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Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 2 — Jan. 27, 2014
  • pp: 2007–2012
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Resonant modes of 12-fold symmetric defect free photonic quasicrystal

Minfeng Chen, Yun-Jing Li, Yuh-Jen Cheng, Yai-Chung Chang, and Chun-Yen Chang  »View Author Affiliations


Optics Express, Vol. 22, Issue 2, pp. 2007-2012 (2014)
http://dx.doi.org/10.1364/OE.22.002007


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Abstract

This work investigates the resonant modes of a 12-fold symmetric defect free photonic quasicrystal (PQC) nanorod array using finite difference time domain (FDTD) simulation. Localized modes can exist in PQC without introducing defects due to the lack of translational symmetry. The resonant modes of the unit cell PQC and the one time expanded PQC from unit cell are systematically examined. The resonant spectrum is that of a single rod modified by the interaction among PQC nanorods. The mode confinement is contributed by guided resonance and destructive interference scattering. The self-scaling similarity of resonant spectrum and mode profile are also investigated.

© 2014 Optical Society of America

1. Introduction

Photonic quasicrystals (PQCs) have generated substantial research interests due to their fascinating properties [1

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

9

9. K. Mnaymneh and R. C. Gauthier, “Mode localization and band-gap formation in defect-free photonic quasicrystals,” Opt. Express 15(8), 5089–5099 (2007). [CrossRef] [PubMed]

]. Unlike photonic crystals, PQCs do not have a periodic structure but follows simple deterministic generation rules. The generated patterns have a long range order and lie somewhere between disorder and periodic structures. They have a high level of rotational symmetry which periodic structures cannot have. Due to the lack of translational symmetry, PQCs do not support extended Bloch waves as in photonic crystals (PCs). The optical modes are critically localized, which means modes are confined in space but decay more weakly than exponentially [10

10. G. Gumbs and M. K. Ali, “Dynamical maps, Cantor spectra, and localization for Fibonacci and related quasiperiodic lattices,” Phys. Rev. Lett. 60(11), 1081–1084 (1988). [CrossRef] [PubMed]

]. Another interesting property of PQC structures is their scaling symmetry, in which the basic quasicrystal pattern repeats itself as the size of the structure expands by geometric scaling factors. There have been interests in exploring the use of PQCs in microcavity and distributed feedback laser applications [11

11. M. Notomi, H. Suzuki, T. Tamamura, and K. Edagawa, “Lasing Action due to the Two-Dimensional Quasiperiodicity of Photonic Quasicrystals with a Penrose Lattice,” Phys. Rev. Lett. 92(12), 123906 (2004). [CrossRef] [PubMed]

17

17. S.-K. Kim, J.-H. Lee, S.-H. Kim, I.-K. Hwang, Y.-H. Lee, and S.-B. Kim, “Photonic quasicrystal single-cell cavity mode,” Appl. Phys. Lett. 86(3), 031101 (2005). [CrossRef]

]. Lasing can be observed from optically focused pumped PQCs without the need of introducing defects [11

11. M. Notomi, H. Suzuki, T. Tamamura, and K. Edagawa, “Lasing Action due to the Two-Dimensional Quasiperiodicity of Photonic Quasicrystals with a Penrose Lattice,” Phys. Rev. Lett. 92(12), 123906 (2004). [CrossRef] [PubMed]

18

18. S. P. Chang, K. P. Sou, C. H. Chen, Y. J. Cheng, J. K. Huang, C. H. Lin, H. C. Kuo, C. Y. Chang, and W. F. Hsieh, “Lasing action in gallium nitride quasicrystal nanorod arrays,” Opt. Express 20(11), 12457–12462 (2012). [CrossRef] [PubMed]

]. This is because localized modes inherently exist in PQCs. The resonant modes of PQC can have fairly complicated mode patterns and resonant peak distribution, which have not been well explored.

