Singular states of resonant nanophotonic lattices Y. H. KO K. J. LEE F. A. SIMLAN AND R. MAGNUSSON Department of Electrical Engineering University of Texas at Arlington Arlington Texas 760 19 USA

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Singular states of resonant nanophotonic lattices

Y. H. KO, K. J. LEE, F. A. SIMLAN AND R. MAGNUSSON*

Department of Electrical Engineering, University of Texas at Arlington, Arlington, Texas 76019, USA

*magnusson@uta.edu

Received XX Month XXXX; revised XX Month, XXXX; accepted XX Month XXXX; posted XX Month XXXX (Doc. ID XXXXX); published XX Month XXXX

Fundamental effects in nanophotonic resonance systems focused on singular states and their properties

are presented. Strongly related to lattice geometry and material composition, there appear resonant

bright chan-nels and non-resonant dark channels in the spectra. The bright state corresponds to high

reflectivity guided-mode resonance (GMR) whereas the dark channel represents a bound state in the

continuum (BIC). Even in simple systems, singular states with tunable bandwidth appear as isolated

spectral lines that are widely sep-arated from other resonance features. Under moderate lattice

modulation, there ensues leaky-band meta-morphosis, merging modal bands and resulting in offset dark

states and reflective BICs along with transmis-sive BICs within a high-reflectance wideband. Rytov-type

effective medium theory (EMT) is shown to be a powerful means to describe, formulate, and understand

the collective GMR/BIC fundamentals in resonant photonic systems. Particularly, the discarded Rytov

analytical solution for asymmetric fields is shown here to predict the dark BIC states essentially exactly

for considerable modulation levels. The propagation con-stant of an equivalent EMT homogeneous film

provides a quantitative evaluation of the eminent, oft-cited embedded BIC eigenvalue. The work

concludes with experimental verification of key effects.

1. INTRODUCTION

Understanding of wave propagation in periodic systems is foun-

dational for their utility in science and engineering. Thus, the

properties of solid-state materials are explained with band

theory modeling propagation of electron waves in periodic

crystal lattices. Elementary Bragg diffraction reveals energy

bands and band gaps and classification of materials as insulators

and conductors [1]. Similar bands appear in three-dimensional

(3D) dielectric lattices called photonic crystals [2]. The band

structure determines how photon propagation is affected by

frequency, polarization, and direction. It may be represented in

the first Brillouin zone with the first and higher band gaps

corresponding to Bragg reflections at increasing frequency [3].

Whereas 3D dielectric periodicity is challenging in experimental

realization, there is much current interest in practical film-based

1D and 2D optical lattices with straightforward fabrication. Key

physical properties of these elements are explained in terms of

the structure of the second (leaky) photonic stopband and its re-

lation to the symmetry of the periodic profile.

When the lattice is confined to a layer thereby forming a peri-

odic waveguide, an incident optical wave may undergo a

guided-mode resonance (GMR) on coupling to a leaky eigen-

mode of the layer system [4-7]. Figure 1(a) models the simplest

resonance system possible, namely a subwavelength 1D peri-

odic lattice or grating. Under normal incidence, counter-propa-

gating leaky modes form a standing wave in the lattice. As the

modes interact with the lattice, they reradiate [8]. A schematic

dispersion diagram is shown in Fig. 1(b). The device works in

the second stop band corresponding to the second-order lattice

[9]. A given evanescent diffraction order can excite not just one

but several leaky modes. To emphasize this point, in Fig. 1(b) we

show the stop bands for the first two TE modes. At each stop

band, a resonance is generated as denoted in Fig. 1(b). The fields

radiated by these leaky modes in a lattice with a symmetric pro-

file can be in phase or out of phase at the edges of the band [10].

At one edge, there is a zero-phase difference, and hence the ra-

diation is enhanced (GMR) while at the other edge, there is a π

phase difference inhibiting the radiation. In this case, if  = 

+ is the complex propagation constant of the leaky mode, 

= 0 at one edge, which implies that no leakage is possible at that

edge marking the condition as a bound state in the continuum

(BIC). For asymmetric lattice profiles, guided-mode resonance

prevails at each band edge. Fundamentally, one-dimensional

(1D) and 2D resonant lattices operate similarly and thus the de-

scription in Fig. 1 applies generally.

Within the domain of nanophotonics, especially relating to

new developments in metamaterials and metasurfaces, the

resonance phenomena explained with Fig. 1 are of major

interest. The leaky (GMR) edge and the nonleaky (BIC) edge are

inherent in resonance systems in this class. Historically, BICs

were proposed in hypothetical quantum systems by von

Neumann and Wigner [11]. In such systems, a completely

bound state exists at an energy level above the lowest

continuum level. Whereas the term BIC appeared in photonics

in 2008 [12], the underlying concept was apparently first

reported by Kazarinov et al. in 1976 [13]. These researchers

derived a formula for the quality factor of a corrugated

waveguide and reported zero radiation loss at the upper band

edge when the second-order Bragg condition was satisfied. In a

later paper, these effects were elaborated with improved clarity

[10]. Analyzing the second-order stop bands, Vincent and

Neviere numerically demonstrated the existence of a non-leaky

edge pertinent to symmetric gratings whereas asymmetric

grating profiles yielded leaky radiant modes at both band edges

[4]. Ding and Magnusson manipulated the separation of the

non-degenerate leaky resonances associated with asymmetric

profiles to engineer the resonant spectral response of periodic

films [14]. Experimentally, the nonleaky edge was brought into

view in 1998 by imposing asymmetry on an otherwise

symmetric 1D periodic structure by variation of the incidence

angle [15]; at the time the BIC terminology was not in use.

