refractive index of the structure and Pexc is the contribution of the 1s excitons to the polarization of a QW [13,17,19]. Equation (1) gives the relationship between the amplitudes of the electric fields at the left and right boundaries of layer j as (E+,E-)(zj+) = Mj(E+,E-)(zj-) where Mj is the transfer matrix through layer j. The reflection spectra in a vth generation DPQW is calculated using the total transfer matrix through the entire structure, MT = MN…M1.
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Received 11 Feb 2015; revised 13 Apr 2015; accepted 18 Apr 2015; published 28 Apr 2015 4 May 2015 | Vol. 23, No. 9 | DOI:10.1364/OE.23.011946 | OPTICS EXPRESS 11948
Figure 1 shows that the maximum values of the reflection spectra for these structures increase when there is an increase in the generation order, namely an increase in the number of QWs. If the generation order is greater than 4, remarkable structural dips occur around the exciton resonance frequency, which is the only qualitative difference from PQWs under the Bragg condition [12,22,24]. If the dips in the middle of the spectra are ignored, the linewidth of the reflection profiles grows linearly as the generation order increases from 4 to 8. This stage is a direct demonstration of superradiance and is referred to as the superradiant regime [12,13,16]. For generation orders greater than 8, the linewidth begins to saturate if there is a sufficiently large number of QWs, which results in an evolution from the superradiant to the photonic crystal regime. These situations are similar to that for PQWs under the Bragg condition. For larger generation orders, the frequency ranges for high reflectance in the reflection spectra of the DPQWs correspond to two symmetrical bandgap regions, which is similar to the situations for FQWs and TMQWs [22,23]. As seen in Figs. 1(a) to 1(d), the bandgap regions in the DPQWs with filling factors of about 0.5 are respectively narrower and wider than those in the DPQWs with F = 0.02 and 0.96. This shows that there are three different conditions that cause the superradiant mode in a DPQW.
Fig. 2. (a) The gap map diagram for a DPQW with v = 9 and D = 0.98λ(ω0). The gray areas denote the major gaps. The green and blue lines show the filling factors for the maximal widths of the major gaps, namely the respective values of FC for regions A and B. (b) FC for regions A, B and C, for v = 9, as a function of D/λ(ω0). The black lines show the condition for (h,h’) = (1,0), (0,1), (0,2).The other parameters are identical to those in Fig. 1, except that Γ = 0.
For a detailed study of the superradiance phenomenon, the gap map diagram is used because there is a close relationship between the band structure and the reflection spectra [12,17,27]. The spacings for the fixed value of v are varied simultaneously since dA = FD and dB = (1-F)D for the fixed value of D, which determines the effect of the filling factor on the bandgaps of the infinite periodic DPQW. If the exciton nonradiative decay is neglected, the band structure and the bandgaps in the material are given by the conditions of the dispersion relation, cos( K Λ ) ≤ 1 and cos( K Λ ) > 1, which respectively determine the allowed band and the forbidden gap, since the waves transport the system for real values of K. The dispersion relation of exciton polaritons propagating through a vth generation DPQW is expressed as cos(KΛ) = (MT11 + MT22)/2, where K is the Bloch wave vector and Λ is the thickness of the entire structure [2,27]. The diagram gives an alternative method of finding the conditions for the occurrence of the superradiant mode in a quasi-periodic QW. Figure 2(a) shows the gap map diagram in a DPQW with v = 9 and D = 0.98λ(ω0), where only the major gaps are shown, for the sake of brevity, although there are also some minor gaps. It is seen that there are two pairs of major gaps in the diagram, because the widths of these major gaps do not decrease, even though the number of forbidden gaps increases as the generation order increases. In this study, the filling factors for the maximal widths of the major gaps are defined as FC, so there are two values of FC that correspond to the major gaps. Therefore, Fig. 2(a) shows that the superradiant mode occurs in DPQWs for two different values of FC. In
#234348 - $15.00 USD © 2015 OSA
Received 11 Feb 2015; revised 13 Apr 2015; accepted 18 Apr 2015; published 28 Apr 2015 4 May 2015 | Vol. 23, No. 9 | DOI:10.1364/OE.23.011946 | OPTICS EXPRESS 11949
