Engineering angle selectivity of extraordinary optical transmission and nonlocal spatial filtering

Hanbyul Chang; Minsu Yeo; Sun-Je Kim; Sun-Je Kim; Yoonchan Jeong; Yoonchan Jeong

doi:10.1364/JOSAB.496551

1. INTRODUCTION

Driven by rapid advances in digital image processing and computer vision, there is currently a considerable demand for numerous ubiquitous applications that require fast and energy-efficient real-time image acquisition and analysis [1]. In particular, small-form-factor mobile devices, including autonomous driving, biomedical surgery, and head-mounted displays, require efficient miniaturized optoelectronic systems capable of image acquisition and analysis to perform specific sensing and reasoning tasks [2–4].

Recently, analog optical computing based on Fourier optics has received renewed attention due to its strong advantages over electronics, including its high speed, parallel processing capability, and low power consumption [5]. Traditional Fourier optics technology was considered to have drawbacks, such as the large volume, heavy weight, and precise alignment requirements of 4f-based systems for 2D spatial frequency filtering and convolution operations. However, the advent of optical metamaterials and nanophotonic thin-film devices has provided new opportunities to overcome such drawbacks through clever design of nanoscale light–matter interactions with large degrees of freedom within sub-micron thicknesses [6–8]. In particular, optical metasurfaces are currently attracting much attention in the nanophotonics community for use in optical computing technologies.

In fact, most of studies using metasurfaces for Fourier domain angular spectrum modulation of the wavefront can be divided into two different strategies: local and nonlocal metasurfaces. A local metasurface modulates an incident optical wave by encoding an abrupt change in amplitude or phase through judicious engineering of a spatially variant array of dielectric or metallic nanoantennas [9–11]. Thus, the modulation of an optical wave is defined in a signal domain as a complex transmission or reflection coefficient function depending on the transverse position on a device. In most cases, the defined transmission or reflection coefficient function is to some extent independent of the angle of incidence of the light. To date, several studies on optical differentiation using local metasurfaces have been reported [12–15]. In contrast, a nonlocal metasurface manipulates the optical wave in the Fourier domain according to the spatial frequency rather than the position in the signal domain. In this case, the complex transmission or reflection coefficient is defined as an optical transfer function of the angle of incidence or spatial frequency [16–22].

In particular, the modulation principle of a transmission-type nonlocal metasurface can be represented simply as [7]

(1)$${\widetilde{U}_{O}}({f}_{x})=t({f}_{x}){\widetilde{U}_{I}}({f}_{x}),$$

where ${\widetilde{U}_{I}}$, ${\widetilde{U}_{O}}$, $t(f_x)$, and $f_x$ denote the angular spectra of the input and output waves, the optical transfer function (angle-selective transmission coefficient), and the spatial frequency in the $x$direction, respectively. Immediately after passing through a nonlocal metasurface, the angular spectrum of an input wave from an object is directly modulated by the optical transfer function, resulting in object plane modulation. From the equation above, it can be deduced that conventional systems that manipulate the transfer function in the angular domain with bulky 4f systems and Fourier plane masks can be replaced by compact ultrathin nonlocal spatial filters. Such compactness of a nonlocal spatial filter promises to reduce the size of various Fourier optical systems. In this context, the main issue is how to design a desired angle-selective optical transfer function with a large degree of freedom. To achieve high angle selectivity and excellent optical efficiency, dielectric materials have often been used to fabricate nonlocal metasurfaces with sharp resonance peaks by means of periodically coupled Mie resonators [16–20], guided mode resonance [21], and quasi-bound states in the continuum with asymmetric nanoantennas [22]. However, little attention has been paid to plasmonic approaches, which can provide a wealth of subwavelength engineering opportunities to explore the nonlocality of metasurfaces.

Since the first report of EOT resonance by Ebbesen et al. [23] to the early stages of EOT research in the 1990s and 2000s, many theoretical studies have been reported to explain the exact mechanism of the resonance [24–26]. In the 1D or 2D plasmonic lattice structures, consisting of an optically thick metallic film with periodic nanoslits or nanoapertures, it has been shown that different mechanisms can occur simultaneously and contribute to sharp transmission peaks [24–26]. In 2014, Yoon et al. reported that the EOT mechanism could be highly complex for a strongly coupled phenomenon of the multiple channels [25]. They proposed a unified theory of EOT, which could provide a semi-analytical approach based on a Fabry–Perot resonator model. Although this kind of approach was very accurate in modeling the resonance spectra, the extraction of each microscopic scattering and coupling coefficient was too complicated to be helpful in providing physical insights to analyze and engineer the angle selectivity of EOT.

In this paper, we focus on the origin principle of angle selectivity and its engineering based on the EOT phenomenon. More specifically, we investigate the surface-plasmon-assisted (SP-assisted) EOT in a 1D silver nanoslit metasurface under TM polarized illumination [23–26] with a rigorous analysis on the angle selectivity of EOT resonances. We present the general engineering recipes for the design of nonlocal EOT spatial filters and verify them through extensive numerical simulations as well as Fourier optics theory. We found that multiple resonances and a large tunability of coupling between them are fully available in plasmonic metasurfaces, which are highly advantageous in demonstrating on-demand lowpass or bandpass spatial filters despite the intrinsic ohmic loss. To the best of our knowledge, a comprehensive, in-depth study of the angle selectivity of EOT has been rarely reported, although there are a handful of papers that partially report the angle-selective dispersion characteristics of EOT [24,27]. The physical origin and design principles of the angle-selective nanoslit metasurface, however, have yet to be elucidated. Furthermore, their application to nonlocal spatial filters located on the object plane remains to be investigated.

Fig. 1. 1D nanoslit metasurface and the three coupled oscillator model. (a) Schematic representation of 1D Ag nanoslit metasurface under oblique TM-polarized illumination. Quartz is chosen as the substrate material. (b) Graphical illustration of hybridization of multiple resonance phenomena, the cavity resonance in nanoslits, front and rear SP excitations, and multiple diffraction orders in the transmission side. (c) Schematic diagram of the three coupled oscillator model that accounts for the angle-selective evolution of the EOT resonance. $k_{1}$, $k_{2}$, and $k_{3}$ refer to the coefficients of the restoring forces of the isolated oscillators. $g_{12}$, $g_{23}$, and $g_{13}$ are, respectively, the coupling coefficients between the cavity and SP modes, the SP and RWA modes, and the cavity and RWA modes.

