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Disorder-immune photonics based on Mie-resonant dielectric metamaterials

Disorder-immune photonics based on Mie-resonant dielectric metamaterials 1;2 3;4 1;5 1 3;6 Changxu Liu , Mikhail V. Rybin , Peng Mao , Shuang Zhang , and Yuri Kivshar School of Physics and Astronomy, University of Birmingham, Birmingham B15 2TT, UK Chair in Hybrid Nanosystems, Nanoinstitute Munich, Faculty of Physics, Ludwig Maximilians University of Munich, 80539 Munich, Germany ITMO University, St Petersburg 197101, Russia Ioffe Institute, St Petersburg 194021, Russia College of Electronic and Optical Engineering and College of Microelectronics, Nanjing University of Posts and Telecommunications, Nanjing 210023, China and Nonlinear Physics Centre, Australian National University, Canberra, ACT 2601, Australia When the feature size of photonic structures becomes comparable or even smaller than the wavelength of light, the fabrication imperfections inevitably introduce disorder that may eliminate many functionalities of subwavelength photonic devices. Here we suggest a novel concept to achieve a robust bandgap which can endure disorder beyond 30% as a result of the transition from photonic crystals to Mie-resonant metamaterials. By utilizing Mie-resonant metamaterials with high refractive index, we demonstrate photonic waveguides and cavities with strong robustness to position disorder, thus providing a novel approach to the bandgap-based nanophotonic devices with new properties and functionalities. The idea of manipulating the electromagnetic waves with subwavelength structures originates from the 19-th century, σ = σ σ >> σ σ = 0 0 0 when Heinrich Hertz managed to control meter-long ra- dio waves through wire-grid polarizer with centimeter spac- ings [1]. As the rapid advancement of the nanotech- nology with fabrication resolution down to micrometer or even nanometers, a plethora of subwavelength systems with structure-induced optical properties are achieved, ranging from photonic crystals to metamaterials [2]. Among them, a disorder photonic crystal (PhC) is a periodic optical structure that has attracted considerable interest for its ability to confine, manip- FIG. 1: Schematic of a photonic structure, composed of dielectric ulate, and guide light [3]. Spatial periodicity of the dielectric nanorods, with an increasing position disorder , respectively. function is essential to obtain a photonic bandgap where the propagation for photons within a certain frequency gap is for- bidden, providing unique features for a variety of applications 43], here we consider photonic structures with the optically ranging from lasers [4, 5], all-optical memories [6] to sensing induced Mie resonances and reveal that they can support [7] and emission control [8]. disorder-immune photonic bandgaps, in a sharp contrast with To achieve unparalleled functionalities with the subwave- PhC where the Bragg resonances require stringent periodicity length structures, a stringent requirement for the fabrication and consequently are not tolerant to disorder. Our numerical accuracy is required. As a result, the impact of disorder on results demonstrate robustness of the optical waveguides un- such photonic structures has extensively been studied, both der intense disorder, suggesting the way towards a new gener- numerically and experimentally [9–38]. When the disorder is ation of disorder-immune photonic devices with cost-effective small enough (up to a few percent of the lattice constant) and fabrication processes. can be treated as a perturbation, the interaction between the We start from an ideal periodic structure composed of order and disorder gives rise to interesting optical transport nanorods arranged in a square lattice, as illustrated in the left phenomena involving multiple light scattering, diffusion and panel of Fig. 1. The lattice constant is a = 500 nm and the localization of light [16–18, 20–24]. As disorder is increased rod radius is r = 125 nm, so that the ratio r=a defines a filling further, the photonic bandgap is destroyed, owing to the ad- fraction of the structure. The permittivity of the nanorod is verse effect to the Bragg reflection [25–33]. For example, only ", with whose value identifying the system either as photonic a few percent of disorder can eliminate the bandgap of inverse crystals (low ") or dielectric metamaterials (high ") [41]. As opal photonic crystals [25, 26]. The only way to achieve ro- i i the first step, we introduce disorder to the rod position (x ; y ) bustness is to utilize nontrivial topological properties [39] in i i i i i i as: x = x + U and y = y + U , where (x ; y ) is the x y 0 0 0 0 waveguides with gyromagnetic materials. However, the ex- original position in the periodic lattice, U and U are random x y ternal magnetic field is a prerequisite to break time-reversal variables distributed uniformly over the interval [1; 1], and symmetry [40], hindering its practical application. the parameter  describes the strength of the disorder. We also Being inspired by the recent studies of the dielectric Mie- consider the normalized disorder strength  defined as the - resonant metamaterials (MMs) and their link to PhCs [41– to-a ratio expressed in %. Since the height of the nanorod is arXiv:1905.13734v2 [physics.app-ph] 25 Sep 2019 2 (a) Photonic crystal (d) Metamaterial -20 -20 -40 -40 -60 -80 -60 η = 0% -100 η = 10% -80 η = 20% -120 η = 40% -100 -140 0.3 0.4 0.5 0.6 0.7 0.8 0.3 0.4 0.5 0.6 0.7 0.8 a/λ a/λ (b) (c) (e) (f) FIG. 2: Transmission spectra of (a) a photonic crystal (with " = 4) and (d) a metamaterial (with " = 25) with different values of the disorder parameter . Black solid curves are perfect structures ( = 0), red dashed curves are weak disorder ( = 10%); green dash-dot curves are moderate disorder ( = 20%); blue dotted curves correspond to a strong disorder ( = 40%). Origin of spectral dips are labeled. Magnetic field distribution in (c,d) photonic crystal (for  = 0 and  = 40%, respectively) and (e,f) metamaterial (with  = 0 and  = 40%, respectively). In each of the panels (c,d) and (e,f), the waves propagate from the bottom to the top. much larger than its radius, a two-dimensional approximation gaps with a stronger effect manifested for the second bandgap in the (x; y) plane is valid. associated with the TE Mie resonances (about 70 dB for even weak disorder  = 50 nm or  = 10%). When disorder is introduced (Fig. 1), the translational sym- metry of the structure becomes broken, making the bandgap In a sharp contrast, in the regime of a metamaterial structure of the spectrum in the reciprocal space to be ill- [Fig. 2(d)], the lowest bandgap survives under even strong dis- defined. As a result, we study properties of these photonic order of  = 200 nm (or  = 40%). In this regime, the Mie structures in the real space assuming that the low-transmission scattering from individual nanorods play a paramount role to spectral regions associated with bandgaps are still observable form the bandgap through the TE Mie resonances, reduc- in the corresponding spectrum. ing strict requirements of periodicity. The field distributions In the analysis of disordered media, the light propaga- shown in Fig. 2(e,f) reveal the effective field suppression by