order to further understand the conditions under which the superradiant mode occurs, the values of FC obtained from the diagrams for different values of D/λ(ω0) near unity are illustrated as a function of D/λ(ω0) in Fig. 2(b). It is observed that the values for FC can be divided into three regions, A, B and C, which correspond to the respective major gaps. Specifically, region A ranges from FC = 0.703 to 0.341 and D/λ(ω0)~0.88 to 1.12, which spans the Bragg condition of D/λ(ω0) = 0.5 for a PQW. In contrast, the ranges of the values of FC in regions B and C respectively equal 0.135 to 0 and 1 to 0.787. In these regions, the respective values of D/λ(ω0) only range from the left and right halves of region A. Moreover, the analytical expression of the condition for superradiance in the DPQW, which is obtained from a generalized Bragg condition 2q = Ghh' where Ghh' is the diffraction vector and h, h' are an integer, is presented by F = [(3h + 2h')λ(ω0)/2D]-1 [22]. It is seen in Fig. 2(b) that the wide black lines given by the condition for (h,h') = (1,0), (0,1) and (0,2) are respectively consistent with the regions A, B and C. However, although the condition can serve as a good reference, the gap map diagram can provide more detailed information about the regions where the superradiant modes occur. This situation is similar to that in [28], where the superradiant modes in the FQW can occur for values of D near nλ(ω0)/2 for n = 2,3…. Likewise, there are also the superradiant modes in the DPQW for values of D close to nλ(ω0)/2 for n = 2,3…. However, for the sake of simplification, we only focus on the DPQW with D/λ(ω0) ≈ 1 in this study.
Fig. 3. The major gap positions in (a) region A and (b) regions B and C, as a function of D/λ(ω0). The black line in (a) indicates the position of the gap for a PQW under the Bragg condition for m = 1. The parameters are identical to those in Fig. 2.
The major gap positions for the maximal widths of the major gaps for regions A, B and C are then shown as a function of D/λ(ω0) in Fig. 3. It is seen that the major gap position for region A is almost symmetrical with respect to D/λ(ω0) = 1. The major gap positions for regions B and C are respectively located at the left and right sides of D/λ(ω0) = 1. For D/λ(ω0) = 1, the width of the major gap for region A is respectively smaller and larger that those for regions B and C. This is explained by considering that the structure for the case of D/λ(ω0) = 1 and FC = 0.5 for region A is identical to the PQW under the Bragg condition for m = 1, which consists of QWs with the exciton radiative damping rate that equals Γ0, and has a gapwidth of 2Δ [12,16]. However, the limits FC→0 for D/λ(ω0) = 1 for region B approaches a PQW under the Bragg condition with m = 2 and an exciton radiative damping rate of 3Γ0, while the limit FC→1 for region C approaches the structure with m = 4 and the damping rate of 3Γ0. Thus, the structures in the limits of FC→0 and 1 for regions B and C respectively have the broader and narrower gapwidths than the PQW under the Bragg condition for m = 2, because of the quasi-periodic arrangement [12,17]. This result means that a comparison of the major gap positions for regions A, B and C for D/λ(ω0)≠1 also shows that the gapwidths of and the distance between the higher and lower major gaps for region A are both respectively smaller and larger than those for regions B and C. In fact, there are several minor gaps between the higher and lower major gaps in regions A, B and C, which prevent superradiant modes in the DPQW. The effect of the minor gaps on the reflection spectra for the DPQWs is shown in Fig. 4. The effect of the opening in the
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Received 11 Feb 2015; revised 13 Apr 2015; accepted 18 Apr 2015; published 28 Apr 2015 4 May 2015 | Vol. 23, No. 9 | DOI:10.1364/OE.23.011946 | OPTICS EXPRESS 11950
photonic stop-band is also attributed to the minor gaps. When the value of D moves farther from λ(ω0), the dramatic increase in the widths of the minor gaps leads to the wider opening between the major gaps, whose widths greatly decrease. This results in an increase in the destruction of the superradiant modes in a DPQW. Figures 4(a) and 4(b) show that some high reflection profiles correspond to the positions of the minor gaps between the higher and lower major gaps in DPQWs with D = 0.95λ(ω0) and 1.05λ(ω0), for region A, when the generation order is high. Therefore, there is no linear growth of the linewidth of the reflection spectra as the generation order increases in the superradiant regime, because of the existence of these minor gaps, which is different from the cases for a FQW and a TMQW [12,22]. In other words, the superradiant modes in DPQWs occur when the values of D are close to λ(ω0), as shown in Fig. 1. Similarly, there is also destruction of the superradiant modes in regions B and C, which is seen in Figs. 4(c) and 4(d). The main difference is that the widths of the major gaps in regions B and C are respectively wider and narrower than that in region A.