Download Full Size | PDF

The rest of this paper has three additional sections. In Section 2, the principle of angle-selective EOTs is studied. In particular, in Section 2.A, angle-selective transmission spectra based on an infinite periodic grating structure of the EOT are analyzed by means of a three coupled oscillator model. In Section 2.B, the origin of angle selectivity is studied based on the strong coupling theory, which is verified by the Fano lineshape fitting. In Section 3, the engineering of the nonlocal spatial filter under consideration is discussed. In Section 3.A, the effects of detuning geometric parameters on the engineering of EOT-based nonlocal plasmonic spatial filters are comprehensively investigated. In Section 3.B, the angle selectivity of the nonlocal spatial filter under consideration is analyzed under the finite size condition, which is required for practical applications. The transmission characteristics obtained by a full numerical calculation are compared to those obtained by an analytical scalar wave approximation using the optical transfer function estimated for the infinite periodic structure together with a pupil function. Finally, in Section 4, the conclusion and outlook are given.

2. PRINCIPLES OF ANGLE-SELECTIVE TRANSMISSION

A. Mechanisms of EOTs and Angle Selectivity

In this research, we focused on the EOT-based angle selectivity in a 1D periodic structure of a nanoslit array of Ag on a quartz substrate. As shown in Fig. 1(a), TM-polarized illumination is assumed. The infinite array is along the $x$ direction, and each of the metallic units and the substrate are infinitely uniform in the $y$ direction.

As shown schematically in Fig. 1(b), there can be multiple resonance channels in the given structure, including the cavity resonance mode formed inside the nanoslits, the propagating SP modes on the front and rear surfaces of the silver film, and the Rayleigh–Wood anomalies (RWAs) [28–31] of different diffraction orders through the substrate side. Figure 1(c) shows the diagram for three dominant resonant modes supporting the coupling phenomena among them, which can be represented by a cavity mode, an SP mode, and an RWA mode. Note that $k_i$ and $g_{ij}$ denote the restoring coefficients of each mode and the coupling coefficient between any two modes. To analyze the angle-selectivity mechanisms of the given system consisting of the three harmonic oscillators, we use the temporal coupled mode theory [32–35]. The Hamiltonian equation of the three coupled oscillator system can be written as

(2)$$\left(\begin{array}{ccc}{E}_{1}&\;\; {g}_{12}&\;\; {g}_{13}\\ {g}_{12}& \;\;{E}_{2}&\;\; {g}_{23}\\ {g}_{13}&\;\; {g}_{23}&\;\; {E}_{3}\end{array}\right)\left(\begin{array}{c}{a}_{1}\\ {a}_{2}\\ {a}_{3}\end{array}\right)=E\left(\begin{array}{c}{a}_{1}\\ {a}_{2}\\ {a}_{3}\end{array}\right),$$

where $E_{1}$, $E_{2}$, $E_{3}$, $a_{1}$, $a_{2}$, $a_{3}$, and $E$ denote the uncoupled resonance energies of the three dominant modes, their corresponding oscillation amplitudes, and the energy eigenvalue of the system, respectively. Note that the decay rate of each mode is neglected here, which will be calculated later in Section 2.B using a phenomenological approach. In fact, $E_{1}$ can readily be calculated based on the Fabry–Perot-like dispersion of the cavity resonance [27], while $E_{2}$ and $E_{3}$ can be analytically determined according to the momentum matching conditions given by

(3)$${k}_{0}{\sin}\theta +m{\big|\vec G\big|}={n}_{{\rm sub},\lambda}{k}_{0}{\sin}{\theta}_{m},$$

(4)$${k}_{0}{\sin}\theta +{m}^{{\prime}}{\big|\vec G\big|}={k}_{0}{n}_{{\rm SP,sub},\lambda},$$

where ${\vec G}$, ${k}_{0}{,n}_{{\rm sub},\lambda}$, ${n}_{{\rm SP,sub},\lambda}$, $m$, $\theta$, and ${\theta}_{m}$ denote the grating vector, the wavenumber in the air, the substrate index at wavelength $\lambda$, the effective index of the SP mode formed between the silver and substrate at wavelength $\lambda$, and, respectively, the mode order, the incidence angle, and the diffraction angle of the $m$-th order mode. The directional angles $\theta$ and ${\theta}_{m}$ are measured as positive for the counterclockwise direction and negative for the clockwise direction. Since the $m$-th RWA mode is the evanescent surface wave of the $m$-th diffracted wave, ${\theta}_{m}$ becomes 90° or $-90^\circ$ for positive or negative values of $m$, respectively. By fitting the transmission spectra with this model, the coupling coefficients can be calculated, which will be explained in detail in Section 2.B.

Before discussing the coupling between the resonant modes in more detail, it is helpful to look at the phenomenological characteristics of the structure under consideration when it appears to exhibit the coupling phenomena mentioned earlier. Figure 2 shows the angular transmission spectra and the transmission curves at specific wavelength conditions under the geometric parameters with a metal thickness of 100 nm, a period of 300 nm, and a slit width of 20 nm, which were obtained by the rigorous coupled-wave analysis [36] based on the improved eigenmode solver [37] and the extended scattering matrix approach [38]. The RF module of the COMSOL Multiphysics software was also used to calculate the field distribution at the periodic unit cell with the Floquet periodic boundary condition, which are shown in the insets of the figures. In this case, the transverse size of the unit cell was set to the period of the structure, i.e., 300 nm. Each of the geometric parameters was chosen based on the parameter sweep, such that the angular spectrum of the structure exhibits a typical and efficient spatial filtering phenomenon with various resonant couplings in the visible wavelength range. In particular, the slit width ($w$) was limited to the range where only the single, i.e., fundamental, cavity mode exists for efficient spatial filtering while the film thickness ($t$) was set to excite the cavity resonance inside the visible spectral range. The material properties of Ag [39] and quartz [40] were taken from the literature.

Fig. 2. Simulation results of the investigated structure. (a) Simulated spectrum of the 0-th order transmission and the dispersion relation of EOT according to the angle of incidence and wavelength. The parameters $w$, $t$, and $p$ were set to be 20, 100, and 300 nm, respectively. The legends in (a) denote the momentum matching conditions of the −1st RWA and 1st and −1st SP excitations on the front or rear sides. The 0-th order transmission spectra according to the angle of incidence at the wavelengths of (b) 572 nm, (c) 518 nm, and (d) 462 nm, respectively. The inset figures in (b)–(d) refer to the magnetic-field amplitude map at the corresponding wavelengths at the angle where the transmission becomes maximum at each wavelength [marked by arrows in (a)]. The color bars indicate the relative H-field magnitude normalized to that of the input H-field, i.e., $|{{\vec H}}_{\text{inc}}|$.

