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

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 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.

Most single-polarization PBG studies in photonic crystals with disorder have been conducted only for TM polarization using lattices decorated with individual dielectric cylinders.Disorder was conventionally found to wipe out energy band gaps and to produce localization and diffusive transport [15][16][17][18].Previous studies on TE-polarization PBG materials were uncommon and have only focused on cases with relatively high refractive-index contrast [19][20][21].
In this study, we construct hyperuniform disordered photonic materials with a wall-network architecture (no cylinders), and demonstrate that TE-polarization band gaps are possible in disordered structures even with low index-of-refraction contrast (1.6:1), a regime where one might expect the disorder to disrupt band-gap formation, and where a tight-binding picture [11] no longer applies.We also show that, in a progression of structures with increasing rotational symmetry, the angular dependence of the stop bands decreases, culminating in the truly isotropic PBG of the disordered system, despite the lack of Bragg scattering.

Design of the hyperuniform disordered structure
Central to the novel designer PBG materials [13] are the concepts of hyperuniformity and stealthiness.A point pattern is hyperuniform if the number variance within a "spherical" sampling window of radius R,   , grows more slowly than the window volume for large R, i.e., more slowly than R d in d dimensions [22,23].Note that, for many 2D random systems (Poisson distribution and molecular liquids) the number variance is proportional to the window area,   2 R R   ; whereas the number variance for hyperuniform structures, grows more slowly than the window area, e. g. for crystals, quasicrystals, and hyperuniform disordered structures used in this study   R R   .Because of this feature, the photonic design pattern has hyperuniform long-range density fluctuations (or, equivalently, a structure factor S(k) approaches zero for wavenumber k approaching zero), similar to crystals [22]; at the same time, the pattern exhibits random positional order, isotropy, and a circularly symmetric diffuse structure factor 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 d-1 .In reciprocal space, hyperuniformity corresponds to having a structure factor 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 C for some positive k C [24] Stealthiness imposes density-fluctuation correlations on intermediate scales.The larger the value of k C , 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] and fabricating the structure using materials that interact resonantly with electromagnetic, electronic, or acoustic excitations.
Our hyperuniform disordered "wall-network" structure was designed using the protocol described in [13]: after generating hyperuniform point patterns within a square of side length 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.

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Our si combinat occur nat more hyp shaped ai defined e TE PBGs.predict th states.As the rotational symmetry of the photonic solid increases, the corresponding stop bands exhibit less and less angular dependence and thus facilitate the formation of a PBG for all directions.For the hyperuniform disordered sample, the structure factor is engineered to be isotropic and stealthy, resulting in a truly isotropic bandgap, as seen in the rotationally invariant deep blue ring in the polarcoordinate transmission plot.This statistical isotropy, an inherent advantage of disordered structures, offers freedom for functional-defect design that is not possible for the crystal symmetries [27,14].

Simulations of band structure and field distributions
In disordered structures, a mobility gap associated with localization due to disorder can also occur, but its nature is rather different from that of a true photonic band gap in which no states, propagating or localized, exist.In order to verify that, for the hyperuniform disordered structure, the transmission gap we observed is due to a real forbidden frequency region (PBG) with the absence of any photonic states, we have carried out numerical simulations to show the band structure for the four samples studied.The theoretical band-structure calculations for the five-fold quasicrystal and the disordered structures were obtained using a supercell approximation and the conventional plane-wave expansion method [28,29].
The supercell used for the hyperuniform disordered structure is a √ a × √ a square, where 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 Figure 4.For the quasicrystals, we use the periodic approximant scheme described at length in [25].Here, we employ the 5/3 periodic approximant, which has a rectangulary shaped unit cell with L x ≈34.27a,L y ≈13.04a, 550 points.The high-symmetry points shown in Figure 4c and 4d are vertices of the irreducible first Brillouin zone of the supercells: k  =0; k X =b 1 /2; k M =(b 1 +b 1 )2; k R =b 2 /2; where b 1 and b 2 are basis vectors of the reciprocal lattice of the supercells.
Comparing Fig. 4 with Fig. 2 clearly shows that the measured and calculated stop bands agree very well in center frequencies, width and angular dependences for the crystals, verifying the validity of our methods.In the quasicrystal and disordered cases, the number of relevant bands increases proportionally with the number of scattering elements in their supercells.For the quasicrystal sample, there is a second, lower frequency gap, whose nature is related to multiple scattering phenomena on longer length scales associated with the quasiperiodicity [30].zoom-in insert shows that the magnetic field is mostly localized inside the air cells of the network (electric fields are concentrated in the dielectric material).For the mode above the gap, the zoom-in insert shows the complementary situation, the magnetic field tending to localize in the higher-index dielectric material (electric fields are concentrated in the air cells).In between these two groups of modes (dielectric bands and air bands), there is a real energy bandgap (forbidden frequency) similar to the TE polarization PBG found in connected photonic crystal networks between the dielectric band and the air band.We next calculate the electric energy concentration factor C F defined by [28]: In Fig. 6, we show variation of the concentration factor as a function of frequency, in the region surrounding the band gap.Clearly, the large difference between the electric field distribution for modes below and above the gap, is responsible for the opening of a sizable photonic band gap in hyperuniform disordered structures.In the disordered sample, some of the modes close to the PBG are localized or diffusive.These modes contribute to the low transmission measured near the gap region.

