## Photonic band gap in isotropic hyperuniform disordered solids with low dielectric contrast |

Optics Express, Vol. 21, Issue 17, pp. 19972-19981 (2013)

http://dx.doi.org/10.1364/OE.21.019972

Acrobat PDF (3835 KB)

### Abstract

We report the first experimental demonstration of a TE-polarization photonic band gap (PBG) in a 2D isotropic hyperuniform disordered solid (HUDS) made of dielectric media with a dielectric index contrast of 1.6:1, very low for PBG formation. The solid is composed of a connected network of dielectric walls enclosing air-filled cells. Direct comparison with photonic crystals and quasicrystals permitted us to investigate band-gap properties as a function of increasing rotational isotropy. We present results from numerical simulations proving that the PBG observed experimentally for HUDS at low index contrast has zero density of states. The PBG is associated with the energy difference between complementary resonant modes above and below the gap, with the field predominantly concentrated in the air or in the dielectric. The intrinsic isotropy of HUDS may offer unprecedented flexibilities and freedom in applications (i. e. defect architecture design) not limited by crystalline symmetries.

© 2013 OSA

## 1. Introduction

1. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. **58**(23), 2486–2489 (1987). [CrossRef] [PubMed]

2. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. **58**(20), 2059–2062 (1987). [CrossRef] [PubMed]

3. J. D. Forster, H. Noh, S. F. Liew, V. Saranathan, C. F. Schreck, L. Yang, J.-G. Park, R. O. Prum, S. G. J. Mochrie, C. S. O’Hern, H. Cao, and E. R. Dufresne, “Biomimetic isotropic nanostructures for structural coloration,” Adv. Mater. **22**(26-27), 2939–2944 (2010). [CrossRef] [PubMed]

6. A. Chutinan, S. John, and O. Toader, “Diffractionless flow of light in all-optical microchips,” Phys. Rev. Lett. **90**(12), 123901 (2003). [CrossRef] [PubMed]

7. Y. S. Chan, C. T. Chan, and Z. Y. Liu, “Photonic band gaps in two dimensional photonic quasicrystals,” Phys. Rev. Lett. **80**(5), 956–959 (1998). [CrossRef]

9. 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). [CrossRef] [PubMed]

10. K. Vynck, M. Burresi, F. Riboli, and D. S. Wiersma, “Photon management in two-dimensional disordered media,” Nat. Mater. **11**(12), 1017–1022 (2012). [PubMed]

11. K. Edagawa, S. Kanoko, and M. Notomi, “Photonic amorphous diamond Structure with a 3D photonic band gap,” Phys. Rev. Lett. **100**(1), 013901 (2008). [CrossRef] [PubMed]

14. W. Man, M. Florescu, E. P. Williamson, Y. He, S. R. Hashemizad, B. Y.C. Leung, D. R. Liner, S. Torquato, P. M. Chaikin, and P. J. Steinhardt, “Isotropic band gaps and freeform waveguides observed in hyperuniform disordered photonic solids, ” (unpublished. under review with Proc. Natl. Acad. Sci.).

13. M. Florescu, S. Torquato, and P. J. Steinhardt, “Designer disordered materials with large, complete photonic band gaps,” Proc. Natl. Acad. Sci. U.S.A. **106**(49), 20658–20663 (2009). [CrossRef] [PubMed]

*high*dielectric contrast (

*ε*= 11.5), a connected-wall network decorated with dielectric cylinders. The presence of a complete PBG in a hyperuniform disordered structure decorated with alumina (

*ε*= 8.76) was recently demonstrated by Man

*et al.*[14

14. W. Man, M. Florescu, E. P. Williamson, Y. He, S. R. Hashemizad, B. Y.C. Leung, D. R. Liner, S. Torquato, P. M. Chaikin, and P. J. Steinhardt, “Isotropic band gaps and freeform waveguides observed in hyperuniform disordered photonic solids, ” (unpublished. under review with Proc. Natl. Acad. Sci.).

