1. Introduction

Subwavelength periodic structures enable versatile applications including label-free optical biosensors [1], high-performance optical filters [2–6], broadband reflectors [7–9], beam-transforming meta-surfaces [10], absorbers [11,12], and high-Q optical resonators [13,14]. In the subwavelength regime with only zero-order external propagating waves, coupling of an input excitation wave with a resonant leaky mode can be modeled as a generic two-port resonance system. In lossless systems, the attendant resonant reflectance is unity regardless of the details of the particular coupling configuration [15]. Therefore, the resonance spectra in reflection and transmission form angular continua on characteristic dispersion bands in the angle-wavelength space [9,16]. In contrast, here we show that non-subwavelength nanogratings with multiple propagative diffraction channels enable an angularly discrete resonance response. Previously it was shown that high first-order diffraction efficiency can be achieved with non-subwavelength resonant gratings. In particular, Parriaux and associates reported near-total first-order reflection from mirror-based waveguide-grating structures [17,18]. There are no prior reports on discrete angular resonance states as provided in this paper.

2. Proposed device

Our device concept is illustrated in Fig. 1(a). The reflection is confined to a narrow, collimated spectral channel even under wideband diverging or converging incident light. Obtaining efficient zero-order reflection in the non-subwavelength regime involves simultaneous suppression of all other propagating orders. We accomplish this by parametric optimization applying the particle-swarm optimization (PSO) method [19]. We use a rigorous coupled-wave analysis (RCWA) [20] algorithm as the forward computational kernel in our PSO code. We model a partially-etched TiO₂ waveguide grating on a glass substrate as shown in Fig. 1(a). The angle-dependent zero-order reflectance (R₀) spectrum under transverse-electric (TE) polarized light incidence in Fig. 1(b) shows the unique discrete angular/spectral reflection state at λ = 852 nm under normal incidence (θ = 0). The reflection peak has 99.8% efficiency and full-width-at-half-maximum (FWHM) bandwidths of Δλ = 0.6 nm in wavelength and Δθ = 0.12° in angle. In the 1550~1700 nm spectral band, this device transits to the subwavelength regime where different functionality prevails. Figure 1(c) shows the corresponding spectrum; it is clearly completely different than the spectrum in Fig. 1(b). The chief objective of this Letter is to explain this dramatic difference and provide experimental verification of the predicted discrete spectra.

 

Fig. 1 Theoretical reflectance showing spatial/spectral delta function characteristics. (a) Device schematic. The geometry is defined by the grating period Λ; fill factor F; total film thickness t and grating depth d. Refractive indices are n_F = 2.52 for the TiO₂ film, n_S = 1.54 for the glass substrate, and n_C = 1.0 for air. (b) Computed angle-dependent zero-order reflectance (R₀) spectrum of an optimized device under TE-polarized light incidence (electric field vector along y). The optimized parameters are Λ = 696 nm, F = 0.33, t = 788 nm, and d = 210 nm. (c) Angle-dependent R₀ spectrum of the same device in the subwavelength regime in the 1550~1700 nm wavelength region.

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The inset in Fig. 2(a) shows the electric field distribution at the reflection peak center and the propagating waves associated with the device. The resonance is induced by a standing TE₀ guided mode coupling to four external waves R₀, T₀, and T _{± 1}. The spectra in Figs. 2(a) and 2(b) show high-efficiency reflectance R₀ even with multiple simultaneous propagating waves that generally reduce the power available to it. The non-resonant background shows weak zero-order intensities R₀ and T₀ with most of the incident energy coupling to the two transmitted first-order substrate waves T _{± 1}. We remark that this effect is not associated with any Rayleigh-type power exchange [21]. The resonant excitation of the standing TE₀ guided mode at λ = 852 nm and θ = 0 exchanges the dominant diffraction channel from T _{± 1} to R₀ while T₀ remains persistently low. This new, intriguing power transfer mechanism requires detailed analysis.

 

Fig. 2 Diffraction efficiency pertinent to the device in Fig. 1. (a) Spectra of all propagating orders under normal incidence. (b) Angular spectra of all propagating orders at wavelength λ = 852 nm. Inset in (a) shows the localized electric field amplitude distribution at the reflection-peak wavelength. The field amplitude is normalized by the incident field amplitude. The y-axis in (a) and (b) are identical.