tion is characterized by a logarithmic-average transmission each nanorod oscillating out-of-phase with the incident wave. instead of average transmission (see Ref. [44] for details). Rigorous model accounts perturbations of coupling constants Figure 2 demonstrates the logarithmic-averaged transmission between neighbor rods [45] due to the position disorder. It re- (averaged over an ensemble of 100 samples) vs. the disor- sults in degradation of the suppression however this affects the der strength  for both photonic crystals and metamaterials. TE Mie gap much weaker than its Bragg counterpart. Re- In addition, we show the results for the wave propagation markably, the position disorder affects all other gaps including through the corresponding structures with a specific disorder higher-order Mie gaps. The reason is that Bragg frequency realization for the lowest bandgap. We observe that the spectra obey the law f / (d cos ) , where d is a lattice spacing and consist of a number of pronounced dips (associated with the  is the propagation angle. The higher-order Mie gaps above spectral gaps) which can be linked to either Mie and Bragg the lowest Bragg gap do not demonstrate robustness, since resonances. The Bragg gaps are observed as symmetric dips, they are not pure Mie gaps but mixtures with Bragg waves for while the Mie gaps have a knife-tip shape. In the regime of certain directions. Thus, in spite of the identical configura- photonic crystals [Fig. 2(a)], we observe a degradation of all tions of the dielectric nanorods in Fig. 2(c) and Fig. 2(f), the Transmission (dB) Bragg gap TE Mie gap TE Mie gap Bragg gap TE Mie gap Bragg gap TE Mie gap TE Mie gap 02 3 1 1 (a) (b) σ (nm) σ (nm) 0 50 100 150 200 250 300 0 50 100 150 200 250 0.8 0.5 90% 0.6 ε=25 Mie 50% 0.4 30% -2 ε=4 -0.5 0.2 16 Mie+Bragg -4 -1 -2 0 2 -2 0 2 0 10 30 50 10% (a) (b) (c) x (μm) x (μm) η (%) 5% Bragg 1 ε=17 3% 0.8 4 4 2% 0 10 20 30 40 50 60 0 10 20 30 40 50 0.6 η (%) η (%) 0.4 Square Hexagonal -2 ε=4 0.2 -4 0 5 10 15 20 -5 0 5 -5 0 5 FIG. 3: Degradation of the reflection at different values of ", illus- x (μm) x (μm) η (%) (d) (e) (f) trating the robustness of the structure for (a) square and (b) hexagonal lattices, respectively. FIG. 4: (a,b) Field distribution H in a straight waveguide created in a square lattice of nanorods with disorder  = 10% ( = 50 nm) for (a) " = 4 and (b) " = 25. (c) The corresponding relative transmis- wave propagation is remarkably different when the dielectric sion T=T vs.  for " = 4 and " = 25, respectively, for a square constant " of each rod changes from low to higher values. lattice of nanorods. (d,e) Field distribution H in a bent waveguide created in a hexagonal lattice with disorder  = 8% ( = 40 nm) for To provide a comprehensive picture of the impact of the (d) " = 4 and (e) " = 17. (f) The corresponding relative transmis- position disorder to the system transforming from the PhC sion T=T vs.  for " = 4 and " = 17, respectively, for a hexagonal to MM regimes, we conduct a series of numerical simu- lattice of nanorods. lations with different values of " ranging from 4 to 25. To quantitatively represent the robustness of the photonic bandgap, we define parameter = (R R)=R ; where 0 0 bandgaps, we analyze the structure robustness for a differ- R = Rd=(  ) is the averaged reflection in the ent geometry, namely a hexagonal lattice of nanorods that can 2 1 bandgap between  and  ;  and  are wavelengths at support a bandgap for the TE waves [46]. In addition, a differ- 1 2 1 2 10% of the transmission minimum. ent ratio r=a = 0:3 is employed for the generality study while The degradation is normalized by R , that is the reflection the lattice constant is kept the same, a = 500 nm. Figure 3(b) in the bandgap without disorder. Consequently, represents shows the value of for varying ". A similar behavior is illus- the deterioration of reflection in the bandgap, with a smaller trated for a square lattice nanorods in Fig. 3(a), demonstrating value demonstrating a better robustness. Figure 3(a) summa- three regimes with different level of robustness to the posi- rizes the value of with different " as the disorder  increases tion disorder. The values of " for achieving strong robustness for the bandgaps with lowest energy (as shown in Figs. 2(a) is ameliorated to a smaller value around 15 due to the optimal and (d). The value of is averaged from three different sets lattice [47]. The typical transmission spectra for both PhC and of the uniform random variables. The relative change to the MM regimes can be found in Supplemental Material [48]. lattice constant is also labeled by the  axis. The diagram can We further investigate the robustness effect for more prac- be unambiguously divided into three regimes. When the value tical structures such as a photonic waveguide. The waveg- of " is small, the system operates as a PhC, corresponding to uide is readily generated by introducing a line defect along the situation shown in Figs. 2(a-c). The bandgap is quite vul- the y-direction. For the PhC case, we select the TE Mie nerable to the disorder, around 10% of the position disorder bandgap with a better light confinement (since the Bragg fre- can break the perfection of the photonic bandgap. As the in- quency has a strong angular dependence). Figures 4(a,b) illus- crement of the permittivity, the system transforms into a new trate the spatial distribution of the magnetic field in a disorder- regime, where the bandgaps is formed by the overlap of Mie impacted waveguide operating as PhC and MM, respectively, and Bragg resonances [41]. The Mie scattering from individ- while the disorder-free example can be found in SM [48]. ual nanorods increases the robustness to the disorder, reducing Here the disorder parameter =50 nm. In spite of the disorder the degradation compared to the previous regime. Further destroying the ideal structure, the light is still well confined in enhancement of " drives the lowest bandgap formed by the the active region [Fig. 4(b)], demonstrating a good robustness Mie scattering, and the system transforms into the effective when operating in the MM regime. With a reduced value of MM structure with a good robustness to the position disor- ", the waveguide losses its function with the transverse dif- der. Under intense disorder, a well-defined stop band persists, fusion under the same position configuration for a PhC, as as illustrated in Figs. 2(d-f). The transition of the robustness shown in Fig. 4a. To quantitatively demonstrate the impact of parameter precisely matches the phase transition from PhC disorder, we calculated the transmission T of the waveguide to MM ”phases” [41], identifying the unique role of the Mie under different , as shown in Fig. 4(c). Similarly, the trans- scattering playing in the disorder-immune photonic bandgaps. mission is normalized to T , the value without disorder for the To illustrate generality of the disorder-immune photonic comparison. A definite improvement of the robustness for the y (μm) y (μm) Amplitude (arb. units) T/T T/T hexagonal square 4 (a) (c) to the disorder, whereas the photonic cavity operating in the MM regime (" = 15) is more robust, only experiencing severe degradation when  reaches 50 nm. With the introduction of moderate degree of