15. A. A. Chabanov and A. Z. Genack, “Photon Localization in Resonant Media,” Phys. Rev. Lett. **87**(15), 153901 (2001). [CrossRef] [PubMed]

18. M. A. Kaliteevski, J. M. Martinez, D. Cassagne, and J. P. Albert, “Disorder-induced modification of the transmission of light in a two-dimensional photonic crystal,” Phys. Rev. B **66**(11), 113101 (2002). [CrossRef]

19. J. Choi, Y. Luo, R. B. Wehrspohn, R. Hillebrand, J. Schilling, and U. Gösele, “Perfect two-dimensional porous alumina photonic crystals with duplex oxide layers,” J. Appl. Phys. **94**(8), 4757–4762 (2003). [CrossRef]

21. M. M. Rahman, J. Ferré-Borrull, J. Pallarès, and L. F. Marsal, “Photonic stop bands of two-dimensional quasi-random structures based on macroporous silicon,” Phys. Status Solidi. C **8**(3), 1066–1070 (2011). [CrossRef]

11. K. Edagawa, S. Kanoko, and M. Notomi, “Photonic amorphous diamond Structure with a 3D photonic band gap,” Phys. Rev. Lett. **100**(1), 013901 (2008). [CrossRef] [PubMed]

## 2. Design of the hyperuniform disordered structure

13. M. Florescu, S. Torquato, and P. J. Steinhardt, “Designer disordered materials with large, complete photonic band gaps,” Proc. Natl. Acad. Sci. U.S.A. **106**(49), 20658–20663 (2009). [CrossRef] [PubMed]

*R*,

*R*, i.e., more slowly than

*R*in

^{d}*d*dimensions [22

22. S. Torquato and F. H. Stillinger, “Local density fluctuations, hyperuniformity, and order metrics,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **68**(4), 041113 (2003). [CrossRef] [PubMed]

23. C. E. Zachary and S. Torquato, “Hyperuniformity in Point Patterns and Two-Phase Random Heterogeneous Media,” J. Stat. Mech. **2009**(12), 12015 (2009). [CrossRef]

*S*(

*k*) approaches zero for wavenumber

*k*approaching zero), similar to crystals [22

22. S. Torquato and F. H. Stillinger, “Local density fluctuations, hyperuniformity, and order metrics,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **68**(4), 041113 (2003). [CrossRef] [PubMed]

*S*(

*k*) similar to that of a glass. The cases considered for our structures have the slowest possible rate of growth, which is proportional to

*R*. In reciprocal space, hyperuniformity corresponds to having a structure factor

^{d-1}*S(k)*that tends to zero as the wavenumber

*|*

*k**|*tends to zero (omitting forward scattering), i.e., the infinite-wavelength density fluctuations vanish. In particular, for generating hyperuniform disordered point patterns, we consider “stealthy” point patterns with a structure factor

*S(k)*that is isotropic, continuous, and equal to zero for a finite range of wavenumbers |

*k**| <k*for some positive

_{C}*k*[24

_{C}24. R. Batten, F. H. Stillinger, and S. Torquato, “Classical disordered ground states: super ideal gases, and stealth and equi-luminous materials,” J. Appl. Phys. **104**(3), 033504 (2008). [CrossRef]

*k*, the stealthier the point pattern is and the more intermediate-range order there is. Hyperuniform materials can then be constructed by first mapping a hyperuniform point pattern onto a network structure using a mathematical protocol [13

_{C}13. M. Florescu, S. Torquato, and P. J. Steinhardt, “Designer disordered materials with large, complete photonic band gaps,” Proc. Natl. Acad. Sci. U.S.A. **106**(49), 20658–20663 (2009). [CrossRef] [PubMed]

**106**(49), 20658–20663 (2009). [CrossRef] [PubMed]

*L*, where

*L*is about 22 times the average inter-particle spacing

*a*, we employ a centroidal tessellation of the point pattern to generate a “relaxed” dual lattice. By construction, the dual-lattice vertices are trihedrally coordinated. We then connect the lattice vertex pairs with dielectric walls of fixed width to generate the trihedral-network photonic architecture [13

**106**(49), 20658–20663 (2009). [CrossRef] [PubMed]

*t*for a given dielectric contrast, the simulations predict the formation of a TE-polarization band gap, a forbidden frequency range with zero density of states.

*Accura® 60*(a clear, polycarbonate-like plastic) from 3D ® Systems Corporation. The resolution is 0.1mm in both lateral and vertical directions. Figure 1 shows these centimeter-scale photonic structures, which are designed to have a nearly circular boundary with a diameter of 135 mm (~22

*a*). The five-fold-symmetry quasicrystal wall-network structure was constructed using the same centroidal tessellation protocol applied to a point pattern consisting of the vertices of a periodic approximant to a Penrose tiling, as described in [25

25. M. Florescu, S. Torquato, and P. J. Steinhardt, “Complete band gaps in two-dimensional photonic quasicrystals,” Phys. Rev. B **80**(15), 155112 (2009). [CrossRef]

*t*was set for each structure separately, so that the total volume filling fraction is about 40.5% for all the samples.