Download Full Size | PDF

The plot shows highly dispersive transmission peaks as a function of both the wavelength and angle of incidence. The solid and dashed lines in Fig. 2(a) denote the previously mentioned phase-matching conditions for the SP and RWA modes, which can be expressed by Eqs. (3) and (4) [25]. As the figure shows, in addition to the fundamental cavity, the −1st rear SP and −1st RWA modes become the dominant modes involved in the coupling phenomena in the range of our interest (i.e., the wavelength range from 450 nm to 700 nm and the angle of incidence range from 0° to 30°). Figures 2(b)–2(d) show the high angle selectivity of the optical transfer function and the magnetic-field profiles around the slit aperture at the selected wavelengths. As Figs. 2(b) and 2(d) show, lowpass filter characteristics are established at the incident wavelengths of, respectively, 572 nm and 462 nm, respectively. In both cases, the transmission reaches its highest value at a normal incidence and decreases drastically as the angle of incidence increases, reaching almost zero transmission at angles of incidence of 18° or 7°, respectively.

The principles underlying these lowpass characteristics are different. From the region of the highest transmission at normal incidence and the bright field distribution in the inset, as given in Fig. 2(b), it can be deduced that the passband results from the cavity mode resonance. It is also clearly shown that the phase matching lines (the −1st rear SP and −1st RWA resonances) close to the initial cavity mode resonance allow coupling between the resonant channels and shifting of the resonance position. In fact, this coupling mechanism allows the structure to act as a lowpass filter. The increase in transmission at the higher angle region shown in Fig. 2(b) is attributed to the presence of the −1st RWA resonance coupled to the −1st rear SP mode. However, as can be seen in Fig. 2(d), the bright periodic field distribution at the rear surface of Ag and the weakened cavity mode field distribution are responsible for the new sharp peak formed by the coupling of the weakened cavity mode with the −1st rear SP and −1st RWA modes, resulting in high transmission only at low angles of incidence.

On the other hand, in the case shown in Fig. 2(c), at the wavelength of 518 nm, a transmission peak is located at an angle of incidence of $\sim 12 ^\circ$ and a sharp decrease is shown as the angle of incidence varies. As a result, the metasurface functions as a spatial bandpass filter at this wavelength. As can be seen in the inset that shows the bright periodic field distribution of the −1st rear SP mode on the rear side of Ag and the oblique propagation of the −1st diffraction mode through the rear side, this bandpass filtering feature is due to the hybridized resonance between the −1st rear SP and −1st RWA modes. The nonzero transmission at high spatial frequencies is also due to the hybridized resonance from the coupling between them, which can be seen in Fig. 2(a). As a result, the three dominant resonant modes involved in the coupling phenomena in the given spectral range can be considered as the fundamental cavity, −1st rear SP, and −1st RWA modes.

B. Strong Coupling and Angle-Selective Fano Resonances

To analyze the design principle of the nonlocal spatial filter under consideration more thoroughly and quantitatively, the shifted resonance positions of the dominant modes, i.e., the cavity, SP, and RWA modes, are calculated based on the temporal coupled-mode theory and the three coupled oscillator model built in the preceding section [32–35].

Figure 3 shows the fitted result of each resonant mode in the angular transmission spectra obtained under the geometrical condition specified in Section 2 and the angle of incidence conditions: 8° in Fig. 3(b), 13° in Fig. 3(c), and 25° in Fig. 3(d), where typical resonant couplings occur significantly. Figure 3(a) shows the fitted resonance dispersion curves along with the angular transmission spectrum at the specified angle. The lines with circled markers in three different colors (i.e., green, blue, and red) denote the dispersion curves for the new eigenmodes, which agree well with the underlying transmission resonances. The calculated resonant frequencies and fitted coupling coefficients of Eq. (2) are listed in Table 1. Since ${E}_{2}$ and ${E}_{3}$ are derived from the momentum-matching conditions by Eqs. (3) and (4), they are given as a function of the angle of incidence $\theta$. On the other hand, since the resonance condition of the cavity mode depends only on the cavity length, ${E}_{1}$ remains constant regardless of the angle of incidence. Although the decay rate of each mode is not specified in the three coupled oscillator model, it can be readily determined in a phenomenological manner, using the double Fano fitting method, which is described in detail below.

Fig. 3. Simulation results of the investigated structure. (a) Transmission spectra with the results of the coupled oscillator fitting. The vertical dashed lines marked as (b), (c), and (d) in (a) refer to the angles of incidence selected: 8°, 13°, and 25°, respectively. CM, cavity mode; SP, surface plasmon; and RWA, Rayleigh–Wood anomaly. (b)–(d) The transmission spectra were fitted with double Fano resonances. The red solid lines and black dashed lines correspond to the simulated transmission and the fitting results, respectively.

Download Full Size | PDF

Table 1. Fitting Parameters Used in the Three Coupled Oscillator Model

View Table | View all tables in this article

It is known that the transmission spectrum of an RWA, as well as that of a cavity resonant EOT can be modeled as Fano resonances with asymmetric sharp lineshapes resulting from interference between direct nonresonant and indirect resonant scattering channels [25,31,41–44]. As can be seen in the case of the angle of incidence of 0° shown in Fig. 3(a), the cavity and RWA modes become dominant in the given regime; thus, the transmission spectrum can be fitted with a superposition of two Lorentzian functions as [41,42]

(5)$$T={\left|a+\frac{b}{i(\omega -{\omega}_{1})+{\gamma}_{1}}+\frac{c{e}^{-i\Phi}}{i(\omega -{\omega}_{2})+{\gamma}_{2}}\right|}^{2},$$

where $b$, $c$, and $\Phi$ are real coefficients; $a$ is a complex coefficient; and $\omega_{1}$, $\omega_{2}$, and $\gamma_{1}$, $\gamma_{2}$ denote, respectively, the resonance angular frequencies and the corresponding decay rates. The decay rates of the cavity and RWA modes are determined to be $2.835\times {10}^{4}$ and $2.092\times {10}^{4}$, respectively.

