Abstract

We present occurrence of the effective Bragg conditions with wide gapwidth and high reflectance in a Fibonacci superlattice, which is a typical one-dimensional quasicrystal. In the Fibonacci material, the number of effective Bragg conditions is two rather than one which appears in traditional periodic structures. Based on the effective Bragg conditions, this study proposes existence of omnidirectional, wideband and high reflectance in the quasiperiodic materials analogous to that in traditional materials.

© 2012 OSA

Since the discovery of quasicrystals, several quasiperiodic materials have been formulated and have attracted intense interest not only in regard to fundamental physical phenomena but also to technological applications [16]. Research has shown that the phenomena of photon propagating in quasiperiodic materials are different from those among periodic and disordered ones [712]. Light propagation can be forbidden in disorder structures by Anderson localization [1315]. In periodic media, high reflectance can also be achieved by Bragg interference. Until now, Bragg’s law has played a major role in the measurement of crystalline solids and the design of several optical devices [1619]. During implementation, a broad bandgap and high reflectance is required to reduce the performance sensitivity due to manufactory error. Based on the Bragg condition, maximum gapwidth can be achieved by multiplying of the quarter-wave length in each layer for the traditional bilayers [20,21]. In particular, with a maximum omnidirectional gap, the highest performance reflectors are found within the first and zero orders Bragg conditions for bilayer dielectric materials [2023] and metamaterials [2428], respectively.

However, for quasiperiodic materials, the broad bandgap and high reflectance of the quarter-wave thickness is destroyed and replaced by self-similar and cycling fracture gaps in the transmission spectra [79]. It seems that the high reflectance condition varies for different generation orders, layer thicknesses and materials in the dielectric Fibonacci superlattice (FSL) [912], which is a typical one-dimensional quasiperiodic material. Thus, it is not convenient to apply quasiperiodic materials in the design of optical devices, which have been easily implemented by using traditional periodical materials based on Bragg’s law. Until now, critical questions remain unanswered. Does broad bandgap and high reflectance exist for quasiperiodic materials similar to the Bragg condition or the quarter-wave stakes for the periodic materials? If the high reflectance exists in quasiperiodic materials, can we obtain the critical condition and express it as a concise law analogous to Bragg’s law for periodical materials.

Here, we show that the broad bandgap and high reflectance does exist in the FSL via gap map diagrams. The high reflectance condition for the FSL can be determined by the midline of the major gaps (MMGs) and described by two concise expressions, called effective Bragg conditions, which are analogous to the Bragg condition for periodic materials. Based on the effective Bragg condition, existence of the highest reflectance in the FSL are determined and described by two critical conditions analogous to the quarter-wave stacks in the traditional bilayers. Moreover, we show that an omnidirectional high reflector for the quasiperiodic materials can be achieved by the effective Bragg condition.

We first consider a FSL composed of two layers, A and B, with the following scheme: Sν={Sν1Sν2} for v≥2, with S0 = {B} and S1 = {A}. The numbers of layer A and B in the Sv superlattice are Nv-1 and Nv-2, respectively. Transmission of light through the FSL is dependent on the light polarization and incident direction and the structure, including the layer thickness, effective refractive index of each layer, and the generation order. The transmission spectra of FSL with arbitrary generation order for arbitrary incident directions of TE and TM polarizations can be calculated by the transfer matrix method or the graph method [26]. The gap map is formed by the bandedges, which are determined by the bandedge formalism to avoid numerical instability [23,26]. Figure 1(a) shows the effect of the generation order on the variants of the bandgaps for the normal incidence, TE 45°, and TM 45° of the FSL, in which the periodic boundary is considered. We see that the number offorbidden gaps within the frequency range increases with increasing generation order. The widths of almost all of the forbidden gaps for each incidence direction and polarization are reduced as the generation order of the system increases. However, the widths of the major gaps indicated by arrow signs in the figure do not reduce significantly with higher generation order.

 

Fig. 1 (a) Photonic bandgap of the normal incidence, TE 45°, and TM 45° in the FSL for generation orders v = 2 to 8. The red, blue and green thick lines correspond to the allowed bands for the normal incidence, TE 45°, and TM 45°, respectively. The arrow signs indicate the major gaps. Parameters of the system are nA = 2.0, nB = 4.0, dA = 0.55μm, and dB = 0.45μm. The normalized frequency Ω is defined by Ω=ωD/(2πc), where D = dA + dB. (b) The transmission spectra of a single cell of the FSL with v = 4, 6 and 8 for normal incidence from air. The gray areas correspond to the region of the major gaps shown in (a).

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As a single FSL bonded by the media of air is considered, the transmission spectra of the normal incident of TE polarization through the FSL for different generation orders is shown in Fig. 1(b). When the generation order is low, such as v = 4, the frequency range of low transmissions cannot be located within the major gaps. However, as the generation order increases, the transmittance within the major gaps becomes lower. For generations greater than 6, the low transmission regions almost fit the forbidden gaps, as in the case of the traditional bilayers, {AB}.Next, we consider the gap map diagram in the S8 FSL for normal incidence, versus the thickness filling factor F, defined by F = dA/(dA + dB) as shown in Fig. 2(a) . The gap map can be divided by zero-gap lines, LA,m and LB,n, obtained by kAdA = mπ and kBdB = nπ, to several regions, each of which has the same pattern. The number of forbidden bands in each region increases as the generation order increases. Thus, the widths of almost all of the forbidden gaps will be reduced for an FSL with a higher generation order. However, the widths of some forbidden gaps within some filling factor ranges do not significantly reduce the gap width for the higher generation order. We define the part of the gap map bounded by LA,m, LA,m+1, LB,n, and LB,n+1 as region (m,n), where m and n are nonnegative integers. With a higher generation order, there are two major forbidden gap holes within each region [11]. We refer to the gap hole with higher frequency as the higher major hole and the other one as the lower major hole.

 

Fig. 2 Effective Bragg condition, major gaps and their centers in the gap map of the FSL. (a) The gap map of the normal incidence in the S8 FSL with nA = 2.0, nB = 4.0. The gray areas mark the major gaps. The green lines present the bandedges. The red lines mark the midline of the major gap holes, MMGs, corresponding to the effective Bragg conditions. The x signs indicate the center of the major gaps. The blue and black dashed lines mark the zero-gap lines, LA,m and LB,n, respectively. (b) the thickness filling factor and (c) the normalized frequency, and (d) the gapwidth to gap center ratio of the CLGs in regions (0,0), (1,0), (0,1) and (1,1) of the gap map for nA = 2.0 and 6.0> nB>3.0. The solid and dashed lines correspond to the maximum gaps and the CLGs, respectively.