disorder, the quality factor is improved with the value Q=Q beyond 1, matching earlier results [29]. (b) (d) 90% 1 70% Mie In addition to fluctuations in the rod positions, we study 50% 0.8 also the robustness to a size disorder and consider a pho- 30% ε=15 tonic structure of different nanorods, see Figs. 5(c-d). We as- 0.6 16 Mie+Bragg sume that the disorder introduces fluctuations in the rod radii, 10% 0.4 r = r +  U , where U are random variables distributed r r r 5% 0.2 Bragg 8 4% uniformly over the interval [1; 1], and the relative disorder ε=4 3% is:  = =a =  =r. Figure 5(c) shows a typical disordered 0 2% 0 4 8 12 16 20 0 10 20 30 40 structure with  = 25%. Similarly, we use the parameter η (%) η (%) to evaluate the structure robustness, as shown in Fig. 5(d) for the regimes transform from PhC (" = 4) to MM (" = 25). FIG. 5: (a) An optical cavity formed by a hexagonal lattice of di- The fluctuations in the nanorod radius cause a variation of the electric rods without disorder (left) and with disorder (right). (b) Mie resonance, consequently inducing deterioration of the ro- Degradation of the relative quality factor Q=Q with disorder  for " = 4 and " = 15. (c) An example of a photonic structure with both bustness compared with the case presented in Fig. 3(a). In ad- position and size disorder, for  = 25%. (d) Robustness parameter ditional, the same fluctuation in size  could bring stronger vs. disorder  and ". disorder to the nanorod with higher permittivity, considering the optical wavelength inside the dielectric  =  = " and consequently increasing the ratio between  =. This effect waveguide is observed for the MM regime with large permit- causes the robustness decreases as the increment of " in some tivity. In addition, a more complicated situation is investigated regions. However, an obvious improvement in robustness is with a bent waveguide embedded into a hexagonal lattice, as observed when the system works in the Mie regime as a meta- shown in Figs. 4(d,e). Despite the vulnerability to the dis- material compared to photonic crystals in Bragg regime. order at the corner where the propagation direction varies, the waveguide operating in the MM regime [Fig. 4(e)] persists the In summary, we have revealed a novel regime for the scat- function under the position disorder, compared with the PhC tering of light in photonic structures with robust bandgaps by regime [Fig. 4(d)]. We implement a quantitative analysis for transforming the structure from a photonic crystal to a dielec- the transmission degradation in Fig. 4(f), demonstrating the tric metamaterial. When the Mie scattering from individual robustness enhancement from permittivity increment similar dielectric elements dominate over the Bragg scattering, both to a straight waveguide in a square lattice of nanorods shown reflection and confinement of light becomes immune to an in- in Fig. 4(c). tense disorder. Our study provides an useful guide for the Besides waveguides, PhCs are widely used as to build op- nanofabrication of different photonic structures by employ- tical cavities for different applications ranging from lasing to ing dielectric metamaterials with high " for achieving the ro- sensing. Consequently, it is crucial to clarify the robustness of bust bandgap regime and also lifting strict requirements on an optical cavity formed by high-index dielectric rods. Here periodicity. For hexagonal lattices, one can achieve robust we create an optical cavity by introducing a point defect in the bandgaps from the visible to infrared spectra for GaAs [49] hexagonal lattice, as shown in Fig. 5(a). Again, we assume and Ge [50]. Importantly, such photonic structures can be re- that disorder is embedded in the rod position, as demonstrated alized with the bottom-up fabrication approach by utilizing in Fig. 5(a) for a perfect cavity ( = 0, upper panel) and a the vertically aligned nanowires [51–53]. In this case, the fluc- disordered cavity ( = 100 nm, lower panel). tuations in position dominates compared with that in size, as The position fluctuations for nanorods defining the cav- shown in Fig.3(a). ity boundary (circles) is intentionally eliminated to provide a fixed cavity shape. For the evaluation of the cavity robust- ness, we analyze the Q factor calculated as Q = ! =! The work has partially been supported by the ERC Con- through the signals from four randomly located points inside solidator Grant (TOPOLOGICAL), the Royal Society, Wolf- the cavity, with ! being the resonant frequency, and ! be- son Foundation, the Ministry of Education and Science of the ing FWHM parameter. Figure 5(b) demonstrates a relation- Russian Federation (3.1500.2017/4.6), the Russian Founda- ship between Q and  for both PhC and MM regimes, re- tion for Basic Research (18-02-00427), the Australian Re- spectively, with the value averaged with three different sets search Council, and the European Union Horizon 2020 re- of random variables. The Q factor is normalized to its value search innovation programme under the Marie Skłodowska- without disorder, Q , to exclude a difference between the two Curie Grant (No. 752102). YK thanks K. Busch, Th. Krauss, cases. We observe that PhC structures (" = 4) are quite fragile and W. Vos for critical comments and useful discussions. Q/Q ε 5 [20] P. D. Garcıa, S. Smolka, S. Stobbe, and P. Lodahl, Density of states controls Anderson localization in disordered photonic crystal waveguides, Phys. Rev. B 82, 165103 (2010). [1] F. Trager ¨ , Springer handbook of lasers and optics (Springer Sci- [21] L. Sapienza, H. Thyrrestrup, S. Stobbe, P. D. Garcia, S. Smolka, ence & Business Media, 2012). and P. Lodahl, Cavity quantum electrodynamics with anderson- [2] P. Cheben, R. Halir, J. H. Schmid, H. A. Atwater, and D. R. localized modes, Science 327, 1352 (2010). Smith, Subwavelength integrated photonics, Nature 560, 565 [22] P. D. Garc´ ıa, R. Sapienza, C. Toninelli, C. Lopez, ´ and D. S. (2018). Wiersma, Photonic crystals with controlled disorder, Phys. [3] J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Rev. A 84, 023813 (2011). Meade, Photonic crystals: molding the flow of light (Princeton [23] O. L. Muskens, A. F. Koenderink, and W. L. Vos, Broadband university press, 2011). coherent backscattering spectroscopy of the interplay between [4] K. Hirose, Y. Liang, Y. Kurosaka, A. Watanabe, T. Sugiyama, order and disorder in three-dimensional opal photonic crystals, and S. Noda, Watt-class high-power, high-beam-quality Phys. Rev. B 83, 155101 (2011). photonic-crystal lasers, Nature Photon. 8, 406 (2014). [24] V. Savona, Electromagnetic modes of a disordered photonic [5] S. Wu, S. Buckley, J. R. Schaibley, L. Feng, J. Yan, D. G. crystal, Phys. Rev. B 83, 085301 (2011). Mandrus, F. Hatami, W. Yao, J. Vuck ˇ ovic, ´ A. Majumdar, [25] Z.-Y. Li and Z.-Q. Zhang, Photonic bandgaps in disordered et al., Monolayer semiconductor nanocavity lasers with ul- inverse-opal photonic crystals, Adv. Mater. 13, 433 (2001). tralow thresholds, Nature 520, 69 (2015). [26] Z.-Y. Li and Z.-Q. Zhang, Fragility of photonic band gaps in [6] E. Kuramochi, K. Nozaki, A. Shinya, K. Takeda, T. Sato, inverse-opal photonic crystals, Phys. Rev. B 62, 1516 (2000). S. Matsuo, H. Taniyama, H. Sumikura, and M. Notomi, Large- [27] M. Bayindir, E. Cubukcu, I. Bulu, T. Tut, E. Ozbay, and scale integration of wavelength-addressable all-optical memo- C. M. Soukoulis, Photonic band gaps, defect characteristics, ries on a photonic crystal chip, Nature Photon. 