Here we report a systematic study of the resonant modes of a 12-fold symmetric defect free PQC. We are interested in the spectral structure and mode pattern. Conventionally, air holes are often placed at the vertices of a lattice pattern. We place pillars at the vertices instead. This allows us to study the changes of the mode field as the structure expands from a single rod to various PQC sizes. We have also investigated the mode localization and scaling symmetry property.

2. Structure model

Figure 1(a)
Fig. 1 (a) Construction of the 12 fold symmetric PQC structure. (b) Illustration of hexagonal rods placed at the PQC unit cell lattice vertices (blue line). Different source locations are used to excite resonant modes. (c) SEM plane view image of the fabricated sample with the PQC lattice pattern overlaid to show its PQC structure.
illustrates the lattice structure of the 12 fold symmetric PQC. It is composed by the vertices of a square-triangle tiling, which is generated from a dodecagonal seed structure by using the Stamfli inflation rule [19

19. M. Oxborrow and C. L. Henley, “Random square-triangle tilings: A model for twelvefold-symmetric quasicrystals,” Phys. Rev. B Condens. Matter 48(10), 6966–6998 (1993). [CrossRef] [PubMed]

]. The dodecagonal seed structure (unit cell) is shown in Fig. 1(a), marked in black circle. The seed structure is then expanded by an inflation factor of 2+3, as show in the dashed red lines. The unit cell is duplicated at each vertex of the inflated dashed red lines. For the convenience of reference, we call this one time expanded structure the first off-spring cell. Figure 1(a) shows a partially filled first off-spring cell, where unit cell has not been duplicated at the most outer vertices. The scaling similarity is inherent in the construction rule as can be seen by comparing the dashed red lines and the blue-line unit cell at the center. The square-triangle tiled dodecagon structure repeats itself as the size expands by the inflation factor. The construction can be iterated till filling the desired space. Figure 1(b) is a schematic of the unit cell with hexagonal pillars placed at the vertices, which consists of 1 central rod + 6 surrounding rods + 12 outer most rods. The dots are the point sources used in simulation to excite the resonant modes of the PQC. The two additional pillars added at the top are only for illustration purpose to describe a point source (4) used in resonant mode calculation for the first off-spring cell. The hexagon is chosen due to the natural crystalline structure of the GaN material used to fabricate a PQC sample. The sample was fabricated from a GaN substrate by patterned top-down etch and epitaxial regrowth [18

18. S. P. Chang, K. P. Sou, C. H. Chen, Y. J. Cheng, J. K. Huang, C. H. Lin, H. C. Kuo, C. Y. Chang, and W. F. Hsieh, “Lasing action in gallium nitride quasicrystal nanorod arrays,” Opt. Express 20(11), 12457–12462 (2012). [CrossRef] [PubMed]

]. Multiple quantum wells (MQWs) were grown on the nanorod surface. The photoluminescent (PL) emission of MQWs is around 460 nm. Figure 1(c) shows a SEM plane view of a fabricated sample. The PQC lattice pattern is overlaid on top for visual guidance. Without the overlaid guidance, the pattern looks almost totally random.

The dimension used in the simulation is as follows. The width of the hexagonal pillar sidewall is 340 nm. The center-to-center pillar spacing is 750 nm. The refractive index of GaN is taken to be 2.4. The fabricated pillar is about 1.2 μm in length standing on a GaN substrate. For simplicity, the simulation is carried out by two-dimensional Finite-Difference-Time-Domain (FDTD) method. Here, we are mainly interested in the general characteristics of the modes of PQC. The finite pillar length is regarded as a second order modification to the simulation results. The electric field polarized along the pillar axis is used in simulation. It is selected because the reflection for electric field parallel to pillar axis is stronger than the perpendicular one. Modes in this polarization experience a stronger confinement by the pillars and therefore are the modes of interest.

