## Abstract

A triple-layer guided-mode resonance Brewster filter consisting of a homogeneous layer and two identical gratings with their refractive indices equal to that of the homogeneous layer is presented. The spectral properties of this filter are analyzed based on the coupling modulation of two identical binary gratings at Brewster angle for a TM-polarized wave. The grating layer between substrate and homogeneous layers can significantly change the linewidth and resonant mode position, which are due to the asymmetric field distribution inside the grating layers. The tunability of the resonance can be altered on different resonant channels and a practical filter can be obtained in TM_{2} waveguide mode. Variation of filling factor can alter the field localization in the grating structure and significantly adjust the linewidth of the filter.

© 2011 OSA

## 1. Introduction

The narrow linewidth of guided-mode resonance (GMR) is typically observed in substrates with geometrically tailored multilayer thin films of grating and waveguide layers [1]. This new type of optical element [2,3] combines principles of diffraction by periodic structures with waveguide properties and antireflection (AR) thin-film characteristics to yield filters with 100% reflectance at a desired wavelength. By appropriate choice of multilayer waveguide-grating parameters, such as thickness and refractive index, the high diffractive order waves yielded by grating layer can be coupled into the guided-mode, which can propagate in the waveguide layer. This interesting anomalous effect resulting in the energy exchange between reflection and transmission wave has been theoretically and experimentally reported in microwave region [4], infrared region [5–8], and visible region [9,10]. Filters based on GMR can be easily extended into several areas, such as optical bistable devices [11], dense wavelength division multiplexing systems in optical communication [12–14] and sensors [15]. The concept of resonant Brewster filters of TM mode was put forward in 1998 by Magnusson *et al* [16]. The filters can obtain high-efficiency reflection at Brewster angle where traditional TM reflection will vanish. The suppression of filter sidebands with absentee layers was advanced by Shin *et al* [17]. Later, the double-layer GMR filter with multiple channels at Brewster angle was presented by Wang *et al* [13] and Sang *et al* [14]. In their studies, the filters consist of homogeneous layer with a refractive index equal to that of grating layer. The fluctuation of the reflectance with variation of homogeneous layer thickness can be distinctly restrained due to equality of the refractive index of grating and homogeneous layer. The resonant center wavelength and linewidth can be controlled without lacking low-reflection sideband features by tuning thickness of homogenous layer, filling factor of grating, and index of substrate layer. Recently, related devices based on tunable characteristic due to the interaction of two GMR elements and the tunability of coupled GMR effects were studied, and wideband tuning range has been demonstrated [18–21]. However, the previous resonant filters based on the interaction of two GMR elements were studied under normal incidence [18–20] or general oblique incidence [21], and less attention is paid to the Brewster angel incidence on this kind of GMR filters.

Besides the simpleness of the GMR filter consisting of single grating layer, it has been proved by *Li* [22] that the truncated propagation equations of the original differential theory in the previous rigorous coupled-wave analysis (RCWA) method [23,24] no longer preserve the continuity of the appropriate field components across the discontinuities of the permittivity function. Using theorems of Fourier factorization, Li proposed three appropriate factorization rules, which are concerned with any numerical work in science that requires Fourier factorization and the rules are called the fast Fourier factorization (FFF) method. In this paper, we proposed a triple-layer guided-mode resonant Brewster filter consisting of two binary identical gratings and a homogeneous layer with refractive index equal to that of the gratings. Using the FFF method, the main properties of reflective spectrum of the Brewster filter were analyzed for TM polarization. The coupling characteristics between two gratings were studied with respect to the thickness of gratings, the thickness of the homogenous layer and the lateral alignment shift between gratings. Results show that adjusting the grating thickness and lateral alignment can alter the linewidth of the spectral response. The dependence of coupling strength on the lateral alignment shift can be affected by the thickness of the homogeneous layer and the grating filling factors. The tunability of the resonance can be altered with respect to different resonant channels and grating filling factors.