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For the Sv FSL, the right end of the lower major hole in region (m,n) is located at the cross point Cm+τv2,v1,n and the left one is located at Cm,n+1, where τv2,v1=Nv2/Nv1. F and Ω at the close point Cp,q are, respectively, given by

F=pσq+pσ,Ω=pσ+q2nBcosθB,
where σ=nBcosθB/nAcosθA. The left end remains in the same position, but the right one moves on the zero-gap line LB,n to a different generation order. F and Ω at the right end of the lower major hole in region (m,n) are (m+τv2,v1)σ/[n+(m+τv2,v1)σ], [n+(m+τv2,v1)σ]/(2nBcosθB). For the higher major hole, the right end is located at the cross points Cm+1,n, which keep the same position for arbitrary generation orders. The left end of the higher major hole is located at Cm+Nv3/Nv1,n+1, which moves for different generation orders. The values of F and Ω at the left end of the top major hole in region (m,n) are (m+τv3,v1)σ/[(n+1+(m+τv3,v1)σ],[n+1+(m+τv3,v1)σ]/(2nBcosθB),respectively. As the generation order of the FSL approaches infinity, we have the relations limvτv2,v1=τ1 and limvτv3,v1=τ2, where τ is the golden ratio, which is defined by τ=(1+5)/21.618. The moving end of each major hole will asymptotically approach a fixed point on the zero-gap line for a higher generation order.

As we know, for the traditional bilayers, there is only one MMG in each region of the gap map [23]. The MMG in region (m,n) corresponds to the Bragg condition described by kAdA+kBdB=(m+n+1)π,which almost corresponds to the center of the lowest transmission region [23]. Similarly, the center of the lowest transmission region also nearly matches the MMG for the FSL. Thus, we can determine the condition of the highest reflectance from the MMGs. The midline of the higher and lower major gap holes in region (m,n) are given by kAdA+τv2,v1kBdB=(m+1+τv2,v1n)π and kAdA+τv2,v1kBdB= [m+τv2,v1(n+1)]π and, respectively, rewritten by

dAnAcosθA+τv2,v1dBnBcosθB=(m+1+τv2,v1n)λ0/2,
dAnAcosθA+τv2,v1dBnBcosθB=[m+τv2,v1(n+1)]λ0/2.
In Fig. 2(a), the MMGs are plotted in the gap map of the S8 FSL. One of the special cases is v→∞, by whichlimvτv2,v1=0.618. For the FSL with a higher generation order, these equations can be approximately described by the infinite case. The other special case is v = 2, which is identical to a bilayer superlattice. For the case of v = 2, The midline of the higher and lower major gap holes in each region are overlapped and described by one line with the same equation as that of the Bragg condition for the bilayers.

In each major hole, we find that the extreme situation of the maximum gapwidth is near the center of the gap hole. Here, the center of the lower major gap hole in region (m,n), CLGm,n, is defined by the intersection of the lines kAdA = (m + 0.5τv-2,v-1)π and kBdB = (n + 0.5)π, which can be rewritten by dAnAcosθA=(2m+τv2,v1)λ0/4 and dBnBcosθB=(2n+1)λ0/4, respectively. Thus, the filling factor and the normalized frequency at the CLGm,n are given by

Fm,n(CLG)=[1+(2n+1)nAcosθA(2m+τv2,v1)nBcosθB]1,
Ωm,n(CLG)=14[(2m+τv2,v1)nAcosθA+(2n+1)nBcosθB].
The center of the higher major hole in the region, CHGm,n, is located on the intersection of the lines kAdA = [m + 0.5(1 + τv-3,v-1)]π and kBdB = (n + 0.5)π and is rewritten by dAnAcosθA=(2m+1+τv3,v1)λ0/4 and dBnBcosθB=(2n+1)λ0/4.The filling factor and normalized frequency at the CHGm,n are
Fm,n(CHG)=[1+(2n+1)nAcosθA(2m+1+τv3,v1)nBcosθB]1,
Ωm,n(CHG)=14[(2m+1+τv3,v1)nAcosθA+(2n+1)nBcosθB].
For the case of v = 2, CHGm,n and CLGm,n are overlapped. The layer thickness of CLGm,n is equal to a multiple of the quarter wave, the same as the Bragg condition for the bilayers. In the fundamental region, the layer thickness of CLG0,0 is identical to the quarter wave stacks [21]. Figure 2(b) shows that the thickness filling factor for the center of the gap holes almost has a linear relation to the refractive index ratio, nA/(nA + nB). The difference between the filling factors of the center of the gap holes and maximum gaps gradually enlarges as the refractive indices of layers A and B get closer. The difference of the thickness filling factors may gradually increase as the refractive index ratio increases. However, the frequency and the gapwidth to gap center ratio for the center of the gap holes are very close to those of the maximum gaps for an arbitrary refractive index ratio, as shown in Figs. 2(c) and 2(d). We find that the gapwidth to gap center ratio almost has a linear relation to the refractive index ratio. As the difference between nA and nB increase, the gapwidth to gap center ratios increase. Of these centers of the gap holes, the maximum gapwidth ratio occurs at CLG0,0. Thus the omni-gap can be determined by only the overlap region of the major holes of the NI and TM1 states. The thickness filling factor and the frequency of the center of the omni-gap can be approximately expressed by the center of the NI and TM1 gap holes. The maximum gapwidth to gap center of the omin-gap is located in the lower major gap of region (0,0), which is called the fundamental omin-gap. In Fig. 3(b) , we see that transmissions for different incident directions and polarizations can be reduced significantly in the range of the omni-gap. Figure 3(c) shows that the gapwidth to gap center ratio of the center of the fundamental omni-gap (CFOG) of the FSL is very close to that of the maximum omnidirectional gap. The almost linear gapwidth to gap center of the CFOG decreases with the increase of the refractive index ratio. Moreover, the relation of the gapwidth to gap center ratios to the refractive index ratio for the CFOG is close that for the maximum omnidirectional gap of the FSL.

 

Fig. 3 Omni-gaps and transmission spectra of the FSL. (a) Sketch of the overlap among the major gaps for NI, TM1, and TE1 states and omni-gap in the gap map. The gray region corresponds to the omni-gap. (b) The transmission spectra of the S8 FSL with nA = 2.0, nB = 4.0 at the CFOG for 0°, 45°, 85° of TE and TM polarizations. The gray region corresponds to the omni-gap. (c) The gapwidth to gap center ratios of the CFOG of the S8 FSL for nA = 1.5, 2.0, 3.0, and 3.0< nB<6.0. The dashed and solid lines correspond to the gapwidth to gap center ratio of the CFOG and the maximum omnidirectional gap, respectively.

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Based on the major gaps of the FSL, high reflection for an arbitrary incident direction, called an omnidirectional gap or omni-gap [2123], can be determined by the overlap of the bandgap for the cases of normal incidence, called NI state and grazing incidence for TE and TM polarizations, as referred to as TE1, and TM1 states, as shown in Fig. 3(a). We find that the ranges of the major holes of TM1 are within the TE1 [11].