8, 474 (2014). and waveguiding in two-dimensional disordered dielectric and [7] C. Fenzl, T. Hirsch, and O. S. Wolfbeis, Photonic crystals for metallic photonic crystals, Phys. Rev. B 64, 195113 (2001). chemical sensing and biosensing, Angew. Chem. Int. Ed. 53, [28] M. A. Kaliteevski, J. M. Martinez, D. Cassagne, and J. P. Al- 3318 (2014). bert, Disorder-induced modification of the transmission of light [8] S. Noda, M. Fujita, and T. Asano, Spontaneous-emission con- in a two-dimensional photonic crystal, Phys. Rev. B 66, 113101 trol by photonic crystals and nanocavities, Nature Photon. 1, (2002). 449 (2007). [29] A. Yamilov and H. Cao, Highest-quality modes in disordered [9] V. N. Astratov, A. M. Adawi, S. Fricker, M. S. Skolnick, D. M. photonic crystals, Phys. Rev. A 69, 031803 (2004). Whittaker, and P. N. Pusey, Interplay of order and disorder in [30] R. Rengarajan, D. Mittleman, C. Rich, and V. Colvin, Effect of the optical properties of opal photonic crystals, Phys. Rev. B disorder on the optical properties of colloidal crystals, Phys. 66, 165215 (2002). Rev. E 71, 016615 (2005). [10] A. F. Koenderink and W. L. Vos, Light exiting from real pho- [31] M. A. Kaliteevski, D. M. Beggs, S. Brand, R. A. Abram, and tonic band gap crystals is diffuse and strongly directional, Phys. V. V. Nikolaev, Stability of the photonic band gap in the pres- Rev. Lett. 91, 213902 (2003). ence of disorder, Phys. Rev. B 73, 033106 (2006). [11] A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, Suppression [32] A. V. Lavrinenko, W. Wohlleben, and R. J. Leyrer, Influence of of left-handed properties in disordered metamaterials, J. Appl. imperfections on the photonic insulating and guiding properties Phys. 97, 113906 (2005). of finite Si-inverted opal crystals, Opt. Express 17, 747 (2009). [12] S. Hughes, L. Ramunno, J. F. Young, and J. E. Sipe, Extrinsic [33] L. A. Dorado and R. A. Depine, Modeling of disorder effects optical scattering loss in photonic crystal waveguides: role of and optical extinction in three-dimensional photonic crystals, fabrication disorder and photon group velocity, Phys. Rev. Lett. Phys. Rev. B 79, 045124 (2009). 94, 033903 (2005). [34] A. Alu ` and N. Engheta, Effect of small random disorders [13] C. Rockstuhl, U. Peschel, and F. Lederer, Correlation between and imperfections on the performance of arrays of plasmonic single-cylinder properties and bandgap formation in photonic nanoparticles, New J. Phys. 12, 013015 (2010). structures, Opt. Lett. 31, 1741 (2006). [35] D. Mogilevtsev, F. A. Pinheiro, R. R. Dos Santos, S. B. Cav- [14] J. Gollub, T. Hand, S. Sajuyigbe, S. Mendonca, S. Cummer, and alcanti, and L. E. Oliveira, Light propagation and Anderson D. R. Smith, Characterizing the effects of disorder in metama- localization in disordered superlattices containing dispersive terial structures, Appl. Phys. Lett. 91, 162907 (2007). metamaterials: effects of correlated disorder, Phys. Rev. B 84, [15] A. A. Asatryan, L. C. Botten, M. A. Byrne, V. D. Freilikher, 094204 (2011). S. A. Gredeskul, I. V. Shadrivov, R. C. McPhedran, and Y. S. [36] E. Maguid, M. Yannai, A. Faerman, I. Yulevich, V. Kleiner, and Kivshar, Suppression of Anderson localization in disordered E. Hasman, Disorder-induced optical transition from spin Hall metamaterials, Phys. Rev. Lett. 99, 193902 (2007). to random Rashba effect, Science 358, 1411 (2017). [16] J. Topolancik, B. Ilic, and F. Vollmer, Experimental observa- [37] M. Jang, Y. Horie, A. Shibukawa, J. Brake, Y. Liu, S. M. Ka- tion of strong photon localization in disordered photonic crystal mali, A. Arbabi, H. Ruan, A. Faraon, and C. Yang, Wavefront waveguides, Phys. Rev. Lett. 99, 253901 (2007). shaping with disorder-engineered metasurfaces, Nature Pho- [17] C. Toninelli, E. Vekris, G. A. Ozin, S. John, and D. S. Wiersma, ton. 12, 84 (2018). Exceptional reduction of the diffusion constant in partially dis- [38] A. N. Poddubny, M. V. Rybin, M. F. Limonov, and Y. S. ordered photonic crystals, Phys. Rev. Lett. 101, 123901 (2008). Kivshar, Fano interference governs wave transport in disor- [18] R. J. P. Engelen, D. Mori, T. Baba, and L. Kuipers, Two regimes dered systems, Nature Commun. 3, 914 ( 2012). of slow-light losses revealed by adiabatic reduction of group [39] Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljaci ˇ c, ´ Ob- velocity, Phys. Rev. Lett. 101, 103901 (2008). servation of unidirectional backscattering-immune topological [19] C. Rockstuhl and F. Lederer, Suppression of the local density electromagnetic states, Nature 461, 772 (2009). of states in a medium made of randomly arranged dielectric [40] J. Lian, J.-X. Fu, L. Gan, and Z.-Y. Li, Robust and disorder- spheres, Phys. Rev. B 79, 132202 (2009). immune magnetically tunable one-way waveguides in a gyro- 6 magnetic photonic crystal, Phys. Rev. B 85, 125108 (2012). supplemental/10.1103/PhysRevLett.???.???? [41] M. V. Rybin, D. S. Filonov, K. B. Samusev, P. A. Belov, Y. S. ?? for details of (S1) field distribution in waveguides; (S2) Kivshar, and M. F. Limonov, Phase diagram for the transi- transmission spectra of structures with a hexagonal lattice; (S3) tion from photonic crystals to dielectric metamaterials, Nature analysis of material losses; (S4) analysis of finite height of Commun. 6, 10102 (2015). rods; (S5) definition of three regimes shown in Fig. 3(a). [42] S. Kruk and Y. Kivshar, Functional meta-optics and nanopho- [49] D. E. Aspnes, S. M. Kelso, R. A. Logan, and R. Bhat, Optical tonics governed by Mie resonances, ACS Photon. 4, 2638 properties of Al Ga As, J. Appl. Phys. 60, 754 (1986). x 1x (2017). [50] G. E. Jellison Jr, Optical functions of GaAs, GaP, and Ge de- [43] Y. Kivshar, All-dielectric meta-optics and non-linear nanopho- termined by two-channel polarization modulation ellipsometry, tonics, National Science Review 5, 144 (2018). Opt. Mater. 1, 151 (1992). [44] I. M. Lifshits, S. A. Gredeskul, and L. A. Pastur, Introduction [51] A. I. Persson, M. W. Larsson, S. Stenstrom, ¨ B. J. Ohlsson, to the theory of disordered systems (Wiley, New York, 1988). L. Samuelson, and L. R. Wallenberg, Solid-phase diffusion [45] A. A. Dmitriev and M. V. Rybin, Combining isolated scatter- mechanism for GaAs nanowire growth, Nature Mater. 3, 677 ers into a dimer by strong optical coupling, Phys. Rev. A 99, (2004). 063837 (2019). [52] S. D. Hersee, X. Sun, and X. Wang, The controlled growth of [46] S. G. Johnson and J. D. Joannopoulos, Introduction to photonic gan nanowires, Nano Lett. 6, 1808 (2006). crystals: Bloch’s theorem, band diagrams, and gaps (but no [53] Z. H. Wu, X. Y. Mei, D. Kim, M. Blumin, and H. E. Ruda, defects), Photonic Crystal Tutorial pp. 1–16 (2003). Growth of Au-catalyzed ordered GaAs nanowire arrays by [47] S. V. Li, Y. S. Kivshar, and M. V. Rybin, Towards silicon-based molecular-beam epitaxy, Appl. Phys. Lett. 81, 5177 (2002). metamaterials, ACS Photon. 5, 4751 (2018). [48] See Supplemental Material at http://link.aps.org/ http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Condensed Matter arXiv (Cornell University)

Disorder-immune photonics based on Mie-resonant dielectric metamaterials

