1. Introduction

Electromagnetic wave propagation in periodic media is governed by the attendant band structure when the period and wavelength are on similar scales. The band gap corresponds to a range of frequencies of electromagnetic waves (i.e., light) that cannot propagate in a certain periodic material or structure along a specific direction [1–3]. In photonics, periodic variations in dielectric constant define the band structure and establish concomitant energy bands and band gaps.

Band engineering of periodic photonic structures is of interest for controlling light at structural nano- or microscales. Engaging lattice resonance effects affords new opportunities for enhancing light-matter interactions in various photonic devices. Particularly, leaky-mode or guided-mode resonance ensues as an incident light wave matches the phase of a lateral Bloch mode supported by an optical lattice. Unlike three-dimensional (3D) photonic crystal Bragg-type stop bands, these lattices operate at the second stop band and possess an out-of-plane radiative energy-coupling channel. Thus, the leaky mode resides above the light line in the Brillouin zone. The physical and spectral properties of the resonance lattice depend on its symmetry. For a symmetric lattice, only one edge of the second stop band exhibits resonance and attendant radiation whereas both edges resonate and radiate for asymmetric lattices. The nonradiant edge is protected by symmetry in the absence of asymmetry and cannot radiate even if its frequency is above the light line and connected with the radiation continuum [4–6]. In 2007, a study of leaky waves and band gaps in optical lattices adopted the term “nonleaky edge” for the symmetry-protected, and thus blocked, resonance and “leaky edge” for the guided-mode resonant and radiant spectral location [6]. In 2008, researchers modeled two parallel resonant lattices, and the nonleaky edge became a “bound state in the continuum” (BIC) [7] by drawing on analogy with bound states in quantum systems [8]. The BIC terminology is now widely adopted, shining much light on lateral guided-mode resonance (GMR) effects associated with the burgeoning fields of metamaterials and metasurfaces. This is because a slight deviation from perfect symmetry, as by turning an incident wave to its oblique state, relieves the BIC blockage thereby yielding a GMR output channel. The physics and potential applications of resonant lattices and their leaky-edge BIC states is of significant scientific interest [9–15].

Prior work suggested the possibility of interchanging the frequency locations of the leaky and nonleaky edges associated with periodic films [6]. Accordingly, when a resonant structure is subject to a perturbation, such as a change in the refractive index or the geometry, the frequency position and shape of its photonic bands can change dramatically. In previous research, it was shown that periodic lattice dielectric functions can have interchanged frequency locations of leaky and nonleaky edges under variation of the duty cycle or dielectric lattice contrast [15]. Via analytical coupled-wave theory models, it is known that the bands close at h₂= Re(h₁) where the coefficients h₁ and h₂ are related to the first and second Fourier harmonics of the dielectric function [6,16,17]. By making design choices, it is possible to achieve closed-band states as well as band flips where leaky and nonleaky edges trade places. This is the subject of the present study.

Resonance lattice band structure and band flip properties have been studied theoretically [15] and experimentally [18] in simple photonic structures including one-dimensional dielectric photonic lattices. In these structures, the band flip behavior can be controlled by tuning the geometrical parameters of the structure such as the lattice contrast and the fill factor. In our previous work [18], we presented an experimental demonstration of band flips and bound-state transitions in one-dimensional dielectric photonic lattices composed of a photoresist grating, Si₃N₄ sublayer, and a glass substrate with distinct fill factors. We explained the physical processes responsible for leaky/blocked (GMR/BIC) edge positioning and band closure point, providing a clear view of the pertinent physics. The photonic lattices were designed, fabricated, and characterized. The corresponding reflectance spectra were measured, and a comparison between theoretical results and experimental data showed a reasonable match. Overall, our study provided valuable insights into the band dynamics of simple dielectric photonic lattices.

In this paper, we investigate formation of multiple simultaneous bands and their properties. For reference, we first present computed bands for a 1D lattice and investigate the band flips relative to the lattice fill factors. Additionally, we explore several properties of the lattice when placed on a substrate and covered with a liquid medium with adjustable boundaries. Experiments are performed with the lattice/substrate confinement covered with index-matching oil and a glass cover. The presence of the liquid layer results in the formation of a relatively thick additional waveguide region that presents multiple modes and resonances.

