Frequency-selective omni-directional scattering of light by weakly disordered periodic planar arrays of dielectric particles

XULIN LIN,1,2,3 CHIA WEI HSU,4 AND GUO PING WANG1,2,3,*

1College of Electronic Science and Technology, Shenzhen University, Shenzhen, 518060, China 2Guangdong Provincial Key Laboratory of Micro/Nano Optomechatronics Engineering, Shenzhen University, Shenzhen, 518060, China 3Key Laboratory of Optoelectronic Devices and Systems of Ministry of Education and Guangdong Province, Shenzhen University, Shenzhen, 518060, China 4Department of Applied Physics, Yale University, New Haven, Connecticut, 06520, USA *gpwang@szu.edu.cn

Abstract: We show that disorder-induced interaction between guided modes and a flat band of low-Q radiating modes in planar arrays of dielectric particles can lead to frequency-selective omni-directional scattering of light. This phenomenon is numerically observed in planar arrays consisting of periodically arranged dielectric particles with small randomness in particle sizes. The flat band of low-Q radiating modes originates from coupling of Mie resonances of individual particles. The small structural randomness enables coupling between the guided modes and the radiating modes with overlapping spectral ranges. Through interaction with the flat band of low-Q radiating modes, the guided modes can be effectively excited by far-field sources from any direction, and make key contribution to the frequency-selective omni-directional scattering. The frequency-selective omni-directional scattering has many potential applications such as display and sensing. © 2016 Optical Society of America

OCIS codes: (350.4238) Nanophotonics and photonic crystals; (290.0290) Scattering; (260.5740) Resonance.

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1. Introduction

Micro-structured materials can greatly modify scattering of light, especially when interference and resonant processes are involved. As one kind of peculiar optical effects supported by micro-structured materials, frequency-selective and angle-insensitive scattering is highly desired for a number of applications, such as angle-independent structural colors, wide-viewing-angle display, and sensing. By frequency-selective and angle-insensitive scattering, we mean light from single direction is scattered to a wide range of directions only within some narrow frequency ranges [1, 2]. Resulting from Mie resonances, a single sub-wavelength particle can support enhanced omni-directional scattering but over a broad frequency range [3, 4]. Multi-layer films [5, 6], periodic planar structures [7–10], and 3D photonic crystals [11] have sharp frequency-selective responses, but can only reflect light to a discrete set of angles due to constraint of wavevector conservation. One apparent method to achieve frequency-selective and angle-insensitive scattering is using dense arrays of randomly distributed scatterers [12–18]. With strong multiple random scattering, the strongly disordered arrays are experienced by light as somewhat isotropic environments, and the constructive interference condition of scattered light can be satisfied within a narrow spectral range and with weak dependence on the orientation. It is worth noting that, however, strong disorder is not a necessary condition of frequency-selective and angle-insensitive scattering. One well-known example is Morpho butterflies' wings [19, 20]. The Morpho butterflies' wings consist of periodically arranged ridges of complex shape and with small randomness, and yet exhibit bright structural colors independent to viewing angle [21–23], thus revealing that the interplay between periodicity and disorder in influencing light scattering is very intriguing.

In this article, we numerically show that in a periodic planar array of dielectric particles imposed with small structural randomness, interaction between guided modes and a flat band of low-Q radiating modes leads to an unusual frequency-selective omni-directional scattering of light. The flat band of radiating modes originates from the coupling of Mie resonances of individual dielectric particles, and covers a broad frequency range and the full angular range [24]. The small structural randomness introduces additional spatial Fourier components and thus enables interaction between guided modes and radiating modes with overlapping spectral ranges and different wavevectors. This interaction provides an indirect pathway for the guided modes to couple to external radiations in almost every direction, and hence gives rise to omni-directional scattering within the frequency ranges overlapped by the guided modes and the radiating modes. The frequency-selective omni-directional scattering does not eliminate the normal reflection and transmission, but is additional to them. Therefore, when the particle array is illuminated by far-field sources, the light goes through two pathways: the direct pathway of the radiating modes responsible for normal reflection and transmission, and the indirect pathway of the guided modes causing omni-directional scattering. Obvious asymmetric spectral oscillations, which manifest effects of interference between the two pathway, are observed inside the particle array.

Light propagation in weakly disordered periodic systems is a long-lasting topic initiated by S. John [25], who proposed that even small randomness would leads to Anderson localization around the band edges, i. e., transportation of light wave is totally suppressed due to coherent multiple random scattering. Inspired by this profound insight, Anderson localization of light has been observed in various periodic structures imposed with weak disorder [26–30]. However, interplay between periodicity and weak disorder becomes more

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interesting when the weak disorder enables interaction between different kinds of modes with distinct physical properties. For example, it has been discovered that disorder-induced interference between Bragg multiple-scattering and single-scattering of Mie resonances or Fabry-Perot resonances may cause unexpected asymmetric rises in transmission spectrum or delocalization effects [31, 32].

