1. Introduction

Guided mode resonance (GMR) filters – also named resonant grating filters – have attracted a great interest since Wang and Magnusson’s proposal in 1990 [1, 2]. This is mainly due to their spectral properties: the peak symmetry, the rejection rate and the narrow bandwidth. These structures are composed of a sub-wavelength grating associated to one or several dielectric layers. The grating is designed in order to control the excitation of eigenmodes in the layers through the grating diffracted orders. In the last fifteen years, different all-dielectric structures have been studied theoretically for notch filtering [3–5] or band-pass filtering [6–8]. Recently, other concepts based on metallo-dielectric structures have been presented [9, 10]. These filters can be used for teledetection and imaging purposes. In this context, the association of a polarizer function with the spectral filtering properties can be useful. Indeed differences between the two polarization states of reflected or scattered light can be used to enhance the image contrast and to detect the shape of objects with a better resolution [11]. This allows for example to distinguish man-made targets in natural or urban scenes [12, 13]. However, polarization independent filters are usually required. In the case of mid- or far-infrared imaging with extremely low photon flux, the 50% of incident light lost by reflection on polarizing filters is a strong drawback. Several studies have been dedicated to the design of polarization independent filters at normal [14–16] and oblique incidences [17–20].

In Ref. [10], we have presented an original concept of band-pass filters based on a sub-wavelength metallic grating deposited on a dielectric free-standing layer. The experimental behaviour of a one dimensional (1D) lamellar structure made of parallel slits was shown at normal and oblique incidence for a polarization where the magnetic field H is parallel to the slits and to the rotation axis of the structure. Here, the complete experimental behavior (angular, spectral, polarization) of three different structures is highlighted: the one dimensional (1D) lamellar grating made of parallel slits, and two different two-dimensional (2D) arrays made of perpendicularly crossed slits with a rectangular or a square pattern. Particularly, experimental and theoretical angularly-resolved transmission spectra of the 1D and 2D structures in oblique incidence with azimuthal polarization are presented. It is worth noticing that this study is usually neglected by papers on 1D structures and yet it is a fundamental problem for applications. We highlight also the non-trivial (especially at oblique incidence) similarity between 2D structures and two crossed-1D structures. Finally, the physical origin of the transmitted bands is demonstrated by the calculation of the guided-modes dispersion curves.

2. 1D and 2D structures for guided-mode resonances filters

The structures consist of gold gratings deposited on a free-standing silicon nitride membrane (SiN_x). The gratings are either one-dimensional, or two-dimensional with rectangular or square patterns as represented in Fig. 1. The components are designed to act as band-pass filters in the mid-infrared wavelength range. Electromagnetic simulations are based on a RCWA modal method [21]. The period d should fulfil two conditions: no diffracted order in free-space, and a coupling between the first-order diffraction wave and the guided-modes in the SiNx layer at the resonance wavelength λ_R. Light is trapped in the SiNx waveguide, and then coupled to the zero-order transmitted beam via the gold grating (Fig. 1(a)). Thus, as the metallic grating has very narrow slits, the structure is highly reflective except at the resonance wavelength λ_R where a sharp transmission peak is obtained thanks to this guided-mode resonance effect. The transmission peak depends mainly on the grating period d and the waveguide thickness t_d. The thickness of the silicon nitride film t_d is set so that the structure exhibits a single transmission peak at normal incidence. For the 1D structure, the metal thickness t_m is chosen sufficiently large and the slit width w sufficiently small to reflect all the transverse electric polarized waves: light is transmitted only when the magnetic field is parallel to the slits [22, 23]. For the 2D structures, the transmission occurs for both polarizations via the slits parallel to the magnetic field.

 

Fig. 1 Bandpass filters based on sub-wavelength metallic gratings deposited on a free-standing dielectric layer (a) Guided mode transmission mechanism. t_m=100 nm, t_d=650 nm, w =200 nm; (b) 1D grating with d_x =2110 nm; 2D grating with rectangular patterns: the grating period are d_x =2110 nm and d_y = 3000 nm; 2D grating with square patterns: d_x = d_y =2110 nm).

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2.1. Fabrication process

The structures have been fabricated following the process described in Ref. [10]. The silicon nitride layer is deposited on a silicon substrate by plasma enhanced chemical vapor deposition. Electron-beam lithography (with a poly(methyl methacrylate) resin), Cr/Au deposition and a lift-off process are used to obtain the grating. The thin chromium layer (1 nm-thick) is used to improve the adhesion of gold on silicon nitride. Last, the substrate is chemically etched on the back-side over a 3mm × 3mm window to leave the grating and the silicon nitride film free-standing. Nominal geometric parameters for the 1D structure are w =200 nm, t_m =100 nm, t_d =650 nm and d =2110 nm. The 2D structures were fabricated with the same nominal geometric parameters as the 1D structure for w, t_m and t_d. For the 2D structure with the rectangular patterns, the grating periods are d_x =2110 nm and d_y =3000 nm whereas for the square patterns, d_x = d_y =2110 nm.