3. Simulation results and discussion

We first study the scattering properties of the PQC by calculating its reciprocal pattern, which is obtained by 2-D Fourier transform of the PQC lattice points. Here, we have only calculated finite size PQCs. The calculated results for a unit cell, the first off-spring cell, and the second off-spring cell are shown in Figs. 2(a)
Fig. 2 The reciprocal pattern of (a) a unit cell, (b) the first off-spring cell, and (c) the second off-spring cell.
2(c), respectively. The reciprocal pattern shows 12-fold rotational symmetry as expected. The pattern complexity increases dramatically as the PQC structure expands. The first off-spring cell is obtained by inflating the unit cell lattice structure and duplicating the unit cell at every vertices of the inflated structure. By the nature of this construction rule, the reciprocal pattern (Fourier transform) of the first off-spring cell is equal to that of the unit cell [(Fig. 2(a))] multiplied by the reduced copy of itself by the inflation factor. This leads to the complicated fine structures as shown in Fig. 2(b). Figure 2(b) carries similar major spots shown in Fig. 2(a), however, with additional fine structures. The same rule is also applicable to Fig. 2(c) with respect to Fig. 2(b).

We then investigate the resonant modes of the PQC pattern. We start with a unit cell. A point source is used to probe the resonant modes and spectrum. A short Gaussian pulse width of 2 fs centered at 450 nm is launched at source locations 1, 2, 3, as shown in Fig. 1(b), and an additional one slightly offset from location 1 (not shown). The 2 fs short pulse width ensures a very broad band excitation. A long enough propagation time is used in the simulation to obtain a steady state E-field distribution. The spectrum is obtained by performing the Fourier transform of the recorded time-domain signal for both electric and magnetic fields, followed by Poynting flux computation. Figure 3(a)
Fig. 3 The calculated resonant spectra of the unit cell obtained from (a) various source locations and (b) various lattice constants. (c) The resonant spectrum of a single rod.
shows the calculated spectra, where the blue, red, green, and gray lines are obtained from the source locations 1, 2, 3, and the offset one, respectively. The resonant peaks do not show obvious regularity. The peaks also changes for different excitation source locations. The center source 1 generates the strongest intensity peaks than the off-center sources. This is because the center source automatically satisfies the rotational symmetry of PQC, therefore can excite the resonant modes better. We consider all the distinct peaks as the resonant modes. To confirm this argument, we also carried out simulations by only changing the lattice constant to 735 and 772 nm using the center source 1. The resonant peaks redshift with increasing rod spacing, as shown in Fig. 3(b). This scaling property with lattice spacing is the evidence of QPC resonance. The change in spectral shape is because the rod dimension and the pulse were not scaled proportionally. To understand the origin of these peaks, we have also calculated the resonant spectrum of a single hexagonal rod by launching an excitation source at its center. Figure 3(c) shows the calculated spectrum. It consists of a broad resonance at 460 nm and a narrow one at 543 nm, which correspond to a fairly lossy and a well confined mode, respectively. The lossy nature of 460 nm resonance implies that it can significantly interact with the resonant modes of the surrounding rods. This results in the multiple resonant peaks around 460 nm in Fig. 3(a). The narrow 543 nm resonance on the other hand is less affected by its surrounding rods due to its well confined nature, therefore keeping its line shape almost unchanged with only very small side peaks added in Fig. 3(a). This may also explain the less position change of 543 nm peak as compared with other peaks in Fig. 3(b). The cavity quality factor of the 460 nm resonance of a single rod is 12. This Q factor is enhanced to 65 for the resonant modes in the unit cell.

Figure 6(a)
Fig. 6 (a) and (c) are the 458 nm and 494 nm resonant mode profiles of the first off-spring cells, respectively. (b) and (d) are respectively the enlarged center portion of (a) and (c). The overlaid dashed red line circles are the unit cells for visual guidance.
shows the mode profile of the λ = 458 nm resonance of the first off-spring cell, obtained by the same calculation approach mentioned previously. The rod contours are overlaid on the mode profile for visual reference. The dashed red line circles indicate the unit cell locations. The high intensity part is mostly confined in the center region consisting of a center rod plus surrounding six rods, as shown in the enlarged Fig. 6(b). Compared with Fig. 4(a), mode pattern similar to the unit cell is maintained. The additional six unit cells surrounding the center one provides better mode confinement by channeling the optical energy back to the center region. It is interesting to see the pseudo wave guiding effect formed by the surrounding rods. Figure 6(c) shows the mode profile of the λ = 494 nm resonance. Figure 6(d) is its enlarged center portion. There is no pseudo wave guiding. Instead, the surrounding rods provide the mode confinement by destructive interference like random scattering.

