1. Introduction

Guided-mode resonance (GMR) refers to a sharp peak in the diffraction efficiency spectrum of waveguide gratings. Physically, this is due to coupling of the externally propagating diffracted fields to the leaky modes of the waveguide [1–4]. Theoretical analysis and experiments show that optical reflection filters can be designed by using GMR to exhibit high efficiency, symmetrical lineshapes, and low non-oscillatory sidebands over extended wavelength regions. GMR filters have been studied for several years, yet their versatility has not been fully realized. Potential applications of GMR filters include such as narrowband reflection and transmission filters [5, 6], tunable filters [7, 8], laser cavity reflectors [9–11], polarization independence filters with large angular tolerance [12], optical switch devices [13], and light modulators [14].

Resonant Brewster filters predicted by Magnusson et al. in 1998 are a type of narrowband filters [15]. The filters can obtain high-efficiency reflection at the Brewster angle at which TM reflection is classically prohibited. Resonant Brewster filter is a type of antireflection structure in essence. As the GMR effect overrides the Brewster effect, high-efficiency resonant reflection can be obtained for a TM-polarized wave at Brewster angle of incidence where classical reflection is prohibited. Low-sideband reflection is thus obtained through the Brewster effect. Later, Shin et al. [16] presented an approach by adding an absentee layer in the structure so as to lower the off-resonance sideband reflection levels of resonant Brewster filters. Previously, we proposed the structure of multichannel resonant Brewster filters [17]. The proposed structure owned two important merits: (1) The structure consisted of only one grating layer, thus the mechanical deflection stress problem was avoided in fabrication [18]. (2) Multiple channels were obtainable by tuning the grating thickness. In succession, we systematically investigated double-layer resonant Brewster filters consisting of a homogenous layer and a grating with equal refractive index [19]. For a double-layer resonant Brewster filter consisting of homogeneous layer with a refractive index equal to that of the grating, the fluctuation of reflectance with the variety of the homogenous layer thickness can be distinctly restrained due to equality of the refractive index of the grating and the homogeneous layer. By tuning the homogeneous layer thickness, instead of the grating thickness, multichannel resonant Brewster filters with fixed grating thickness can be obtained by this way. Very naturally, some problems emerge. How to obtain double-layer surface-relief resonant Brewster filters consisting of a homogenous layer and a grating with equal refractive index? How to control the filter linewidth?

Facing these problems, in this paper, we investigate the linewidth properties of double-layer surface-relief resonant Brewster filters consisting of a homogenous layer and a grating with equal refractive index. By tuning the grating filling factor, double-layer surface-relief resonant Brewster filters consisting of a homogenous layer and a grating with equal refractive index can be obtained, and linewidth properties of the filter are intensively investigated. It is shown the filter linewidth can be significantly changed by varying the grating filling factor and the substrate refractive index, but the etch depth errors do not alter the filter linewidth. Moreover, the coupling strength can be appreciately affected by the homogeneous layer thickness, and one can obtain different linewidths at the same operating wavelength by selecting different homogeneous layer thickness with other parameters maintained.

2. Fundamental principle

A schematic diagram of a double-layer waveguide grating structure under TM illumination is illustrated in Fig. 1. This structure consists of a grating layer and a homogeneous layer. For a zero-order waveguide grating, the effective index of the grating layer can be approximately estimated by the zero-order permittivity of the effective media theory (EMT) [20], i.e.,

Where f is the grating filling factor, and n_1H and n_1L are the high and low refractive index materials of the grating layer, respectively.

 

Fig. 1. Schematic diagram of a double layer zero-order dielectric grating under TM illumination with rectangular refractive-index profile of high (n_1H) and low (n_1L) index with period Λ and thickness d_g, the incident angle is θ, and the grating filling factor is f. The homogeneous layer’s thickness and its refractive index are d_l and n₂, respectively. The refractive index of cover and substrate are n_c and n_s, respectively.

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In order to obtain double-layer resonant Brewster filters consisting of a homogenous layer and a grating with equal refractive index, the refractive index of high refractive index material of the grating layer should be higher than that of the homogenous layer (i.e., n_1H>n₂=n_eff). For given high refractive index and low refractive index material of the grating layer, the required grating filling factor can be obtained from Eq. (1), i.e.,

Thus double-layer surface-relief resonant Brewster filters consisting of a homogenous layer and a grating with equal refractive index can be obtained by adjusting the grating filling factor.