2.2. Optical characterization at normal incidence

Measured transmission spectra of the three structures are shown in Fig. 2 for normal incidence. They have been measured with a Fourier-transform infrared spectrometer (spectral resolution of 5 cm⁻¹) [24]. Figure 2(a) represents the transmission through the 1D structure for the two polarization states. It confirms that the light is transmitted only when the magnetic field, H, is parallel to the slits. The transmission peak reaches 78% at 2.97 μm, which represents an eight-fold enhancement compared to the geometrical transmission of the grating ( $\frac{w}{d}\approx 0.1$). The rectangular pattern structure exhibits two distinct peaks, at 3.12 μm and 3.96 μm, with respect to the polarization. Spectral position can be selected with the use of a polarizer (Fig. 2(b)). As light can be transmitted only when the magnetic field is parallel to the slits (Fig. 2(a)), a 2D structure can be considered as two crossed-1D structures. The square patterns transmission results are shown in Fig. 2(c). This structure is a particular case of rectangular patterns structure where the two transmission peaks coincide. Thus a polarization-independent behavior, with a single peak at 3.12 μm is obtained (component equivalent to two identical crossed-1D structures).

 

Fig. 2 Transmission spectra measured at normal incidence. The light is polarized with the H field parallel to the x-axis (dashed dark line) or parallel to the y-axis (red line). (a) 1D structure ; (b) 2D structure with rectangular patterns; (c) 2D structure with square patterns. Insets: Scanning electron microscope images of the samples (right), and wavevectors of the propagative diffracted orders in the SiNx layer at the resonance wavelength (left).

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Fabricated with identical d_x, the peak position of the three structures (red solid curves) should be nearly identical. We attribute the small discrepancies between the 1D and 2D cases to different SiN_x thicknesses, due to slightly different fabrication conditions.

3. Angular behavior and mechanism of guided-mode resonances

3.1. 1D structure

Figure 3, shows the absolute transmission intensity diagrams T (σ,k_//) of the 1D structure, where σ = 1/λ is the wavenumber, and k_// is the component of incident wavevector parallel to the (x0y)-plane. Depending on whether the rotation axis of the sample is y or x, we have respectively: $k_{//}=k_x^{(0)}=2\pi \mathit{sin}({\theta }_x)/\lambda$ or $k_{//}=k_y^{(0)}=2\pi \mathit{sin}({\theta }_y)/\lambda$ (see sketches of Fig. 3 for (xy) orientation). Figures 3(a) and 3(b) show the measured transmission diagrams of the 1D structure. Transmission spectra have been determined experimentally by angle-resolved measurements, performed with a Fourier-transform infrared spectrometer (spectral resolution of 5 cm⁻¹ and beam convergence angle of ±0.5° [24]). The incidence angles θ_x (or θ_y) range from 0° to 40° in 0.5° increments. Figures 3(c) and 3(d) present the simulation of the transmission calculated with RCWA modal method [21] with a SiN_x refractive index of 2, and a Drude model for the refractive index of gold: ɛ(λ) = 1 − [(λ_p/λ + iγ)λ_p/λ]⁻¹ with λ_p = 159 nm and γ = 0.0077. Calculated and experimental transmission diagrams are in qualitative agreement, reproducing the same features and dispersion properties. The small discrepancies (position and width of transmission resonances) are attributed to opto-geometrical parameters. It will be presented elsewhere after further investigation.

 

Fig. 3 Angle-resolved transmission through the 1D structure as a function of the wavenumber, σ =1/λ and of the incident wavevector. (a) and (b) Measurements. (c) and (d) Calculations. (a) and (c) The rotation axis is parallel to the H field: the incident wavevector is $k_x^{(0)}=2\pi \mathit{sin}({\theta }_x)/\lambda$; (b) and (d) The rotation axis is parallel to the E field: the incident wavevector is $k_y^{(0)}=2\pi \mathit{sin}({\theta }_y)/\lambda$. Solid and dashed lines: calculated dispersion curves related to the guided modes k_gTM and k_gTE respectively (for a continuous gold layer)

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In order to study the resonant transmission mechanisms, we first determine the wavevectors k_gTM(λ) and k_gTE(λ) of the transverse magnetic (TM) and transverse electric (TE) guided mode respectively. TM and TE polarizations are defined according to the guided wave propagation plane in the SiN_x film. In a first-order approximation, these guided modes can be calculated in the case of a thick continuous gold layer deposited on a SiN_x waveguide of thickness t_d. They are calculated by solving a modal equation for each polarization state. In the following, we show that resonances occur when a diffracted order couples to one of these modes [2, 25].