We now turn to the self scaling similarity property, which is inherent in the inflation construction rule. To test this property, the center wavelength of the broad band excitation source was redshifted to ~1300 nm. The results are depicted in Fig. 7
Fig. 7 (a) The calculated spectrum from 1100 to 1700 nm. (b) The calculated mode profile. The large hexagons represent the locations of unit cells.
. The resonant spectral pattern shown in Fig. 7(a) is similar to that of the unit cell shown in Fig. 3(a). The 1285 nm and 1398 nm peaks correspond to the 458 nm and 494 nm peaks in the unit cell, respectively. Figure 7(b) shows the calculated mode pattern of the λ = 1285 nm peak. The mode profile is similar to that of the unit cell shown in Fig. 4(a). The individual rod dimension is negligible at this wavelength. The rods collective together in a unit cell structure, denoted by the large hexagons, determines the mode pattern as individual rod does in Fig. 4(a). This demonstrates the interesting self-scaling similarity property inherent in the PQC construction rule.

4. Summary

PQC represents an interesting class of pattern that is between ordered and completely random structure. Localized resonant modes can exist without introducing any defect due to lack of translational symmetry. The resonant mode can be well confined in the first-offspring cell, which is just the first expansion from the unit cell. The resonant spectrum can be viewed as that of a single rod modified by the scattering interaction among rods in PQC pattern. The interaction can form pseudo wave guiding channels and destructive interference like random scattering to provide the mode confinement. The self scaling similarity inherent in the PQC construction rule is also demonstrated in the resonant mode and spectrum calculation.

Acknowledgments

We thank the National Science Council of Taiwan for the support of this research under contract NSC 100-2112-M-001-021-MY3 and Sinica nano project.

References and links

1.

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

2.

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]

3.

T. Hattori, N. Tsurumachi, S. Kawato, and H. Nakatsuka, “Photonic dispersion relation in a one-dimensional quasicrystal,” Phys. Rev. B Condens. Matter 50(6), 4220–4223 (1994). [CrossRef] [PubMed]

4.

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]

5.

W. Man, M. Megens, P. J. Steinhardt, and P. M. Chaikin, “Experimental measurement of the photonic properties of icosahedral quasicrystals,” Nature 436(7053), 993–996 (2005). [CrossRef] [PubMed]

6.

A. Ledermann, L. Cademartiri, M. Hermatschweiler, C. Toninelli, G. A. Ozin, D. S. Wiersma, M. Wegener, and G. von Freymann, “Three-dimensional silicon inverse photonic quasicrystals for infrared wavelengths,” Nat. Mater. 5(12), 942–945 (2006). [CrossRef] [PubMed]

7.

T. Matsui, A. Agrawal, A. Nahata, and Z. V. Vardeny, “Transmission resonances through aperiodic arrays of subwavelength apertures,” Nature 446(7135), 517–521 (2007). [CrossRef] [PubMed]

8.

S. Walter and S.-W. Daniel, “Photonic and phononic quasicrystals,” J. Phys. D Appl. Phys. 40(13), R229–R247 (2007). [CrossRef]

9.

K. Mnaymneh and R. C. Gauthier, “Mode localization and band-gap formation in defect-free photonic quasicrystals,” Opt. Express 15(8), 5089–5099 (2007). [CrossRef] [PubMed]

10.

G. Gumbs and M. K. Ali, “Dynamical maps, Cantor spectra, and localization for Fibonacci and related quasiperiodic lattices,” Phys. Rev. Lett. 60(11), 1081–1084 (1988). [CrossRef] [PubMed]

11.