## 3. Results and discussion

### 3.1. Measurements of microwave transmission

8. 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]

*a*apart to produce approximately plane wavefronts at the samples. The samples were aligned so that the incident beam was perpendicular to the vertical axes. We rotated each sample about its vertical axis and recorded the transmission every two degrees. The transmission is defined as the ratio between detected intensities with and without the sample in place.

**that resides on the plane defined by a reciprocal lattice vector**

*k***is Bragg-scattered by**

*G***. Such a wavevector hits a Brillouin-zone boundary, and its wavenumber**

*G**k*= |

**| satisfies the condition |**

*k***| = |**

*k*

*G**|*/(

*2cos*(

*θ*)). When the dielectric contrast is low, the center frequency of a stop band due to Bragg scattering is, to the lowest-order approximation,

*f~c|k|/*(

*n2π*), which is inversely proportional to

*cos*(

*θ*), where

*c*is the speed of light in vacuum, and

*n*is the Bruggeman effective medium index [26

26. X. C. Zeng, D. J. Bergman, P. M. Hui, and D. Stroud, “Effective-medium theory for weakly nonlinear composites,” Phys. Rev. B Condens. Matter **38**(15), 10970–10973 (1988). [CrossRef] [PubMed]

**and –**

*k***. Hence, its spectrum repeats every 36 degrees, and the change between the furthest symmetry points (from 0 to 18 degrees) is much smaller than that found in the crystalline structures, enabling PBGs to form. For the hyperuniform disordered sample, the transmission spectrum is truly isotropic without any angular dependence; hence the stop bands overlap to form a TE PBG for all directions.**

*k*8. 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]

*k*values are too small to be visible at this resolution. The transmission plot of the five-fold quasicrystal sample shows two sets of ten straight lines associated with Bragg scattering planes arranged with 10-fold symmetry. Along both the two sets, it is apparent that stop bands overlap to form bandgaps. Quasicrystals are self-similar and quasiperiodic, so they can, in principle, have many band gaps separated by mean wavenumbers with irrational ratios.

14. W. Man, M. Florescu, E. P. Williamson, Y. He, S. R. Hashemizad, B. Y.C. Leung, D. R. Liner, S. Torquato, P. M. Chaikin, and P. J. Steinhardt, “Isotropic band gaps and freeform waveguides observed in hyperuniform disordered photonic solids, ” (unpublished. under review with Proc. Natl. Acad. Sci.).

27. M. Florescu, P. J. Steinhardt, and S. Torquato, “Optical cavities and waveguides in hyperuniform disordered photonic solids,” Phys. Rev. B **87**(16), 165116 (2013). [CrossRef]

### 3.2. Simulations of band structure and field distributions

29. S. Johnson and J. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express **8**(3), 173–190 (2001). [CrossRef] [PubMed]

*N*= 500 is the number of points in the pattern and a is the average point separation. The convergence of the results for larger supercell sizes has been confirmed. We solve the vectorial Maxwell equations, assuming the structure is infinitely long in the direction perpendicular to the 2D plane, and the results are presented in Fig. 4. For the quasicrystals, we use the periodic approximant scheme described at length in [25

25. M. Florescu, S. Torquato, and P. J. Steinhardt, “Complete band gaps in two-dimensional photonic quasicrystals,” Phys. Rev. B **80**(15), 155112 (2009). [CrossRef]

*L*≈34.27

_{x}*a*,

*L*≈13.04

_{y}*a*, containing 550 points. The high-symmetry points shown in Fig. 4(c) and 4(d) are vertices of the irreducible first Brillouin zone of the supercells:

**= 0;**

*k*_{Γ}**=**

*k*_{X}**/2;**

*b*_{1}**= (**

*k*_{M}**)2;**

*b*_{1}+ b_{1}**=**

*k*_{R}**/2; where**

*b*_{2}**and**

*b*_{1}**are basis vectors of the reciprocal lattice of the supercells.**

*b*_{2}### 3.3. Discussion

11. K. Edagawa, S. Kanoko, and M. Notomi, “Photonic amorphous diamond Structure with a 3D photonic band gap,” Phys. Rev. Lett. **100**(1), 013901 (2008). [CrossRef] [PubMed]

