1. Introduction

The coupling of a freely propagating electromagnetic wave to a state of confinement at a surface is presently the subject of considerable research activity. Periodic structures with subwavelength features provide effective means of achieving such coupling. The resulting strong localization of energy at a metallic surface or in a dielectric layer with corresponding high-amplitude near fields is of interest for numerous applications including biosensors, light sources, nonlinear frequency converters, and particle traps [1]. In this paper, we apply the guided-mode resonance effect in periodically modulated dielectric layers to realize a variety of passive optical elements. These are narrowline bandpass filters with extensive low sidebands operating with TE and TM polarized light, wideband reflectors for TE and TM polarization, wideband antireflective resonant layer, a wideband polarizer, and a wideband polarization-independent element. All of the elements presented in this paper can be fabricated with only a single binary layer with one-dimensional periodicity on a suitable low-index substrate.

When an incident optical wave is coupled by a second-order grating to a leaky waveguide mode supported by the thin-film system, pronounced resonance reflections can occur. For small modulation amplitudes, the resonance linewidths are narrow with high resonance Q factors. Much past work has focused on narrow-line resonance elements which have been amply demonstrated experimentally [1–6] with the measured diffraction efficiencies generally agreeing well with numerical calculations. Increasing the modulation strength broadens the resonance peaks and reduces the local field amplitudes. Gale et al. experimentally observed broad (~50–100 nm full-width half-maximum (FWHM) linewidth) resonance peaks with Lorentzian shape using an embedded structure with modulation strength of Δε=1.75 where we define Δε=${\mathrm{n}}_{\mathrm{h}}^2$-${\mathrm{n}}_{\mathrm{l}}^2$ with ε denoting the dielectric constant and n_h and n_l denoting the high and low refractive index, respectively, within the period [7]. Brundrett et al. designed and fabricated a silicon-on-sapphire guided-mode resonance grating with Δε~11 resulting in a computed FWHM linewidth of ~200 nm and measured linewidth of ~70–100 nm and also demonstrating the high sensitivity of the linewidth to the grating parameters; in particular to grating fill factor and thickness [8]. These results [7, 8] were obtained with TE-polarized incident light such that the optical electric field vector is along the grating lines and normal to the plane of incidence. Recently, using Δε~11, Mateus et al. reported resonance elements with very large linewidths of several hundred nanometers and flattop profiles operating in TM polarization [9]; they subsequently fabricated a high-quality reflector with reflectance exceeding 98% over a 500 nm range and excellent agreement with numerical simulations [10]. These experimental results were obtained with elements with simple one-dimensional (1D) grating profiles [2, 4–10]. Polarization independent resonance elements can be realized by using two-dimensional (2D) gratings or photonic crystal slabs [3, 11].

An alternate method to realize broadband flattop high reflectors was presented by Liu and Magnusson where several distinct diffraction orders operate in a two-waveguide geometry with an interceding coupling layer [12]. Here, a moderate modulation of Δε~1.4 yields a bandstop filter with a nearly flat top and steep filter sidewalls and linewidth of ~7 nm. In principle, this method can be applied to design flattop filters over arbitrarily wide spectral regions. Additionally, using a resonance element where adjacent leaky-mode excitation by several evanescent diffraction orders occurs within a single layer, bandpass filters based on excitation of such resonant leaky-mode pairs are found [13]. The bandpass filter examples in [13] utilize simple symmetric periodic elements; the transmission peak is supported by adjacent, wideband (exceeding 50 nm on each side) high reflectance (>99%) spectral regions. This concept is basic to the single- and multi-layer bandpass filters with extensive, flat reflection bands surrounding the resonance transmission peaks predicted by theoretical modeling in [14].