In this article, we pay attention to another aspect of light propagation that can be influenced by disorder, i. e., out-of-plane scattering of periodic planar dielectric structures. It is well known that periodic planar dielectric structures can support guided modes which are totally confined within the periodic plane [33]. In the presence of obvious disorder or defects, some fraction of guided modes energy is scattered out of the periodic plane [34, 35], but direct coupling between guided modes and external radiations is still very weak due to severer momentum (wavevector) mismatch. Our main finding of this article is that in periodic planar arrays of dielectric particles which simultaneously support guided modes and low-Q radiating modes with overlapping spectral ranges, a very weak disorder can lead to frequency-selectively enhanced out-of-plane scattering by enabling coupling of the guided modes and the low-Q radiating modes. More importantly, through interaction with a flat band of low-Q radiating modes, the guided modes can be effectively excited by far-field sources from any direction and make notable contribution to the far-field scattering spectrum. We believe the frequency-selective omni-directional scattering can be considered as another interesting phenomenon arising from the interplay between periodicity and weak disorder.

2. Model and theory

The random array we consider throughout this article consists of dielectric cuboids arranged in a square lattice [inset of Fig. 1(a)]. Side lengths of the cuboids in x and y direction, denoted as l, vary randomly and uniformly from 0.475a to 0.525a, where a is the unit length of the square lattice. Heights of the cuboids in z direction are h = 0.4a. x' and y' denote the directions of diagonal lines of the square lattice. Dielectric constant of the cuboids is ε = 12.0 (which corresponds to Si or GaAs at optical wavelengths), and the background medium is air (n0 = 1.0). To show that the frequency-selective omni-directional scattering is directly related to the small randomness in particle sizes, we also consider a periodic planar array consisting of dielectric cuboids with uniform size, i.e., l = 0.50a; other parameters of the periodic array are identical to those of the random array.

We first show that the periodic array supports a continuous flat band of low-Q radiating modes whose frequency range overlaps with that of the guided modes. It is known that Mie resonances of individual dielectric particles make important contributions to the collective properties of photonic crystals in complex ways [36–39]; they can mix with bands from Bragg resonances, form separating bands, or widen the bandgaps. Here we consider the case that in periodic planar arrays of dielectric particles, strong coupling of magnetic dipolar (MD) Mie resonances gives rise to a continuous flat band of low-Q radiating modes, as shown in Fig. 1(a). We choose the MD Mie resonance because it is the lowest order Mie resonance with the broadest spectral width.

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Fig. 1. (a) Spectra of electromagnetic modes supported by the periodic array along the ΓM axis, excited by a magnetic dipole source placed at the center of the dielectric cuboid with the magnetic moment in y’ direction. The slant dashed line is the light

line2 2

0/ (2 )

x yf c n k kπ= + separating the radiating modes and the guided modes. The two

horizontal dashed lines indicate the spectral range of the radiating modes. Inset shows

schematic of the particle array. (b)-(e) Magnetic field '

yH distributions in the central xy plane

within one unit cell for spectral positions labeled in (a). The dashed square shows cross-section of the dielectric cuboid in xy plane.

Electromagnetic (EM) modes supported by the periodic array are computed using finite-difference time-domain (FDTD) simulations [40]. The computational region is one unit cell of the periodic array bounded by Bloch boundary conditions in x and y direction, and perfectly-matched layers (PMLs) in z direction. A pulsed magnetic dipole source with the magnetic moment in y’ direction is placed at the center of the cuboid, and dozens of point-monitors randomly distributed within the computational region are used to record time sequences of magnetic field after the source is off. Spectra of EM modes supported by the periodic array are obtained by applying Fourier transformation to the recorded time sequences of magnetic field. Figure 1(a) shows normalized magnetic spectra of radiating modes and trajectories of resonant frequencies of guided modes. Magnetic spectra of the radiating modes are normalized with respect to the maximum value of each wavevector k. The guided modes are represented by solid lines since their spectral widths are infinitesimal in theory.