27. M. Florescu, P. J. Steinhardt, and S. Torquato, “Optical cavities and waveguides in hyperuniform disordered photonic solids,” Phys. Rev. B **87**(16), 165116 (2013). [CrossRef]

31. K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, “Photonic band gaps in three dimensions: New layer-by-layer periodic structures,” Solid State Commun. **89**(5), 413–416 (1994). [CrossRef]

32. K. Ishizaki, M. Koumura, K. Suzuki, K. Gondaira, and S. Noda, “Realization of three-dimensional guiding of photons in photonic crystals,” Nat. Photonics **7**(2), 133–137 (2013). [CrossRef]

27. M. Florescu, P. J. Steinhardt, and S. Torquato, “Optical cavities and waveguides in hyperuniform disordered photonic solids,” Phys. Rev. B **87**(16), 165116 (2013). [CrossRef]

**106**(49), 20658–20663 (2009). [CrossRef] [PubMed]

33. W. Man, M. Florescu, K. Matsuyama, P. Yadak, S. Torquato, P. J. Steinhardt, and P. Chaikin, “Experimental observation of photonic bandgaps in Hyperuniform disordered materials,” presented at the Conference on Lasers and Electro-Optics, San Jose, United States, 16–21 May. 2010. [CrossRef]

**106**(49), 20658–20663 (2009). [CrossRef] [PubMed]

34. J. Duplat, B. Bossa, and E. Villermaux, “On two-dimensional foam ageing,” J. Fluid Mech. **673**, 147–179 (2011). [CrossRef]

35. H. Cao, Y. G. Zhao, S. T. Ho, E. W. Seeling, Q. H. Wang, and R. P. H. Chang, “Random laser action in semiconductor powder,” Phys. Rev. Lett. **82**(11), 2278–2281 (1999). [CrossRef]

36. Y. B. Guo, C. Divin, A. Myc, F. L. Terry Jr, J. R. Baker Jr, T. B. Norris, and J. Y. Ye, “Sensitive molecular binding assay using a photonic crystal structure in total internal reflection,” Opt. Express **16**(16), 11741–11749 (2008). [CrossRef] [PubMed]

37. S. Noda, A. Chutinan, and M. Imada, “Trapping and emission of photons by a single defect in a photonic bandgap structure,” Nature **407**(6804), 608–610 (2000). [CrossRef] [PubMed]

6. A. Chutinan, S. John, and O. Toader, “Diffractionless flow of light in all-optical microchips,” Phys. Rev. Lett. **90**(12), 123901 (2003). [CrossRef] [PubMed]

## 4. Conclusions

**87**(16), 165116 (2013). [CrossRef]

10. K. Vynck, M. Burresi, F. Riboli, and D. S. Wiersma, “Photon management in two-dimensional disordered media,” Nat. Mater. **11**(12), 1017–1022 (2012). [PubMed]

38. I. Schnitzer, E. Yablonovitch, C. Caneau, T. J. Gmitter, and A. Scherer, “30% external quantum efficiency from surface textured, thin film light emitting diodes,” Appl. Phys. Lett. **63**(16), 2174–2176 (1993). [CrossRef]

## Acknowledgments

## References and links

1. | S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. |

2. | E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. |

3. | J. D. Forster, H. Noh, S. F. Liew, V. Saranathan, C. F. Schreck, L. Yang, J.-G. Park, R. O. Prum, S. G. J. Mochrie, C. S. O’Hern, H. Cao, and E. R. Dufresne, “Biomimetic isotropic nanostructures for structural coloration,” Adv. Mater. |

4. | H. Altug, D. Englund, and J. Vučković, “Ultrafast photonic crystal nanocavity laser,” Nat. Phys. |

5. | I. El-Kady, M. M. Reda Taha, and M. F. Su, “Application of photonic crystals in submicron damage detection and quantification,” Appl. Phys. Lett. |

6. | A. Chutinan, S. John, and O. Toader, “Diffractionless flow of light in all-optical microchips,” Phys. Rev. Lett. |

7. | Y. S. Chan, C. T. Chan, and Z. Y. Liu, “Photonic band gaps in two dimensional photonic quasicrystals,” Phys. Rev. Lett. |

8. | W. Man, M. Megens, P. J. Steinhardt, and P. M. Chaikin, “Experimental measurement of the photonic properties of icosahedral quasicrystals,” Nature |

9. | M. C. Rechtsman, H.-C. Jeong, P. M. Chaikin, S. Torquato, and P. J. Steinhardt, “Optimized structures for photonic quasicrystals,” Phys. Rev. Lett. |