The novelty of the current study lies in the design and experimental quantification of new resonant lattice systems that support multiple leaky modes. Implementing an adjustable interface within the multimode waveguide structure allows us to manipulate the number of modes propagating within the structure. With this adjustability, we can modulate the mode propagation properties, resulting in the emergence or suppression of specific resonant modes. Unlike our previous work [18], which focused on a single resonant mode, the current study investigates the considerably more complex behavior of multiple simultaneous bands and the related properties of each resonant mode. Thereby, we provide new insights into the control and manipulation of multiple modes within the structure. The band flips and bound-state transitions observed in this work can find applications in areas such as optical filters and sensors. Particularly, in biosensing, a key measure is sensitivity defined as S = Δλ/ΔN where Δλ is change of resonance wavelength for a change ΔN in refractive index of an adjacent biolayer or immersion solution. The sensitivity S relates to the mode structure and local field properties. In relation to the current study, the crosspoints (θ, l) near or at band closure denote overlapped mode loci with interfering mode fields. Crosspoints are of interest in optimizing, via high S, the various embodiments of resonant chemical and biological sensors. Understanding the underlying the fundamental physics and optimizing the design parameters can harness these resonance phenomena to achieve tailored light manipulation for improved resonance devices for this and other uses.

2. Theory and numerical results

Figure 1 shows a conceptual membrane device made of rectangular Si₃N₄ rods with a refractive index (n) of 1.8, a thickness (d_g) of 900 nm, and a period (Λ) of 1200 nm. Corresponding band structures and band flips are illustrated using numerical methods based on RCWA by calculating the zero-order spectral reflectance as a function of incident angle [19]. The band flip is achieved for fill factors (F) of 0.28, 0.32, and 0.36, as shown in Fig. 1(a). When the F is 0.28, the band gap is open with a nonleaky (BIC) edge and leaky (GMR) edge at the upper and lower frequency bands, respectively. Raising F to 0.32, the band closes and then reopens at 0.36 with the BIC-GMR edges interchanged. Analogous results are obtained when the lattice with d_g= 2000nm is immersed in index-matching oil with refractive index of 1.53 and placed on a glass substrate with refractive index of 1.5. Figure 1(b) shows the band flip process for sequential fill factors of 0.4, 0.45, and 0.5. The resonance linewidth of the lattice in Fig. 1(a) is broader than that in Fig. 1(b) due to the larger refractive index contrast. Specifically, the first lattice has a refractive index contrast Δn of 0.8 while the second lattice has a Δn of 0.27. These differences in Δn lead to the distinct photonic properties of the two lattices. These findings emphasize the importance of refractive index contrast in determining the photonic behavior of a given lattice.

 

Fig. 1. Band flip demonstration in Si₃N₄ gratings with distinct fill factors. (a) The zero-order spectral reflectance as a function of incident angle for a Si₃N₄ grating in air with d_g= 900 nm, Λ= 1200 nm, n = 1.8 for F = 0.28, F = 0.32, and F = 0.36. (b) The zero-order spectral reflectance as a function of incident angle for a Si₃N₄ grating on top of a glass substrate immersed in index-matching oil with d_g= 2000nm, Λ= 1000 nm, n = 1.8, n_s= 1.5, n_c= 1.53 with F = 0.4, F = 0.45, and F = 0.5. The incoming wave is polarized in the TE state and its electric field vector is aligned with the grating ridges. It should be noted that the upper and lower edges refer to shorter and longer wavelengths, respectively, as is conventionally understood in frequency terms.

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Band flipping is not limited to a specific polarization; it can also be observed in TM polarization. Figure 2 illustrates the band dynamics for TM polarization using the same structure depicted in Fig. 1(a) and Fig. 1(b). In Fig. 2(b), the resonance linewidth is exceedingly small resulting in the faint lines on the wavelength scales used.