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3. Pathway and intermodal effects analysis

The discrete angular/spectral reflection state observed in Fig. 2 can be explained by invoking two distinct interference mechanisms. First, interference between resonant and non-resonant diffraction pathways plays an important role. Figure 3(a) illustrates the two different diffraction pathways. The incident wave couples to four propagating diffraction orders. These are the zero-order reflected wave, zero-order transmitted wave, and two first-order transmitted waves. The second-order diffracted wave in the TiO₂ layer is totally reflected at the bottom TiO₂-glass interface and drives a TE₀-type mode yielding a guided-mode resonance mechanism [2]. Therefore, re-radiation of the mode to each outgoing diffraction order constitutes the resonant pathway. As indicated in Fig. 3(a), ρ₀^(R), τ₀^(R), and τ _{± 1}^(R) refer to the resonant zero-order reflection, zero-order transmission, and first-order transmission amplitudes, respectively. Consequently, the non-resonant pathway corresponds to all other diffraction pathways which do not involve the second-order diffraction in the TiO₂ layer. The non-resonant diffraction amplitudes are thus denoted by ρ₀^(N), τ₀^(N), and τ _{± 1}^(N). Analogous interference picture applies to reflection resonances in the zero-order regime [22]. The intensity of the outgoing diffraction orders is expressed by a superposition such that R₀ = |ρ₀^(N) + ρ₀^(R)|², T₀ = |τ₀^(N) + τ₀^(R)|², and T _{± 1} = |τ _{± 1}^(N) + τ _{± 1}^(R)|². The non-resonant amplitudes contribute the sideband intensity in the spectra. In the spectra in Figs. 2(a) and 2(b), we see that ρ₀^(N) ≈0, τ₀^(N) ≈0, and τ₀^(R) ≈0 such that R₀ ≈|ρ₀^(R)|² and T₀ ≈0. At the reflection peak center at λ = 852 nm and θ = 0 in Figs. 2(a) and 2(b) we see that T _{± 1} ≈0 implying a nearly complete destructive interference between τ _{± 1}^(N) and τ _{± 1}^(R), meaning that there is a π phase difference between the two pathways with equal amplitudes, i.e., τ _{± 1}^(N) ≈–τ _{± 1}^(R). By energy conservation this implies R₀ ≈|ρ₀^(R)|² ≈1. The optimized structure is designed so that complete destructive pathway interference occurs in the first-order transmission channels at θ = 0.

 

Fig. 3 Interference effects associated with the discrete angular/spectral reflection. (a) Radiation pathways hypothesized to explain the observed spectra. The non-resonant pathway is formed by direct diffraction of the incident light. The resonant pathway is defined by leakage radiation stimulated by a TE₀ mode coupling through second-order diffraction. Integer numbers label dominant coupling diffraction orders for waves indicated by the arrows. (b) Leakage radiation probabilities pertaining to a traveling TE₀ mode at λ = 848 nm and θ = 0.565°. (c) Leakage radiation probabilities associated with counter-propagating leaky modes at λ = 852 nm and θ = 0.

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Under off-normal incidence, the destructive pathway interference in the first-order transmission becomes incomplete due to asymmetry in τ₊₁^(R) and τ_–1^(R). In Fig. 3(b), we indicate the leakage radiation probabilities associated with a TE₀ guided mode resonating at λ = 848 nm at a small off-normal angle (θ = 0.565°). The leakage radiation probabilities are calculated by the absorbance analysis method proposed in [23] with following equation for peak absorbance

The enhancement of R₀ on resonance at θ = 0 involves interference between secondary waves radiated by counter-propagating leaky modes. In Fig. 3(c), we show the radiation probabilities and coupling orders for the standing TE₀ mode at λ = 852 nm and θ = 0 corresponding to the resonance center of the discrete angular/spectral reflection state. We confirm that the radiation probability towards zero-order reflection is remarkably enhanced to 42.7% for the standing-mode case relative to 13.3% for the traveling-mode case in Fig. 3(b). In contrast, as noted in Fig. 3(c), the net radiation probability to the first-order transmission channels is substantially reduced. In Fig. 3(c), the TE₀( + ) mode traveling in the + x direction couples to the T₊₁ channel through first-order diffraction while the TE₀(–) mode traveling along –x contributes to this channel by third-order diffraction. We calculate the phase difference between these two contributions finding ~1.09π. The close-to-π phase difference provides destructive interference between the two constitutive components generated by the modes running along ± x. This reduces the amplitude of τ _{± 1}^(R) sufficiently to enable their final extinction by interference with τ _{± 1}^(N).