10. | K. Vynck, M. Burresi, F. Riboli, and D. S. Wiersma, “Photon management in two-dimensional disordered media,” Nat. Mater. |

11. | K. Edagawa, S. Kanoko, and M. Notomi, “Photonic amorphous diamond Structure with a 3D photonic band gap,” Phys. Rev. Lett. |

12. | S. Imagawa, K. Edagawa, K. Morita, T. Niino, Y. Kagawa, and M. Notomi, “Photonic band-gap formation, light diffusion, and localization in photonic amorphous diamond structures,” Phys. Rev. B |

13. | M. Florescu, S. Torquato, and P. J. Steinhardt, “Designer disordered materials with large, complete photonic band gaps,” Proc. Natl. Acad. Sci. U.S.A. |

14. | W. Man, M. Florescu, E. P. Williamson, Y. He, S. R. Hashemizad, B. Y.C. Leung, D. R. Liner, S. Torquato, P. M. Chaikin, and P. J. Steinhardt, “Isotropic band gaps and freeform waveguides observed in hyperuniform disordered photonic solids, ” (unpublished. under review with Proc. Natl. Acad. Sci.). |

15. | A. A. Chabanov and A. Z. Genack, “Photon Localization in Resonant Media,” Phys. Rev. Lett. |

16. | E. Lidorikis, M. M. Sigalas, E. N. Economou, and C. M. Soukoulis, “Gap deformation and classical wave localization in disordered two-dimensional photonic-band-gap materials,” Phys. Rev. B |

17. | A. A. Asatryan, P. A. Robinson, L. C. Botten, R. C. McPhedran, N. A. Nicorovici, and C. M. de Sterke, “Effects of geometric and refractive index disorder on wave propagation in two-dimensional photonic crystals,” Phys. Rev. E . |

18. | M. A. Kaliteevski, J. M. Martinez, D. Cassagne, and J. P. Albert, “Disorder-induced modification of the transmission of light in a two-dimensional photonic crystal,” Phys. Rev. B |

19. | J. Choi, Y. Luo, R. B. Wehrspohn, R. Hillebrand, J. Schilling, and U. Gösele, “Perfect two-dimensional porous alumina photonic crystals with duplex oxide layers,” J. Appl. Phys. |

20. | Y. Su, G. T. Fei, Y. Zhang, H. Li, P. Yan, G. L. Shang, and L. D. Zhang, “Anodic alumina photonic crystal heterostructures,” J. Opt. Soc. Am. B |

21. | M. M. Rahman, J. Ferré-Borrull, J. Pallarès, and L. F. Marsal, “Photonic stop bands of two-dimensional quasi-random structures based on macroporous silicon,” Phys. Status Solidi. C |

22. | S. Torquato and F. H. Stillinger, “Local density fluctuations, hyperuniformity, and order metrics,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

23. | C. E. Zachary and S. Torquato, “Hyperuniformity in Point Patterns and Two-Phase Random Heterogeneous Media,” J. Stat. Mech. |

24. | R. Batten, F. H. Stillinger, and S. Torquato, “Classical disordered ground states: super ideal gases, and stealth and equi-luminous materials,” J. Appl. Phys. |

25. | M. Florescu, S. Torquato, and P. J. Steinhardt, “Complete band gaps in two-dimensional photonic quasicrystals,” Phys. Rev. B |

26. | X. C. Zeng, D. J. Bergman, P. M. Hui, and D. Stroud, “Effective-medium theory for weakly nonlinear composites,” Phys. Rev. B Condens. Matter |

27. | M. Florescu, P. J. Steinhardt, and S. Torquato, “Optical cavities and waveguides in hyperuniform disordered photonic solids,” Phys. Rev. B |

28. | J. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Mead, |

29. | S. Johnson and J. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express |

30. | A. Della Villa, S. Enoch, G. Tayeb, V. Pierro, V. Galdi, and F. Capolino, “Band Gap Formation and Multiple Scattering in Photonic Quasicrystals with a Penrose-Type Lattice,” Phys. Rev. Lett. |

31. | K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, “Photonic band gaps in three dimensions: New layer-by-layer periodic structures,” Solid State Commun. |

32. | K. Ishizaki, M. Koumura, K. Suzuki, K. Gondaira, and S. Noda, “Realization of three-dimensional guiding of photons in photonic crystals,” Nat. Photonics |