M. Notomi, H. Suzuki, T. Tamamura, and K. Edagawa, “Lasing Action due to the Two-Dimensional Quasiperiodicity of Photonic Quasicrystals with a Penrose Lattice,” Phys. Rev. Lett. 92(12), 123906 (2004). [CrossRef] [PubMed]

12.

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). [CrossRef] [PubMed]

13.

K. Nozaki and T. Baba, “Lasing Characteristics of 12-Fold Symmetric Quasi-periodic Photonic Crystal Slab Nanolasers,” Jpn. J. Appl. Phys. 45(8A), 6087–6090 (2006). [CrossRef]

14.

K. Nozaki and T. Baba, “Quasiperiodic photonic crystal microcavity lasers,” Appl. Phys. Lett. 84(24), 4875–4877 (2004). [CrossRef]

15.

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). [CrossRef] [PubMed]

16.

L. Mahler, A. Tredicucci, F. Beltram, C. Walther, J. Faist, H. E. Beere, D. A. Ritchie, and D. S. Wiersma, “Quasi-periodic distributed feedback laser,” Nat. Photonics 4(3), 165–169 (2010). [CrossRef]

17.

S.-K. Kim, J.-H. Lee, S.-H. Kim, I.-K. Hwang, Y.-H. Lee, and S.-B. Kim, “Photonic quasicrystal single-cell cavity mode,” Appl. Phys. Lett. 86(3), 031101 (2005). [CrossRef]

18.

S. P. Chang, K. P. Sou, C. H. Chen, Y. J. Cheng, J. K. Huang, C. H. Lin, H. C. Kuo, C. Y. Chang, and W. F. Hsieh, “Lasing action in gallium nitride quasicrystal nanorod arrays,” Opt. Express 20(11), 12457–12462 (2012). [CrossRef] [PubMed]

19.

M. Oxborrow and C. L. Henley, “Random square-triangle tilings: A model for twelvefold-symmetric quasicrystals,” Phys. Rev. B Condens. Matter 48(10), 6966–6998 (1993). [CrossRef] [PubMed]

20.

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60(8), 5751–5758 (1999). [CrossRef]

OCIS Codes
(030.4070) Coherence and statistical optics : Modes
(160.5298) Materials : Photonic crystals

ToC Category:
Photonic Crystals

History
Original Manuscript: September 17, 2013
Revised Manuscript: December 13, 2013
Manuscript Accepted: January 7, 2014
Published: January 23, 2014

Citation
Minfeng Chen, Yun-Jing Li, Yuh-Jen Cheng, Yai-Chung Chang, and Chun-Yen Chang, "Resonant modes of 12-fold symmetric defect free photonic quasicrystal," Opt. Express 22, 2007-2012 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-2-2007