In summary, analogous to one Bragg condition for the traditional bilayers, the broad bandgap and high reflectance for the FSL is found by the midline of the major gaps and expressed by two concise critical conditions, called the effective Bragg conditions. Moreover, the highest performance of the reflector for the FSL is determined by the centers of the effective Bragg conditions, which are analogous to quarter wave condition for the traditional bilayers. Similar to the highly omnidirectional reflection properties of the quarter wave stack in bilayers [2023], the existence of omnidirectional reflection in the FSL is found by the center of the omni-gap based on the effective Bragg conditions. From this concept, we have potential to deal with novel quasicrystals for design and application in various fields and potential to reach higher performance than traditional periodical crystals.

Acknowledgments

The authors acknowledge the support in part by the National Science Council of Taiwan under grant numbers NSC 100-2221-E-002-021 and NSC 101-2221-E-002-030.

References and links

1. D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, “Metallic phase with long-range orientational order and no translational symmetry,” Phys. Rev. Lett. 53(20), 1951–1953 (1984). [CrossRef]  

2. D. Levine and P. J. Steinhardt, “Quasicrystals: A new class of ordered structures,” Phys. Rev. Lett. 53(26), 2477–2480 (1984). [CrossRef]  

3. E. Abe, Y. Yan, and S. J. Pennycook, “Quasicrystals as cluster aggregates,” Nat. Mater. 3(11), 759–767 (2004). [CrossRef]   [PubMed]  

4. J. Mikhael, J. Roth, L. Helden, and C. Bechinger, “Archimedean-like tiling on decagonal quasicrystalline surfaces,” Nature 454(7203), 501–504 (2008). [CrossRef]   [PubMed]  

5. K. Ueda, T. Dotera, and T. Gemma, “Photonic band structure calculations of two-dimensional Archimedean tiling patterns,” Phys. Rev. B 75(19), 195122 (2007). [CrossRef]  

6. G. J. Parker, M. E. Zoorob, M. D. B. Charlton, J. J. Baumberg, and M. C. Netti, “Complete photonic bandgaps in 12-fold symmetric quasicrystals,” Nature 404(6779), 740–743 (2000). [CrossRef]   [PubMed]  

7. M. Kohmoto, B. Sutherland, and K. Iguchi, “Localization of optics: Quasiperiodic media,” Phys. Rev. Lett. 58(23), 2436–2438 (1987). [CrossRef]   [PubMed]  

8. W. Gellermann, M. Kohmoto, B. Sutherland, and P. C. Taylor, “Localization of light waves in Fibonacci dielectric multilayers,” Phys. Rev. Lett. 72(5), 633–636 (1994). [CrossRef]   [PubMed]  

9. S. W. Wang, X. Chen, W. Lu, M. Li, and H. Wang, “Fractal independently tunable multichannel filters,” Appl. Phys. Lett. 90(21), 211113 (2007). [CrossRef]  

10. L. Dal Negro, C. J. Oton, Z. Gaburro, L. Pavesi, P. Johnson, A. Lagendijk, R. Righini, M. Colocci, and D. S. Wiersma, “Light transport through the band-edge states of Fibonacci quasicrystals,” Phys. Rev. Lett. 90(5), 055501 (2003). [CrossRef]   [PubMed]  

11. W. J. Hsueh, S. J. Wun, and C. W. Tsao, “Branching features of photonic bandgaps in Fibonacci dielectric heterostructures,” Opt. Commun. 284(7), 1880–1886 (2011). [CrossRef]  

12. A. N. Poddubny and E. L. Ivchenko, “Photonic quasicrystalline and aperiodic structures,” Physica E 42(7), 1871–1895 (2010). [CrossRef]  

13. P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. 109(5), 1492–1505 (1958). [CrossRef]  

14. S. John, “Electromagnetic absorption in a disordered medium near a photon mobility edge,” Phys. Rev. Lett. 53(22), 2169–2172 (1984). [CrossRef]  

15. D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, “Localization of light in a disordered medium,” Nature 390(6661), 671–673 (1997). [CrossRef]  

16. Y. V. Shvyd’kol, S. Stoupin, A. Cunsolo, A. H. Said, and X. Huang, “High-reflectivity high-resolution X-ray crystal optics with diamonds,” Nat. Phys. 6, 196–199 (2010).

17. J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science 264(5158), 553–556 (1994). [CrossRef]   [PubMed]  

18. B. S. Williams, “Terahertz quantum-cascade lasers,” Nat. Photonics 1(9), 517–525 (2007). [CrossRef]  

19. Y. Taniyasu, M. Kasu, and T. Makimoto, “An aluminium nitride light-emitting diode with a wavelength of 210 nanometres,” Nature 441(7091), 325–328 (2006). [CrossRef]   [PubMed]  

20. E. Yablonovitch, “Engineered omnidirectional external-reflectivity spectra from one-dimensional layered interference filters,” Opt. Lett. 23(21), 1648–1649 (1998). [CrossRef]   [PubMed]  

21. Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282(5394), 1679–1682 (1998). [CrossRef]   [PubMed]  

22. R. G. DeCorby, H. T. Nguyen, P. K. Dwivedi, and T. J. Clement, “Planar omnidirectional reflectors in chalcogenide glass and polymer,” Opt. Express 13(16), 6228–6233 (2005). [CrossRef]   [PubMed]  

23. W. J. Hsueh and S. J. Wun, “Simple expressions for the maximum omnidirectional bandgap of bilayer photonic crystals,” Opt. Lett. 36(9), 1581–1583 (2011). [CrossRef]   [PubMed]  

24. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000). [CrossRef]   [PubMed]  

25. P. V. Parimi, W. T. Lu, P. Vodo, and S. Sridhar, “Photonic crystals: Imaging by flat lens using negative refraction,” Nature 426(6965), 404 (2003). [CrossRef]   [PubMed]  

26. 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). [CrossRef]  

27. J. Li, L. Zhou, C. T. Chan, and P. Sheng, “Photonic band gap from a stack of positive and negative index materials,” Phys. Rev. Lett. 90(8), 083901 (2003). [CrossRef]   [PubMed]  

28. N. M. Litchinitser, A. I. Maimistov, I. R. Gabitov, R. Z. Sagdeev, and V. M. Shalaev, “Metamaterials: electromagnetic enhancement at zero-index transition,” Opt. Lett. 33(20), 2350–2352 (2008). [CrossRef]   [PubMed]  