Fig. 4. The reflection spectra for DPQWs with (a) D = 0.95λ(ω0) and FC = 0.58, (b) D = 1.05λ(ω0) and F = 0.43 for region A, (c) D = 0.95λ(ω0) and FC = 0.05 for region B and (d) D = 1.05λ(ω0) and FC = 0.91 for region C for v = 6 and 10. The shaded areas indicate the major gap positions for the respective cases. The other parameters are the same as those in Fig. 1.
In conclusion, the existence of superradiant modes in resonant quasi-periodic DPQWs is discovered. This was not studied in pervious research because of a lack of Bragg peaks. The gap map diagrams show that the values for FC are divided into three regions, A, B and C, for values of D around λ(ω0). The reflection spectra for these three regions show that there is a linear growth in the linewidth of the reflection profiles as the generation order increases from 4 to 8, if the structural dips in the middle are neglected, which is a straightforward demonstration of the superradiant regime. For generation orders greater than 8, there is a transition from the superradiant to the photonic crystal regime because the linewidth becomes saturated. However, the linear increase in the linewidth of the reflection spectra is prevented by the minor gaps between the major gaps when the value of D moves farther from λ(ω0). Therefore, the superradiant modes in a DPQW occur for values of D that are near λ(ω0). Compared to PQWs, FQWs or TMQWs, this study verifies a similar superradiance phenomenon in a DPQW, which provides more flexibility and selectivity for applications in optical devices. Acknowledgments The authors acknowledge the support in part by the National Science Council of Taiwan under grant numbers MOST 103-2221-E-002-118 and MOST 103-3113-E-002-001.
#234348 - $15.00 USD © 2015 OSA
Received 11 Feb 2015; revised 13 Apr 2015; accepted 18 Apr 2015; published 28 Apr 2015 4 May 2015 | Vol. 23, No. 9 | DOI:10.1364/OE.23.011946 | OPTICS EXPRESS 11951
Abstract: This paper firstly proposes the existence of superradiant modes in resonant quasi-periodic double-period quantum wells (QWs), which has not been observed from analyzing the structure factor by traditional methods. Using the gap map method, the reflection spectra under the relevant conditions show that there are dips in the middle and the linewidth grows linearly, despite the dips, as the number of QWs increases, which is a direct demonstration of superradiance. It is also found that the relevant conditions are divided into three regions, each of which has a different width of bandgaps. ©2015 Optical Society of America OCIS codes: (140.6630) Superradiance, superfluorescence; (240.5420) Polaritons; (230.5590) Quantum-well, -wire and -dot devices; (160.5293) Photonic bandgap materials.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
Z. V. Vardeny, A. Nahata, and A. Agrawal, “Optics of photonic quasicrystals,” Nat. Photonics 7(3), 177–187 (2013). Y. H. Cheng, C. H. Chang, C. H. Chen, and W. J. Hsueh, “Bragg-like interference in one-dimensional doubleperiod quasicrystals,” Phys. Rev. A 90(2), 023830 (2014). M. C. Rechtsman, H.-C. Jeong, P. M. Chaikin, S. Torquato, and P. J. Steinhardt, “Optimized structures for photonic quasicrystals,” Phys. Rev. Lett. 101(7), 073902 (2008). M. E. Zoorob, M. D. B. Charlton, G. J. Parker, J. J. Baumberg, and M. C. Netti, “Complete photonic bandgaps in 12-fold symmetric quasicrystals,” Nature 404(6779), 740–743 (2000). C. H. Chen and W. J. Hsueh, “Enhancement of tunnel magnetoresistance in magnetic tunnel junction by a superlattice barrier,” Appl. Phys. Lett. 104(4), 042405 (2014). 