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10.1103/PhysRevLett.123.163901
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1;2 3;4 1;5 1 3;6 Changxu Liu , Mikhail V. Rybin , Peng Mao , Shuang Zhang , and Yuri Kivshar School of Physics and Astronomy, University of Birmingham, Birmingham B15 2TT, UK Chair in Hybrid Nanosystems, Nanoinstitute Munich, Faculty of Physics, Ludwig Maximilians University of Munich, 80539 Munich, Germany ITMO University, St Petersburg 197101, Russia Ioffe Institute, St Petersburg 194021, Russia College of Electronic and Optical Engineering and College of Microelectronics, Nanjing University of Posts and Telecommunications, Nanjing 210023, China and Nonlinear Physics Centre, Australian National University, Canberra, ACT 2601, Australia When the feature size of photonic structures becomes comparable or even smaller than the wavelength of light, the fabrication imperfections inevitably introduce disorder that may eliminate many functionalities of subwavelength photonic devices. Here we suggest a novel concept to achieve a robust bandgap which can endure disorder beyond 30% as a result of the transition from photonic crystals to Mie-resonant metamaterials. By utilizing Mie-resonant metamaterials with high refractive index, we demonstrate photonic waveguides and cavities with strong robustness to position disorder, thus providing a novel approach to the bandgap-based nanophotonic devices with new properties and functionalities. The idea of manipulating the electromagnetic waves with subwavelength structures originates from the 19-th century, σ = σ σ >> σ σ = 0 0 0 when Heinrich Hertz managed to control meter-long ra- dio waves through wire-grid polarizer with centimeter spac- ings [1]. As the rapid advancement of the nanotech- nology with fabrication resolution down to micrometer or even nanometers, a plethora of subwavelength systems with structure-induced optical properties are achieved, ranging from photonic crystals to metamaterials [2]. Among them, a disorder photonic crystal (PhC) is a periodic optical structure that has attracted considerable interest for its ability to confine, manip- FIG. 1: Schematic of a photonic structure, composed of dielectric ulate, and guide light [3]. Spatial periodicity of the dielectric nanorods, with an increasing position disorder , respectively. function is essential to obtain a photonic bandgap where the propagation for photons within a certain frequency gap is for- bidden, providing unique features for a variety of applications 43], here we consider photonic structures with the optically ranging from lasers [4, 5], all-optical memories [6] to sensing induced Mie resonances and reveal that they can support [7] and emission control [8]. disorder-immune photonic bandgaps, in a sharp contrast with To achieve unparalleled functionalities with the subwave- PhC where the Bragg resonances require stringent periodicity length structures, a stringent requirement for the fabrication and consequently are not tolerant to disorder. Our numerical accuracy is required. As a result, the impact of disorder on results demonstrate robustness of the optical waveguides un- such photonic structures has extensively been studied, both der intense disorder, suggesting the way towards a new gener- numerically and experimentally [9–38]. When the disorder is ation of disorder-immune photonic devices with cost-effective small enough (up to a few percent of the lattice constant) and fabrication processes. can be treated as a perturbation, the interaction between the We start from an ideal periodic structure composed of order and disorder gives rise to interesting optical transport nanorods arranged in a square lattice, as illustrated in the left phenomena involving multiple light scattering, diffusion and panel of Fig. 1. The lattice constant is a = 500 nm and the localization of light [16–18, 20–24]. As disorder is increased rod radius is r = 125 nm, so that the ratio r=a defines a filling further, the photonic bandgap is destroyed, owing to the ad- fraction of the structure. The permittivity of the nanorod is verse effect to the Bragg reflection [25–33]. For example, only ", with whose value identifying the system either as photonic a few percent of disorder can eliminate the bandgap of inverse crystals (low ") or dielectric metamaterials (high ") [41]. As opal photonic crystals [25, 26]. The only way to achieve ro- i i the first step, we introduce disorder to the rod position (x ; y ) bustness is to utilize nontrivial topological properties [39] in i i i i i i as: x = x + U and y = y + U , where (x ; y ) is the x y 0 0 0 0 waveguides with gyromagnetic materials. However, the ex- original position in the periodic lattice, U and U are random x y ternal magnetic field is a prerequisite to break time-reversal variables distributed uniformly over the interval [1; 1], and symmetry [40], hindering its practical application. the parameter  describes the strength of the disorder. We also Being inspired by the recent studies of the dielectric Mie- consider the normalized disorder strength  defined as the - resonant metamaterials (MMs) and their link to PhCs [41– to-a ratio expressed in %. Since the height of the nanorod is arXiv:1905.13734v2 [physics.app-ph] 25 Sep 2019 2 (a) Photonic crystal (d) Metamaterial -20 -20 -40 -40 -60 -80 -60 η = 0% -100 η = 10% -80 η = 20% -120 η = 40% -100 -140 0.3 0.4 0.5 0.6 0.7 0.8 0.3 0.4 0.5 0.6 0.7 0.8 a/λ a/λ (b) (c) (e) (f) FIG. 2: Transmission spectra of (a) a photonic crystal (with " = 4) and (d) a metamaterial (with " = 25) with different values of the disorder parameter . Black solid curves are perfect structures ( = 0), red dashed curves are weak disorder ( = 10%); green dash-dot curves are moderate disorder ( = 20%); blue dotted curves correspond to a strong disorder ( = 40%). Origin of spectral dips are labeled. Magnetic field distribution in (c,d) photonic crystal (for  = 0 and  = 40%, respectively) and (e,f) metamaterial (with  = 0 and  = 40%, respectively). In each of the panels (c,d) and (e,f), the waves propagate from the bottom to the top. much larger than its radius, a two-dimensional approximation gaps with a stronger effect manifested for the second bandgap in the (x; y) plane is valid. associated with the TE Mie resonances (about 70 dB for even weak disorder  = 50 nm or  = 10%). When disorder is introduced (Fig. 1), the translational sym- metry of the structure becomes broken, making the bandgap In a sharp contrast, in the regime of a metamaterial structure of the spectrum in the reciprocal space to be ill- [Fig. 2(d)], the lowest bandgap survives under even strong dis- defined. As a result, we study properties of these photonic order of  = 200 nm (or  = 40%). In this regime, the Mie structures in the real space assuming that the low-transmission scattering from individual nanorods play a paramount role to spectral regions associated with bandgaps are still observable form the bandgap through the TE Mie resonances, reduc- in the corresponding spectrum. ing strict requirements of periodicity. The field distributions In the analysis of disordered media, the light propaga- shown in Fig. 2(e,f) reveal the effective field suppression by tion is characterized by a