In this paper, we demonstrate, by numerical computations, that various optical elements can be implemented with resonant periodic waveguide films with binary profiles. The distribution of the materials within the period, by choice of the various fill factors, affords control of Fourier harmonic content and, thus, pertinent leaky-mode resonance linewidths. Furthermore, as shown in [15], the profile symmetry influences the guided-mode resonance (GMR) separation and their number and overlap, thereby decisively affecting their mutual interaction and the resulting external spectral characteristics. These ideas can be applied to precisely tune the resonance characteristics of such elements to yield useful spectra. In fact, in [15] we showed that single-layer bandstop and bandpass filters with versatile spectral attributes can be implemented with modulated films possessing asymmetric grating profiles. A key point presented there is that, besides modulation strength, the separation of the GMRs is controllable through the modulation profile. Moreover, inducing asymmetry can double the number of available resonances. Thus, with certain modulation strength available (i. e. given materials), various spectra can be obtained by utilizing the interaction between multiple GMRs. In this paper, this idea is used to design new resonance devices with either symmetric or asymmetric (i.e. lacking reflection symmetry in a plane erected normal to the grating vector) grating profiles with binary, easily fabricated, form. It is effectively an extension of [15] presenting many new device designs and their computed spectral, angular, and modal characteristics.

In this paper, only single-layer structures modulated with one-dimensional (1D) binary profiles will be considered. For simplicity, it is assumed that the gratings are transversely infinite and that the materials are lossless and dispersion free. The dispersion curves are calculated with the method introduced by Peng et al. [16]. The spectra and field profiles are calculated with computer codes based on rigorous coupled-wave analysis of wave propagation in periodic media [17, 18].

2. Second stopband characteristics

 

Fig. 1. Brillouin diagram illustrating the second stopband for a single-layer waveguide grating with an asymmetric profile. Structural parameters for one period are shown in the inserted schematic where Λ denotes the period, d is the thickness, and n is the refractive index in the various regions (c=cover, s=substrate, l=low, h=high). The fill factors for the grating materials are shown beneath the layer. In the figure, k₀=2π/λ where λ is the wavelength in free space, K=2π/Λ, and β_R (β_I) is the real (imaginary) part of the propagation constant of the leaky mode. The dispersion curve is associated with the TE₀ mode and has been transferred to the first Brillouin zone. The dashed curves show the resonance spectrum at normal incidence (not to scale).

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A guided-mode resonance (GMR) occurs when the incident wave is coupled to a leaky waveguide mode by phase-matching with a second-order grating. The connection between leaky modes and the second stopband has been addressed in several papers [15, 19–21, 4]. A guided-mode resonance will appear at one edge of the stopband for a symmetric grating, while GMR peaks will appear at each edge of the stopband for a grating without reflection symmetry as discussed in some detail in [15]. In comparison with the first stopband, the bandedge of the leaky (second) stopband is not clearly defined on the dispersion diagram. As shown in Fig. 1, even for a weakly modulated waveguide, β_R (the real part of the propagation constant) in the second stopband is not constant, and β_I (the imaginary part of the propagation constant) outside the stopband is not zero.

Apart from the ambiguity at the bandedge, Fig. 1 gives a clear picture of the stopband because β has different curvatures in the band than outside the band. The numerical results in Fig. 2 show that the locations of the two minima of |∂k₀/∂β_R| and |∂k₀/∂β_I| match exactly, and they also coincide with two GMRs. Of the two derivatives, |∂k₀/∂β_R| bears the form of a group velocity. Even though the group velocity of a leaky wave is not well defined [22] because its propagation constant is complex, it is appropriate to interpret |∂k₀/∂β_R| as group velocity in this case. The incident wave couples to a slow resonant mode that locates at the bandedge.

 

Fig. 2. Numerically computed derivative of the propagation constant in Fig. 1.

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Fig. 3. Second stop band for a structure with two different modulations. (a) β_R and β_I for structure with nh=2.8. Only half of the first Brillouin zone is shown; (b) |∂k₀/∂β_R| and reflectance at normal incidence for structure with n_h=2.8 (not to scale). (c) β_R and β_I for structure with n_h=3.3. Only half of the first Brillouin zone is shown; (d) |∂k₀/∂β_R| and reflectance at normal incidence for structure with n_h=3.3 (not to scale). The parameters are: d=0.67µm, Λ=1µm, n_c=1, n_s=1.48, n_avg=2.445, and modulation profile n_h/n_l/n_h/n_l of 0.397/0.103/0.051/0.449.

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While the example given in Figs. 1 and 2 refers to a structure with small modulation, Figs. 3(a)–(d) show results computed for structures with increased modulation (Δε~5.8 and Δε~10.3). As the modulation grows, the Brillouin diagram increasingly deviates from the clear picture of the leaky stopband [23] shown in Fig. 1. This type of behavior was reported previously in [21]. While the bandedges are now less clear in the dispersion diagrams, they can be accentuated by displaying the derivative |∂k₀/∂β_R| as shown in Fig. 3(b) and 3(d).