Discussion
The formation of PBGs is well understood for crystals and quasicrystals using Bloch's theorem [28]: a periodic modulation of the dielectric constant mixes degenerate waves propagating in opposite directions and leads to standing waves with high electric field intensity in the low-dielectric region for states just above the gap and in the high-dielectric region for states just below the gap.Long-range periodic order, as evidenced by Bragg peaks, is necessary for this picture to hold.Recently, the tightbinding approximation has been applied to explain apparent gaps in 3D amorphous diamond photonic structures, in analogy to amorphous Si electronic bandgaps [11].In contrast, although our hyperuniform disordered structure does not have translational order or Bragg peaks, it still exhibits a real gap (with the absence of any photonic states) even at low index contrast, where the tight-binding approximation condition is no longer valid.
The presence of a real energy bandgap in hyperuniform disordered materials characterized by the absence of any states of a particular polarization, propagating or localized, is highly relevant for various applications, such as functional defects, i.e., cavities and waveguides [14,27].For the present case of symmetric 2D slabs of the desired height, polarization is preserved.Devices can be constructed for guiding or otherwise controlling either TE or TM modes.Thus the observation of an isotropic PBG for TE modes directly enables a number of applications suitable for PBG materials.The intrinsic anisotropy associated with the periodicity of photonic crystals can greatly limit the scope of PBG applications and places a major constraint on device design.For example, even though three-dimensional (3D) photonic crystals with complete PBGs have been fabricated for two decades [31] 3D waveguiding continues to be a challenge.Recently, the first successful demonstration of 3D waveguiding has been reported [32].However, in the 3D woodpile photonic crystal reported, it was proved that waveguiding is possible only along certain major symmetry directions, due to the mismatch of the propagation modes in line defects along various orientations.The intrinsic isotropy associated with our engineered hyperuniform disordered structures offers unprecedented freedom in defect-architecture design, unlimited by crystalline or quasicrystalline symmetries, including light guiding with arbitrary bending angles along freeform paths [14,27].
In this study, we have focused on "wall-network" structures for TE polarization PBGs.Similar design principles [13] can be applied to obtain TM-polarization PBGs [33] or complete PBGs [13,14] and can be extended to 3D as well.These results are applicable to all wavelengths.Hence, this novel class of disordered photonic bandgap materials may be used for many applications envisioned for photonic crystals.Deep reactive ion etching on silicon can be used to construct similar hyperuniform disordered network structures with a much wider TE PBG in the infrared or optical regimes.Since this singlepolarization PBG exists even at the low index contrast of 1.6:1, it becomes feasible to use soft-matter materials and self-assembly to generate similar hyperuniform disordered network structures for TEpolarization PBG, for example, by polymerizing the continuous phase of a foam or emulsion [34].The intrinsic isotropy of these hyperuniform disordered materials offers defect design freedom, not available in photonic crystals or quasicrystals, that may play an essential role in the development of flexible optical insulator platforms and isotropic light and thermal radiation sources.It also offers advantages in PBG-enhanced technologies, e.g., displays, lasers [35], sensors [36], telecommunication devices [37], and optical micro-circuits [6].

Conclusions.
We have designed and tested an engineered hyperuniform disordered network structure with a refractive-index contrast of 1.6:1, which exhibits an isotropic TE-polarization PBG verified in both measurement and simulation.For comparison, we have studied similar "wall-network" photonic crystals or quasicrystals of square, triangular and Penrose lattices.In a series of structures with increasing rotational symmetry, culminating in our disordered structure, the photonic stop bands exhibit progressively less angular dependence, with only the disordered structure forming a truly isotopic PBG.The intrinsic isotropy of our disordered structure is an inherent advantage associated with the absence of limitations of orientational order.The combination of hyperuniformity, uniform local topology, and short-range geometric order appear to be crucial for PBG formation.
In summary, we have extended the creation of PBG media with disordered structure to low indexcontrast regime, while illustrating the role of isotropy in PBG formation.This isotropy may give disordered structure significant advantages over crystalline photonic materials limited by their periodicities.Potential applications include novel architectures for cavities and waveguides displaying arbitrary bending angles [14,27] and highly efficient thin-film solar cells [10] and light-emitting diodes [38].

Acknowledgments
This work was partially supported by the Research Corporation for Science Advancement (Grant 10626 to W. M.), the San Francisco State University internal award to W. M., the University of Surrey's support to M. F. (FRSF and Santander awards), and the National Science Foundation (NSF DMR-1105417 and NYU-MRSEC DMR-0820341 to P.M.C, DMR-0606415 to ST, and ECCS-1041083 to P.S.J and M.F.).We thank Roger Bland for helpful discussions.

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Fig 6.Calculated field concentration factor as a function of frequency.The two vertical dashed lines represent the calculated bandgap boundaries.