4. Experimental results

Proof-of-concept experiments verify the proposed concept. We fabricate devices using sputtered TiO₂ films on microscope slides applying UV-laser interference lithography and reactive-ion etching. Shown in Figs. 4(a) and 4(b) are scanning electron micrographs of the fabricated device. From these micrographs, the measured device parameters are period Λ = 753 nm, fill factor F = 0.25, TiO₂-film thickness t = 602 nm and grating depth d = 210 nm. The measured R₀ spectrum is shown in Fig. 4(c). We clearly observe a discrete reflection peak at λ = 812 nm and θ = 0. The RCWA calculation assuming the device structure indicated by the red-dashed lines in Fig. 4(b) shows excellent quantitative agreement with the experimental results as indicated in Figs. 4(d) and 4(e). In the experiment, the resonance peak attains 80.1% efficiency and FWHM bandwidth of 1.42 nm in wavelength and 0.24° in angle.

 

Fig. 4 Experimental demonstration of a discrete angular and spectral reflector. (a) Top-view and (b) cross-sectional scanning electron micrographs of the fabricated device. Red-dashed lines in (b) indicate the structure used in the numerical calculation (RCWA) for comparison. (c) Measured angle-dependent R₀ spectrum. (d) Measured spectra and comparison with numerical calculation under normal incidence. (e) Angular spectrum at wavelength λ = 812 nm.

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5. Spatial-filtering application

An interesting potential use of the narrow angular selectivity is focus-free spatial filtering. In many applications, spatial filters are necessary to eliminate spatial noise associated with random laser fluctuation and scattering. Figure 5(a) shows conventional spatial filtering applying a pair of lenses and a pinhole. In this arrangement, high-accuracy optical alignment is required. Moreover, high-power applications are limited by potential pinhole thermal failure. Our device avoids these limitations owing to its focus-free spatial filtering property as illustrated in Fig. 5(b). In this example, the spatial filter rejects high-frequency noise components without a focusing lens and pinhole.

 

Fig. 5 Spatial filtering with a discrete angular reflector element. (a) Conventional spatial filter using a pinhole/lens pair. (b) Proposed spatial/spectral filter fashioned with a resonant nanograting. (c) Reflectance spectrum of the spatial/spectral filter. The device parameters are Λ = 696 nm, F = 0.33, t = 2210 nm, and d = 210 nm. (d) Theoretical performance of the spatial filter with an input super-Gaussian laser beam with FWHM width of 16 mm carrying computer-generated pseudo-random noise.

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We numerically demonstrate example performance of a focus-free spatial filter in this class. We design an ultra-sharp angular reflector with peak reflectance ~99.9% at wavelength λ = 874.069 nm and bandwidths of Δθ = 0.005° and Δλ = 23 pm as shown in Fig. 5(c). This angular bandwidth corresponds to a conventional spatial filter using a pinhole with a 1.7-μm diameter and an objective lens with focal length of 10 mm. The top TiO₂ grating of this design is identical to the design in Fig. 1 but the new design has a thicker TiO₂ sublayer. The filtering performance of this element is presented in Fig. 5(d). We assume a 16-mm-wide TE-polarized super-Gaussian beam carrying computer-generated pseudo-random noise as input light. The electric field of the beam is given by

6. Conclusions

To conclude, we provide a new spatial/spectral filter concept based on guided-mode resonance effects in the non-subwavelength regime. We show that complex inter-modal and pathway interference processes enable these unique angular and spectral properties that are not feasible in the subwavelength regime. The experimental results show excellent quantitative agreement with theoretical predictions. We numerically demonstrate focus-free spatial filtering with the proposed device concept. We remark that the pathway and inter-modal interference effects invoked in the physical explanation of the device operation represent general aspects of guided-mode resonance phenomena in the non-subwavelength regime. Therefore, we expect high-efficiency spatial/spectral selectivity to be feasible in other diffraction channels including in zero-order transmission T₀. Developing such elements in connection with appropriate tuning methods may enable innovative solutions for directional and spectral beam management in solid-state light emitters without bulky external components.