References

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  1. D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, “Metallic phase with long-range orientational order and no translational symmetry,” Phys. Rev. Lett. 53(20), 1951–1953 (1984).
    [CrossRef]
  2. D. Levine and P. J. Steinhardt, “Quasicrystals: A new class of ordered structures,” Phys. Rev. Lett. 53(26), 2477–2480 (1984).
    [CrossRef]
  3. E. Abe, Y. Yan, and S. J. Pennycook, “Quasicrystals as cluster aggregates,” Nat. Mater. 3(11), 759–767 (2004).
    [CrossRef] [PubMed]
  4. J. Mikhael, J. Roth, L. Helden, and C. Bechinger, “Archimedean-like tiling on decagonal quasicrystalline surfaces,” Nature 454(7203), 501–504 (2008).
    [CrossRef] [PubMed]
  5. K. Ueda, T. Dotera, and T. Gemma, “Photonic band structure calculations of two-dimensional Archimedean tiling patterns,” Phys. Rev. B 75(19), 195122 (2007).
    [CrossRef]
  6. G. J. Parker, M. E. Zoorob, M. D. B. Charlton, J. J. Baumberg, and M. C. Netti, “Complete photonic bandgaps in 12-fold symmetric quasicrystals,” Nature 404(6779), 740–743 (2000).
    [CrossRef] [PubMed]
  7. M. Kohmoto, B. Sutherland, and K. Iguchi, “Localization of optics: Quasiperiodic media,” Phys. Rev. Lett. 58(23), 2436–2438 (1987).
    [CrossRef] [PubMed]
  8. W. Gellermann, M. Kohmoto, B. Sutherland, and P. C. Taylor, “Localization of light waves in Fibonacci dielectric multilayers,” Phys. Rev. Lett. 72(5), 633–636 (1994).
    [CrossRef] [PubMed]
  9. S. W. Wang, X. Chen, W. Lu, M. Li, and H. Wang, “Fractal independently tunable multichannel filters,” Appl. Phys. Lett. 90(21), 211113 (2007).
    [CrossRef]
  10. L. Dal Negro, C. J. Oton, Z. Gaburro, L. Pavesi, P. Johnson, A. Lagendijk, R. Righini, M. Colocci, and D. S. Wiersma, “Light transport through the band-edge states of Fibonacci quasicrystals,” Phys. Rev. Lett. 90(5), 055501 (2003).
    [CrossRef] [PubMed]
  11. W. J. Hsueh, S. J. Wun, and C. W. Tsao, “Branching features of photonic bandgaps in Fibonacci dielectric heterostructures,” Opt. Commun. 284(7), 1880–1886 (2011).
    [CrossRef]
  12. A. N. Poddubny and E. L. Ivchenko, “Photonic quasicrystalline and aperiodic structures,” Physica E 42(7), 1871–1895 (2010).
    [CrossRef]
  13. P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. 109(5), 1492–1505 (1958).
    [CrossRef]
  14. S. John, “Electromagnetic absorption in a disordered medium near a photon mobility edge,” Phys. Rev. Lett. 53(22), 2169–2172 (1984).
    [CrossRef]
  15. D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, “Localization of light in a disordered medium,” Nature 390(6661), 671–673 (1997).
    [CrossRef]
  16. Y. V. Shvyd’kol, S. Stoupin, A. Cunsolo, A. H. Said, and X. Huang, “High-reflectivity high-resolution X-ray crystal optics with diamonds,” Nat. Phys. 6, 196–199 (2010).
  17. J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science 264(5158), 553–556 (1994).
    [CrossRef] [PubMed]
  18. B. S. Williams, “Terahertz quantum-cascade lasers,” Nat. Photonics 1(9), 517–525 (2007).
    [CrossRef]
  19. Y. Taniyasu, M. Kasu, and T. Makimoto, “An aluminium nitride light-emitting diode with a wavelength of 210 nanometres,” Nature 441(7091), 325–328 (2006).
    [CrossRef] [PubMed]
  20. E. Yablonovitch, “Engineered omnidirectional external-reflectivity spectra from one-dimensional layered interference filters,” Opt. Lett. 23(21), 1648–1649 (1998).
    [CrossRef] [PubMed]
  21. Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282(5394), 1679–1682 (1998).
    [CrossRef] [PubMed]
  22. R. G. DeCorby, H. T. Nguyen, P. K. Dwivedi, and T. J. Clement, “Planar omnidirectional reflectors in chalcogenide glass and polymer,” Opt. Express 13(16), 6228–6233 (2005).
    [CrossRef] [PubMed]
  23. W. J. Hsueh and S. J. Wun, “Simple expressions for the maximum omnidirectional bandgap of bilayer photonic crystals,” Opt. Lett. 36(9), 1581–1583 (2011).
    [CrossRef] [PubMed]
  24. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
    [CrossRef] [PubMed]
  25. P. V. Parimi, W. T. Lu, P. Vodo, and S. Sridhar, “Photonic crystals: Imaging by flat lens using negative refraction,” Nature 426(6965), 404 (2003).
    [CrossRef] [PubMed]
  26. 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).
    [CrossRef]
  27. J. Li, L. Zhou, C. T. Chan, and P. Sheng, “Photonic band gap from a stack of positive and negative index materials,” Phys. Rev. Lett. 90(8), 083901 (2003).
    [CrossRef] [PubMed]
  28. N. M. Litchinitser, A. I. Maimistov, I. R. Gabitov, R. Z. Sagdeev, and V. M. Shalaev, “Metamaterials: electromagnetic enhancement at zero-index transition,” Opt. Lett. 33(20), 2350–2352 (2008).
    [CrossRef] [PubMed]

2011 (2)

W. J. Hsueh, S. J. Wun, and C. W. Tsao, “Branching features of photonic bandgaps in Fibonacci dielectric heterostructures,” Opt. Commun. 284(7), 1880–1886 (2011).
[CrossRef]

W. J. Hsueh and S. J. Wun, “Simple expressions for the maximum omnidirectional bandgap of bilayer photonic crystals,” Opt. Lett. 36(9), 1581–1583 (2011).
[CrossRef] [PubMed]

2010 (2)

A. N. Poddubny and E. L. Ivchenko, “Photonic quasicrystalline and aperiodic structures,” Physica E 42(7), 1871–1895 (2010).
[CrossRef]

Y. V. Shvyd’kol, S. Stoupin, A. Cunsolo, A. H. Said, and X. Huang, “High-reflectivity high-resolution X-ray crystal optics with diamonds,” Nat. Phys. 6, 196–199 (2010).