Because the structure under consideration can support two waveguide modes (TE₀ and TE₁), two leaky stopbands are shown in Fig. 3. Since the structure has an asymmetric profile, there will be two resonances associated with each mode, one at each bandedge. This results in four resonances that are labeled GMR#1-4, respectively, with GMR#1 being at the lowest frequency.

The separation (in wavelength or frequency) of the two GMRs associated with a particular leaky mode is related to the width of the bandgap, which can be increased by increasing the modulation strength Δε. As shown in Fig. 3(b) and 3(d), when modulation increases, GMR#2 and #3 approach each other while the separation between GMRs#1 and #2 as well as GMRs #3 and #4 increases. Note that GMR#4 in Fig. 3(d) has shifted to a higher frequency outside the range shown. If one further increases the modulation strength, GMRs#2 and #3 will get even closer, partially merge, and interact more intensely.

 

Fig. 4. Standing-wave pattern (electric-field amplitude) of the leaky mode at resonance when the field is close to maximum. (a) GMR#3 in Fig. 3(d); (b) GMR#2 in Fig. 3(d). The size of region is ~1.5µm×2µm. The excitation wave has unit amplitude and is incident from the left as shown.

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Figure 4 presents examples of the electric field profile associated with the resonant leaky modes. Figure 4(a) shows that the resonant mode generating GMR#3 in Fig. 3(d) is a reasonably clean TE₁ mode. Figure 4(b) demonstrates that the interaction of GMRs #2 and #3 results in a mixed mode pattern consisting of both TE₀-like and TE₁-like modes. The possible occurrence of complicated resonant leaky-mode patterns was discussed in our previous paper [15]. Common to Figs. 4(a) and 4(b) are the standing-wave patterns formed by two counter-propagating leaky modes at resonance. Because the grating layer is used as both waveguide and phase matching element in this case, the maximum field value is located in the grating layer with the evanescent tails gradually penetrating into the substrate and cover. It is known for periodic elements with simple profiles that the mode at the upper bandedge (with higher frequency) will concentrate its energy in the low-ε region, and the mode at the lower bandedge (with lower frequency) will concentrate its energy in the high-ε region [24] as do the fields illustrated in Figs. 4(a) and 4(b). As GMR#3 arises at the lower edge of the TE1 second stopband, its electric field concentrates mostly in the high-ε region. On the other hand, the electric field of GMR#2 concentrates mostly in the low-ε region because it is at the upper edge of the TE₀ second stopband.

3. Characteristics of resonant leaky-mode devices

Numerous new optical devices enabled by resonant leaky-modes are presented in this section. These are narrowband bandpass filters operating with TE and TM polarized light, wideband reflectors for TE and TM polarization, wideband antireflective resonant layer, a wideband polarizer, and a wideband polarization-independent element. To attain similar functionalities with traditional thin-film optics based on homogeneous layers may require a significantly larger number layers. Furthermore, these elements possess new properties generally not realized in other ways. All of the devices presented contain only a single periodic layer on a substrate with lower refractive index.

Bandpass filters

The first device presented is a bandpass filter for TE polarization. The parameters of the filter and the corresponding spectra are shown in Figs. 5(a) and 5(b). As in [14], the passband lies within an extensive (~300 nm) region of high reflectance and exhibits a substantial angular aperture. The inset shows the device parameters and the incident wave, reflected wave (R), and transmitted wave (T). The grating profile has an asymmetrical distribution of the materials within the period. The low transmission background is formed by two differentiated TE_1,1 modes that locate on the edges of the second stopband (i.e. nondegenerate case as presented in [15]). We denote the resonance by TE_m,ν, where m represents the evanescent diffraction order exciting the ν-th mode; thus TE_1,1 is the resonance formed by the interaction between the TE₁ mode and the first evanescent diffracted order.

 

Fig. 5. Spectra of a single-layer bandpass filter for TE polarization. (a) Reflectance of the filter with the parameters shown in the inserted schematic. (b) Angular spectra at the central wavelength.