3. Linewidth properties

For a homogeneous layer under a TM-polarized plane wave illumination, assumed the refractive index of the cover, the homogeneous layer, and the substrate are 1.0, 1.63, and 1.46, respectively. The incident wavelength is 650 nm, and the layer thickness is 199 nm. The Brewster angle is 56.72° calculated by using transfer matrix method of thin film [21]. Replacing the upper 100 nm homogeneous layer with a grating that possesses the same effective index as the homogeneous layer produces a low-sideband resonant Brewster filter.

For the structure of double-layer surface-relief resonant Brewster filter, assumed n_1H=2.05, and n_1L=1.0, the required grating filling factor is 0.82 obtained by using Eq. (2).

 

Fig. 2. Response of a double-layer surface-relief resonant Brewster filter consisting of a homogeneous layer with a refractive index equal to that of the grating for an incident TM-polarized wave at 650-nm wavelength. The parameters are n_c=n_1L=1.0, n_1H=2.05, n₂=n_eff=1.63, n_s=1.46, f=0.82, d_g=100 nm, d_l=99 nm, and Λ=281.51 nm.

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Figure 2 illustrates a numerically calculated angular resonance occurring at the Brewster angle (56.72°) for the operating wavelength of 650 nm in a double-layer surface-relief resonant Brewster filter. We position the resonance peak in the reflection minimum by setting the grating period to Λ=281.51 nm. The numerical results, as well as others to appear in this paper, are computed based on rigorous coupled-wave analysis [22]. In this example, the resonance location occurs at the Brewster reflection minimum, but angular resonance linewidth exceeds slightly the width of the Brewster reflection minimum in angular spectrum due to high grating modulation index (defined as (n ² _1H-n ² _1L)/n ² _eff, used in the filter). A Brewster-like minimum is generally obtained independent of the particular value of the layer, but Brewster angle calculated by using the zero-order approximation of the EMT may deviate from that of the filters appreciately as the grating modulation index keeps increasing. However, it can still be used as a starting point which to locate the Brewster angle in the case of strong modulation. In order to obtain high contrast filter, the resonance angle should be located near the Brewster-like reflection minimum of the waveguide grating. This can be implemented by proper change of grating period.

 

Fig. 3. Effects of underetching and overetching, and other parameters are the same as in Fig. 2. (a) Etch depth error on spectral response for the TM-polarized wave incident at the Brewster angle (56.72°). (b) The spectral shifts are adjusted by using angle tuning. Underetching 10 nm and 20 nm correspond to the incident angle of 56.41° and 56.09°, respectively. Overetching 10 nm and 20 nm correspond to the incident angle of 56.97° and 57.22°, respectively.

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For double-layer resonant Brewster filters consisting of a homogeneous layer and a grating with equal refractive index, grating thickness is a key parameter to control the linewidth, and one can obtain desired linewidth by changing the grating thicknesss with the total thickness (d_g+d_l) maintained [19]. But in practice, etch depth error is always a problem that can not be ignored, thus we first investigate the effect of etch depth fabrication error, distinguishing between two types of errors: an underetched grating (i.e., part of the thin n_1H layer remains on the homogeneous layer n₂), and an overetched grating (i.e., the homogeneous layer n₂ is slightly etched). As can be seen in Fig. 3(a), the linewidth is insensitive to both types of errors. Deviation from the design value of 100 nm grating thickness (±20% in this example) results in slight modification of resonance wavelength and lineshape as well as sideband levels, and the filter linewidth and peak reflection are not changed. In the case of underching, the remaining thin layer of n_1H (a dense media in this example) increases the refractive index of waveguide, resulting in increase of resonance wavelength. In the case of overetching, resonance wavelength shifts to the shorter wavelength due to decrease of the waveguide refractive index. Since the two types of errors do not change the filter linewidth, the spectral shifts can be adjusted by angle tuning with the filter linewidth maintained. In Fig. 3(b), the central wavelengths are all adjusted to 650 nm by using angle tuning. As can be seen in Fig. 3(b), the filter linewidths are kept the same, but the lineshape becomes asymmetrical and the sideband level is increased in both cases of etch depth error. In the case of underetching, the sideband level is increased more in contrast with that of overetching, thus an accurate etch depth control to avoid underetching is necessary to obtain good performance from these devices.

 

Fig. 4. Spectral response as a function of the grating filling factor (f) at the Brewster angle (56.72°), and other parameters are the same as in Fig. 2.