2008 (3)

J. Mikhael, J. Roth, L. Helden, and C. Bechinger, “Archimedean-like tiling on decagonal quasicrystalline surfaces,” Nature 454(7203), 501–504 (2008).
[CrossRef] [PubMed]

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).
[CrossRef]

N. M. Litchinitser, A. I. Maimistov, I. R. Gabitov, R. Z. Sagdeev, and V. M. Shalaev, “Metamaterials: electromagnetic enhancement at zero-index transition,” Opt. Lett. 33(20), 2350–2352 (2008).
[CrossRef] [PubMed]

2007 (3)

B. S. Williams, “Terahertz quantum-cascade lasers,” Nat. Photonics 1(9), 517–525 (2007).
[CrossRef]

K. Ueda, T. Dotera, and T. Gemma, “Photonic band structure calculations of two-dimensional Archimedean tiling patterns,” Phys. Rev. B 75(19), 195122 (2007).
[CrossRef]

S. W. Wang, X. Chen, W. Lu, M. Li, and H. Wang, “Fractal independently tunable multichannel filters,” Appl. Phys. Lett. 90(21), 211113 (2007).
[CrossRef]

2006 (1)

Y. Taniyasu, M. Kasu, and T. Makimoto, “An aluminium nitride light-emitting diode with a wavelength of 210 nanometres,” Nature 441(7091), 325–328 (2006).
[CrossRef] [PubMed]

2005 (1)

2004 (1)

E. Abe, Y. Yan, and S. J. Pennycook, “Quasicrystals as cluster aggregates,” Nat. Mater. 3(11), 759–767 (2004).
[CrossRef] [PubMed]

2003 (3)

L. Dal Negro, C. J. Oton, Z. Gaburro, L. Pavesi, P. Johnson, A. Lagendijk, R. Righini, M. Colocci, and D. S. Wiersma, “Light transport through the band-edge states of Fibonacci quasicrystals,” Phys. Rev. Lett. 90(5), 055501 (2003).
[CrossRef] [PubMed]

P. V. Parimi, W. T. Lu, P. Vodo, and S. Sridhar, “Photonic crystals: Imaging by flat lens using negative refraction,” Nature 426(6965), 404 (2003).
[CrossRef] [PubMed]

J. Li, L. Zhou, C. T. Chan, and P. Sheng, “Photonic band gap from a stack of positive and negative index materials,” Phys. Rev. Lett. 90(8), 083901 (2003).
[CrossRef] [PubMed]

2000 (2)

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
[CrossRef] [PubMed]

G. J. Parker, M. E. Zoorob, M. D. B. Charlton, J. J. Baumberg, and M. C. Netti, “Complete photonic bandgaps in 12-fold symmetric quasicrystals,” Nature 404(6779), 740–743 (2000).
[CrossRef] [PubMed]

1998 (2)

E. Yablonovitch, “Engineered omnidirectional external-reflectivity spectra from one-dimensional layered interference filters,” Opt. Lett. 23(21), 1648–1649 (1998).
[CrossRef] [PubMed]

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282(5394), 1679–1682 (1998).
[CrossRef] [PubMed]

1997 (1)

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, “Localization of light in a disordered medium,” Nature 390(6661), 671–673 (1997).
[CrossRef]

1994 (2)

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science 264(5158), 553–556 (1994).
[CrossRef] [PubMed]

W. Gellermann, M. Kohmoto, B. Sutherland, and P. C. Taylor, “Localization of light waves in Fibonacci dielectric multilayers,” Phys. Rev. Lett. 72(5), 633–636 (1994).
[CrossRef] [PubMed]

1987 (1)

M. Kohmoto, B. Sutherland, and K. Iguchi, “Localization of optics: Quasiperiodic media,” Phys. Rev. Lett. 58(23), 2436–2438 (1987).
[CrossRef] [PubMed]

1984 (3)

D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, “Metallic phase with long-range orientational order and no translational symmetry,” Phys. Rev. Lett. 53(20), 1951–1953 (1984).
[CrossRef]

D. Levine and P. J. Steinhardt, “Quasicrystals: A new class of ordered structures,” Phys. Rev. Lett. 53(26), 2477–2480 (1984).
[CrossRef]

S. John, “Electromagnetic absorption in a disordered medium near a photon mobility edge,” Phys. Rev. Lett. 53(22), 2169–2172 (1984).
[CrossRef]

1958 (1)

P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. 109(5), 1492–1505 (1958).
[CrossRef]

Abe, E.

E. Abe, Y. Yan, and S. J. Pennycook, “Quasicrystals as cluster aggregates,” Nat. Mater. 3(11), 759–767 (2004).
[CrossRef] [PubMed]

Anderson, P. W.

P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. 109(5), 1492–1505 (1958).
[CrossRef]

Bartolini, P.

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, “Localization of light in a disordered medium,” Nature 390(6661), 671–673 (1997).
[CrossRef]

Baumberg, J. J.

G. J. Parker, M. E. Zoorob, M. D. B. Charlton, J. J. Baumberg, and M. C. Netti, “Complete photonic bandgaps in 12-fold symmetric quasicrystals,” Nature 404(6779), 740–743 (2000).
[CrossRef] [PubMed]

Bechinger, C.

J. Mikhael, J. Roth, L. Helden, and C. Bechinger, “Archimedean-like tiling on decagonal quasicrystalline surfaces,” Nature 454(7203), 501–504 (2008).
[CrossRef] [PubMed]

Blech, I.

D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, “Metallic phase with long-range orientational order and no translational symmetry,” Phys. Rev. Lett. 53(20), 1951–1953 (1984).
[CrossRef]

Cahn, J. W.

D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, “Metallic phase with long-range orientational order and no translational symmetry,” Phys. Rev. Lett. 53(20), 1951–1953 (1984).
[CrossRef]

Capasso, F.

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science 264(5158), 553–556 (1994).
[CrossRef] [PubMed]

Chan, C. T.

J. Li, L. Zhou, C. T. Chan, and P. Sheng, “Photonic band gap from a stack of positive and negative index materials,” Phys. Rev. Lett. 90(8), 083901 (2003).
[CrossRef] [PubMed]

Charlton, M. D. B.

G. J. Parker, M. E. Zoorob, M. D. B. Charlton, J. J. Baumberg, and M. C. Netti, “Complete photonic bandgaps in 12-fold symmetric quasicrystals,” Nature 404(6779), 740–743 (2000).
[CrossRef] [PubMed]

Chen, C.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282(5394), 1679–1682 (1998).
[CrossRef] [PubMed]

Chen, C. H.

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).
[CrossRef]

Chen, C. T.

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).
[CrossRef]

Chen, X.

S. W. Wang, X. Chen, W. Lu, M. Li, and H. Wang, “Fractal independently tunable multichannel filters,” Appl. Phys. Lett. 90(21), 211113 (2007).
[CrossRef]

Cho, A. Y.

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science 264(5158), 553–556 (1994).
[CrossRef] [PubMed]

Clement, T. J.

Colocci, M.