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(20), 15382–15387 (2008). C. H. Chang, C. H. Chen, and W. J. Hsueh, “Strong photoluminescence emission from resonant Fibonacci quantum wells,” Opt. Express 21(12), 14656–14661 (2013). M. Gross and S. Haroche, “Superradiance: An essay on the theory of collective spontaneous emission,” Phys. Rep. 93(5), 301–396 (1982). M. O. Scully and A. A. Svidzinsky, “Physics. The super of superradiance,” Science 325(5947), 1510–1511 (2009). L. Ostermann, H. Zoubi, and H. Ritsch, “Cascaded collective decay in regular arrays of cold trapped atoms,” Opt. Express 20(28), 29634–29645 (2012). T. V. Teperik and A. Degiron, “Superradiant optical emitters coupled to an array of nanosize metallic antennas,” Phys. Rev. Lett. 108(14), 147401 (2012). A. N. Poddubny, L. Pilozzi, M. M. Voronov, and E. L. Ivchenko, “Resonant Fibonacci quantum well structures in one dimension,” Phys. Rev. B 77(11), 113306 (2008). L. Pilozzi, A. D’Andrea, and K. Cho, “Spatial dispersion effects on the optical properties of a resonant Bragg reflector,” Phys. Rev. B 69(20), 205311 (2004). M. Hübner, J. P. Prineas, C. Ell, P. Brick, E. S. Lee, G. Khitrova, H. M. Gibbs, and S. W. Koch, “Optical lattices achieved by excitons in periodic quantum well structures,” Phys. Rev. Lett. 83(14), 2841–2844 (1999). D. Goldberg, L. I. Deych, A. A. Lisyansky, Z. Shi, V. M. Menon, V. Tokranov, M. Yakimov, and S. Oktyabrsky, “Exciton-lattice polaritons in multiple-quantum-well-based photonic crystals,” Nat. Photonics 3(11), 662–666 (2009). T. Ikawa and K. Cho, “Fate of the superradiant mode in a resonant Bragg reflector,” Phys. Rev. B 66(8), 085338 (2002).
#234348 - $15.00 USD © 2015 OSA
Received 11 Feb 2015; revised 13 Apr 2015; accepted 18 Apr 2015; published 28 Apr 2015 4 May 2015 | Vol. 23, No. 9 | DOI:10.1364/OE.23.011946 | OPTICS EXPRESS 11946
17. 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(19), 195106 (2004). 18. H. M. Gibbs, G. Khitrova, and S. W. Koch, “Exciton-polariton light-semiconductor coupling effects,” Nat. Photonics 5(5), 273–282 (2011). 19. S. M. Sadeghi, W. Li, X. Li, and W.-P. Huang, “Photonic electromagnetically induced transparency and collapse of superradiant modes in Bragg multiple quantum wells,” Phys. Rev. B 74(16), 161304 (2006). 20. 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(24), 241106 (2006). 21. Z. S. Yang, N. H. Kwong, R. Binder, and A. L. Smirl, “Distortionless light pulse delay in quantum-well Bragg structures,” Opt. Lett. 30(20), 2790–2792 (2005). 22. A. N. Poddubny, L. Pilozzi, M. M. Voronov, and E. L. Ivchenko, “Exciton-polaritonic quasicrystalline and aperiodic structures,” Phys. Rev. B 80(11), 115314 (2009). 23. 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(8), 6813–6828 (2009). 24. W. J. Hsueh, C. H. Chang, and C. T. Lin, “Exciton photoluminescence in resonant quasi-periodic Thue-Morse quantum wells,” Opt. Lett. 39(3), 489–492 (2014). 25. M. S. Vasconcelos, P. W. Mauriz, F. F. de Medeiros, and E. L. Albuquerque, “Photonic band gaps in quasiperiodic photonic crystals with negative refractive index,” Phys. Rev. B 76(16), 165117 (2007). 26. E. Liviotti, “A study of the structure factor of Thue-Morse and period-doubling chains by wavelet analysis,” J. Phys. Condens. Matter 8(27), 5007–5015 (1996). 27. W. J. Hsueh, C. T. Chen, and C. H. Chen, “Omnidirectional band gap in Fibonacci photonic crystals with metamaterials using a band-edge formalism,” Phys. Rev. A 78(1), 013836 (2008). 28. C. H. Chang, C. W. Tsao, and W. J. Hsueh, “Superradiant modes in Fibonacci quantum wells under resonant conditions,” New J. Phys. 16(11), 113069 (2014).