In Fig. 1(a) we can see one band of radiating modes and two bands of guided modes, separated by the light line [the slant dashed line in Fig. 1(a)]. The radiating modes with broad spectral widths and almost-constant resonant frequencies originate from the coupling of MD Mie resonances of individual dielectric particles, as their symmetric mode patterns [Fig. 1(b)] and spectral ranges are very similar to the MD Mie resonance of single dielectric particle (see the Appendix). When moving close to the light line, spectral widths of the radiating modes are significantly reduced, and finally evolve into a flat guided band below the light line. Frequency ranges of the radiating modes [indicated by two horizontal dashed lines in Fig. 1(a)] overlap with the whole upper guided band and the upper part of the lower guided band. Frequency ranges overlapped by the radiating modes and guided modes are [0.487,0.506]( / )f c a∈ and [0.532,0.543]( / )f c a∈ .

To further show that the flat bands of radiating modes and guided band results from the coupling of individual Mie resonances, we also compute band diagrams of periodic planar arrays with smaller and larger lattice units (see the Appendix). For the configuration with smaller lattice unit, coupling of Mie resonances becomes stronger as distances between neighboring particles are reduced. For this reason, the spectral widths of the radiating modes become broader, and the upper guided modes become flatter. For the configuration with larger lattice unit, due to weaker coupling of Mie resonances, the spectral widths of the

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radiating modes become narrower, and the upper guided band has a significant slope; here the frequency ranges of the radiating modes no longer overlap with the guided modes.

Since the magnitude of randomness in particle sizes ( [ 0.025 ,0.025 ]l a aΔ ∈ − ) introduced

into the random array is far smaller than the wavelengths of interest, one can expect that EM modes supported by the random array are generally identical to those in the periodic array. Nevertheless, the small structural randomness influences the scattering properties of the random array by enabling the interaction between guided modes and radiating modes. In the periodic array, coupling of EM modes with different wavevectors is forbidden by the conservation of wavevector. In the random array, however, the small structural randomness effectively adds additional spatial Fourier components, which bridge coupling between the guided modes and the radiating modes. Interaction between the guided mode and the radiating modes in the random array results in frequency-selective omni-directional scattering, as will be shown below. The small randomness also causes coupling among radiating modes with different wavevectors, which also results in scattering over diverse directions. However, inner-coupling among radiating modes is much weaker than cross-coupling between guided modes and radiating modes because the latter one is enhanced by the much longer lifetime of the guided modes.

3. Transmissive FDTD simulations

Fig. 2. Frequency-selective leakage of guided modes in the random array. (a) Schematic of FDTD simulation setup. The in-plane incident light (the red arrow) is linearly polarized with magnetic field in y' direction. The particle array is bounded by periodic boundary conditions in y' direction and PMLs in x' and z directions. (b) Magnetic field spectra at the center point of the periodic arrays (blue line) and the random array (red line). The dash-dotted line shows

spectrum of the radiating mode at '

0x

k = , and spectral range of the radiating mode is

indicated by two vertical dashed lines. (c)-(f) Cross-sections of Hy’ distribution in the central x'z plane for spectral positions labeled in (b). (c) and (e) correspond to the periodic case, and (d) and (f) correspond to the random case.

In the following, we run FDTD simulations to demonstrate the effects of interaction between guided modes and radiating modes in the random array. Corresponding results of the periodic array are also provided for comparison. Configurations we use in the FDTD simulations are schematically shown in Fig. 2(a). Periodic boundary conditions are imposed in y’ direction, so that we can concentrate on modes on the ΓM axis shown in Fig. 1(a). We place PMLs in x'

and z directions. Lengths of the particle arrays in x' direction and in y' direction are 30 2a

and 2a , respectively.

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3.1 In-plane planewave source

We first show that guided modes in the random array radiate to free space in diverse directions through coupling with the radiating modes. As schematically shown in Fig. 2(a), an in-plane planewave source with the magnetic field in y' direction is launched from the left side of the particle array. Figure 2(b) shows the 'yH spectrum at the center point of the

periodic array (the periodic case) and that at the center point of the random array (the random case). In Fig. 2(b) we also show spectrum of the radiating mode at ' 0xk = , and its

spectral range is indicated by two vertical dashed lines. As can be seen in Fig. 2(b), spectral ranges of the periodic case and the random case are generally consistent with that of the guided modes in Fig. 1(a). The radiating modes are also excited by the planewave source, but they radiate to free space quickly due to their low Q. Outside the spectral range of the radiating modes, magnetic field magnitudes of the guides modes of the random case is almost identical to those of the periodic case, indicating that the small structural randomness alone has little influence on the guided modes. Within the spectral range of the radiating modes, magnitudes of the guided modes of the random case are obviously smaller than those of the periodic case, indicating that some energy of the guided modes leaks to free space due to coupling with the radiating modes.