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Figure 4 shows the spectral response of the structure in Fig. 2 as the grating filling factor (f) is varied. As can be seen in Fig. 4, the linewidth, the resonance wavelength, the lineshape, and the sideband levels are influenced significantly as the grating filling factor is varied. Note GMR vanishes as f→ 1 and f<0.6, which appear similar in phenomenon, but their physical mechanisms are different in essence. GMR does not occur for f→ 1 because the filling factor controls the coupling-loss coefficient in a sinusoidal form, thus the linewildth tends to zero as f approaches 1 [23]. GMR does not occur for f<0.6 because the waveguide (the homogeneous layer) is too thin to support the leaky mode as the filling factor decreases. Thus the filling factor can be an important physical parameter to control the linewidth. Moreover, a modification of the filling factor is associated with a change in the effective refractive index of the grating layer, resulting in a shift of resonance wavelength. Furthermore, the modification of the filling factor is also associated with a change of the grating modulation index and the breakdown of Brewster’s no-reflection effect, thus the lineshape and sideband levels are affected appreciately as the filling factor is varied.

 

Fig. 5. Spectral response as a function of the substrate refractive index (n_s) at the Brewster angle (56.72°), and other parameters are the same as in Fig. 2.

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Figure 5 shows the spectral response of the structure in Fig. 2 as the substrate refractive index (n_s) is varied. As can be seen in Fig. 5, change in the substrate refractive index can shift the resonance, and the filter linewidth is altered appreciately with slight change in sideband levels as the substrate refractive is varied. Selecting the substrate refractive index of 1.38, 1.46, and 1.52 can obtain the linewidth of 1.47 nm, 1.05 nm, and 0.24 nm, respectively. The linewidth increases as the substrate refractive index decreases, and the linewidth decreases as the substrate refractive index increases. This is because the degree of mode confinement [3] (refractive-index difference at boundaries) decreases as substrate refractive index increases, resulting in decrease of the filter linewidth. As the refractive index of substrate approaches the refractive index of waveguide (n_s→1.63), other parameters remain but the waveguide nearly vanishes. As the waveguide nearly vanishes, the incident wave can not be phase-matched to the leaky mode of the filter, and the linewidth tends to zero as a result.

 

Fig. 6. Reflectance as a function of the homogeneous layer thickness at the Brewster angle of 56.72° for an incident TM-polarized wave at 650 nm wavelength, and other parameters are the same as in Fig. 2.

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Figure 6 illustrates reflectance under variation of the homogeneous layer thickness at the Brewster angle of 56.72° for an incident TM-polarized wave at 650 nm wavelength, and other parameters are the same as in Fig. 2. Three resonance peaks appear in the range of 0-1.2 µm, these resonance location are in the position of 99 nm, 562 nm, and 1025 nm, which correspond the second stop bands of TM₀, TM₁ and TM₂ waveguide modes of the filter, respectively [24]. The fluctuation of reflectance with the varieties of the layer thickness is distinctly restrained duo to Brewster effect. For this type of filter, increasing layer thickness has the effect of admitting additional leaky modes, thus it exhibits unique signature of multimode resonance [25]. The coupling strength can be appreciately affected by the homogeneous layer thickness in these devices. For the same resonance wavelength, the coupling strength decreases as leaky mode order increases, resulting in decrease of the spectral linewidth. Thus one can obtain different linewidths at the same operating wavelength by selecting different homogeneous layer thickness with other parameters maintained.

 

Fig. 7. Spectral response as a function of the homogenous layer thickness (d_l) at the Brewster angle (56.72°), and other parameters are the same as in Fig. 2.

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Fig. 7 shows the spectral response of the structure in Fig. 2 by selecting the homogeneous layer thickness of 99 nm, 562 nm, and 1025 nm, which correspond the TM₀, TM₁ and TM₂ leaky modes of the filter, respectively. As can be seen in Fig. 7, GMR occurs at the same operating wavelength without using angle tuning or grating period tuning, and different linewidths can be obtained. Note that the filter linewidth decreases as leaky mode order increases and that linshape and sideband levels are nearly preserved. The spectral possess narrow linewidth, high peak reflectance, and low sideband levels simultaneously, thus they exhibit good filtering features.

4. Conclusion

In conclusion, double-layer surface-relief resonant Brewster filters consisting of a homogenous layer and a grating with equal refractive index can be obtained by adjusting the grating filling factor, and linewidth properties of these devices are investigated. It is shown the filter linewidth can be significantly changed by varying the grating filling factor and the substrate refractive index, but etch depth error will not change the filter linewidth except the lineshape symmetry and sideband levels. The coupling strength can be appreciately affected by the homogeneous layer thickness, and one can obtain different linewidths at the same operating wavelength by selecting different homogeneous layer thickness with other physical parameters maintained.