Since the concept of quasicrystals was first proposed, quasi-periodic materials have been the subject of many studies, because of their extensive experimental and theoretical applications [1–5]. Recently, studies have concentrated on an investigation of the propagation of exciton polaritons in multiple quantum well structures that are arranged in different quasi-periodic sequences. Previous studies have concentrated on the differences between the polaritonic properties in quasi-periodic and periodic structures [6,7]. A phenomenon that is analogous to the Dicke superradiance, which is a cooperative radiation phenomenon that was first proposed by Dicke, has been of particular interest [8–11]. For periodic quantum wells (PQWs), the superradiant mode occurs when the structures satisfy the Bragg condition, q(ω0) = mπ, where q(ω0) is the light wave vector at the exciton resonance frequency, d is the periodic thickness and m is a non-zero positive integer. The evolution of the reflection spectrum for the superradiant mode has two stages, depending on the increase in the number of QWs, N: superradiant and photonic crystal regimes [12–16]. When N is small, the reflection spectrum has a Lorentzian shape and the linewidth grows linearly as the number of QWs increases. This also results in a decay rate that is proportional to the number of QWs in the time domain, which is a direct demonstration of Dicke superradiance. If N is sufficiently large, there is a transition from the superradiant to the photonic crystal regime occurs. The frequency range for a high reflectance that is close to unity is also consistent with the bandgap in an infinite periodic structure. The width of the bandgap is given by 2Δ / m , where Δ = 2Γ 0ω0 / π and Γ0 is the exciton radiative damping rate in a single QW [12,17]. Because of these unique properties, PQWs allow the applications in the design of various optical devices [18–21]. In terms of quasi-periodic QWs, the relevant studies demonstrate that there are the resonant Bragg conditions in Fibonacci and Thue-More QWs (FQWs and TMQWs) that are analogous to the Bragg condition in PQWs [22,23]. The resonant Bragg conditions are determined by the structure factors in one-dimensional quasicrystals, where the diffraction vectors depend on the different quasi-periodic structures. Under the resonant Bragg conditions, there are also the similar superradiant and photonic crystal behaviors in the
#234348 - $15.00 USD © 2015 OSA
Received 11 Feb 2015; revised 13 Apr 2015; accepted 18 Apr 2015; published 28 Apr 2015 4 May 2015 | Vol. 23, No. 9 | DOI:10.1364/OE.23.011946 | OPTICS EXPRESS 11947
reflection spectra for a FQW and a TMQW. The only qualitative distinction between the reflection spectra in quasi-periodic and periodic QWs is that significant structural dips near the exciton resonance frequency occur only in the former [12,24]. For quasi-crystalline schemes, the double-period sequence is another typical structure, as well as Fibonacci and Thue-Morse sequences [25,26]. Previous studies do not give detailed evidence of the existence of the superradiant mode in a double-period QW (DPQW), because the decrease in the structure factor of the system, as N increases, leads to the lack of Bragg peaks at arbitrary wave vectors, except for q = 0 [22,26]. However, this study further determines whether the superradiant mode also occurs in a DPQW using the gap map diagram, which allows a more comprehensive study of this phenomenon. If the superradiant mode in a DPQW was verified, it would give more flexibility and selectivity in the design of optical devices.
Fig. 1. The reflection spectra for DPQWs with (a) D = 0.98λ(ω0) and F = 0.53, (b) D = 1.02λ(ω0) and F = 0.47, (c) D = 0.98λ(ω0) and F = 0.02 and (d) D = 1.02λ(ω0) and F = 0.96, as the generation order increases. The shaded areas show the bandgap regions for v = 9 for the respective cases, which are later discussed in detail. The parameters are as follows: ħω0 = 1.533 eV, ħΓ0 = 50μeV, ħΓ = 100 μeV, and nb = 3.55, where Γ is the exciton nonradiative damping rate in a single QW. The normalized frequency, Ω, is defined as Ω = (ω-ω0)D/(2πc), where D = dA + dB.