logarithmic-average transmission each nanorod oscillating out-of-phase with the incident wave. instead of average transmission (see Ref. [44] for details). Rigorous model accounts perturbations of coupling constants Figure 2 demonstrates the logarithmic-averaged transmission between neighbor rods [45] due to the position disorder. It re- (averaged over an ensemble of 100 samples) vs. the disor- sults in degradation of the suppression however this affects the der strength  for both photonic crystals and metamaterials. TE Mie gap much weaker than its Bragg counterpart. Re- In addition, we show the results for the wave propagation markably, the position disorder affects all other gaps including through the corresponding structures with a specific disorder higher-order Mie gaps. The reason is that Bragg frequency realization for the lowest bandgap. We observe that the spectra obey the law f / (d cos ) , where d is a lattice spacing and consist of a number of pronounced dips (associated with the  is the propagation angle. The higher-order Mie gaps above spectral gaps) which can be linked to either Mie and Bragg the lowest Bragg gap do not demonstrate robustness, since resonances. The Bragg gaps are observed as symmetric dips, they are not pure Mie gaps but mixtures with Bragg waves for while the Mie gaps have a knife-tip shape. In the regime of certain directions. Thus, in spite of the identical configura- photonic crystals [Fig. 2(a)], we observe a degradation of all tions of the dielectric nanorods in Fig. 2(c) and Fig. 2(f), the Transmission (dB) Bragg gap TE Mie gap TE Mie gap Bragg gap TE Mie gap Bragg gap TE Mie gap TE Mie gap 02 3 1 1 (a) (b) σ (nm) σ (nm) 0 50 100 150 200 250 300 0 50 100 150 200 250 0.8 0.5 90% 0.6 ε=25 Mie 50% 0.4 30% -2 ε=4 -0.5 0.2 16 Mie+Bragg -4 -1 -2 0 2 -2 0 2 0 10 30 50 10% (a) (b) (c) x (μm) x (μm) η (%) 5% Bragg 1 ε=17 3% 0.8 4 4 2% 0 10 20 30 40 50 60 0 10 20 30 40 50 0.6 η (%) η (%) 0.4 Square Hexagonal -2 ε=4 0.2 -4 0 5 10 15 20 -5 0 5 -5 0 5 FIG. 3: Degradation of the reflection at different values of ", illus- x (μm) x (μm) η (%) (d) (e) (f) trating the robustness of the structure for (a) square and (b) hexagonal lattices, respectively. FIG. 4: (a,b) Field distribution H in a straight waveguide created in a square lattice of nanorods with disorder  = 10% ( = 50 nm) for (a) " = 4 and (b) " = 25. (c) The corresponding relative transmis- wave propagation is remarkably different when the dielectric sion T=T vs.  for " = 4 and " = 25, respectively, for a square constant " of each rod changes from low to higher values. lattice of nanorods. (d,e) Field distribution H in a bent waveguide created in a hexagonal lattice with disorder  = 8% ( = 40 nm) for To provide a comprehensive picture of the impact of the (d) " = 4 and (e) " = 17. (f) The corresponding relative transmis- position disorder to the system transforming from the PhC sion T=T vs.  for " = 4 and " = 17, respectively, for a hexagonal to MM regimes, we conduct a series of numerical simu- lattice of nanorods. lations with different values of " ranging from 4 to 25. To quantitatively represent the robustness of the photonic bandgap, we define parameter = (R R)=R ; where 0 0 bandgaps, we analyze the structure robustness for a differ- R = Rd=(  ) is the averaged reflection in the ent geometry, namely a hexagonal lattice of nanorods that can 2 1 bandgap between  and  ;  and  are wavelengths at support a bandgap for the TE waves [46]. In addition, a differ- 1 2 1 2 10% of the transmission minimum. ent ratio r=a = 0:3 is employed for the generality study while The degradation is normalized by R , that is the reflection the lattice constant is kept the same, a = 500 nm. Figure 3(b) in the bandgap without disorder. Consequently, represents shows the value of for varying ". A similar behavior is illus- the deterioration of reflection in the bandgap, with a smaller trated for a square lattice nanorods in Fig. 3(a), demonstrating value demonstrating a better robustness. Figure 3(a) summa- three regimes with different level of robustness to the posi- rizes the value of with different " as the disorder  increases tion disorder. The values of " for achieving strong robustness for the bandgaps with lowest energy (as shown in Figs. 2(a) is ameliorated to a smaller value around 15 due to the optimal and (d). The value of is averaged from three different sets lattice [47]. The typical transmission spectra for both PhC and of the uniform random variables. The relative change to the MM regimes can be found in Supplemental Material [48]. lattice constant is also labeled by the  axis. The diagram can We further investigate the robustness effect for more prac- be unambiguously divided into three regimes. When the value tical structures such as a photonic waveguide. The waveg- of " is small, the system operates as a PhC, corresponding to uide is readily generated by introducing a line defect along the situation shown in Figs. 2(a-c). The bandgap is quite vul- the y-direction. For the PhC case, we select the TE Mie nerable to the disorder, around 10% of the position disorder bandgap with a better light confinement (since the Bragg fre- can break the perfection of the photonic bandgap. As the in- quency has a strong angular dependence). Figures 4(a,b) illus- crement of the permittivity, the system transforms into a new trate the spatial distribution of the magnetic field in a disorder- regime, where the bandgaps is formed by the overlap of Mie impacted waveguide operating as PhC and MM, respectively, and Bragg resonances [41]. The Mie scattering from individ- while the disorder-free example can be found in SM [48]. ual nanorods increases the robustness to the disorder, reducing Here the disorder parameter =50 nm. In spite of the disorder the degradation compared to the previous regime. Further destroying the ideal structure, the light is still well confined in enhancement of " drives the lowest bandgap formed by the the active region [Fig. 4(b)], demonstrating a good robustness Mie scattering, and the system transforms into the effective when operating in the MM regime. With a reduced value of MM structure with a good robustness to the position disor- ", the waveguide losses its function with the transverse dif- der. Under intense disorder, a well-defined stop band persists, fusion under the same position configuration for a PhC, as as illustrated in Figs. 2(d-f). The transition of the robustness shown in Fig. 4a. To quantitatively demonstrate the impact of parameter precisely matches the phase transition from PhC disorder, we calculated the transmission T of the waveguide to MM ”phases” [41], identifying the unique role of the Mie under different , as shown in Fig. 4(c). Similarly, the trans- scattering playing in the disorder-immune photonic bandgaps. mission is normalized to T , the value without disorder for the To illustrate generality of the disorder-immune photonic comparison. A definite improvement of the robustness for the y (μm) y (μm) Amplitude (arb. units) T/T T/T hexagonal square 4 (a) (c) to the disorder, whereas the photonic cavity operating in the MM regime (" = 15) is more robust, only experiencing severe degradation when  reaches 50 nm. With the introduction of moderate degree of disorder, the quality factor is