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References

  1. M. Kohmoto, B. Sutherland, K. Iguchi, “Localization of optics: quasiperiodic media,” Phys. Rev. Lett. 58(23), 2436–2438 (1987). [CrossRef] [PubMed]
  2. W. Gellermann, M. Kohmoto, B. Sutherland, P. C. Taylor, “Localization of light waves in Fibonacci dielectric multilayers,” Phys. Rev. Lett. 72(5), 633–636 (1994). [CrossRef] [PubMed]
  3. T. Hattori, N. Tsurumachi, S. Kawato, H. Nakatsuka, “Photonic dispersion relation in a one-dimensional quasicrystal,” Phys. Rev. B Condens. Matter 50(6), 4220–4223 (1994). [CrossRef] [PubMed]
  4. L. Dal Negro, C. J. Oton, Z. Gaburro, L. Pavesi, P. Johnson, A. Lagendijk, R. Righini, M. Colocci, D. S. Wiersma, “Light transport through the band-edge states of Fibonacci quasicrystals,” Phys. Rev. Lett. 90(5), 055501 (2003). [CrossRef] [PubMed]
  5. W. Man, M. Megens, P. J. Steinhardt, P. M. Chaikin, “Experimental measurement of the photonic properties of icosahedral quasicrystals,” Nature 436(7053), 993–996 (2005). [CrossRef] [PubMed]
  6. A. Ledermann, L. Cademartiri, M. Hermatschweiler, C. Toninelli, G. A. Ozin, D. S. Wiersma, M. Wegener, G. von Freymann, “Three-dimensional silicon inverse photonic quasicrystals for infrared wavelengths,” Nat. Mater. 5(12), 942–945 (2006). [CrossRef] [PubMed]
  7. T. Matsui, A. Agrawal, A. Nahata, Z. V. Vardeny, “Transmission resonances through aperiodic arrays of subwavelength apertures,” Nature 446(7135), 517–521 (2007). [CrossRef] [PubMed]
  8. S. Walter, S.-W. Daniel, “Photonic and phononic quasicrystals,” J. Phys. D Appl. Phys. 40(13), R229–R247 (2007). [CrossRef]
  9. K. Mnaymneh, R. C. Gauthier, “Mode localization and band-gap formation in defect-free photonic quasicrystals,” Opt. Express 15(8), 5089–5099 (2007). [CrossRef] [PubMed]
  10. G. Gumbs, M. K. Ali, “Dynamical maps, Cantor spectra, and localization for Fibonacci and related quasiperiodic lattices,” Phys. Rev. Lett. 60(11), 1081–1084 (1988). [CrossRef] [PubMed]
  11. M. Notomi, H. Suzuki, T. Tamamura, K. Edagawa, “Lasing Action due to the Two-Dimensional Quasiperiodicity of Photonic Quasicrystals with a Penrose Lattice,” Phys. Rev. Lett. 92(12), 123906 (2004). [CrossRef] [PubMed]
  12. M. E. Zoorob, M. D. B. Charlton, G. J. Parker, J. J. Baumberg, M. C. Netti, “Complete photonic bandgaps in 12-fold symmetric quasicrystals,” Nature 404(6779), 740–743 (2000). [CrossRef] [PubMed]
  13. K. Nozaki, T. Baba, “Lasing Characteristics of 12-Fold Symmetric Quasi-periodic Photonic Crystal Slab Nanolasers,” Jpn. J. Appl. Phys. 45(8A), 6087–6090 (2006). [CrossRef]
  14. K. Nozaki, T. Baba, “Quasiperiodic photonic crystal microcavity lasers,” Appl. Phys. Lett. 84(24), 4875–4877 (2004). [CrossRef]
  15. J. Hendrickson, B. C. Richards, J. Sweet, G. Khitrova, A. N. Poddubny, E. L. Ivchenko, M. Wegener, H. M. Gibbs, “Excitonic polaritons in Fibonacci quasicrystals,” Opt. Express 16(20), 15382–15387 (2008). [CrossRef] [PubMed]
  16. L. Mahler, A. Tredicucci, F. Beltram, C. Walther, J. Faist, H. E. Beere, D. A. Ritchie, D. S. Wiersma, “Quasi-periodic distributed feedback laser,” Nat. Photonics 4(3), 165–169 (2010). [CrossRef]
  17. S.-K. Kim, J.-H. Lee, S.-H. Kim, I.-K. Hwang, Y.-H. Lee, S.-B. Kim, “Photonic quasicrystal single-cell cavity mode,” Appl. Phys. Lett. 86(3), 031101 (2005). [CrossRef]
  18. S. P. Chang, K. P. Sou, C. H. Chen, Y. J. Cheng, J. K. Huang, C. H. Lin, H. C. Kuo, C. Y. Chang, W. F. Hsieh, “Lasing action in gallium nitride quasicrystal nanorod arrays,” Opt. Express 20(11), 12457–12462 (2012). [CrossRef] [PubMed]
  19. M. Oxborrow, C. L. Henley, “Random square-triangle tilings: A model for twelvefold-symmetric quasicrystals,” Phys. Rev. B Condens. Matter 48(10), 6966–6998 (1993). [CrossRef] [PubMed]
  20. S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, L. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60(8), 5751–5758 (1999). [CrossRef]

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