1. Introduction

Although the optical characteristics of periodic films have been studied for more than 100 years, they remain an active field of research. Their properties yield functionalities that are not easily realized with other means. As nanoscale patterning and processing technology continues to advance, there will be steady associated progress in passive and active optical elements engineered with spatially modulated films. Much current research in diffractive optics or photonic crystals emphasizes light control near the first Bragg condition thereby applying the first stopband. However, additional interesting possibilities for functional devices arise near the second stopband where leaky modes can be generated in suitable geometries. A recent paper highlights progress in analysis and experiment of leaky mode devices including optical filters and sensors utilizing resonant leaky modes [1].

We have reported bandpass filters based on excitation of a leaky-mode resonance pair with one resonance providing a broad low transmission band with the other supplying the transmission peak through its asymmetrical line shape [2]. There, it is demonstrated that interacting leaky modes can be applied to render a desired filter response using excitation by higher-order evanescent waves in a single modulated layer [2]. A related idea was presented previously where the distinct diffraction orders operated in a two-waveguide geometry with an interceding coupling layer [3]. In this paper, we show that bandstop and bandpass filters with versatile spectral attributes can be implemented with modulated films possessing an asymmetric grating profile. This breaks the mode degeneracy at normal incidence and permits interaction of the resulting differentiated modes with interesting consequences as explained in the paper.

2. Profile asymmetry and the second stopband

Guided-mode resonance (GMR) occurs when the incident wave is phase-matched to a leaky waveguide mode. Guided-mode resonance excitation at normal incidence results in two leaky modes counterpropagating along the grating layer. In this case, if the leaky modes are excited by the first diffracted order, the structure will operate at the second stopband. The connection between leaky modes and the second stopband has been studied in several papers [4–7]. Kazarinov and Henry (KH) provided perturbation analysis that shows that the radiated fields generated by the two counterpropagating leaky modes can be in phase or out of phase at the edges of the second stopband [4]. At one edge, there is zero phase difference and radiation is enhanced whereas at the other edge there is π phase difference quenching the radiation. Kazarinov and Henry were primarily interested in applying this idea in DFB laser design. Rosenblatt et al. applied the KH model to resonant waveguide gratings and were able to solve the scattering problem including an incident wave excitation both at normal and nonnormal incidence [5]. Brundrett et al. investigated the properties of resonant scattering near the second stop band for both weakly and strongly modulated gratings using rigorous numerical methods [6]. In particular, they found that the resonance falls at the edges of the stopband and that the fill factor of the grating defines at which edge the resonance locates, consistent with the KH and Rosenblatt models. The results presented in these papers [4–6] refer to symmetric grating profiles.

The focus of the current paper is on spectral filters enabled by resonant elements with asymmetric grating profiles. It was first demonstrated by Vincent and Neviere that there exists a selection rule at the second stop band associated with the symmetry of the grating profile [7]. For symmetric structures, consistent with the KH model, this provides β_I=0 at one edge, where β = β_R + jβ_I is the complex propagation constant. These authors then showed how to defeat the selection rule by use of asymmetric structures such that β_I≠0 at both edges [7]. In this paper we apply such profile asymmetry to shape the spectral bands provided by single-layer modulated films illustrated in Fig. 1. The asymmetry works to remove the leaky-mode degeneracy at normal incidence. Consistent with the arguments just presented, it is shown in Fig. 1(a) that GMR will only appear at one edge of the second stopband [6] for a symmetric grating while resonance peaks will appear at each edge of the band for a grating without reflection symmetry as shown in Fig. 1(b). In the figure, the value of the average refractive index is the same for both structures for comparison. Thus, the low grating index used in the computation is

where F=F₁+F₂ for the type II profile. The dispersion curves are calculated with the method introduced by Peng et al. [8]. The spectra and field profiles are calculated with computer codes based on rigorous coupled-wave analysis of wave propagation in periodic media [9,10].