L. Dal Negro, C. J. Oton, Z. Gaburro, L. Pavesi, P. Johnson, A. Lagendijk, R. Righini, M. Colocci, and D. S. Wiersma, “Light transport through the band-edge states of Fibonacci quasicrystals,” Phys. Rev. Lett. 90(5), 055501 (2003).
[CrossRef] [PubMed]

Cunsolo, A.

Y. V. Shvyd’kol, S. Stoupin, A. Cunsolo, A. H. Said, and X. Huang, “High-reflectivity high-resolution X-ray crystal optics with diamonds,” Nat. Phys. 6, 196–199 (2010).

Dal Negro, L.

L. Dal Negro, C. J. Oton, Z. Gaburro, L. Pavesi, P. Johnson, A. Lagendijk, R. Righini, M. Colocci, and D. S. Wiersma, “Light transport through the band-edge states of Fibonacci quasicrystals,” Phys. Rev. Lett. 90(5), 055501 (2003).
[CrossRef] [PubMed]

DeCorby, R. G.

Dotera, T.

K. Ueda, T. Dotera, and T. Gemma, “Photonic band structure calculations of two-dimensional Archimedean tiling patterns,” Phys. Rev. B 75(19), 195122 (2007).
[CrossRef]

Dwivedi, P. K.

Faist, J.

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science 264(5158), 553–556 (1994).
[CrossRef] [PubMed]

Fan, S.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282(5394), 1679–1682 (1998).
[CrossRef] [PubMed]

Fink, Y.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282(5394), 1679–1682 (1998).
[CrossRef] [PubMed]

Gabitov, I. R.

Gaburro, Z.

L. Dal Negro, C. J. Oton, Z. Gaburro, L. Pavesi, P. Johnson, A. Lagendijk, R. Righini, M. Colocci, and D. S. Wiersma, “Light transport through the band-edge states of Fibonacci quasicrystals,” Phys. Rev. Lett. 90(5), 055501 (2003).
[CrossRef] [PubMed]

Gellermann, W.

W. Gellermann, M. Kohmoto, B. Sutherland, and P. C. Taylor, “Localization of light waves in Fibonacci dielectric multilayers,” Phys. Rev. Lett. 72(5), 633–636 (1994).
[CrossRef] [PubMed]

Gemma, T.

K. Ueda, T. Dotera, and T. Gemma, “Photonic band structure calculations of two-dimensional Archimedean tiling patterns,” Phys. Rev. B 75(19), 195122 (2007).
[CrossRef]

Gratias, D.

D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, “Metallic phase with long-range orientational order and no translational symmetry,” Phys. Rev. Lett. 53(20), 1951–1953 (1984).
[CrossRef]

Helden, L.

J. Mikhael, J. Roth, L. Helden, and C. Bechinger, “Archimedean-like tiling on decagonal quasicrystalline surfaces,” Nature 454(7203), 501–504 (2008).
[CrossRef] [PubMed]

Hsueh, W. J.

W. J. Hsueh, S. J. Wun, and C. W. Tsao, “Branching features of photonic bandgaps in Fibonacci dielectric heterostructures,” Opt. Commun. 284(7), 1880–1886 (2011).
[CrossRef]

W. J. Hsueh and S. J. Wun, “Simple expressions for the maximum omnidirectional bandgap of bilayer photonic crystals,” Opt. Lett. 36(9), 1581–1583 (2011).
[CrossRef] [PubMed]

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).
[CrossRef]

Huang, X.

Y. V. Shvyd’kol, S. Stoupin, A. Cunsolo, A. H. Said, and X. Huang, “High-reflectivity high-resolution X-ray crystal optics with diamonds,” Nat. Phys. 6, 196–199 (2010).

Hutchinson, A. L.

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science 264(5158), 553–556 (1994).
[CrossRef] [PubMed]

Iguchi, K.

M. Kohmoto, B. Sutherland, and K. Iguchi, “Localization of optics: Quasiperiodic media,” Phys. Rev. Lett. 58(23), 2436–2438 (1987).
[CrossRef] [PubMed]

Ivchenko, E. L.

A. N. Poddubny and E. L. Ivchenko, “Photonic quasicrystalline and aperiodic structures,” Physica E 42(7), 1871–1895 (2010).
[CrossRef]

Joannopoulos, J. D.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282(5394), 1679–1682 (1998).
[CrossRef] [PubMed]

John, S.

S. John, “Electromagnetic absorption in a disordered medium near a photon mobility edge,” Phys. Rev. Lett. 53(22), 2169–2172 (1984).
[CrossRef]

Johnson, P.

L. Dal Negro, C. J. Oton, Z. Gaburro, L. Pavesi, P. Johnson, A. Lagendijk, R. Righini, M. Colocci, and D. S. Wiersma, “Light transport through the band-edge states of Fibonacci quasicrystals,” Phys. Rev. Lett. 90(5), 055501 (2003).
[CrossRef] [PubMed]

Kasu, M.

Y. Taniyasu, M. Kasu, and T. Makimoto, “An aluminium nitride light-emitting diode with a wavelength of 210 nanometres,” Nature 441(7091), 325–328 (2006).
[CrossRef] [PubMed]

Kohmoto, M.

W. Gellermann, M. Kohmoto, B. Sutherland, and P. C. Taylor, “Localization of light waves in Fibonacci dielectric multilayers,” Phys. Rev. Lett. 72(5), 633–636 (1994).
[CrossRef] [PubMed]

M. Kohmoto, B. Sutherland, and K. Iguchi, “Localization of optics: Quasiperiodic media,” Phys. Rev. Lett. 58(23), 2436–2438 (1987).
[CrossRef] [PubMed]

Lagendijk, A.

L. Dal Negro, C. J. Oton, Z. Gaburro, L. Pavesi, P. Johnson, A. Lagendijk, R. Righini, M. Colocci, and D. S. Wiersma, “Light transport through the band-edge states of Fibonacci quasicrystals,” Phys. Rev. Lett. 90(5), 055501 (2003).
[CrossRef] [PubMed]

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, “Localization of light in a disordered medium,” Nature 390(6661), 671–673 (1997).
[CrossRef]

Levine, D.

D. Levine and P. J. Steinhardt, “Quasicrystals: A new class of ordered structures,” Phys. Rev. Lett. 53(26), 2477–2480 (1984).
[CrossRef]

Li, J.

J. Li, L. Zhou, C. T. Chan, and P. Sheng, “Photonic band gap from a stack of positive and negative index materials,” Phys. Rev. Lett. 90(8), 083901 (2003).
[CrossRef] [PubMed]

Li, M.

S. W. Wang, X. Chen, W. Lu, M. Li, and H. Wang, “Fractal independently tunable multichannel filters,” Appl. Phys. Lett. 90(21), 211113 (2007).
[CrossRef]

Litchinitser, N. M.

Lu, W.