Table 2. Fitting Parameters Used for the Double Fano Resonances

View Table | View all tables in this article

Table 3. Fitted Angular Frequencies by the Three Coupled Oscillator Model and Double Fano Resonances

View Table | View all tables in this article

From the fitted coupling coefficients, it can be deduced that the coupling between the cavity mode and the rear SP mode is stronger than those of the others, and that the coupling between the SP mode and the RWA mode is almost absent (i.e., $g_{23} = 0$). In particular, $g_{13}$ (i.e., the coupling between the cavity and RWA modes) is analyzed in more detail to assess the strength of the coupling. “Strong coupling” in a two oscillator coupled system conventionally refers to an inter-resonance interaction with a coupling coefficient greater than the difference in decay rates multiplied by $\hbar/2$ [45]. In other words, when two oscillators are strongly coupled, the new eigenmodes undergo decayed Rabi oscillations at the shifted frequency positions. On the contrary, when weakly coupled, their eigenmodes do not change [45]. The decay rate difference multiplied by $\hbar/2$ ($3.922\times {10}^{-31}$) is obviously smaller than the coupling coefficient given in Table 1. Thus, both nonzero couplings eventually turn out to be strong couplings, with the relatively stronger one (i.e., the coupling between the cavity and SP modes) having a coupling coefficient about six times larger than that of the weaker one (i.e., the coupling between the cavity and RWA modes). Although the coupling between the cavity and SP modes are strong, the coupling between the SP and RWA modes is almost absent (i.e., $g_{23} = 0$). Thus, two transmission peaks are formed because of two eigen-resonances over the wide angular ranges, as shown in Fig. 3(a).

In fact, the double Fano resonance function given in Eq. (5) is used to conduct the fitting of the transmission spectra marked at 8°, 13°, and 25°, respectively, as marked in Fig. 3(a) as (b), (c), and (d). Figures 3(b)–3(d) show that the double Fano fitting curves agree well with the full numerical results, supporting our qualitative analysis given above. The fitting parameters used for Figs. 3(b)–3(d) are listed in Table 2. In addition, the eigen-frequencies obtained using the three coupled oscillator model and the double Fano fitting are compared in Table 3, which are also in good agreement.

3. ENGINEERING NONLOCAL PLASMONIC SPATIAL FILTERS

A. Effects of Detuning Geometric Parameters on the Angle Selectivity of the EOT

Since the overall spatial filtering characteristics and efficiency crucially depend on the structural geometrical parameters of the structure, the evolution of the transmission spectra is examined while varying the geometric parameters of $t, p$, and $w$. In Fig. 4, the transmission spectra are calculated for $t = 80$, 120 nm [Figs. 4(a) and 4(b)], $p = 250$, 350 nm [Figs. 4(c) and 4(d)], and $w = 10$, 30 nm [Figs. 4(e) and 4(f)], while keeping the other parameters fixed at certain values from the initial parameter set, i.e., $t = 100\; {\rm nm}$, $p = 300\; {\rm nm}$, and $w = 20\; {\rm nm}$.

Fig. 4. Effects of changes in (a)–(b) $t$, (c)–(d) $p$, and (e)–(f) $w$. The transmission spectra were calculated when $t$ was (a) reduced to 80 nm and (b) increased to 120 nm compared to 100 nm, respectively. The transmission spectra were calculated when $p$ was (c) reduced to 250 nm and (d) increased to 350 nm compared to 300 nm, respectively. The transmission spectra were calculated when $w$ was (e) reduced to 10 nm and (f) increased to 30 nm compared to 20 nm, respectively. When $w$, $t$, or $p$ was varied, other geometric parameters were fixed as in the case of Fig. 2(a) ($t$: 100 nm, $p$: 300 nm, and $w$: 20 nm). The legends denote the momentum-matching conditions of the −1st RWA and 1st and −1st SP excitations on the front or rear sides.

Download Full Size | PDF

In Figs. 4(a) and 4(b), as $t$ is increased, the cavity resonance wavelength at around 520 nm is increased to nearly 630 nm with the overall increase in transmission as shown, which is due to the fact that the increase or decrease in $t$ leads to the red or blue shift of the cavity resonance, respectively. Next, in Fig. 4(c) and 4(d), as $p$ is increased, the slopes of the −1st SP and −1st RWA lines become steeper, which is because the magnitude of the grating vector is inversely proportional to $p$ while the position and strength of the cavity resonance is not varied much. Finally, in Figs. 4(e) and 4(f), as $w$ is increased, the cavity resonance wavelength is blueshifted from about 660 nm to 550 nm with a slight decrease in transmission. Due to the increase in $w$, the dispersion relation of the cavity mode changes in such a way that the effective index of the cavity mode decreases, resulting in a decrease in the resonance wavelength. In summary, variations in the thickness and width mainly affect the cavity resonance wavelength and transmission, while the grating period controls the cavity resonance wavelength spacing and hence the angle selectivity.

B. Effect of the Finite Number of Periods on the Nonlocality of the EOT

As can be seen from the results above, the structure under consideration can be used as a lowpass and bandpass spatial filter over a wide range of visible wavelengths. Note that these results have been calculated under the condition of an infinite periodic structure. However, from an engineering point of view, this condition cannot be met in most practical situations, as the periodic structure should somehow be implemented in a finite number of unit cells. It is therefore necessary to investigate the angle selectivity of such a finite periodic structure. Thus, a finite periodic structure in the $x$ direction with the same geometric form as the infinite periodic structure discussed in the previous section is further investigated, assuming that the region outside the finite periodic structure is simply covered with silver of the same thickness as the periodic structure and that the structure is still infinitely uniform in the $y$ direction. If this structure is illuminated by a plane wave or an optical wave with a beam size much larger than the dimension of the structure at a given angle of incidence, its angle-selective transmission can be characterized by the beam intensity distribution formed at a distance, i.e., the far-zone intensity distribution of the transmitted wave, because the transmitted wave will certainly have a finite beam width in the direction of the finite periodic structure, i.e., in the $x$ direction, as shown in Fig. 1(a). Therefore, the far-zone intensity distribution of the transmitted wave through the given structure is mainly investigated in this section.

This finite periodic structure can readily be analyzed if the full vector-field analysis based on the finite element method (FEM) is implemented. Once the far-field pattern of the wave transmitted through the structure is obtained from an input wave at a given angle of incidence, an additional Fourier analysis on it will eventually lead to the angle-selective transmission characteristics of the structure.

Alternatively, what we believe is a novel analytical scalar wave approximation can be considered to formulate the angle-selective transmission characteristics of a finite periodic structure, using the optical transfer function already found for the infinite periodic case along with the introduction of an appropriate pupil or aperture function to deal with the finite length of the structure. Although this analytical approximation is not an exact and rigorous way to analyze the situation, as the nonlocal responses of the virtual unit cells outside the finite periodic structure are unnecessarily included in the given optical transfer function, it can provide a qualitative estimate of the response change due to the reduction of the dimension of the structure to a finite length. This analytical approximation is therefore described below, and the simulation results of both the FEM-based full numerical calculation and the analytical approximation are compared.