 

Fig. 2. Band flip demonstration in Si₃N₄ gratings with distinct fill factors. (a) The zero-order spectral reflectance as a function of incident angle for a Si₃N₄ grating in air with d_g= 900 nm, Λ= 1200 nm, n = 1.8 for F = 0.3, F = 0.35, and F = 0.4. (b) The zero-order spectral reflectance as a function of incident angle for a Si₃N₄ grating on top of a glass substrate immersed in index-matching oil with d_g= 2000nm, Λ= 1000 nm, n = 1.8, n_s= 1.5, n_c= 1.53 with F = 0.4, F = 0.46, and F = 0.5. The incoming wave is polarized in the TM state and its electric field vector is perpendicular to the grating ridges. Again, the upper and lower edges refer to shorter and longer wavelengths, respectively, as is conventionally understood in frequency terms.

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3. Analysis of band flips in two-mode propagation

The results depicted in Fig. 1 focus on the band dynamics of a single supported leaky mode. However, when a diffraction grating is illuminated by an incident wave, it scatters the light into a number of diffracted orders. Each diffraction order has the potential to excite multiple leaky modes, not just one, due to the evanescent coupling between a diffracted order and attendant quasi-guided modes. In this context, we present our analysis of the band structure and band flip results for a subwavelength grating that enables the propagation of two TE modes. Our design comprises a Si₃N₄ layer with a refractive index of 1.8 and a grating thickness of 170 nm, deposited on a glass substrate with a refractive index of 1.5 immersed in a 3-µm-thick oil layer with a refractive index of 1.53, approximately matching the glass index. A glass half-space is also employed as the cover for this structure.

Our results which are shown in Fig. 3, reveal that by modifying the fill factor of the grating, a band flip occurs not only in the band structure associated with the fundamental mode TE₀ but also for the TE₁ mode. Thus, by changing the geometry of the grating, we can induce a transition for the two distinct modes of propagation.

 

Fig. 3. Band flip analysis of a subwavelength Si₃N₄ grating structure supporting two TE modes. The structure is immersed in oil with d_oil= 3 µm, and n_oil= 1.53. The grating thickness is d_g= 170 nm, n = 1.8, and (a) F = 0.35, (b) F = 0.48, (c) F = 0.6. Glass is used as a substrate as well as a cover medium.

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To better understand the two-mode propagation in the proposed structure, we calculate the stop bands for the first two TE modes in detail. Figure 4 indicates that at each stop band, a resonance is generated. As the fill factor is varied, the amplitude of the leaky modes at the edges of the Brillouin zone changes, leading to a change in the relative phase of the radiation. In the physical picture offered in [16], when the phase difference between the leaky modes at the band edge is zero, constructive interference occurs, leading to a resonance and an increase in the radiated power. Conversely, when the phase difference is π, destructive interference occurs, leading to inhibition of the radiated power.

 

Fig. 4. Stop bands and resonances for the first two TE modes in a conceptual subwavelength lattice structure are shown for (a) F = 0.35, (b) F = 0.48, and (c) F = 0.6.

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4. Experimental band flip of multimode propagation

The study utilizes Si₃N₄ (n = 1.8) with a thickness of d_g= 250 nm deposited on a glass (n_s= 1.5) substrate along with index-matching oil (n_oil= 1.53) and thickness of d_oil= 10 µm to achieve experimental band dynamics. Following the use of suitable solutions to clean the quartz substrate wafer, a thin layer of Si₃N₄ is deposited through plasma-enhanced chemical vapor deposition (PECVD). Then, 1D Si₃N₄ grating patterns are produced using UV laser interference lithography and a dry etching process. A spin-coated photoresist (PR, Shipley 1813) layer is exposed to a coherent beam (λ=266 nm) through a classic Lloyd’s mirror to attain each desired fill factor. The Si₃N₄ layer is subsequently etched using a reactive-ion etcher (Oxford Plasmalab 80 Plus) with CHF₃ + SF₆ gas mixture. A Si₃N₄ etch recipe has been developed for this work to optimize the etch rate, selectivity, and anisotropy of the device profile. Following the reactive-ion etching, a thin PR layer remains on the grating, which is eliminated by utilizing O₂ plasma ashing for 8 minutes. After residual PR removal, a Si₃N₄ 1D grating is formed on a glass substrate.