S. W. Wang, X. Chen, W. Lu, M. Li, and H. Wang, “Fractal independently tunable multichannel filters,” Appl. Phys. Lett. 90(21), 211113 (2007).
[CrossRef]

Lu, W. T.

P. V. Parimi, W. T. Lu, P. Vodo, and S. Sridhar, “Photonic crystals: Imaging by flat lens using negative refraction,” Nature 426(6965), 404 (2003).
[CrossRef] [PubMed]

Maimistov, A. I.

Makimoto, T.

Y. Taniyasu, M. Kasu, and T. Makimoto, “An aluminium nitride light-emitting diode with a wavelength of 210 nanometres,” Nature 441(7091), 325–328 (2006).
[CrossRef] [PubMed]

Michel, J.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282(5394), 1679–1682 (1998).
[CrossRef] [PubMed]

Mikhael, J.

J. Mikhael, J. Roth, L. Helden, and C. Bechinger, “Archimedean-like tiling on decagonal quasicrystalline surfaces,” Nature 454(7203), 501–504 (2008).
[CrossRef] [PubMed]

Netti, M. C.

G. J. Parker, M. E. Zoorob, M. D. B. Charlton, J. J. Baumberg, and M. C. Netti, “Complete photonic bandgaps in 12-fold symmetric quasicrystals,” Nature 404(6779), 740–743 (2000).
[CrossRef] [PubMed]

Nguyen, H. T.

Oton, C. J.

L. Dal Negro, C. J. Oton, Z. Gaburro, L. Pavesi, P. Johnson, A. Lagendijk, R. Righini, M. Colocci, and D. S. Wiersma, “Light transport through the band-edge states of Fibonacci quasicrystals,” Phys. Rev. Lett. 90(5), 055501 (2003).
[CrossRef] [PubMed]

Parimi, P. V.

P. V. Parimi, W. T. Lu, P. Vodo, and S. Sridhar, “Photonic crystals: Imaging by flat lens using negative refraction,” Nature 426(6965), 404 (2003).
[CrossRef] [PubMed]

Parker, G. J.

G. J. Parker, M. E. Zoorob, M. D. B. Charlton, J. J. Baumberg, and M. C. Netti, “Complete photonic bandgaps in 12-fold symmetric quasicrystals,” Nature 404(6779), 740–743 (2000).
[CrossRef] [PubMed]

Pavesi, L.

L. Dal Negro, C. J. Oton, Z. Gaburro, L. Pavesi, P. Johnson, A. Lagendijk, R. Righini, M. Colocci, and D. S. Wiersma, “Light transport through the band-edge states of Fibonacci quasicrystals,” Phys. Rev. Lett. 90(5), 055501 (2003).
[CrossRef] [PubMed]

Pendry, J. B.

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
[CrossRef] [PubMed]

Pennycook, S. J.

E. Abe, Y. Yan, and S. J. Pennycook, “Quasicrystals as cluster aggregates,” Nat. Mater. 3(11), 759–767 (2004).
[CrossRef] [PubMed]

Poddubny, A. N.

A. N. Poddubny and E. L. Ivchenko, “Photonic quasicrystalline and aperiodic structures,” Physica E 42(7), 1871–1895 (2010).
[CrossRef]

Righini, R.

L. Dal Negro, C. J. Oton, Z. Gaburro, L. Pavesi, P. Johnson, A. Lagendijk, R. Righini, M. Colocci, and D. S. Wiersma, “Light transport through the band-edge states of Fibonacci quasicrystals,” Phys. Rev. Lett. 90(5), 055501 (2003).
[CrossRef] [PubMed]

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, “Localization of light in a disordered medium,” Nature 390(6661), 671–673 (1997).
[CrossRef]

Roth, J.

J. Mikhael, J. Roth, L. Helden, and C. Bechinger, “Archimedean-like tiling on decagonal quasicrystalline surfaces,” Nature 454(7203), 501–504 (2008).
[CrossRef] [PubMed]

Sagdeev, R. Z.

Said, A. H.

Y. V. Shvyd’kol, S. Stoupin, A. Cunsolo, A. H. Said, and X. Huang, “High-reflectivity high-resolution X-ray crystal optics with diamonds,” Nat. Phys. 6, 196–199 (2010).

Shalaev, V. M.

Shechtman, D.

D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, “Metallic phase with long-range orientational order and no translational symmetry,” Phys. Rev. Lett. 53(20), 1951–1953 (1984).
[CrossRef]

Sheng, P.

J. Li, L. Zhou, C. T. Chan, and P. Sheng, “Photonic band gap from a stack of positive and negative index materials,” Phys. Rev. Lett. 90(8), 083901 (2003).
[CrossRef] [PubMed]

Shvyd’kol, Y. V.

Y. V. Shvyd’kol, S. Stoupin, A. Cunsolo, A. H. Said, and X. Huang, “High-reflectivity high-resolution X-ray crystal optics with diamonds,” Nat. Phys. 6, 196–199 (2010).

Sirtori, C.

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science 264(5158), 553–556 (1994).
[CrossRef] [PubMed]

Sivco, D. L.

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science 264(5158), 553–556 (1994).
[CrossRef] [PubMed]

Sridhar, S.

P. V. Parimi, W. T. Lu, P. Vodo, and S. Sridhar, “Photonic crystals: Imaging by flat lens using negative refraction,” Nature 426(6965), 404 (2003).
[CrossRef] [PubMed]

Steinhardt, P. J.

D. Levine and P. J. Steinhardt, “Quasicrystals: A new class of ordered structures,” Phys. Rev. Lett. 53(26), 2477–2480 (1984).
[CrossRef]

Stoupin, S.

Y. V. Shvyd’kol, S. Stoupin, A. Cunsolo, A. H. Said, and X. Huang, “High-reflectivity high-resolution X-ray crystal optics with diamonds,” Nat. Phys. 6, 196–199 (2010).

Sutherland, B.

W. Gellermann, M. Kohmoto, B. Sutherland, and P. C. Taylor, “Localization of light waves in Fibonacci dielectric multilayers,” Phys. Rev. Lett. 72(5), 633–636 (1994).
[CrossRef] [PubMed]

M. Kohmoto, B. Sutherland, and K. Iguchi, “Localization of optics: Quasiperiodic media,” Phys. Rev. Lett. 58(23), 2436–2438 (1987).
[CrossRef] [PubMed]

Taniyasu, Y.

Y. Taniyasu, M. Kasu, and T. Makimoto, “An aluminium nitride light-emitting diode with a wavelength of 210 nanometres,” Nature 441(7091), 325–328 (2006).
[CrossRef] [PubMed]

Taylor, P. C.