If the periodic structure is reduced to have a finite length, the resulting effect should be reflected in two aspects: One is to impose a limit on the length of the structure itself, and the other is to impose a limit on the size of the input wave incident on the finite-length structure. To impose the limits, a pupil or aperture function based on a rectangular function of finite length and unity height can be used. Thus, for a finite periodic structure, the nonlocal response of the infinite periodic structure defined by Eq. (1) is analytically modified to

(6)$$\begin{split}{\widetilde{U}_{O}}({f}_{x})&=t^{\prime}({f}_{x}){\widetilde{U_{I}^{\prime}}}({f}_{x})\\&\equiv \left({\tilde p}({f}_{x})\otimes t({f}_{x})\right)\cdot \left({\tilde p}({f}_{x})\otimes {\widetilde{{U}_{I}}}({f}_{x})\right),\end{split}$$

where ${f}_{x}$, $t^{\prime}({f}_{x})$, ${\tilde{U_{I}^{\prime}}}({f}_{x})$, ${\widetilde{U}_{I}}({f}_{x})$, ${\widetilde{U}_{O}}({f}_{x})$, $t({f}_{x})$, and ${\tilde p}({f}_{x})$ denote the spatial frequency in the $x$ direction, the effective optical transfer function modified from the optical transfer function of the infinite periodic structure, the effective input wave’s spatial-frequency spectrum modified from, respectively, the spatial-frequency spectra of the input wave and the pupil function, the spatial-frequency spectra of the input and output waves, the optical transfer function of the infinite periodic structure, and the Fourier transform of the pupil function. The first convolution product represents the modulated optical transfer function of the infinite periodic structure to account for the effect of a finite length of the structure, while the second convolution product represents the modulated input wave due to the limited size of the input wave incident on the finite-length structure. Furthermore, the product of the two convolution products indicates that the inverse Fourier transform of ${\widetilde{U}_{O}}({f}_{x})$ is given by the convolution product of the two corresponding spatial-domain functions, which means that the resulting response is nonlocal. In addition, the integral of the Fourier transform of the pupil function is always reduced to unity, i.e., ${\int}_{-\infty}^{{\infty}}{\tilde p}({f}_{x}){\rm d}{f}_{x}=1$, because the rectangular function has a height of unity. This consequence implies that for a local response structure, i.e., for the case of $t({f}_{x})={t}_{\text{local}}=\text{constant}$, the first convolution product in Eq. (6) is eventually reduced to ${t}_{\text{local}}$, i.e., $t^{\prime}({f}_{x})={\tilde p}({f}_{x}) \otimes t({f}_{x})={t}_{\text{local}}$. Therefore, ${\widetilde{U}_{O}}({f}_{x})$ simply becomes the convolution product of ${\tilde p}({f}_{x})$ and ${\widetilde{U}_{I}}({f}_{x})$ multiplied by ${t}_{\text{local}}$. In other words, it is verified that the expression given by Eq. (6) is fully compatible with a local response structure governed by Fourier optics. Also note that the expression given by Eq. (6) is obviously reduced to that of Eq. (1) if the length of the periodic structure goes to infinity, i.e., the pupil function is given by a rectangular function of infinite length. In fact, this analytical approximation approach can be extended to the case of 2D nonlocal structures.

Fig. 5. Normalized far-zone intensity distributions of the transmitted waves calculated by (a)–(c) the full numerical calculation and (d)–(f) the analytical approximation with a pupil function when TM-polarized plane waves were incident at different angles of incidence for the three different wavelengths (572 nm, 518 nm, and 462 nm). The full width of the Ag nanoslit metasurface structure was set as 3 µm (10 periods). The angle of transmission $\phi$ was defined as $\mathrm{atan}(x/z)$ [see Eq. (7)]. The legends in the plots indicate the angles of incidence under the given conditions.

Download Full Size | PDF

Based on the output function given by Eq. (6), the far-field pattern can also be calculated using the Fraunhofer diffraction theory, which can be expressed as

(7)$${U}_{O,{\rm far}}(\xi )=\frac{{e}^{\textit{jkz}}{e}^{j\frac{k}{2z}{\xi}^{2}}}{j\sqrt{\lambda z}}\iint_{-\infty}^{{\infty}}{U}_{O}(x){\exp}\!\left(-j\frac{2\pi}{\lambda z}\xi x\right){\rm d}x,$$

where ${U}_{O,{\rm far}}(\xi)$, ${U}_{O}(x)$, $k$, $\lambda$, and $z$ denote, respectively, the spatial distributions of the output wave in the far-field and the output planes, the wavenumber and wavelength of the input wave in the substrate, and the distance between the output and far-field planes along the direction transverse to the $xy$ plane. Note that the finite periodic structure is assumed to lie on the $xy$ plane, as shown in Fig. 1, and that, apart from the multiplicative factors placed in front of the integral, the result of Eq. (7) is equal to the Fourier transform of ${U}_{O}(x)$ as a function of the spatial frequency represented by $\frac{\xi}{\lambda z}$.

The finite periodic structure under consideration was assumed here to be based on a unit cell of $t = 100\; {\rm nm}$, $p = 300\; {\rm nm}$, and $w = 20\; {\rm nm}$ as the same as discussed in the preceding section and to have 10 periods, resulting in a length of 3 µm in the $x$ direction. For the FEM-based full numerical calculation, COMSOL Multiphysics software was used with the simulation space set to 106.7 µm in the $x$ direction and 26.67 µm in the $z$ direction. The far-zone intensity distributions were then obtained using the far-field calculation provided by the software package, which was, in fact, based on the Stratton–Chu formula [46]. For the analytical approximation with a pupil function, the pupil size was obviously set to 3 µm and the far-zone plane was placed at $z = 1\; {\rm mm}$, which is much longer than the Rayleigh lengths of the incident waves in the given geometry. For both methods, the incident waves were assumed to be plane waves and the far-zone intensities were normalized to those of the incident waves.