## 2. Structure and theory

A schematic diagram of the triple-layer waveguide grating structure under TM polarization light at oblique incident angle is depicted in Fig. 1 . The resonant part above the substrate layer consists of two identical grating layers and a homogeneous layer between the gratings. According to the effective media theory (EMT) [25], the second order of effective refractive index of the grating layer under TM illumination for $\Lambda >{\lambda}_{0}$ can be written in the following form

*Λ*and ${\lambda}_{0}$ are the grating period and the central resonant wavelength of the structure, respectively.

*F*is the grating filling factor and ${\epsilon}_{H}$ and ${\epsilon}_{L}$ are high and low permittivity of the grating materials, respectively. The zero-order permittivities under TE and TM-polarization conditions in Eq. (1) are given by

For a high-spatial frequency waveguide grating ($\Lambda /{\lambda}_{0}\to 0$), the second term of the polynomial expression on right side of Eq. (1) is neglectable and the expression of the effective index of the grating under TM-polarization can be approximately reduced as

For a fixed filling factor, the triple-layer resonant Brewster filter can be obtained through appropriate choosing of the homogeneous layer’s refractive index, which should be equal to the effective refractive indices of gratings calculated by Eq. (3).

In Fourier space, the basic equations of differential theory of gratings can be expressed as follows [24]

*ω*is the circular frequency and ${\mu}_{0}$ is the permeability of vacuum).

*E*and

_{x}*H*are the

_{z}*x*and

*z*components of electric and magnetic field, respectively.

*k*

^{2}is a periodic function ${k}^{2}(x,y)={k}_{0}^{2}\epsilon (x,y)$, where

*k*

_{0}is the modulus of the wave vector in vacuum.

*α*is a diagonal matrix with elements ${\alpha}_{n}={\alpha}_{0}+2\pi n/\Lambda $, where ${\alpha}_{0}={k}_{0}\mathrm{sin}({\theta}_{i})$ and $n\in \mathbb{Z}$. $\u301af\u301b$ denotes the Toeplitz matrix generated by the Fourier coefficients of

*f*such that its (

*p*,

*q*) ($p,\text{}q\in {\mathbb{N}}^{\ast}$) element is

*f*, and −1 denotes the matrix inverse [22]. The numerical calculation based on differential Eq. (4) can be implemented to produce high accurate results with rapid convergence by preserving fewer harmonic waves.

_{p-q}## 3. Results and discussion

By using the transfer matrix method of thin film, the calculated Brewster angle of the triple-layer filter is 57.13°. The angular response of the triple-layer waveguide grating structure is shown in Fig. 2(a) with the operating wavelength of 800 nm. For locating the resonant peak at Brewster angle, the periods of the identical gratings are set to $\Lambda =333.19\text{nm}$. The resonance is induced because the coupling of the first evanescent diffraction order to a leaky waveguide mode replaces the classical Brewster angle zero reflection effect, thus the zeroth reflected order is reradiated. Here, the period of the gratings satisfies the high-spatial frequency condition ($\Lambda /{\lambda}_{0}=0.42$). The deviation of the zero-order permittivity of the grating from the second order one is 0.02, which can be ignored if the homogeneous material is chosen. The spectral response in Fig. 2(b) with the incident wave at Brewster angle exhibits a resonant peak at $\lambda =800\text{nm}$ with a linewidth of ~0.05 nm, which is much narrower than the linewidth in the previous work [14]. The linewidth of the low sideband reflection is over ~200 nm. The resonant response with a reflectance peak over 99.9% is zoomed in Fig. 2(b). Result shows that the line shape of the spectral response is symmetrical as the total thickness of the triple-layer structure satisfies the half-wavelength condition.

For multilayer resonant Brewster filters consisting of grating layers and homogeneous layer with same effective refractive index, the line shape and linewidth of the reflection response can be controlled by altering the thickness of the grating layer while fixing the total thickness (d_{g}_{1} + d* _{u}* + d

_{g}_{2}). In the triple-layer presented in this paper, the thickness of the homogeneous layer is thinner than that of the grating layer, and the field inside the two identical gratings is strongly coupled. Consequently, the reflection response will be greatly affected by varying the grating thickness. Besides, the distribution of electromagnetic field in the grating layers is different as constructive and destructive interference of the internal diffracted fields, which are reflected from the top and bottom surfaces of the gratings due to the different adjacent mediums of the two identical gratings. Therefore, the dependence of the reflection response on the thickness of upper and lower gratings should be analyzed while fixing the thickness of (d