This paper studies a DPQW that has two different spacings, A and B, and which follows a double-period scheme: A→AB and B→AA [2,25]. The generation orders of the system, v = 1, 2, 3 and 4, correspond to the different structures, A, AB, ABAA and ABAAABAB. Since there is a QW in each of the spacings, the number of QWs in a DPQW equals N = 2(v-1). Each QW is assumed to be the same. The DPQWs are sandwiched between two semi-infinite barriers. The dielectric contrast between the QW and the barrier is assumed to be negligible. It is also assumed that the barrier is sufficiently thick to avoid interaction between excitons in the QWs, so the excitons only couple with the electromagnetic field. First of all, the reflection spectra in the DPQWs for increasing generation orders demonstrate the superradiance phenomenon, as shown in Fig. 1. The different spacing thicknesses are given by the specific values of the thickness filling factor, defined as F = dA/(dA + dB). These values are determined from an analysis of the gap map diagrams, which is later explained in detail. Maxwell’s equations give that the coupling between the electric field and the excitons for normal incidence on a single layer, where a QW is situated at z = 0 and within the left and right boundaries, z− and z+, is governed by: d 2 E( z) 4πω 2 + q 2 (ω ) E ( z ) = − 2 Pexc , 2 dz c
(1)
where the coordinate, z, is set to the growth direction, q = ωnb/c is the light wave vector, nb is the background refractive index of the structure and Pexc is the contribution of the 1s excitons to the polarization of a QW [13,17,19]. Equation (1) gives the relationship between the amplitudes of the electric fields at the left and right boundaries of layer j as (E+,E-)(zj+) = Mj(E+,E-)(zj-) where Mj is the transfer matrix through layer j. The reflection spectra in a vth generation DPQW is calculated using the total transfer matrix through the entire structure, MT = MN…M1.
#234348 - $15.00 USD © 2015 OSA
Received 11 Feb 2015; revised 13 Apr 2015; accepted 18 Apr 2015; published 28 Apr 2015 4 May 2015 | Vol. 23, No. 9 | DOI:10.1364/OE.23.011946 | OPTICS EXPRESS 11948
Figure 1 shows that the maximum values of the reflection spectra for these structures increase when there is an increase in the generation order, namely an increase in the number of QWs. If the generation order is greater than 4, remarkable structural dips occur around the exciton resonance frequency, which is the only qualitative difference from PQWs under the Bragg condition [12,22,24]. If the dips in the middle of the spectra are ignored, the linewidth of the reflection profiles grows linearly as the generation order increases from 4 to 8. This stage is a direct demonstration of superradiance and is referred to as the superradiant regime [12,13,16]. For generation orders greater than 8, the linewidth begins to saturate if there is a sufficiently large number of QWs, which results in an evolution from the superradiant to the photonic crystal regime. These situations are similar to that for PQWs under the Bragg condition. For larger generation orders, the frequency ranges for high reflectance in the reflection spectra of the DPQWs correspond to two symmetrical bandgap regions, which is similar to the situations for FQWs and TMQWs [22,23]. As seen in Figs. 1(a) to 1(d), the bandgap regions in the DPQWs with filling factors of about 0.5 are respectively narrower and wider than those in the DPQWs with F = 0.02 and 0.96. This shows that there are three different conditions that cause the superradiant mode in a DPQW.
Fig. 2. (a) The gap map diagram for a DPQW with v = 9 and D = 0.98λ(ω0). The gray areas denote the major gaps. The green and blue lines show the filling factors for the maximal widths of the major gaps, namely the respective values of FC for regions A and B. (b) FC for regions A, B and C, for v = 9, as a function of D/λ(ω0). The black lines show the condition for (h,h’) = (1,0), (0,1), (0,2).The other parameters are identical to those in Fig. 1, except that Γ = 0.