Figure 5 shows the normalized far-zone intensity distributions obtained by the full-numerical calculation and the analytical approximation as a function of the angle of transmission $\phi$ for the specific wavelengths considered in Section 2 at some different angles of incidence. The angle of transmission $\phi$ was defined as $\mathrm{atan}(\frac{x}{z})$, as shown in Eq. (7). Note that the reduction in the normalized intensity is due to both the inherent transmittance of the structure and the diffraction effect of the output wave transmitted through the structure. Although the angle-selective transmission characteristics analyzed by both methods agree qualitatively, there are some discrepancies. This is because, in the case of the analytical approximation with a pupil function, the modulated transfer function could not be perfectly equivalent to that of the real finite one. In other words, the nonlocal responses of the virtual unit cells outside the finite periodic structure were unnecessarily included in the given optical transfer function obtained for the infinite periodic structure, whereas they were not included in the case of the full numerical calculation. Moreover, the transverse interaction between the surrounding silver walls and the nonlocal metasurface structure via surface plasmon modes on both ends of the structure occurring in the real finite periodic structure could not be considered by the simple analytical modeling via Eq. (6). In other words, the limitation of the analytical approximation arose from the fact that the finite-sized nonlocal metasurface structure is not a space-invariant system.

The differences between the full numerical results and the approximation results are more pronounced for the cases of the 30° angle of incidence at 572 nm and the 10° angle of incidence at 518 nm. This is because, as discussed in Section 2, the given situations are particularly related with the region where the −1st SP and −1st RWA modes resonate more strongly than the cavity mode. In fact, the former tends to be more affected by the number of unit cells involved in the resonances and couplings than the latter. On the other hand, the relatively strong signal intensities in the negative radiation angle region for 0° to −45° in Figs. 5(b) and 5(c) are due to the generation of positive order diffraction modes caused by the finite size of the periodic structure, which propagate in the direction opposite to the direction of the 0-th order diffraction mode.

Despite the presence of the differences above, the calculated results by the two methods show a similar overall spatial filtering performance. Furthermore, these discrepancies can be taken as evidence that the size of the structure is important in the spatial filter design. The discrepancies are expected to decrease as the number of periods increases because they arose from the inherited difference in the number of unit cells involved in the modal/intermodal resonances and couplings. It was verified that the lowpass filtering performance at wavelengths of 572 nm and 462 nm and the bandpass filtering performance at a wavelength of 518 nm can be exploited even under the finite-size condition to be considered for miniaturized applications such as digital image processing devices, head-mounted displays, and metasurface-based holography.

4. CONCLUSION

In this paper, for the first time, to the best of our knowledge, a comprehensive analysis of the angle selectivity and nonlocality of the EOT resonance has been carried out based on the temporal coupled mode theory for the three resonance modes, which include the fundamental cavity, −1st rear SP, and −1st RWA modes. It is noteworthy that an anticrossing phenomenon, because of the strong coupling, plays a significant role in the manifestation of an angle-selective transmission of the infinite periodic plasmonic nanoslit metasurface structure. It has also been shown that a finite-size EOT metasurface could be designed as a nonlocal lowpass or bandpass spatial filter, which has been verified by both the FEM-based full numerical calculation and the analytical scalar wave approximation based on the convolution theorem together with the optical transfer function obtained for the infinite periodic structures. To the best of our knowledge, this is the first time that the analytical scalar wave approximation on the nonlocal metasurface structure of finite size has been formulated and investigated compared to the full numerical calculation. For further research, multi-objective optimization of nonlocal spatial filters in terms of transmission efficiency, angle selectivity, finite-size-induced degradation, and operating wavelength could be carried out to demonstrate high-performance metasurfaces when accompanied by efficient iterative algorithms [47–49]. We envisage that the proposed EOT-based nonlocal spatial filtering will be useful to develop ultrathin wavefront sensing and holographic devices as well as to gain new physical insights into the field of nonlocal flat optics.

Funding

National Research Foundation of Korea (2021R1A5A1032937, 2021R1F1A1062368); Brain Korea 21 Four Program.

Acknowledgment

The authors acknowledge Mr. Hyunwoo Son for his advice on multiple resonator fitting.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

REFERENCES

1. G. H. Granlund and H. Knutsson, Signal Processing for Computer Vision (Springer, 2013).

2. N. Li, C. P. Ho, I. T. Wang, P. Pitchappa, Y. H. Fu, Y. Zhu, and L. Y. T. Lee, “Spectral imaging and spectral LIDAR systems: moving toward compact nanophotonics-based sensing,” Nanophotonics 10, 1437–1467 (2021). [CrossRef]

3. H. Neumann and R. Bisschops, “Artificial intelligence and the future of endoscopy,” Dig. Endosc. 31, 389–390 (2019). [CrossRef]

4. J. H. Song, J. van de Groep, S. J. Kim, and M. L. Brongersma, “Non-local metasurfaces for spectrally decoupled wavefront manipulation and eye tracking,” Nat. Nanotechnol. 16, 1224–1230 (2021). [CrossRef]

5. J. W. Goodman, Introduction to Fourier Optics (Roberts and Company, 2005).

6. F. Zangeneh-Nejad, D. L. Sounas, A. Alù, and R. Fleury, “Analogue computing with metamaterials,” Nat. Rev. Mater. 6, 207–225 (2021). [CrossRef]

7. L. Wesemann, T. J. Davis, and A. Roberts, “Meta-optical and thin film devices for all-optical information processing,” Appl. Phys. Rev. 8, 031309 (2021). [CrossRef]

8. S. He, R. Wang, and H. Luo, “Computing metasurfaces for all-optical image processing: a brief review,” Nanophotonics 11, 1083–1108 (2022). [CrossRef]

9. N. Yu, P. Genevet, M. Kats, A. M. F. Aieta, J. P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334, 333–337 (2011). [CrossRef]

10. A. I. Kuznetsov, A. E. Miroshnichenko, M. L. Brongersma, Y. S. Kivshar, and B. Luk’yanchuk, “Optically resonant dielectric nanostructures,” Science 354, aag2472 (2016). [CrossRef]

11. N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nat. Mater. 13, 139–150 (2014). [CrossRef]

12. W. Fu, D. Zhao, Z. Li, S. Liu, C. Tian, and K. Huang, “Ultracompact meta-imagers for arbitrary all-optical convolution,” Light Sci. Appl. 11, 62 (2022). [CrossRef]

13. Y. Kim, G. Y. Lee, J. Sung, J. Jang, and B. Lee, “Spiral metalens for phase contrast imaging,” Adv. Funct. Mater. 32, 2106050 (2022). [CrossRef]

14. J. Zhou, H. Qian, C. F. Chen, J. Zhao, G. Li, Q. Wu, H. Luo, S. Wen, and Z. Liu, “Optical edge detection based on high-efficiency dielectric metasurface,” Proc. Natl. Acad. Sci. USA 116, 11137–11140 (2019). [CrossRef]