_{g}_{1}+ d

*) or (d*

_{u}*+ d*

_{u}

_{g}_{2}), which are shown in Fig. 3 . As can be seen, altering thickness of the grating layers will result in the change of the linewidth as well as the shifts of resonant peak. The resonant peak has a red shift as increasing the thickness of the grating layers for two cases, but the linewidth changes distinctly as varying the lower grating thickness (see in Fig. 3(a)). The key parameters of the spectral response in Fig. 3(a) are listed in Table 1 . The data in Table 1 indicates that the effect of the lower grating layer on the spectral characteristic is larger than that of the upper one because boundary conditions of phase matching for the two gratings are different [26], which results in the different distribution of electromagnetic field inside the grating layers [21]. An additional interesting point is that the central wavelength can be adjusted to desired wavelength by slightly altering the incident angle while keeping the linewidth, line shape and sideband features almost unchanged for same grating thickness. Figures 3(b) and (c) demonstrate that the resonant peak can be tuned to the same resonant wavelength by slightly varying the incident angle. The resonant incident angle can be changed by altering the thicknesses of upper and lower grating layers while keeping the spectral characteristics unchanged. In contrast to the previous work [14], the thickness of the lower grating can be adjusted to obtain different linewidths at the central wavelength of 800 nm while keeping the feature of low-reflection sideband, as shown in Fig. 3(c). Considering the practical applications, we can simultaneously adjust the upper and lower grating layer to obtain spectral response of the Brewster filter for desiring linewidth and operating wavelength.

As reported in previous works, multiple channels of the resonant Brewster filter can be obtained by using multiple resonances [12,13]. For simplification of the fabrication process, multiple resonances can be achieved by adjusting the thickness of the homogeneous layer due to the equality of refractive indices of the gratings and the homogeneous layer. We calculate the reflectance at Brewster angle with varying the thickness of the homogeneous layer, as shown in Fig. 4
. The thicknesses of the grating layers are maintained at the same time. As can be seen, three resonant peaks (30.0, 355.2 and 681.1 nm) appear almost periodically in the range of 0-800 nm, which correspond to the second stop band of the TM_{0}, TM_{1} and TM_{2} waveguide modes of the filter, respectively [27]. When the thickness of the homogeneous layer is tuned to these three values, the single, double or triple channels can be obtained in the Brewster filter, respectively.

The coupling strength between the electromagnetic fields in the gratings and homogeneous layer can be appropriately affected by the homogeneous layer thickness and lateral alignment condition between two gratings in this kind of devices [14]. Meanwhile, the tunable range and ability of the lateral alignment shift along transverse direction, which could be applied to alter the line shape and linewidth of spectral response of the Brewster filter, is also dependent on the characteristic of the field inside the homogeneous layer [18]. Below, the tunable range of the resonant mode location and the characteristic of spectral response of the Brewster filter due to the lateral alignment shift are studied near the same resonant wavelength of 800 nm for three different thicknesses of the homogeneous layer. The reflectance of the Brewster filter as the function of the lateral alignment shift and wavelength for TM_{0}, TM_{1} and TM_{2} waveguide modes are shown in Fig. 5(a), (b) and (c)
, respectively. For the three waveguide modes, the lateral alignment condition does not almost cause any shift for the resonant peak but the linewidth and reflectance peak. The resonant peaks of the TM_{0} and TM_{1} modes shift toward short wavelength as S increases from 0 to 0.5. For example, the wavelength of resonant peak shifts from 800 to 799.12 nm in Fig. 5(a) and from 800 to 799.57 nm in Fig. 5(b). The reason is that the field inside the Brewster filter is mostly confined in the grating layer due to their relative large thicknesses, and the lateral alignment shift S can affect the coupling strength between the two gratings [18]. With increasing of S from 0 to 0.5, the linewidth of the TM_{0} mode increases, however, that of the TM_{1} mode changes more complicated because of the comparable thicknesses of two grating layers to the homogeneous layer. The linewidth of the TM_{2} guided-mode decreases sharply as S varies from 0 to 0.5, but the location of the resonant peak is invariable. These characteristics are in good agreement with the coupled-mode theory [20].