improved with the value Q=Q beyond 1, matching earlier results [29]. (b) (d) 90% 1 70% Mie In addition to fluctuations in the rod positions, we study 50% 0.8 also the robustness to a size disorder and consider a pho- 30% ε=15 tonic structure of different nanorods, see Figs. 5(c-d). We as- 0.6 16 Mie+Bragg sume that the disorder introduces fluctuations in the rod radii, 10% 0.4 r = r +  U , where U are random variables distributed r r r 5% 0.2 Bragg 8 4% uniformly over the interval [1; 1], and the relative disorder ε=4 3% is:  = =a =  =r. Figure 5(c) shows a typical disordered 0 2% 0 4 8 12 16 20 0 10 20 30 40 structure with  = 25%. Similarly, we use the parameter η (%) η (%) to evaluate the structure robustness, as shown in Fig. 5(d) for the regimes transform from PhC (" = 4) to MM (" = 25). FIG. 5: (a) An optical cavity formed by a hexagonal lattice of di- The fluctuations in the nanorod radius cause a variation of the electric rods without disorder (left) and with disorder (right). (b) Mie resonance, consequently inducing deterioration of the ro- Degradation of the relative quality factor Q=Q with disorder  for " = 4 and " = 15. (c) An example of a photonic structure with both bustness compared with the case presented in Fig. 3(a). In ad- position and size disorder, for  = 25%. (d) Robustness parameter ditional, the same fluctuation in size  could bring stronger vs. disorder  and ". disorder to the nanorod with higher permittivity, considering the optical wavelength inside the dielectric  =  = " and consequently increasing the ratio between  =. This effect waveguide is observed for the MM regime with large permit- causes the robustness decreases as the increment of " in some tivity. In addition, a more complicated situation is investigated regions. However, an obvious improvement in robustness is with a bent waveguide embedded into a hexagonal lattice, as observed when the system works in the Mie regime as a meta- shown in Figs. 4(d,e). Despite the vulnerability to the dis- material compared to photonic crystals in Bragg regime. order at the corner where the propagation direction varies, the waveguide operating in the MM regime [Fig. 4(e)] persists the In summary, we have revealed a novel regime for the scat- function under the position disorder, compared with the PhC tering of light in photonic structures with robust bandgaps by regime [Fig. 4(d)]. We implement a quantitative analysis for transforming the structure from a photonic crystal to a dielec- the transmission degradation in Fig. 4(f), demonstrating the tric metamaterial. When the Mie scattering from individual robustness enhancement from permittivity increment similar dielectric elements dominate over the Bragg scattering, both to a straight waveguide in a square lattice of nanorods shown reflection and confinement of light becomes immune to an in- in Fig. 4(c). tense disorder. Our study provides an useful guide for the Besides waveguides, PhCs are widely used as to build op- nanofabrication of different photonic structures by employ- tical cavities for different applications ranging from lasing to ing dielectric metamaterials with high " for achieving the ro- sensing. Consequently, it is crucial to clarify the robustness of bust bandgap regime and also lifting strict requirements on an optical cavity formed by high-index dielectric rods. Here periodicity. For hexagonal lattices, one can achieve robust we create an optical cavity by introducing a point defect in the bandgaps from the visible to infrared spectra for GaAs [49] hexagonal lattice, as shown in Fig. 5(a). Again, we assume and Ge [50]. Importantly, such photonic structures can be re- that disorder is embedded in the rod position, as demonstrated alized with the bottom-up fabrication approach by utilizing in Fig. 5(a) for a perfect cavity ( = 0, upper panel) and a the vertically aligned nanowires [51–53]. In this case, the fluc- disordered cavity ( = 100 nm, lower panel). tuations in position dominates compared with that in size, as The position fluctuations for nanorods defining the cav- shown in Fig.3(a). ity boundary (circles) is intentionally eliminated to provide a fixed cavity shape. For the evaluation of the cavity robust- ness, we analyze the Q factor calculated as Q = ! =! The work has partially been supported by the ERC Con- through the signals from four randomly located points inside solidator Grant (TOPOLOGICAL), the Royal Society, Wolf- the cavity, with ! being the resonant frequency, and ! be- son Foundation, the Ministry of Education and Science of the ing FWHM parameter. Figure 5(b) demonstrates a relation- Russian Federation (3.1500.2017/4.6), the Russian Founda- ship between Q and  for both PhC and MM regimes, re- tion for Basic Research (18-02-00427), the Australian Re- spectively, with the value averaged with three different sets search Council, and the European Union Horizon 2020 re- of random variables. The Q factor is normalized to its value search innovation programme under the Marie Skłodowska- without disorder, Q , to exclude a difference between the two Curie Grant (No. 752102). YK thanks K. Busch, Th. Krauss, cases. We observe that PhC structures (" = 4) are quite fragile and W. Vos for critical comments and useful discussions. Q/Q ε 5 [20] P. D. Garcıa, S. Smolka, S. Stobbe, and P. Lodahl, Density of states controls Anderson localization in disordered photonic crystal waveguides, Phys. Rev. B 82, 165103 (2010). [1] F. Trager ¨ , Springer handbook of lasers and optics (Springer Sci- [21] L. Sapienza, H. Thyrrestrup, S. Stobbe, P. D. Garcia, S. Smolka, ence & Business Media, 2012). and P. Lodahl, Cavity quantum electrodynamics with anderson- [2] P. Cheben, R. Halir, J. H. Schmid, H. A. Atwater, and D. R. localized modes, Science 327, 1352 (2010). Smith, Subwavelength integrated photonics, Nature 560, 565 [22] P. D. Garc´ ıa, R. Sapienza, C. Toninelli, C. Lopez, ´ and D. S. (2018). Wiersma, Photonic crystals with controlled disorder, Phys. [3] J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Rev. A 84, 023813 (2011). Meade, Photonic crystals: molding the flow of light (Princeton [23] O. L. Muskens, A. F. Koenderink, and W. L. Vos, Broadband university press, 2011). coherent backscattering spectroscopy of the interplay between [4] K. Hirose, Y. Liang, Y. Kurosaka, A. Watanabe, T. Sugiyama, order and disorder in three-dimensional opal photonic crystals, and S. Noda, Watt-class high-power, high-beam-quality Phys. Rev. B 83, 155101 (2011). photonic-crystal lasers, Nature Photon. 8, 406 (2014). [24] V. Savona, Electromagnetic modes of a disordered photonic [5] S. Wu, S. Buckley, J. R. Schaibley, L. Feng, J. Yan, D. G. crystal, Phys. Rev. B 83, 085301 (2011). Mandrus, F. Hatami, W. Yao, J. Vuck ˇ ovic, ´ A. Majumdar, [25] Z.-Y. Li and Z.-Q. Zhang, Photonic bandgaps in disordered et al., Monolayer semiconductor nanocavity lasers with ul- inverse-opal photonic crystals, Adv. Mater. 13, 433 (2001). tralow thresholds, Nature 520, 69 (2015). [26] Z.-Y. Li and Z.-Q. Zhang, Fragility of photonic band gaps in [6] E. Kuramochi, K. Nozaki, A. Shinya, K. Takeda, T. Sato, inverse-opal photonic crystals, Phys. Rev. B 62, 1516 (2000). S. Matsuo, H. Taniyama, H. Sumikura, and M. Notomi, Large- [27] M. Bayindir, E. Cubukcu, I. Bulu, T. Tut, E. Ozbay, and scale integration of wavelength-addressable all-optical memo- C. M. Soukoulis, Photonic band gaps, defect characteristics, ries on a photonic crystal chip, Nature Photon. 