The Si₃N₄ grating is later immersed in refractive index-matching oil (Cargille Lab. Series A), which is a type of liquid that is used to match the refractive index of two materials with different embodiments. Subsequently, a glass cover is used to encapsulate the design. Because the oil has a refractive index that is higher than the glass index, an effective waveguiding film is created. This waveguide layer admits additional quasi-guided leaky modes to the structure. To illustrate the fabrication process of the Si₃N₄/oil guided-mode resonant structure, a schematic diagram is provided in Fig. 5.

 

Fig. 5. Schematic diagram illustrating the fabrication process of the multi-mode guided-mode resonant structure.

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After the device is fabricated, we employ atomic force microscopy (AFM) to characterize the properties of our resonant structures. This involves measuring the grating period, thickness, and fill factor. Figure 6 displays the measured parameters for a typical device with red, green, and blue arrows indicating the period, fill factor, and grating thickness, respectively. As shown in Fig. 6, this device has a period (Λ) of 1016 nm, a fill factor (F) of 0.48, and a grating thickness (d_g) of 250 nm.

 

Fig. 6. Atomic force microscopy (AFM) image of a resonant structure used in the study, showing measured parameters of period, fill factor, and grating thickness indicated by red, green, and blue arrows, respectively.

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We earlier [18] demonstrated the experimental setup used to measure the optical transmission of the resonant grating, which involves sending light from a supercontinuum laser source through an optical fiber and a collimator, and then through a polarizer and an optical aperture before reaching the lattice. A detector and an optical spectrum analyzer (OSA) quantify the transmitted light, and data is collected for wavelengths spanning 1480 nm to 1580 nm and angles of incidence ranging from -1° to 1° on a grid with Δθ = 0.01°. The reflection spectrum is formed by using R₀= 1 − T₀, and the data is processed in MATLAB. These measurements are performed for all three fabricated devices.

Figure 7 displays both the theoretical and corresponding experimental results for the fabricated devices. The simulation results for F = 0.353, F = 0.48, and F = 0.6 are presented in Figs. 7(a)–7(c), respectively. Meanwhile, Figs. 7(d)–7(f) present the experimental data for the same structures. The figure illustrates that the fabricated devices exhibit band structure and band flip effects that broadly agree with the simulation. Additionally, the experimental data illustrates the transition from the nonleaky BIC state to the leaky GMR state over the bandgap.

 

Fig. 7. Theoretical and experimental results for devices with varying F values. (a)–(c) show simulation results for F values of 0.353, 0.48, and 0.6, respectively, while (d)–(f) show corresponding experimental data. The figure demonstrates multi-mode band structure and band flip effects observed in both simulation and experimentation, as well as the transition from the nonleaky BIC state to the leaky GMR state across the bandgap.

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5. Discussion

The zero-order reflection map as a function of wavelength and angle is obtained for the symmetric Si₃N₄ grating immersed in index-matching oil. The maps exhibit the multiple band structures that arise from the additional interface introduced by the oil layer and the attendant formation of a waveguide region. The important point of the results in Fig. 7 is the appearance of the multiple experimental resonant leaky mode bands. The main resonance lines with highest efficiencies are due to the fundamental mode TE₀ that interacts most strongly with the high-index film and thus have the largest linewidth. There is approximate agreement in the wavelength locations of the experimental and theoretical spectra for this mode. The lines for the higher modes are narrower and the spectra are weaker limited by the experimental setup and its resolution of Δλ=0.05 nm. The incident Gaussian beam has 1 mm diameter and attendant divergence additionally limiting the narrow-line observability somewhat.