15. L. Wan, D. Pan, S. Yang, W. Zhang, A. A. Potapov, X. Wu, W. Liu, T. Feng, and Z. Li, “Optical analog computing of spatial differentiation and edge detection with dielectric metasurfaces,” Opt. Lett. 45, 2070–2073 (2020). [CrossRef]

16. A. Komar, R. A. Aoni, L. Xu, M. Rahmani, A. E. Miroshnichenko, and D. N. Neshev, “Edge detection with Mie-resonant dielectric metasurfaces,” ACS Photon. 8, 864–871 (2021). [CrossRef]

17. A. Cordaro, H. Kwon, D. Sounas, A. F. Koenderink, A. Alù, and A. Polman, “High-index dielectric metasurfaces performing mathematical operations,” Nano Lett. 19, 8418–8423 (2019). [CrossRef]

18. H. Kwon, D. Sounas, A. Cordaro, A. Polman, and A. Alù, “Nonlocal metasurfaces for optical signal processing,” Phys. Rev. Lett. 121, 173004 (2018). [CrossRef]

19. H. Kwon, A. Cordaro, D. Sounas, A. Polman, and A. Alù, “Dual-polarization analog 2D image processing with nonlocal metasurfaces,” ACS Photon. 7, 1799–1805 (2020). [CrossRef]

20. Y. Zhou, H. Zheng, I. I. Kravchenko, and J. Valentine, “Flat optics for image differentiation,” Nat. Photonics 14, 316–323 (2020). [CrossRef]

21. A. Saba, M. R. Tavakol, P. Karimi-Khoozani, and A. Khavasi, “Two-dimensional edge detection by guided mode resonant metasurface,” IEEE Photon. Technol. Lett. 30, 853–856 (2018). [CrossRef]

22. A. C. Overvig, S. C. Malek, and N. Yu, “Multifunctional nonlocal metasurfaces,” Phys. Rev. Lett. 125, 017402 (2020). [CrossRef]

23. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998). [CrossRef]

24. H. Liu and P. Lalanne, “Microscopic theory of the extraordinary optical transmission,” Nature 452, 728–731 (2008). [CrossRef]

25. J. W. Yoon, J. H. Lee, S. H. Song, and R. Magnusson, “Unified theory of surface-plasmonic enhancement and extinction of light transmission through metallic nanoslit arrays,” Sci. Rep. 4, 5683 (2014). [CrossRef]

26. F. van Beijnum, C. Rétif, C. B. Smiet, H. Liu, P. Lalanne, and M. P. van Exter, “Quasi-cylindrical wave contribution in experiments on extraordinary optical transmission,” Nature 492, 411–414 (2012). [CrossRef]

27. L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, “Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114–1117 (2001). [CrossRef]

28. A. A. Rahman, P. Majewski, and K. Vasilev, “Extraordinary optical transmission: coupling of the Wood–Rayleigh anomaly and the Fabry–Perot resonance,” Opt. Lett. 37, 1742–1744 (2012). [CrossRef]

29. A. S. Vengurlekar, “Optical properties of metallo-dielectric deep trench gratings: role of surface plasmons and Wood–Rayleigh anomaly,” Opt. Lett. 33, 1669–1671 (2008). [CrossRef]

30. J. M. McMahon, J. Henzie, T. W. Odom, G. C. Schatz, and S. K. Gray, “Tailoring the sensing capabilities of nanohole arrays in gold films with Rayleigh anomaly-surface plasmon polaritons,” Opt. Express 15, 18119–18129 (2007). [CrossRef]

31. M. Sarrazin, J. P. Vigneron, and J. M. Vigoureux, “Role of Wood anomalies in optical properties of thin metallic films with a bidimensional array of subwavelength holes,” Phys. Rev. B 67, 085415 (2003). [CrossRef]

32. A. I. Väkeväinen, R. J. Moerland, H. T. Rekola, A. P. Eskelinen, J. P. Martikainen, D. H. Kim, and P. Törmä, “Plasmonic surface lattice resonances at the strong coupling regime,” Nano Lett. 14, 1721–1727 (2014). [CrossRef]

33. C. Qian, S. Wu, F. Song, K. Peng, X. Xie, J. Yang, S. Xiao, M. J. Steer, I. G. Tayne, C. Tang, Z. Zuo, K. Jin, C. Gu, and X. Xu, “Two-photon Rabi splitting in a coupled system of a nanocavity and exciton complexes,” Phys. Rev. Lett. 120, 213901 (2018). [CrossRef]

34. P. Jiang, G. Song, Y. Wang, C. Li, L. Wang, and L. Yu, “Tunable strong exciton–plasmon–exciton coupling in WS₂–J-aggregates–plasmonic nanocavity,” Opt. Express 27, 16613–16623 (2019). [CrossRef]

35. H. Zhang, B. Abhiraman, Q. Zhang, J. Miao, K. Jo, S. Roccasecca, M. W. Knight, A. R. Davoyan, and D. Jariwala, “Hybrid exciton-plasmon-polaritons in van der Waals semiconductor gratings,” Nat. Commun. 11, 3552 (2020). [CrossRef]

36. J. P. Hugonin and P. Lalanne, “Reticolo software for grating analysis,” arXiv, 2101.00901 (2021). [CrossRef]

37. P. Lalanne and G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779–784 (1996). [CrossRef]

38. J. P. Hugonin, M. Besbes, and P. Lalanne, “Hybridization of electromagnetic numerical methods through the G-matrix algorithm,” Opt. Lett. 33, 1590–1592 (2008). [CrossRef]

39. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972). [CrossRef]

40. I. H. Malitson, “Interspecimen comparison of the refractive index of fused silica,” J. Opt. Soc. Am. 55, 1205–1208 (1965). [CrossRef]

41. F. Shanhui, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. 20, 569–572 (2003). [CrossRef]

42. K. X. Wang, Z. Yu, S. Sandhu, V. Liu, and S. Fan, “Condition for perfect antireflection by optical resonance at material interface,” Optica 1, 388–395 (2014). [CrossRef]

43. S.-J. Kim, I. Kim, S. Choi, H. Yoon, C. Kim, Y. Lee, J. Son, Y. W. Lee, J. Rho, and B. Lee, “Reconfigurable all-dielectric Fano metasurfaces for strong intensity modulations of visible light,” Nanoscale Horiz. 5, 1088–1095 (2020). [CrossRef]