8, 474 (2014). and waveguiding in two-dimensional disordered dielectric and [7] C. Fenzl, T. Hirsch, and O. S. Wolfbeis, Photonic crystals for metallic photonic crystals, Phys. Rev. B 64, 195113 (2001). chemical sensing and biosensing, Angew. Chem. Int. Ed. 53, [28] M. A. Kaliteevski, J. M. Martinez, D. Cassagne, and J. P. Al- 3318 (2014). bert, Disorder-induced modification of the transmission of light [8] S. Noda, M. Fujita, and T. Asano, Spontaneous-emission con- in a two-dimensional photonic crystal, Phys. Rev. B 66, 113101 trol by photonic crystals and nanocavities, Nature Photon. 1, (2002). 449 (2007). [29] A. Yamilov and H. Cao, Highest-quality modes in disordered [9] V. N. Astratov, A. M. Adawi, S. Fricker, M. S. Skolnick, D. M. photonic crystals, Phys. Rev. A 69, 031803 (2004). Whittaker, and P. N. Pusey, Interplay of order and disorder in [30] R. Rengarajan, D. Mittleman, C. Rich, and V. Colvin, Effect of the optical properties of opal photonic crystals, Phys. Rev. B disorder on the optical properties of colloidal crystals, Phys. 66, 165215 (2002). Rev. E 71, 016615 (2005). [10] A. F. Koenderink and W. L. Vos, Light exiting from real pho- [31] M. A. Kaliteevski, D. M. Beggs, S. Brand, R. A. Abram, and tonic band gap crystals is diffuse and strongly directional, Phys. V. V. Nikolaev, Stability of the photonic band gap in the pres- Rev. Lett. 91, 213902 (2003). ence of disorder, Phys. Rev. B 73, 033106 (2006). [11] A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, Suppression [32] A. V. Lavrinenko, W. Wohlleben, and R. J. Leyrer, Influence of of left-handed properties in disordered metamaterials, J. Appl. imperfections on the photonic insulating and guiding properties Phys. 97, 113906 (2005). of finite Si-inverted opal crystals, Opt. Express 17, 747 (2009). [12] S. Hughes, L. Ramunno, J. F. Young, and J. E. Sipe, Extrinsic [33] L. A. Dorado and R. A. Depine, Modeling of disorder effects optical scattering loss in photonic crystal waveguides: role of and optical extinction in three-dimensional photonic crystals, fabrication disorder and photon group velocity, Phys. Rev. Lett. Phys. Rev. B 79, 045124 (2009). 94, 033903 (2005). [34] A. Alu ` and N. Engheta, Effect of small random disorders [13] C. Rockstuhl, U. Peschel, and F. Lederer, Correlation between and imperfections on the performance of arrays of plasmonic single-cylinder properties and bandgap formation in photonic nanoparticles, New J. Phys. 12, 013015 (2010). structures, Opt. Lett. 31, 1741 (2006). [35] D. Mogilevtsev, F. A. Pinheiro, R. R. Dos Santos, S. B. Cav- [14] J. Gollub, T. Hand, S. Sajuyigbe, S. Mendonca, S. Cummer, and alcanti, and L. E. Oliveira, Light propagation and Anderson D. R. Smith, Characterizing the effects of disorder in metama- localization in disordered superlattices containing dispersive terial structures, Appl. Phys. Lett. 91, 162907 (2007). metamaterials: effects of correlated disorder, Phys. Rev. B 84, [15] A. A. Asatryan, L. C. Botten, M. A. Byrne, V. D. Freilikher, 094204 (2011). S. A. Gredeskul, I. V. Shadrivov, R. C. McPhedran, and Y. S. [36] E. Maguid, M. Yannai, A. Faerman, I. Yulevich, V. Kleiner, and Kivshar, Suppression of Anderson localization in disordered E. Hasman, Disorder-induced optical transition from spin Hall metamaterials, Phys. Rev. Lett. 99, 193902 (2007). to random Rashba effect, Science 358, 1411 (2017). [16] J. Topolancik, B. Ilic, and F. Vollmer, Experimental observa- [37] M. Jang, Y. Horie, A. Shibukawa, J. Brake, Y. Liu, S. M. Ka- tion of strong photon localization in disordered photonic crystal mali, A. Arbabi, H. Ruan, A. Faraon, and C. Yang, Wavefront waveguides, Phys. Rev. Lett. 99, 253901 (2007). shaping with disorder-engineered metasurfaces, Nature Pho- [17] C. Toninelli, E. Vekris, G. A. Ozin, S. John, and D. S. Wiersma, ton. 12, 84 (2018). Exceptional reduction of the diffusion constant in partially dis- [38] A. N. Poddubny, M. V. Rybin, M. F. Limonov, and Y. S. ordered photonic crystals, Phys. Rev. Lett. 101, 123901 (2008). Kivshar, Fano interference governs wave transport in disor- [18] R. J. P. Engelen, D. Mori, T. Baba, and L. Kuipers, Two regimes dered systems, Nature Commun. 3, 914 ( 2012). of slow-light losses revealed by adiabatic reduction of group [39] Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljaci ˇ c, ´ Ob- velocity, Phys. Rev. Lett. 101, 103901 (2008). servation of unidirectional backscattering-immune topological [19] C. Rockstuhl and F. Lederer, Suppression of the local density electromagnetic states, Nature 461, 772 (2009). of states in a medium made of randomly arranged dielectric [40] J. Lian, J.-X. Fu, L. Gan, and Z.-Y. Li, Robust and disorder- spheres, Phys. Rev. B 79, 132202 (2009). immune magnetically tunable one-way waveguides in a gyro- 6 magnetic photonic crystal, Phys. Rev. B 85, 125108 (2012). supplemental/10.1103/PhysRevLett.???.???? [41] M. V. Rybin, D. S. Filonov, K. B. Samusev, P. A. Belov, Y. S. ?? for details of (S1) field distribution in waveguides; (S2) Kivshar, and M. F. Limonov, Phase diagram for the transi- transmission spectra of structures with a hexagonal lattice; (S3) tion from photonic crystals to dielectric metamaterials, Nature analysis of material losses; (S4) analysis of finite height of Commun. 6, 10102 (2015). rods; (S5) definition of three regimes shown in Fig. 3(a). [42] S. Kruk and Y. Kivshar, Functional meta-optics and nanopho- [49] D. E. Aspnes, S. M. Kelso, R. A. Logan, and R. Bhat, Optical tonics governed by Mie resonances, ACS Photon. 4, 2638 properties of Al Ga As, J. Appl. Phys. 60, 754 (1986). x 1x (2017). [50] G. E. Jellison Jr, Optical functions of GaAs, GaP, and Ge de- [43] Y. Kivshar, All-dielectric meta-optics and non-linear nanopho- termined by two-channel polarization modulation ellipsometry, tonics, National Science Review 5, 144 (2018). Opt. Mater. 1, 151 (1992). [44] I. M. Lifshits, S. A. Gredeskul, and L. A. Pastur, Introduction [51] A. I. Persson, M. W. Larsson, S. Stenstrom, ¨ B. J. Ohlsson, to the theory of disordered systems (Wiley, New York, 1988). L. Samuelson, and L. R. Wallenberg, Solid-phase diffusion [45] A. A. Dmitriev and M. V. Rybin, Combining isolated scatter- mechanism for GaAs nanowire growth, Nature Mater. 3, 677 ers into a dimer by strong optical coupling, Phys. Rev. A 99, (2004). 063837 (2019). [52] S. D. Hersee, X. Sun, and X. Wang, The controlled growth of [46] S. G. Johnson and J. D. Joannopoulos, Introduction to photonic gan nanowires, Nano Lett. 6, 1808 (2006). crystals: Bloch’s theorem, band diagrams, and gaps (but no [53] Z. H. Wu, X. Y. Mei, D. Kim, M. Blumin, and H. E. Ruda, defects), Photonic Crystal Tutorial pp. 1–16 (2003). Growth of Au-catalyzed ordered GaAs nanowire arrays by [47] S. V. Li, Y. S. Kivshar, and M. V. Rybin, Towards silicon-based molecular-beam epitaxy, Appl. Phys. Lett. 81, 5177 (2002). metamaterials, ACS Photon. 5, 4751 (2018). [48] See Supplemental Material at http://link.aps.org/

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Condensed Matter – arXiv (Cornell University)

Published: May 31, 2019

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