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Insight into the formation of the filter characteristics can be gained by tracking the spectra of the structure with different modulations. Figure 6(a) illustrates the development of the spectra as function of spatial modulation strength. When the modulation is not very strong, i.e., n_h=2.8, the two GMRs are localized and only impact the nearby spectrum. With increasing modulation strength, both resonances expand and change the background of the TE_1,1 resonance at the shorter wavelength, and eventually lead to the formation of a low transmission background surrounding the bandpass filter peak. The transmission peak is provided by the asymmetrical lineshape of TE1,1 at the upper stopband edge which generates the corresponding minima in the η_R curve near λ=1.65 µm as seen in Fig. 6(a).

 

Fig. 6. Spectra of the bandpass filter for TE polarization. (a) Reflectance spectra for different modulations with the effective index of the waveguide kept constant. (b) Transmittance of the filter on a logarithmic scale.

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Because of the dominance of the broadband low-transmission background, the resonance locations are indistinct on the linear scale used in Figs. 5(a) and 6(a) for the larger modulation values. For structures using lossless materials, the resonance is associated with zero transmission; thus the GMRs can be accentuated by plotting the transmission spectrum on a logarithmic scale as shown in Fig. 6(b). This filter has a ~300 nm background with η_R>99% (indicated as a horizontal dashed line in Fig. 6(b)) and FWHM passband of 3.5 nm.

Figures 7(a) and 7(b) show the leaky-mode patterns associated with the two TE_1,1 GMRs appearing at each bandedge for the modulation with n_h=2.8. As the modulation is not very strong, the TE₁ mode profile can be unambiguously identified. Figure 7(a) presents the GMR at the upper edge of TE₁ second stopband as its electric field concentrates in the low-ε region within the period, while Fig. 7(b) presents the GMR at the lower edge of TE₁ second stopband as the electric field concentrates in the high-ε region. The amplitude of the standing wave indicates the Q-factor of the resonance as well as linewidth. With high maximum amplitude, the GMR in Fig. 7(a) has a higher Q-factor and smaller linewidth as shown in Fig. 6(a). As indicated above, the standing wave pattern at resonance may exhibit a mixed state for the higher levels of modulation. Figures 7(c) and 7(d) provide mode patterns for the two resonances contributing to the final bandpass spectrum showing a mixed picture formed by TE₁- and TE₂-like modes. As the linewidths of these low-Q GMRs are broad, the field amplitudes are much smaller than in Figs. 7(a) and 7(b) as measured by the field scales in Fig. 7. The field patterns shown in this paper represent the total fields including the superimposed zero order wave.

 

Fig. 7. Standing-wave pattern (electric-field amplitude) of the leaky mode at resonance when the field is close to maximum. (a) GMR TE_1,1 at short wavelength (upper bandedge) for n_h=2.8; (b) GMR TE_1,1 at long wavelength (lower bandedge) for n_h=2.8. (c) GMR TE_1,1 at short wavelength for n_h=3.48; (d) GMR TE_1,1 at long wavelength for n_h=3.48. The size of region is ~2.5µm×2µm.

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Fig. 8. Spectra of a bandpass filter for TM polarization. (a) Reflectance of the filter. Profile parameters are shown in the inserted schematic. (b) Angular spectra at central wavelength.

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Fig. 9. Standing-wave pattern (magnetic-field amplitude) of the leaky mode at resonance when the field is close to maximum. (a) GMR TM_1,1 at short wavelength; (b) GMR TM_1,1 at long wavelength. The size of region is ~2.5µm×2µm. The incident beam with unit amplitude approaches from the left side.

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Figure 8 shows the spectra of a bandpass filter for TM polarization. The low transmission background is formed by two TM_1,1 resonances located at the edges of the TM₁ second stopband. As in the previous TE case, the transmission peak stems from the asymmetrical lineshape of the GMR at the upper stopband edge. Figures 9(a) and 9(b) show mode patterns for the two GMRs contributing to the bandpass spectrum. In contrast to the TE case in Fig. 7(c) and 7(d), the two GMRs have dissimilar patterns. The GMR at short wavelength shows a mixture of TM₁ and TM₂ modes, whereas the GMR at long wavelength bears the form of TM₁ and TM₀ modes.