To further identify the multiple bands, we consider the case in Fig. 7(a) in some detail. Figure 8(a) shows the computed reflection map of the structure as a function of wavelength and angle. To provide a clear visual representation of the resonant modes in the structure, we plot the reflection spectrum as a function of wavelength, showing the intensity of the zero-order reflection for each resonance. The graph in Fig. 8(b) taken at θ=0° reveals multiple resonances, each corresponding to a different mode in the structure.

 

Fig. 8. Reflection maps and resonance wavelengths of a multimode guided-mode resonant structure with Si₃N₄ grating on a glass substrate immersed in index-matching oil. (a) Zero-order reflection map as a function of wavelength and angle, showing multiple bands due to the presence of the oil interface. (b) Reflection spectra at normal incidence angle, plotted as a function of resonance wavelengths for the points (1), (2), (3), (4) on the reflection map emphasizing the presence of multiple resonances in the structure.

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To provide a better understanding of the nature of the modes and their behavior within the guided-mode resonant structure, we calculate the electric field distribution for each of the resonant wavelengths in Fig. 8. The electric field (E_y) profile shows the spatial distribution of the electromagnetic field for each mode, allowing us to visualize how the light is interacting with the structure and how it is being confined and guided within it. Figure 9 presents the field intensity profiles for each of the resonance wavelengths. Based on the reflection map and the 1D graph of the resonance wavelengths shown in Fig. 8, we anticipate specific characteristics for the electric field distribution. At the first resonance wavelength λ₁, Fig. 9(a) shows that the lateral mode field is primarily propagating in the oil layer with a standing wave character typical for counterpropagating modes at normal incidence and having a TE₃ mode shape. Analogous properties are observed at the second λ₂ and third resonance λ₃ wavelengths in Fig. 9(b) and Fig. 9(c) with respective TE₂ and TE₁ mode character. In contrast, at the last resonance wavelength λ₄, the electric field distribution in Fig. 9(d) shows that the field is highly confined within the Si₃N₄ grating. A standing wave, generated by the interference between the input wave and the perfectly reflecting resonant wave, is clearly seen in the oil layer. The confinement of the electric field within the Si₃N₄ grating at this wavelength contributes to the attendant wideband resonance reflection.

 

Fig. 9. Modal electric field distributions. The electric field amplitude is shown as a color map on the structure’s cross-section at each resonance wavelength (a) λ₁, (b) λ₂, (c) λ₃, (d) λ₄ calculated with RSoft [19]. The resonances correspond to the peaks in the reflection spectrum shown in Fig. 8. The TE₀, TE₁, TE₂, and TE₃ modes are excited and propagate in the structure.

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Overall, the electric field distribution provides valuable information about the behavior of the guided modes and the role of each layer in guiding and confining the light within the structure. Our structure supports the propagation of multiple modes, including the fundamental TE₀ mode at λ₄, as well as the TE₁, TE₂, and TE₃ modes at λ₃, λ₂, and λ₁ respectively, as shown in Fig. 9.

6. Conclusions

In summary, we present an analysis of the band structure and band flip for a subwavelength silicon-nitride lattice integrated with a liquid film with an adjustable boundary. The study reveals the appearance of multiple resonant leaky mode bands due to the additional interface introduced by the liquid layer and the attendant formation of a waveguide region. It is shown that a band flip occurs not only in the band structure associated with the fundamental mode but also in the higher-order modes. By modifying the fill factor of the grating, the proposed structure can induce a transition for each propagating mode. The photonic lattices are shown to support the propagation of the fundamental TE₀ mode, as well as the TE₁, TE₂, and TE₃ modes. We illustrate that the main resonance lines with the highest efficiencies are due to the fundamental mode TE₀, which interacts most strongly with the high-index film. The modal electric field distribution was calculated for each mode at the resonant wavelengths, revealing the behavior of the guided modes and the role of each layer in guiding and confining the light within the structure. The experimental demonstration of the band flip for multimode propagation agrees well with the numerical results provided. Our study not only contributes to the fundamental understanding of band flip phenomena but also highlights their potential for real-world applications. The manipulation of resonant modes and the control of light propagation in subwavelength photonic lattices integrated with liquid films offer exciting opportunities for innovative optical devices and technologies.