33. | W. Man, M. Florescu, K. Matsuyama, P. Yadak, S. Torquato, P. J. Steinhardt, and P. Chaikin, “Experimental observation of photonic bandgaps in Hyperuniform disordered materials,” presented at the Conference on Lasers and Electro-Optics, San Jose, United States, 16–21 May. 2010. [CrossRef] |

34. | J. Duplat, B. Bossa, and E. Villermaux, “On two-dimensional foam ageing,” J. Fluid Mech. |

35. | H. Cao, Y. G. Zhao, S. T. Ho, E. W. Seeling, Q. H. Wang, and R. P. H. Chang, “Random laser action in semiconductor powder,” Phys. Rev. Lett. |

36. | Y. B. Guo, C. Divin, A. Myc, F. L. Terry Jr, J. R. Baker Jr, T. B. Norris, and J. Y. Ye, “Sensitive molecular binding assay using a photonic crystal structure in total internal reflection,” Opt. Express |

37. | S. Noda, A. Chutinan, and M. Imada, “Trapping and emission of photons by a single defect in a photonic bandgap structure,” Nature |

38. | I. Schnitzer, E. Yablonovitch, C. Caneau, T. J. Gmitter, and A. Scherer, “30% external quantum efficiency from surface textured, thin film light emitting diodes,” Appl. Phys. Lett. |

**OCIS Codes**

(160.0160) Materials : Materials

(160.5293) Materials : Photonic bandgap materials

(160.5298) Materials : Photonic crystals

**ToC Category:**

Materials

**History**

Original Manuscript: June 26, 2013

Revised Manuscript: August 3, 2013

Manuscript Accepted: August 7, 2013

Published: August 16, 2013

**Citation**

Weining Man, Marian Florescu, Kazue Matsuyama, Polin Yadak, Geev Nahal, Seyed Hashemizad, Eric Williamson, Paul Steinhardt, Salvatore Torquato, and Paul Chaikin, "Photonic band gap in isotropic hyperuniform disordered solids with low dielectric contrast," Opt. Express **21**, 19972-19981 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-17-19972

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### References

- S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett.58(23), 2486–2489 (1987). [CrossRef] [PubMed]
- E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett.58(20), 2059–2062 (1987). [CrossRef] [PubMed]
- J. D. Forster, H. Noh, S. F. Liew, V. Saranathan, C. F. Schreck, L. Yang, J.-G. Park, R. O. Prum, S. G. J. Mochrie, C. S. O’Hern, H. Cao, and E. R. Dufresne, “Biomimetic isotropic nanostructures for structural coloration,” Adv. Mater.22(26-27), 2939–2944 (2010). [CrossRef] [PubMed]
- H. Altug, D. Englund, and J. Vučković, “Ultrafast photonic crystal nanocavity laser,” Nat. Phys.2(7), 484–488 (2006). [CrossRef]
- I. El-Kady, M. M. Reda Taha, and M. F. Su, “Application of photonic crystals in submicron damage detection and quantification,” Appl. Phys. Lett.88(25), 253109 (2006). [CrossRef]
- A. Chutinan, S. John, and O. Toader, “Diffractionless flow of light in all-optical microchips,” Phys. Rev. Lett.90(12), 123901 (2003). [CrossRef] [PubMed]
- Y. S. Chan, C. T. Chan, and Z. Y. Liu, “Photonic band gaps in two dimensional photonic quasicrystals,” Phys. Rev. Lett.80(5), 956–959 (1998). [CrossRef]
- W. Man, M. Megens, P. J. Steinhardt, and P. M. Chaikin, “Experimental measurement of the photonic properties of icosahedral quasicrystals,” Nature436(7053), 993–996 (2005). [CrossRef] [PubMed]
- 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). [CrossRef] [PubMed]
- K. Vynck, M. Burresi, F. Riboli, and D. S. Wiersma, “Photon management in two-dimensional disordered media,” Nat. Mater.11(12), 1017–1022 (2012). [PubMed]
- K. Edagawa, S. Kanoko, and M. Notomi, “Photonic amorphous diamond Structure with a 3D photonic band gap,” Phys. Rev. Lett.100(1), 013901 (2008). [CrossRef] [PubMed]
- S. Imagawa, K. Edagawa, K. Morita, T. Niino, Y. Kagawa, and M. Notomi, “Photonic band-gap formation, light diffusion, and localization in photonic amorphous diamond structures,” Phys. Rev. B82(11), 115116 (2010). [CrossRef]
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