For a detailed study of the superradiance phenomenon, the gap map diagram is used because there is a close relationship between the band structure and the reflection spectra [12,17,27]. The spacings for the fixed value of v are varied simultaneously since dA = FD and dB = (1-F)D for the fixed value of D, which determines the effect of the filling factor on the bandgaps of the infinite periodic DPQW. If the exciton nonradiative decay is neglected, the band structure and the bandgaps in the material are given by the conditions of the dispersion relation, cos( K Λ ) ≤ 1 and cos( K Λ ) > 1, which respectively determine the allowed band and the forbidden gap, since the waves transport the system for real values of K. The dispersion relation of exciton polaritons propagating through a vth generation DPQW is expressed as cos(KΛ) = (MT11 + MT22)/2, where K is the Bloch wave vector and Λ is the thickness of the entire structure [2,27]. The diagram gives an alternative method of finding the conditions for the occurrence of the superradiant mode in a quasi-periodic QW. Figure 2(a) shows the gap map diagram in a DPQW with v = 9 and D = 0.98λ(ω0), where only the major gaps are shown, for the sake of brevity, although there are also some minor gaps. It is seen that there are two pairs of major gaps in the diagram, because the widths of these major gaps do not decrease, even though the number of forbidden gaps increases as the generation order increases. In this study, the filling factors for the maximal widths of the major gaps are defined as FC, so there are two values of FC that correspond to the major gaps. Therefore, Fig. 2(a) shows that the superradiant mode occurs in DPQWs for two different values of FC. In
#234348 - $15.00 USD © 2015 OSA
Received 11 Feb 2015; revised 13 Apr 2015; accepted 18 Apr 2015; published 28 Apr 2015 4 May 2015 | Vol. 23, No. 9 | DOI:10.1364/OE.23.011946 | OPTICS EXPRESS 11949
order to further understand the conditions under which the superradiant mode occurs, the values of FC obtained from the diagrams for different values of D/λ(ω0) near unity are illustrated as a function of D/λ(ω0) in Fig. 2(b). It is observed that the values for FC can be divided into three regions, A, B and C, which correspond to the respective major gaps. Specifically, region A ranges from FC = 0.703 to 0.341 and D/λ(ω0)~0.88 to 1.12, which spans the Bragg condition of D/λ(ω0) = 0.5 for a PQW. In contrast, the ranges of the values of FC in regions B and C respectively equal 0.135 to 0 and 1 to 0.787. In these regions, the respective values of D/λ(ω0) only range from the left and right halves of region A. Moreover, the analytical expression of the condition for superradiance in the DPQW, which is obtained from a generalized Bragg condition 2q = Ghh' where Ghh' is the diffraction vector and h, h' are an integer, is presented by F = [(3h + 2h')λ(ω0)/2D]-1 [22]. It is seen in Fig. 2(b) that the wide black lines given by the condition for (h,h') = (1,0), (0,1) and (0,2) are respectively consistent with the regions A, B and C. However, although the condition can serve as a good reference, the gap map diagram can provide more detailed information about the regions where the superradiant modes occur. This situation is similar to that in [28], where the superradiant modes in the FQW can occur for values of D near nλ(ω0)/2 for n = 2,3…. Likewise, there are also the superradiant modes in the DPQW for values of D close to nλ(ω0)/2 for n = 2,3…. However, for the sake of simplification, we only focus on the DPQW with D/λ(ω0) ≈ 1 in this study.
Fig. 3. The major gap positions in (a) region A and (b) regions B and C, as a function of D/λ(ω0). The black line in (a) indicates the position of the gap for a PQW under the Bragg condition for m = 1. The parameters are identical to those in Fig. 2.