44. S. J. Kim, I. Kim, S. Choi, H. Yoon, C. Kim, Y. Lee, C. Choi, J. Son, Y. Lee, J. Rho, and B. Lee, “Dynamic phase-change metafilm absorber for strong designer modulation of visible light,” Nanophotonics 10, 713–725 (2021). [CrossRef]

45. H. Benisty, J. J. Greffet, and P. Lalanne, Introduction to Nanophotonics (Oxford University, 2022).

46. J. A. Stratton and L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56, 99–107 (1939). [CrossRef]

47. M. Zhou, D. Liu, S. W. Belling, H. Cheng, M. A. Kats, S. Fan, M. L. Povinelli, and Z. Yu, “Inverse design of metasurfaces based on coupled-mode theory and adjoint optimization,” ACS Photon. 8, 2265–2273 (2021). [CrossRef]

48. P. R. Wiecha, A. Arbouet, C. Girard, and O. L. Muskens, “Deep learning in nano-photonics: inverse design and beyond,” Photon. Res. 9, B182–B200 (2021). [CrossRef]

49. C. M. Lalau-Keraly, S. Bhargava, O. D. Miller, and E. Yablonovitch, “Adjoint shape optimization applied to electromagnetic design,” Opt. Express 21, 21693–21701 (2013). [CrossRef]

Uncoupled Energy Eigenvalues	Values [rad/s]	Coupling Coefficients	Values [1/m]
$E_{1} / ℏ$	$3.366 \times 10^{15}$	$g_{12}$	$1.870 \times 10^{- 21}$
$E_{2} / ℏ$	$2 π c / (q_{1} θ^{3} + q_{2} θ^{2} + q_{3} θ + q_{4})$ ^a	$g_{13}$	$3.000 \times 10^{- 21}$
$E_{3} / ℏ$	$2 π c / (r_{1} θ^{3} + r_{2} θ^{2} + r_{3} θ + r_{4})$ ^a	$g_{23}$	$0.000 \times 10^{- 20}$

Parameters	3(b)	3(c)	3(d)
$a$	$0.1336 + 0.0220 j$	$0.1191 - 0.0680 j$	$0.0230 - 0.1165 j$
$b$	$1.379 \times 10^{- 9}$	$3.175 \times 10^{- 9}$	$11.14 \times 10^{- 9}$
$c$	$9.317 \times 10^{- 9}$	$7.782 \times 10^{- 9}$	$3.844 \times 10^{- 9}$
$Φ$ [°]	8.037	−5.317	13.52
$ω_{1}$ [rad/s]	$3.703 \times 10^{15}$	$3.618 \times 10^{15}$	$3.460 \times 10^{15}$
$ω_{2}$ [rad/s]	$3.277 \times 10^{15}$	$3.248 \times 10^{15}$	$3.109 \times 10^{15}$
$γ_{1}$ [Hz]	$1.240 \times 10^{4}$	$1.824 \times 10^{4}$	$7.192 \times 10^{4}$
$γ_{2}$ [Hz]	$4.002 \times 10^{4}$	$3.441 \times 10^{4}$	$2.059 \times 10^{4}$

	3(b)		3(c)		3(d)
Fitting Method	$ω_{1}$ [rad/s]	$ω_{2}$ [rad/s]	$ω_{1}$ [rad/s]	$ω_{2}$ [rad/s]	$ω_{1}$ [rad/s]	$ω_{2}$ [rad/s]
Three coupled oscillator model	$3.674 \times 10^{15}$	$3.263 \times 10^{15}$	$3.584 \times 10^{15}$	$3.222 \times 10^{15}$	$3.335 \times 10^{15}$	$3.057 \times 10^{15}$
Double Fano resonances	$3.703 \times 10^{15}$	$3.277 \times 10^{15}$	$3.618 \times 10^{15}$	$3.248 \times 10^{15}$	$3.460 \times 10^{15}$	$3.109 \times 10^{15}$

Uncoupled Energy Eigenvalues	Values [rad/s]	Coupling Coefficients	Values [1/m]
$E_{1} / ℏ$	$3.366 \times 10^{15}$	$g_{12}$	$1.870 \times 10^{- 21}$
$E_{2} / ℏ$	$2 π c / (q_{1} θ^{3} + q_{2} θ^{2} + q_{3} θ + q_{4})$ ^a	$g_{13}$	$3.000 \times 10^{- 21}$
$E_{3} / ℏ$	$2 π c / (r_{1} θ^{3} + r_{2} θ^{2} + r_{3} θ + r_{4})$ ^a	$g_{23}$	$0.000 \times 10^{- 20}$

Parameters	3(b)	3(c)	3(d)
$a$	$0.1336 + 0.0220 j$	$0.1191 - 0.0680 j$	$0.0230 - 0.1165 j$
$b$	$1.379 \times 10^{- 9}$	$3.175 \times 10^{- 9}$	$11.14 \times 10^{- 9}$
$c$	$9.317 \times 10^{- 9}$	$7.782 \times 10^{- 9}$	$3.844 \times 10^{- 9}$
$Φ$ [°]	8.037	−5.317	13.52
$ω_{1}$ [rad/s]	$3.703 \times 10^{15}$	$3.618 \times 10^{15}$	$3.460 \times 10^{15}$
$ω_{2}$ [rad/s]	$3.277 \times 10^{15}$	$3.248 \times 10^{15}$	$3.109 \times 10^{15}$
$γ_{1}$ [Hz]	$1.240 \times 10^{4}$	$1.824 \times 10^{4}$	$7.192 \times 10^{4}$
$γ_{2}$ [Hz]	$4.002 \times 10^{4}$	$3.441 \times 10^{4}$	$2.059 \times 10^{4}$

Engineering angle selectivity of extraordinary optical transmission and nonlocal spatial filtering

Abstract

1. INTRODUCTION

2. PRINCIPLES OF ANGLE-SELECTIVE TRANSMISSION

A. Mechanisms of EOTs and Angle Selectivity

B. Strong Coupling and Angle-Selective Fano Resonances

3. ENGINEERING NONLOCAL PLASMONIC SPATIAL FILTERS

A. Effects of Detuning Geometric Parameters on the Angle Selectivity of the EOT

B. Effect of the Finite Number of Periods on the Nonlocality of the EOT

4. CONCLUSION

Funding

Acknowledgment

Disclosures

Data availability

REFERENCES

Data availability

Cited By

Figures (5)

Tables (3)

Equations (7)

Journal of the Optical Society of America B