Only single-layer structures modulated with rectangular profiles will be considered in this paper. The two profiles under study, denoted type I and type II, are illustrated in Fig. 1. It is assumed that the gratings are transversely infinite and that the materials are lossless. The incident wave will be taken as being TE polarized (electric vector normal to the page) and at normal incidence.

 

Fig. 1 Brillouin diagram showing second stopband detail for a single-layer waveguide grating. Both gratings have cover index n_c=1, substrate index n_s=1.48, average grating index n_avg=2, high grating index n_h=2.05, period Λ=0.3μm, thickness d=0.14μm. (a) Grating with symmetric profile (Type I) and fill factor F=0.33; (b) Grating with asymmetric profile (Type II). F₁=0.2, F₂=0.13, and M=0.5. In the figure, k₀=2π/λ where λ is the wavelength in free space, K=2π/Λ, and β_R is the real part of the propagation constant of the leaky mode. Note that these dispersion curves are associated with the TE₀ mode and have 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 key point of this paper is that, besides modulation, the separation of two GMRs is controllable through the modulation profile by inducing asymmetry and thus modal nondegeneracy. Therefore, with certain modulation strength available (i.e., given materials), it is still possible to design both wide-band and narrow-band filter devices with geometric manipulations as shown in Section 4.

3. Nondegenerate modes via profile asymmetry

The multiple guided-mode resonances (GMRs) resulting from interaction between several modes and multiple evanescent diffracted orders can be utilized to condition the spectral response. This method has been used to design three-layer wideband bandstop filters [3] and single-layer bandpass filters [2] using symmetric grating profiles. One difficulty in using this approach stems from the fact that the resulting resonance locations may be relatively widely separated. Figure 2 illustrates this point for a chosen set of parameters. The figure is constructed with the help of following two relationships: 1) homogeneous planar waveguide eigenvalue equation; 2) phase-matching condition needed for coupling [11]. In the figure, the order-mode resonance connection is indicated by TE_m,v, where m represents the evanescent diffraction order exciting the v-th mode; for example, the resonance formed by the interaction between TE₂ and the 1^st evanescent diffracted order is written as TE_1,2. The resonances falling below the horizontal line λ/Λ=1.48 will have diffracted orders other than the 0^th orders radiating. Because higher orders will draw power and decrease the diffraction efficiency of the zero orders, this case is not desirable and will not be considered.

For an asymmetric structure, as described in the previous section, there will be two resonances associated with each mode, one on each side of the curves in Fig. 2. As an example, for a structure with d/Λ=0.65 in Fig. 2, there will be four such GMRs of interest. Nondegenerate GMRs #1 & #2 are associated with leaky mode TE₀, while GMRs #3 & #4 are associated with leaky mode TE₁.

 

Fig. 2. Estimated resonance locations based on the eigenfunction of the equivalent homogeneous waveguide. Material parameters are indicated on the figure.

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The separation (in wavelength or frequency) of the two GMRs associated with a particular nondegenerate leaky mode is related to the width of the bandgap in Fig. 1. Thus, the spectral separation of GMRs#1 & 2 indicates the width of the TE₀ stopband and that of GMRs# 3 and 4 shows the width for the TE₁ band. The width of the stopband can be increased by increasing the modulation strength Δε=${{\mathrm{n}}_{\mathrm{h}}}^2$-${{\mathrm{n}}_{\mathrm{l}}}^2$ [11] and by profile design. By inspection of Fig. 2, one can see possible interaction between GMRs #2 & 3 as Δε increases. Since each GMR is associated with 100% reflection, placing two GMRs near each other opens the possibility of a flat reflection band.

4. Example filter spectra

In this section, we show three example filters utilizing this concept. All structures have the type II profile shown in Fig. 1.