The major gap positions for the maximal widths of the major gaps for regions A, B and C are then shown as a function of D/λ(ω0) in Fig. 3. It is seen that the major gap position for region A is almost symmetrical with respect to D/λ(ω0) = 1. The major gap positions for regions B and C are respectively located at the left and right sides of D/λ(ω0) = 1. For D/λ(ω0) = 1, the width of the major gap for region A is respectively smaller and larger that those for regions B and C. This is explained by considering that the structure for the case of D/λ(ω0) = 1 and FC = 0.5 for region A is identical to the PQW under the Bragg condition for m = 1, which consists of QWs with the exciton radiative damping rate that equals Γ0, and has a gapwidth of 2Δ [12,16]. However, the limits FC→0 for D/λ(ω0) = 1 for region B approaches a PQW under the Bragg condition with m = 2 and an exciton radiative damping rate of 3Γ0, while the limit FC→1 for region C approaches the structure with m = 4 and the damping rate of 3Γ0. Thus, the structures in the limits of FC→0 and 1 for regions B and C respectively have the broader and narrower gapwidths than the PQW under the Bragg condition for m = 2, because of the quasi-periodic arrangement [12,17]. This result means that a comparison of the major gap positions for regions A, B and C for D/λ(ω0)≠1 also shows that the gapwidths of and the distance between the higher and lower major gaps for region A are both respectively smaller and larger than those for regions B and C. In fact, there are several minor gaps between the higher and lower major gaps in regions A, B and C, which prevent superradiant modes in the DPQW. The effect of the minor gaps on the reflection spectra for the DPQWs is shown in Fig. 4. The effect of the opening in the
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Received 11 Feb 2015; revised 13 Apr 2015; accepted 18 Apr 2015; published 28 Apr 2015 4 May 2015 | Vol. 23, No. 9 | DOI:10.1364/OE.23.011946 | OPTICS EXPRESS 11950
photonic stop-band is also attributed to the minor gaps. When the value of D moves farther from λ(ω0), the dramatic increase in the widths of the minor gaps leads to the wider opening between the major gaps, whose widths greatly decrease. This results in an increase in the destruction of the superradiant modes in a DPQW. Figures 4(a) and 4(b) show that some high reflection profiles correspond to the positions of the minor gaps between the higher and lower major gaps in DPQWs with D = 0.95λ(ω0) and 1.05λ(ω0), for region A, when the generation order is high. Therefore, there is no linear growth of the linewidth of the reflection spectra as the generation order increases in the superradiant regime, because of the existence of these minor gaps, which is different from the cases for a FQW and a TMQW [12,22]. In other words, the superradiant modes in DPQWs occur when the values of D are close to λ(ω0), as shown in Fig. 1. Similarly, there is also destruction of the superradiant modes in regions B and C, which is seen in Figs. 4(c) and 4(d). The main difference is that the widths of the major gaps in regions B and C are respectively wider and narrower than that in region A.
Fig. 4. The reflection spectra for DPQWs with (a) D = 0.95λ(ω0) and FC = 0.58, (b) D = 1.05λ(ω0) and F = 0.43 for region A, (c) D = 0.95λ(ω0) and FC = 0.05 for region B and (d) D = 1.05λ(ω0) and FC = 0.91 for region C for v = 6 and 10. The shaded areas indicate the major gap positions for the respective cases. The other parameters are the same as those in Fig. 1.
In conclusion, the existence of superradiant modes in resonant quasi-periodic DPQWs is discovered. This was not studied in pervious research because of a lack of Bragg peaks. The gap map diagrams show that the values for FC are divided into three regions, A, B and C, for values of D around λ(ω0). The reflection spectra for these three regions show that there is a linear growth in the linewidth of the reflection profiles as the generation order increases from 4 to 8, if the structural dips in the middle are neglected, which is a straightforward demonstration of the superradiant regime. For generation orders greater than 8, there is a transition from the superradiant to the photonic crystal regime because the linewidth becomes saturated. However, the linear increase in the linewidth of the reflection spectra is prevented by the minor gaps between the major gaps when the value of D moves farther from λ(ω0). Therefore, the superradiant modes in a DPQW occur for values of D that are near λ(ω0). Compared to PQWs, FQWs or TMQWs, this study verifies a similar superradiance phenomenon in a DPQW, which provides more flexibility and selectivity for applications in optical devices. Acknowledgments The authors acknowledge the support in part by the National Science Council of Taiwan under grant numbers MOST 103-2221-E-002-118 and MOST 103-3113-E-002-001.
#234348 - $15.00 USD © 2015 OSA
Received 11 Feb 2015; revised 13 Apr 2015; accepted 18 Apr 2015; published 28 Apr 2015 4 May 2015 | Vol. 23, No. 9 | DOI:10.1364/OE.23.011946 | OPTICS EXPRESS 11951