The projection of the (±1) diffracted wavevectors in the (x0y)-plane is:

We first consider the case where the incidence angle θ_x varies only in the (x0z) plane. Hence, $k_y^{(0)}=0$ and Eq. (1) can be simplified as:

These two dispersion equations, Eq. (3) and (4) are represented as solid line in Fig. 3(c). The coupling occurs only with the TM-polarized mode because in this case, the magnetic field remains on the y direction whereas the propagation direction is along the x axis. We can see that the approximation of the continuous gold layer fits quite well: calculated transmission bands are parallel to the dispersion curves. The discrepancies are attributed to the perturbation on the guided waves induced by the slits of the actual gold structure.

In the other configuration, the incidence angle θ_y varies in the (y0z) plane: hence, $k_x^{(0)}=0$ and Eq. (1) can be simplified as:

The agreement between the calculated dispersion curves (in the case of a continuous gold layer) and the calculated transmission band shows that the transmission peaks are due to guided mode resonance effects. Although a subwavelength metallic grating is here concerned, the effects of vertical and horizontal plasmonic resonances, as defined in Ref. [26], have no influence on the high transmission of the component. On the one hand, grating is too thin to exhibit vertical plasmonic resonances (see Ref. [27]) and is not in an optically symmetric environment as in Ref. [27, 28]. On the other hand, the horizontal surface plasmon dispersion curves don’t fit the calculated transmission band (see Ref. [10], where Rayleigh anomalies dispersion curves, very near form the SPP dispersion curves, are shown). The metallic grating here aims at obtaining a structure highly reflective, except at the resonance wavelength λ_R.

3.2. 2D structures

Figure 4(a) and 4(b) represent experimental optical transmission diagrams T(σ,k_//) obtained with a 2D-square pattern. The rotation axis of the sample is y or x (see sketches). They are similar to the diagrams of the 1D structure (Fig. 3).

 

Fig. 4 Angle-resolved transmission measurements through a structure with 2D square patterns as a function of σ=1/λ, the wavenumber, and of the incident wavevector. (a) The incident wavevector is $k_x^{(0)}=2\pi \mathit{sin}({\theta }_x)/\lambda$; (b) The incident wave vector is $k_y^{(0)}=2\pi \mathit{sin}({\theta }_y)/\lambda$.

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Experimental transmission diagrams of 2D-rectangular component represented on Fig. 5 are also a combination of transmission diagrams of 1D structures with periods of d_x =2110 nm and d_y =3000 nm: Fig. 5(a) and 5(b) are similar to Fig. 3(a) and 4(a), except that Fig. 5(b) has a shifted resonance peak at normal incidence. Indeed in this case, the magnetic field H is parallel to the slits separated by a period of 3000 nm. This confirms the similarity between 1D and 2D structures, even at oblique incidence. These results highlight the main diffracted orders involved in the resonance process, according to the magnetic field orientation. In Fig. 2, the colors of the arrows show the orders which are mainly diffracted according to the magnetic field orientation. This is confirmed by calculations: for the square pattern, diffraction efficiency in (0, ±1) diffracted orders is 1 to 3 orders of magnitude lower than in the (±1, 0) orders. These calculations prove that diffraction is only efficient in one direction: the one perpendicular to the magnetic field orientation. The similarity between 2D and two-crossed 1D structures results from this.

 

Fig. 5 Angle-resolved transmission measurements through a structure with 2D rectangular patterns as a function of σ=1/λ, the wavenumber, and of the incident wavevector. (a) The H field is parallel to the slits separated by a period of 2110 nm and the incident wave vector is $k_y^{(0)}=2\pi \mathit{sin}({\theta }_y)/\lambda$; (b) The H field is parallel to the slits separated by a period of 3000 nm and the incident wavevector is $k_x^{(0)}=2\pi \mathit{sin}({\theta }_x)/\lambda$.

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4. Conclusion

We have presented three structures which can be used as transmission band-pass filters with different polarization behaviors. The 1D structure is a filter with polarization selectivity, the 2D structure with rectangular patterns is a tunable spectral filter when used in combination with a polarizer (it exhibits two different spectral positions according to the polarization state of the incident light) and the 2D structure with square patterns is polarization independent at normal incidence. These filters have a high efficiency: the experimental transmission peak reaches 78% for the 1D structure (68% for the 2D structures), which represents an eight-fold (seven-fold respectively) enhancement compared to the geometrical transmission of the grating. The spectral position of these filters can be adapted over a wide wavelength range by changing the grating period. Moreover, for a given period, the bandwidth and transmission maximum can be adjusted with the slit width. The 1D structure has different angular sensitivity, depending on the orientation of the incident plane relative to the slits. We explained this particular behavior with a simplified model based on the guided mode resonance mechanisms. In particular, diffracted orders in the SiN_x layer have different expressions depending on whether the plane of incidence is parallel or perpendicular to the slits. Besides, we highlighted the similarities between 2D structures and two crossed-1D structures. The flexibility of polarization and spectral behaviors allows to adapt these filters to numerous applications, such as sensing, dense wavelength division multiplexing networks or multispectral imaging [29].