The frequency-selective leakage of the guided modes in the random array can be clearly seen by inspecting spatial distributions of 'yH at the central x'z plane [Figs. 2(c)-2(f)]. For

Fig. 2(d) corresponding to the random case but outside the spectral range of the radiating modes, the mode energy of guided modes is tightly confined by the particle array, similar to the periodic case in Figs. 2(c) and 2(e). In Fig. 2(f) corresponding to the random case and within the spectral range of the radiating modes, there are complicated radiation patterns above and below the particle array. The radiation fields are clear evidence that the guided modes radiate to the free space above and below the particle array.

3.2 Vertical Gaussian beam source

We next show that when illuminated by far-field sources, disorder-induced interaction between guided modes and radiating modes leads to frequency-selective omni-directional scattering of light (Fig. 3). The far-field source we use here is a normally launched Gaussian beam. Waist radius of the Gaussian beam is set to be 0 15r a= , so that most of the beam

profile is covered by the particle array. Figures 3(a) and 3(b) are reflection spectra of the periodic array and the random array, respectively. Figure 3(b) is generally identical to Fig. 3(a), except for the small fluctuations within frequency ranges overlapped by the guided modes and the radiating modes (shaded by gray areas). Angular distributions of the reflected light in far-field are calculated using the well-established near-to-far-field transformation techniques [40], as shown in Figs. 3(c) and 3(d). In Fig. 3(c) corresponding to the periodic case, we can only see normally reflected light concentrating around the normal direction ( 0θ = ). Higher order diffraction is missing as the frequency range of interest is below 1.0(c/ a)f = , the minimum frequency for diffraction to take place. In Fig. 3(d)

corresponding to the random case, in addition to the normally reflected light, we can also see scattered lights spreading in diverse directions. Most importantly, in frequency ranges overlapped by the guided modes and the radiating modes, the scattered lights spread through the full angular range [ 90 ,90 ]θ ∈ − .

With the band diagram shown in Fig. 1(a), the process of frequency-sensitive omni-directional scattering can be understood as follows. First, the incident light couples to radiating modes around ' 0xk = according to the conservation of the wavevector. Then, a

small fraction of the energy transfers from the radiating modes to the guided modes due to the presence of small structural randomness. Finally, the guided modes couple again with the

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radiating modes and radiate to free space below and above the particle array. In the third step, as the radiating modes constitute a continuous flat band, the guided modes can couple to radiating modes in all directions, thus resulting in omni-directional scattering. By applying spatial and temporal Fourier transformations to 'yH distribution at the center x'y' plane of the

particle arrays, we obtain spectra of EM modes excited by the vertically launched Gaussian beam, as shown in Figs. 3(e) and 3(f). In Fig. 3(f) corresponding to the random case, we can see that the guided modes below the light line are excited within the spectral ranges of the radiating modes (indicated by red arrows), and the mode energy of guided modes spreads to the radiating modes above the light line (indicated by green arrows), thus proving that the guided modes are indirectly excited by the far-field source and play an important role in causing frequency-selective omni-directional scattering.

Fig. 3. Comparison of responses to a vertically launched gaussian beam between the periodic array and the random array. (a) and (b) Reflection spectra. Insets show schematics of FDTD

simulation setup. (c) and (d) Far-field projections2( , )E f θ of the reflected light along x' axis.

The far-field intensities are normalized to their maximum value (unity reflection for 0.552( / )f c a= and 0θ = ). (e) and (f) Results of temporal and spatial Fourier

transformations of 'yH in the central x'y' plane of the particle arrays. (g) and (h) 'yH spectra

within the center unit of the particle arrays. The vertical or horizontal dashed lines in each subfigure indicate spectral range of the radiating modes. The slant dashed lines in (e) and (f) are the light line which separates radiating modes and guided modes. Gray areas in (a), (b), (g), and (h) are frequency ranges overlapped by the guided modes and the radiating modes.

We note that the small structural randomness also enables inner-coupling of radiating modes with different 'xk , which can also cause scattering in widely diverse directions within

the whole spectral range of the radiating modes. However, as can be seen in Figs. 3(b) and

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3(d), fluctuations in the reflection spectrum are observed only within the frequency ranges overlapped by the guided modes and the radiating modes, and scattered light within the frequency ranges overlapped by the guided modes and the radiating modes is significantly brighter than that in frequency ranges covered only by the radiating modes; these indicate that the inner-coupling of radiating modes is a much smaller contribution compared to the cross-coupling between guided modes and radiating modes. The latter one is significantly enhanced by the much longer lifetime of the guided modes. We also note that the light transferred from the guided modes to the radiating modes makes up only a small proportion of the total energy of the guided modes; a large part of the guided mode leaks out of the particle arrays through the open boundaries in x' direction.