Figure 5(d) shows the variety of reflectance peak around 800 nm as a function of the lateral alignment shift S for TM_{0} (squares), TM_{1} (circles) and TM_{2} (up triangles) waveguide modes. For the TM_{0} and TM_{1} guided-modes, the electromagnetic field of the excited guided-mode inside the filter can be greatly affected due to oblique incident angle when the two gratings are not perfectly aligned. Thus, as the lateral alignment shift S changes, the reflectance peak varies in the range of 0.2-1 for TM_{0} mode and 0.75-1 for TM_{1} mode, respectively. For the TM_{2} waveguide mode, the field distribution in the homogeneous layer reduced the difference between the diffractive characteristics of the two gratings when the lateral alignment S varies and the condition of phase matching is satisfied. Thus, the spectral reflectance peak of the TM_{2} waveguide mode in Fig. 5(d) is almost unchanged. This configuration can be considered as a perfect GMR filter at Brewster incident angle for practical applications of narrowing the linewidth of reflective spectrum. The spectral responses under particular alignment conditions (S = 0, 0.25 and 0.5) TM_{0}, TM_{1} and TM_{2} waveguide modes are shown in Fig. 6(a), (b) and (c)
, respectively. As can be seen, by adjusting the lateral alignment condition, TM_{0} and TM_{2} guided-modes can be respectively used for expanding or narrowing the linewidth of the spectral response of the Brewster filter with symmetric line shape and low sideband features maintained.

Since the field inside the GMR filter is mostly confined in the medium with high refractive index [19], therefore, the tunable range of the resonant mode location and the characteristic of spectral response of the Brewster filter due to the lateral alignment shift could be affected by the filling factors of the gratings. Figures 7(a) and (b)
show the spectrums of the Brewster filter with a grating filling factor of *F* = 0.1 and *F* = 0.9, respectively, which correspond to two modulated grating structures. For locating the resonant wavelength at 800 nm, the periods of the gratings are adjusted to 342.10 nm and 321.71 nm in Fig. 7(a) and (b), respectively. The other parameters are the same as those in Fig. 5(a). In Fig. 7(a), results show that the GMR effect does not vanish under the perfect alignment condition (S = 0). The range of the spectral linewidth is still tunable as S varies, since the field in the gratings is extremely confined in the narrow high refractive index region. In Fig. 7(a), the effective refractive indices of the identical gratings are close to that of the substrate. Therefore, the location of the resonant peak cannot reach the minimum value of the tunable range when S = 0.5. When the filling factor is set to 0.9, a strong coupling strength between the fields in the gratings is kept. Therefore, the linewidth of the spectral response is almost unchanged as varying of S from 0 to 0.5, which is shown in Fig. 7(b).

## 4. Conclusion

In summary, a triple-layer GMR Brewster filter can be fabricated by selecting a homogeneous layer with refractive index equal to two identical gratings with filling factors of 0.5. It is shown that the linewidth of the spectrum can be significantly changed by altering the thickness of the lower grating layer, but the upper grating layer mainly affects the resonant mode location and sideband levels. Higher order of guided-mode can be excited when the thickness of homogeneous layer increase and different line shape of spectral response can be obtained by selecting different homogeneous layer thicknesses. The tunable range and ability of the lateral alignment shift along the transverse direction is also dependent on the thickness of the homogeneous layer. Different dependence of the linewidth and reflectance peak on the lateral alignment shift S can be obtained at TM_{0} and TM_{2} waveguide modes, the later case can be used as a perfect GMR filter with the reflectance peak unchanged as S varies. For practical applications, the triple-layer Brewster filter can still be used to obtain extremely narrow linewidth of spectral response when the filling factor is set to 0.1. Meanwhile, the resonant mode location has a certain tunable range when the filling factor is set to 0.9.

## Acknowledgments

This work is supported by the Chinese National Key Basic Research Special Fund (grant 2011CB922003), the Natural Science Foundation of China (grant 60678025 and 61008002), 111 Project (grant B07013), the Specialized Research Fund for the Doctoral Program of Higher Education (grant 20100031120005), and the Fundamental Research Funds for the Central Universities.

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