Resonant leaky-mode TE and TM high reflectors

 

Fig. 10. Spectra of a wideband reflector for TE polarization. (a) Reflectance of the filter. Profile parameters are shown in the inserted schematic. (b) Angular spectra at central wavelength. (c) Reflectance spectra for differing modulation strengths with the effective index of the waveguide kept constant. (d) Transmittance of the filter on a log scale.

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Figure 10 presents the spectra of a wideband TE reflector that has a ~600 nm band with η_R>99%. As wideband resonance structures are associated with wide angular aperture, it is not surprising to see from Fig. 10(b) that the structure has a Δθ_FWHM>40° at the central wavelength. Again, insight into the formation of the filter characteristics can be gained by tracking the spectra of the structure with different modulations. Figure 10(c) illustrates the development of the spectra as spatial modulation strength increases. When the modulation is not very strong, i.e., n_h=2.8, the two GMRs are localized. Because the profile is symmetric in this example, only one GMR will exist for each stopband [15]. Since the total effective fill factor is less than 0.5 (F=0.45), the GMR appears at the lower bandedge and concentrates in the high-ε region. Using effective-medium theory and the eigenfunction of the equivalent homogeneous waveguide, as in [15], the resonance locations are initially estimated. By subsequently numerically tracking the resonance locations, the two GMRs are found to be TE_1,0 and TE_1,1 consistent with computed mode patterns. With increasing modulation strength, both resonances expand and eventually lead to the formation of the broad reflectance band. The contribution of these two GMRs to the final spectrum is obvious in Fig. 10(d). The experimental wideband reflector reported by Mateus et al. [10] can be explained using similar physical arguments.

Figure 11 shows a wideband reflector with a ~600 nm band with η_R>99% for TM polarization. Like the TE reflector, the device exhibits both a wide flattop and a wide angular aperture as shown in Figs. 11(a) and 11(b). The spectrum obtained with n_h=2.8 in Fig. 11(c) reveals distinct TM₀ and TM₁ modes. Note further that since the device has a symmetrical grating profile, only two GMRs (one TM_1,0 and one TM_1,1) appear (one for each stopband) for this modulation. As the modulation increases, both resonances are seen to expand and move to longer wavelengths. At n_h=3.48, the TM_1,0 resonance is isolated on account of its wide spectral separation from its neighbor and consequently there appears a dip in the final spectrum at λ~2.3 µm. The final reflection band is supported by the interaction of TM_1,1 and two mixed modes that we denote TM_1,1&2 as displayed in Fig. 11(d).

 

Fig. 11. Spectra of a wideband reflector for TM polarization. (a) Reflectance of the filter. Profile parameters are shown in the inserted schematic. (b) Angular spectra at central wavelength. (c) Reflectance spectra for different modulation with the effective index of the waveguide kept constant. (d) Transmittance of the filter on a log scale.

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Magnetic field patterns associated with each resonance in Fig. 11(d) are presented in Fig. 12. The three GMRs that contribute to the flattop are seen to have a mixed-mode character. Figures 12(a) and 12(b) illustrate that the two GMRs at shorter wavelengths have mode profiles resembling superimposed TM₁ and TM₂ modes denoted as TM_1,1&2, while Fig. 12(c) shows that the leaky mode of TM_1,1 actually has a profile between TM₀ and TM₁. The profile for TM_1,0 is presented in Fig. 12(d); since TM_1,0 is not strongly coupled to other resonances, its mode profile shows a TM₀ mode unambiguously.

 

Fig. 12. Magnetic field patterns of the leaky modes at resonance. (a) Pattern of TM_1,1&2, short wavelength side (b) Pattern of TM_1,2, long wavelength side (c) Pattern of TM_1,1 (d) Pattern of TM_1,0.

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Polarizers

 

Fig. 13. Spectra of a wideband polarizer. (a) Reflectance of the polarizer. Profile parameters are shown in the inserted schematic. (b) Angular spectra at central wavelength. (c) TE reflectance spectra for different modulation with the effective index of the waveguide kept constant. (d) TM transmittance of the polarizer on a log scale.