W. Gellermann, M. Kohmoto, B. Sutherland, and P. C. Taylor, “Localization of light waves in Fibonacci dielectric multilayers,” Phys. Rev. Lett. 72(5), 633–636 (1994).
[CrossRef] [PubMed]

Thomas, E. L.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282(5394), 1679–1682 (1998).
[CrossRef] [PubMed]

Tsao, C. W.

W. J. Hsueh, S. J. Wun, and C. W. Tsao, “Branching features of photonic bandgaps in Fibonacci dielectric heterostructures,” Opt. Commun. 284(7), 1880–1886 (2011).
[CrossRef]

Ueda, K.

K. Ueda, T. Dotera, and T. Gemma, “Photonic band structure calculations of two-dimensional Archimedean tiling patterns,” Phys. Rev. B 75(19), 195122 (2007).
[CrossRef]

Vodo, P.

P. V. Parimi, W. T. Lu, P. Vodo, and S. Sridhar, “Photonic crystals: Imaging by flat lens using negative refraction,” Nature 426(6965), 404 (2003).
[CrossRef] [PubMed]

Wang, H.

S. W. Wang, X. Chen, W. Lu, M. Li, and H. Wang, “Fractal independently tunable multichannel filters,” Appl. Phys. Lett. 90(21), 211113 (2007).
[CrossRef]

Wang, S. W.

S. W. Wang, X. Chen, W. Lu, M. Li, and H. Wang, “Fractal independently tunable multichannel filters,” Appl. Phys. Lett. 90(21), 211113 (2007).
[CrossRef]

Wiersma, D. S.

L. Dal Negro, C. J. Oton, Z. Gaburro, L. Pavesi, P. Johnson, A. Lagendijk, R. Righini, M. Colocci, and D. S. Wiersma, “Light transport through the band-edge states of Fibonacci quasicrystals,” Phys. Rev. Lett. 90(5), 055501 (2003).
[CrossRef] [PubMed]

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, “Localization of light in a disordered medium,” Nature 390(6661), 671–673 (1997).
[CrossRef]

Williams, B. S.

B. S. Williams, “Terahertz quantum-cascade lasers,” Nat. Photonics 1(9), 517–525 (2007).
[CrossRef]

Winn, J. N.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282(5394), 1679–1682 (1998).
[CrossRef] [PubMed]

Wun, S. J.

W. J. Hsueh, S. J. Wun, and C. W. Tsao, “Branching features of photonic bandgaps in Fibonacci dielectric heterostructures,” Opt. Commun. 284(7), 1880–1886 (2011).
[CrossRef]

W. J. Hsueh and S. J. Wun, “Simple expressions for the maximum omnidirectional bandgap of bilayer photonic crystals,” Opt. Lett. 36(9), 1581–1583 (2011).
[CrossRef] [PubMed]

Yablonovitch, E.

Yan, Y.

E. Abe, Y. Yan, and S. J. Pennycook, “Quasicrystals as cluster aggregates,” Nat. Mater. 3(11), 759–767 (2004).
[CrossRef] [PubMed]

Zhou, L.

J. Li, L. Zhou, C. T. Chan, and P. Sheng, “Photonic band gap from a stack of positive and negative index materials,” Phys. Rev. Lett. 90(8), 083901 (2003).
[CrossRef] [PubMed]

Zoorob, M. E.

G. J. Parker, M. E. Zoorob, M. D. B. Charlton, J. J. Baumberg, and M. C. Netti, “Complete photonic bandgaps in 12-fold symmetric quasicrystals,” Nature 404(6779), 740–743 (2000).
[CrossRef] [PubMed]

Appl. Phys. Lett. (1)

S. W. Wang, X. Chen, W. Lu, M. Li, and H. Wang, “Fractal independently tunable multichannel filters,” Appl. Phys. Lett. 90(21), 211113 (2007).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

(a) Photonic bandgap of the normal incidence, TE 45°, and TM 45° in the FSL for generation orders v = 2 to 8. The red, blue and green thick lines correspond to the allowed bands for the normal incidence, TE 45°, and TM 45°, respectively. The arrow signs indicate the major gaps. Parameters of the system are nA = 2.0, nB = 4.0, dA = 0.55μm, and dB = 0.45μm. The normalized frequency Ω is defined by Ω=ωD/(2πc), where D = dA + dB. (b) The transmission spectra of a single cell of the FSL with v = 4, 6 and 8 for normal incidence from air. The gray areas correspond to the region of the major gaps shown in (a).

Fig. 2
Fig. 2

Effective Bragg condition, major gaps and their centers in the gap map of the FSL. (a) The gap map of the normal incidence in the S8 FSL with nA = 2.0, nB = 4.0. The gray areas mark the major gaps. The green lines present the bandedges. The red lines mark the midline of the major gap holes, MMGs, corresponding to the effective Bragg conditions. The x signs indicate the center of the major gaps. The blue and black dashed lines mark the zero-gap lines, LA,m and LB,n, respectively. (b) the thickness filling factor and (c) the normalized frequency, and (d) the gapwidth to gap center ratio of the CLGs in regions (0,0), (1,0), (0,1) and (1,1) of the gap map for nA = 2.0 and 6.0> nB>3.0. The solid and dashed lines correspond to the maximum gaps and the CLGs, respectively.

Fig. 3
Fig. 3

Omni-gaps and transmission spectra of the FSL. (a) Sketch of the overlap among the major gaps for NI, TM1, and TE1 states and omni-gap in the gap map. The gray region corresponds to the omni-gap. (b) The transmission spectra of the S8 FSL with nA = 2.0, nB = 4.0 at the CFOG for 0°, 45°, 85° of TE and TM polarizations. The gray region corresponds to the omni-gap. (c) The gapwidth to gap center ratios of the CFOG of the S8 FSL for nA = 1.5, 2.0, 3.0, and 3.0< nB<6.0. The dashed and solid lines correspond to the gapwidth to gap center ratio of the CFOG and the maximum omnidirectional gap, respectively.

Equations (7)

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F= pσ q+pσ , Ω= pσ+q 2 n B cos θ B ,
d A n A cos θ A + τ v2,v1 d B n B cos θ B =(m+1+ τ v2,v1 n) λ 0 /2,
d A n A cos θ A + τ v2,v1 d B n B cos θ B =[m+ τ v2,v1 (n+1)] λ 0 /2.
F m,n (CLG) = [1+ (2n+1) n A cos θ A (2m+ τ v2,v1 ) n B cos θ B ] 1 ,
Ω m,n (CLG) = 1 4 [ (2m+ τ v2,v1 ) n A cos θ A + (2n+1) n B cos θ B ].
F m,n (CHG) = [1+ (2n+1) n A cos θ A (2m+1+ τ v3,v1 ) n B cos θ B ] 1 ,
Ω m,n (CHG) = 1 4 [ (2m+1+ τ v3,v1 ) n A cos θ A + (2n+1) n B cos θ B ].

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