Now we know that for the random array, there are two pathways coupling to external radiations. The first pathway is the direct pathway of the radiating modes, which is responsible for normal transmission and reflection. The second pathway is the indirect pathway through coupling to the guided modes and back, which causes omni-directional scattering. According to the concept of Fano resonances, interference between two pathways would give rise to asymmetric spectral lines [41]. Fano resonances characterized by sharp asymmetric dips or rises in transmission and reflection spectra have been found in various planar dielectric structures [41–47]. However, in our case, the interference effect in far-field is very weak: first because the magnitude of the scattered light is smaller than that of the normally reflected light, and second because they propagate in different directions. Instead, we can observe more obvious asymmetric oscillations in the magnetic field spectrum inside the particle array where lights in the two pathways spatially overlap with each other [Fig. 3(h)]. Phases of the guided modes are modulated by Fabry-Perot resonances [48] caused by multiple-reflection at the boundaries of the particle array, which renders interference between the guided modes and the radiating modes transforming from constructive one to destructive one, or vice versa, quickly with respect to frequency, thus giving rise to the asymmetric spectral oscillations.

Finally, we note that the frequency-selective omni-directional scattering can also be excited by oblique incident light, since the continuous radiating band contains radiating modes that can directly couple with external radiations from any direction.

4. Conclusions

In summary, we have numerically shown that frequency-selective omni-directional scattering can be supported by periodic planar arrays of dielectric particles imposed with weak disorder. Firstly, we show that due to strong coupling of Mie resonances of individual dielectric particles, periodic planar arrays of dielectric particles can support a flat band of low-Q radiating modes which partially overlap with guided modes in spectral ranges. Then we show that out-of-plane leakage of the guided modes in the presence of weak disorder is selectively enhanced due to coupling with the low-Q radiating modes. Finally, we show that when illuminated by a vertically launched Gaussian beam, disorder-induced interaction between the guided modes and the flat band of low-Q radiating modes leads to frequency-selective omni-directional scattering. The frequency-selective omni-directional scattering represents an indirect pathway for the guided modes to couple with external radiations, in addition to the direct pathway of the radiating modes. Unlike Fano resonances previously found in planar dielectric structures, interference between the two pathways outside the particle array in our case is very weak, but we can observe obvious asymmetric spectral oscillations caused by the interference inside the particle array.

Due to our limited computation resources and also for simplicity of discussion, we pay special attention to radiating modes and guided modes with wavevectors at the ΓM axis, and the disorder-induced interaction between them is demonstrated by using a simplified one-dimensional structure with periodic boundary conditions. For this reason, the frequency-selective omni-directional scattering is restricted in a vertical plane with respect to the particle

Vol. 24, No. 20 | 3 Oct 2016 | OPTICS EXPRESS 23144

array. Real omni-directional scattering is expected to be observed using a true two-dimensional planar array of dielectric particles. For the configuration we discuss in this article, the magnitude of omni-directional scattering is weak compared to the normal reflection and transmission, which may hinder its actual applications. Nevertheless, we expect that the magnitude of the frequency-selective omni-directional scattering can be significantly enhanced in optimally designed configurations, which we leave for future work.

Appendix

In the following, we are going to show magnetic dipolar Mie resonance supported by a single dielectric particle (Fig. 4) and influence of lattice unit length to band diagram of the periodic planar array of dielectric particles (Fig. 5).

Fig. 4. Magnetic dipolar Mie resonance of single dielectric particle obtained by FDTD simulation. The dielectric particle we used here is identical to the dielectric cuboids which comprise the periodic array in the main text. The magnetic dipolar Mie resonances is excited by a magnetic dipole source placed at the center point of the cuboid with the magnetic moment in y' direction. (a) Magnetic field spectrum of the magnetic dipolar Mie resonance. (b)

'yH distribution at the resonant frequency of the magnetic dipolar Mie resonance in the

central xy plane of the cuboid. The dashed square shows cross-section of the dielectric cuboid in xy plane.

Fig. 5. Band diagrams of periodic planar particle arrays with different lattice unit lengths: (a)

10.75a a= , (b)

21.25a a= . Other parameters are identical to the periodic array in the main

text.

Funding

National Natural Science Foundation of China (NSFC) (Grant Nos.11274247, 11574218, and 11504243), Natural Science Foundation of Guangdong Province, China (Grant Nos. 2016A030313042, 2015A030310400).

Acknowledgments

We thank Yichen Shen for helpful discussions.

Vol. 24, No. 20 | 3 Oct 2016 | OPTICS EXPRESS 23145