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Figure 13 presents the characteristics of a polarizer that features comparatively wideband spectra with a wide angular aperture. The TE spectrum is generated by the interaction between the TE_1,1 resonance and the TE_1,0 resonance where, in contrast to the high reflectors above, a low reflection band is formed. Figure 13(c) illustrates how the band develops with increasing modulation. As shown in Fig. 13(d), the high reflectance band at 1.5~1.6 µm for TM polarization is supported by two TM_1,1 GMRs. The associated leaky-mode magnetic field patterns in Fig. 14 appear as distorted TM₁-like modes. The key to this device is application of simultaneous TE and TM resonant leaky modes in the same spectral band.

 

Fig. 14. Resonant leaky-mode magnetic field pattern at resonance. (a) Pattern for TM_1,1, short wavelength side (b) Pattern for TM_1,1, long wavelength side.

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Polarization independent element

 

Fig. 15. Spectra of a polarization-independent reflector with asymmetric grating profile. (a) Reflectance with TE and TM incidence with n_h=3.48. (b) Transmittance for TE and TM incidence with n_h=3.48. (c) TE spectrum with n_h=2.8. (d) TM spectrum with n_h=2.8.

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Figure 15 shows the spectra associated with a polarization-independent bandstop filter with a wide band. Figure 15(a) gives the final spectral properties indicating a reflection band that is independent of polarization. The GMRs contributing to the TE and TM reflection bands are marked in Fig. 15(b). The resonances associated with different leaky modes are displayed in Figs. 15(c) and 15(d) for a comparatively small modulation, i.e. n_h=2.8. As the modulation increases, the GMRs approach each other and their linewidths increase with resulting interaction. When n_h increases to 3.48, a polarization independent 20 dB reflection band centered at 1.6 µm with ~20 nm flattop bandwidth is formed; the TM reflection band is much wider (~400 nm) than the TE band which is limiting in this case.

Antireflection element

Figure 16 presents the spectra of an antireflection element for TM polarization. As shown in Fig. 16(b), the element has a wide angular aperture. The broad passband forms by interaction of TM_1,1 and TM_1,0 resonances whose locations are given in Fig. 16(c). Figures 16(c) and 16(d) show that the passband performance improves by approximately an order of magnitude with the modulated layer as compared to a single homogeneous layer.

 

Fig. 16. Spectra of a wideband antireflector for TM polarization. (a) Reflectance of the element. Profile parameters are shown in the inserted schematic. (b) Angular spectra at central wavelength. (c) Reflectance of the filter on a log scale. (d) Angular spectra on a log scale

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4. Conclusions

In conclusion, it has been shown that guided-mode resonance elements possessing asymmetric and symmetric grating profiles are candidates for optical elements with diverse spectral features. The separation of the resonances arising at the edges of the second stopband can be manipulated by controlling the bandgap via the grating modulation amplitude and profile fill factors. Thus, the spectral spacing and level of interaction between adjacent resonant modes is controllable. This provides interesting new dimensions in design of resonant photonic elements as the example devices included here show.

The spectral properties of leaky-mode resonance elements rival those obtained with traditional homogeneous thin-film layer systems. Thin optical films are applied to design and fabricate elements with diverse spectral features which are widely used in optical systems and thus have high economic significance [25]. Guided-mode resonance elements possess analogous spectral versatility but are governed by different physics and are realized with periodic layers. Thus, new possibilities in function and applications can be envisioned. For example, problems with multilayer interface scattering buildup and adhesion of dissimilar materials may be reduced.

The elements presented in this paper require only one 1D layer for their construction. Enhanced features are expected on integration of such periodic films with additional homogeneous layers as suggested previously [15]. This will be an interesting topic for future research. Other research topics to be addressed include symmetric and asymmetric 2D resonance elements where polarization independence will be an important consideration.

While some features in the binary profiles of the elements presented are small, the thicknesses used are moderate providing moderate aspect ratios in general. The achievable aspect ratio is a key consideration in practical fabrication. If replication is contemplated, the aspect ratios become a critical consideration [26]. Use of single-layer binary profiles generally simplifies fabrication as compared to other grating profiles.

Finally, it is known that the spectral response of guided-mode resonance elements is highly sensitive to variations in the structural parameters and loss, especially when operated at high Q as shown in [27]. Maintaining the binary grating profiles, layer thickness, and refractive index values within prescribed limits will be necessary to realize specific spectral characteristics.