Reviewing briefly recent works, Marinica et al. proposed a

symmetric double-grating structure to support embedded

photonic bound states by coupling between two identical

resonant grating layers [12]. Hsu et al. experimentally showed a

diverging radiation Q factor as a signature of embedded bound

states in a 2D modulated layer of silicon nitride [16]. By tuning

the structural symmetry or coupling strength between different

resonance channels, quasi-BICs can be generated possessing

ultranarrow linewidths [17-19]. Such high-Q resonances

neighbouring BIC points enable ultra-sharp transmission and

reflection spectra, yielding giant near-field enhancement and

various promising applications including BIC-based chirality

[20-22], lasing [23-25], nonlinearity [26-29], modulation [30]

and sensing in various spectral regions [31]. In addition to the

symmetry protected BICs in the Brillouin zone center at a 

point, there exist off- BIC states under non-normal incidence,

sometimes called accidental BICs or quasi-BICs [32-38]; such

BIC states provide additional degrees of freedom and

application possibilities.

In this paper, we treat leaky band metamorphosois where

evolution of the leaky band structure with lattice modulation

strength is evaluated. The resulting singular state is then shown

to emerge as an isolated resonance feature brought to perfect

narrowband reflection under broken symmetry. Effective-

medium theory (EMT) based on the Rytov symmetric and

Fig. 1. (a) A schematic view of the simplest subwavelength resonance

system. The model lattice has thickness (), fill factor (), period (),

and refractive indices of background and lattice material (, ). When

phase matching occurs between evanescent diffraction orders and a

waveguide mode, a guided-mode resonance occurs. , , and  denote

the incident wave with wavelength , reflectance, and transmittance, re-

spectively. (b) A schematic dispersion diagram of a resonant lattice at

the second stop band. For a symmetric lattice, the leaky edge supports

guided-mode resonant radiation while the non-leaky edge hosts a non-

radiant bound state. This picture applies to both TE (electric field vector

normal to the plane of incidence and pointing along the grating grooves)

and TM (magnetic field vector normal to the plane of incidence) polari-

zation states. Here, the grating vector has magnitude  = 2π/Λ,  =

2π/λ, and  denotes a propagation constant of a leaky mode.

asymmetric formalisms is shown to model the resonant and BIC

states with high precision. The veracity of the Rytov EMT in

delivering homogeneous waveguide slabs that contain the

GMR/BIC resonance spectral properties is shown for practical

modulation levels up to 3. Experimental results support the

main theoretical conclusions.

2. LEAKY-BAND METAMORPHOSIS

As in Fig. 1(b) pertaining to a symmetric lattice, each resonant

mode has a clear GMR leaky edge and a corresponding BIC non-

leaky edge. This always holds for “weakly” modulated lattices.

On increase of modulation, the band deviates and the GMR-BIC

pairing is obscured. Key aspects of the band transformation can

be brought out via the simple model in Fig. 2(a). We compute

incidence-angle () dependent zero-order reflectance () in

TE polarization. Figure 2(b) displays the (, ) map for =1.4

where the Rayleigh lines (=  ) are marked by white

dashed lines. At normal incidence, a single  peak appears at

the upper band edge generated by symmetry-allowed GMR

marked as s-GMR with the attendant

(a) I

FΛ

TE0

TE1

Symmetric

system Asymmetric

system

(b) BIC

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SingularstatesofresonantnanophotoniclatticesY.H.KO,K.J.LEE,F.A.SIMLANANDR.MAGNUSSON*DepartmentofElectricalEngineering,UniversityofTexasatArlington,Arlington,Texas76019,USA*magnusson@uta.eduReceivedXXMonthXXXX;revisedXXMonth,XXXX;acceptedXXMonthXXXX;postedXXMonthXXXX(Doc.IDXXXXX);publishedXXMonthXXXX...

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Singular states of resonant nanophotonic lattices Y. H. KO K. J. LEE F. A. SIMLAN AND R. MAGNUSSON Department of Electrical Engineering University of Texas at Arlington Arlington Texas 760 19 USA.pdf

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Singular states of resonant nanophotonic lattices Y. H. KO K. J. LEE F. A. SIMLAN AND R. MAGNUSSON Department of Electrical Engineering University of Texas at Arlington Arlington Texas 760 19 USA

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