Example 1: Bandstop filter with narrow flattop

Figure 3 shows the spectra pertinent to a bandstop filter with narrow flattop. Near 1.8μm wavelength, the structure has a ~ -35dB transmission dip with bandwidth about 2nm. The associated flat reflection top is formed by interacting GMRs# 2 & 3, whose locations are seen in Fig. 3(b). Thus this example filter is enabled by interaction of the differentiated TE₀ and TE₁ modes as indicated in Fig. 2.

 

Fig. 3. Spectra of a narrowband reflection filter. The parameters are: F₁=0.397, F₂=0.051, M=0.5, d=0.67μm, Λ=1μm, n_c=1, n_h=3.48, n_s=1.48, and n_avg=2.445. η_R is the reflectance, η_T is the transmittance.

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Figures 4(a) and 4(b) show the leaky-mode field profiles associated with GMR#3 and #2, respectively, for a structure with n_h=2.8 while n_avg is kept unchanged at 2.445. The differentiation of the modes by the profile asymmetry is discernible (albeit not on the scale used in Fig. 4) but small with the amplitudes of leaky modes S₁ and S_-1 nearly overlapping; GMR#3 is associated with a TE₁ -like mode while GMR#2 is with a TE₀ mode as evident in Figs. 4(a) and 4(b). As the modulation increases, there is some mixing of the modes on account of the resonance interaction; this is shown in Figs. 4(c) and 4(d). It is noted that as the modulation grows, higher evanescent diffraction orders (S_±2 and S_±3 shown) can contribute to the mode picture.

 

Fig. 4 Field profiles of the excitation wave S₀ and leaky modes S_±1 pertaining to the resonant filter in Example 1.

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Example 2: Bandstop filter with wide flattop

The second example is a bandstop structure with a wide flattop simulating a dielectric stack mirror. Its spectra are shown in Fig. 5. As in Example 1, the flat reflection band is formed by GMRs#2 & 3 yielding a linewidth of ~150nm with central wavelength near λ=1.75μm. A corresponding wide angular spectrum is also found.

 

Fig. 5. Spectra of a wideband reflection structure. The parameters are: F₁=0.35, F₂=0.1, M=0.52, d=0.45μ, Λ=1μ, n_c=1, n_h=3.48, n_s=1.48, and n_avg=2.45.

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Example 3: Bandpass filter

The final example is a bandpass structure with spectra shown in Fig. 6. There is a narrow transmission peak at ~1.6μm and a wide transmission band between 1.9~2.3μm as indicated by the corresponding low reflectance region in Fig. 6(a). The background of the narrow transmission peak at ~1.6μm is provided by GMRs #2 and #3, while the peak is due to the asymmetrical lineshape associated with GMR#2. The wide transmission band between 1.9~2.3μm is formed by the interaction between GMRs #1 and #2 shown in Fig. 2 in combination with a λ/4 antireflection effect near λ=2.1 μm.

 

Fig. 6. Spectra of a transmission structure. The parameters are: F₁=0.5, F₂=0.05, M=0.55, d=0.39μm, Λ=1μm, n_c=1, n_h=3.48, n_s=1.48, and n_avg=2.667.

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In the examples above, a high index material (n_h=3.48) is used to produce large modulation strength. Lower refractive indices can also be used. Control of the width of the second stopband (i. e., the nondegenerate resonance locations at each edge) involves tradeoff between the value of the modulation amplitude and the profile geometry.

5. Conclusions

It has been shown that guided-mode resonance elements possessing asymmetric grating profiles are candidates for bandstop and bandpass filters with diverse spectral features. The separation of the nondegenerate 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 conveniently controllable. This provides interesting new dimensions in design of resonant photonic elements. Example filters given in the paper include wideband and narrowband flattop bandstop and bandpass filters. Whereas the discussion has been limited to TE polarization and single-layer structures, it is expected that additional layers will enhance the filter features. The ideas presented are generally applicable to 2D layered photonic crystal lattices; for example, by including a sublattice that is properly offset from the main lattice, the degeneracy can be broken.