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Two-groove narrowband transmission filter integrated in a slab waveguide

We propose a simple integrated narrowband filter consisting of two grooves on the surface of a slab waveguide. Spectral filtering is performed in transmission at oblique incidence due to excitation of an eigenmode of the structure localized at a ridge cavity between the grooves. For the considered parameters, strictly zero reflectance and unity transmittance are achieved at resonant conditions. Width and location of the transmittance peak can be controlled by changing the widths of the grooves and of the ridge, respectively. The proposed filter may find application in waveguide-integrated spectrometers.


INTRODUCTION
Spectral and spatial filters that selectively transmit or reflect input light are indispensable in various optical devices including spectrum analyzers, image sensors, wavelength demultiplexers, liquid crystal displays and remote sensing systems. In a wide class of planar (integrated) optoelectronic systems, spectral or spatial filtering is performed in a slab waveguide [1][2][3][4][5][6][7]. Such geometry corresponds to the "insulator-on-insulator" platform and is suitable for the creation of fully integrated optical devices. In Refs. [1,[8][9][10][11][12], planar Bragg gratings (BGs) and phase-shifted Bragg gratings (PSBGs) for spectral filtering of optical radiation propagating along the waveguide were proposed. A planar BG corresponds to a single-mode slab waveguide with an appropriate periodically corrugated surface and acts as a reflection filter centered at the Bragg wavelength. A change in the waveguide thickness leads to a change in the effective refractive index of the guided mode and makes it possible to encode the required refractive index distribution analogous to a free-space BG. A planar PSBG consists of two or more symmetric BGs separated by defects (phase-shift regions) and enables to obtain a narrow transmission peak at the center of the stopband [1,11,12].
Recently, novel structures that can be applied in integrated spectrometers, the so-called digital planar holograms (DPHs), were proposed [3][4][5][6]. DPHs correspond to a sophisticated superposition of narrowband curvilinear BG-based reflectors fabricated on the surface of the waveguide layer. The curvilinear (e.g. elliptic) shape of the reflectors enables to perform spatiallyresolved spectral filtering: radiation with different wavelengths is focused at different points. It is important to note that the methods for designing DPHs are a know-how of the authors and are not described in detail in the published works.
In this work, we propose a planar spectral filter consisting of two grooves on the surface of a slab waveguide and operating in transmission. In contrast to the PSBG-based filters, our filter has a much simpler geometry and is substantially more compact.

GEOMETRY OF THE STRUCTURE
The proposed filter is a planar version of dielectric three-layer resonant structure [ Fig. 1(a)], which was studied in Ref. [13] for achieving large optical field enhancement over a large volume. A similar dielectric-metal-dielectric structure has been demonstrated to tunnel light through metal layers [14]. The structure in Fig. 1(a) consists of a central layer (refractive index wg n , thickness wg h ) and two identical cladding layers (refractive index cl wg nn  , thickness cl h ). If the refractive index of the surrounding medium sup n satisfies the condition sup cl nn  , at a certain angle of incidence the structure may exhibit unity resonant transmission due to coupling of the incident light to a waveguide mode localized at the central layer, which in this case acts as a resonant cavity [13]. Let us note that the unity resonant transmission condition can be strictly satisfied only if the resonant structure has a horizontal symmetry plane sp zz  [ Fig. 1(a)] [15,16]. In this case, the structure corresponds to a so-called W-type waveguide [17]. At significantly large cladding thickness cl h , the leaky mode of the W-type structure can be approximately described by the dispersion relation of the modes of a slab waveguide, which becomes exact at infinite cl h . In this case, the excitation of the mode of the structure of Fig. 1(a) will occur in the vicinity of the frequency 0   and the tangential component ,0 xx kk  of the wave vector, which satisfy the dispersion relation of a slab waveguide [18]. Therefore, at resonant conditions the reflection coefficient will vanish at certain  and x k values close to 0  and ,0 x k , respectively. It is worth noting that the quality factor of the resonance (the width of the resonant transmission peak) is determined by the thickness of the cladding layer cl h [13]. The presented analysis allows us to expect that the structure presented in Fig. 1(a) may be used as a spectral or spatial (angular) narrowband transmission filter. However, despite the simple geometry of the structure, it has several disadvantages concerning its practical use. In particular, due to the condition sup cl nn  , such a filter requires the use of two high-index prisms for incoupling and outcoupling of the incident optical beam, and thus cannot be highly integrated. At the same time, this structure can serve as a prototype for designing a simple planar filter integrated in a slab waveguide. The considered waveguide has the following parameters: dielectric permittivity and the thickness of the core layer f  and inc w , respectively, and the dielectric permittivities of the claddings u  and l  . As a planar analogue of the three-layer structure of Fig.   1(a), we propose to use the structure consisting of two grooves on the waveguide surface shown in Fig. 1(c). The lengths of the grooves and of the ridge between them are equal to the thicknesses of the layers of the conventional structure for free-space radiation.
Let us denote by

RESULTS AND DISCUSSION
Let us numerically study the performance of the proposed twogroove planar filter (TGPF). In all the simulations, the following parameters were used: free-space wavelength  (Fig. 2). As shown below, at these groove widths the angular and spectral widths of the resonant peaks differ by approximately 10 times. The width of the ridge is fixed and equals 100 nm wg h  . The spectra were calculated using an in-house implementation of the aperiodic rigorous coupled-wave analysis (aRCWA) technique [19]. RCWA, also called the Fourier modal method, is an established numerical technique for solving Maxwell's equations. The angular spectra in Fig. 2(a)     are evanescent. Thus, the energy of the incident TE-polarized mode will be divided only between two modes (reflected and transmitted) having the same polarization. Due to this, strictly zero reflectance and unity transmittance are achieved in the structure at resonant conditions [15].
The wavelength spectra in Fig. 2(b) Fig. 2(a), the inset in Fig. 2(b) shows the FWHM of the peak vs. the width cl h . To further investigate the resonances associated with the transmittance peaks in Fig. 2 . Since the spectra in Fig. 2 Figure 3 shows the distributions of the  Figs. 3(a) and 3(b). At 630 nm, a leaky eigenmode is excited at the ridge cavity between the grooves (Fig.  3(c)), and resonant unity transmittance is achieved. The incident and transmitted waves are not visible in Fig. 3(c) due to the field enhancement at the cavity. It is worth noting that the field distribution in Fig. 3(c) is visually identical to the field distribution of the leaky eigenmode of the structure. In the design of micro-spectrometers and hyperspectral sensors, so-called linear variable filters (LVFs) are widely used. LVF is a bandpass filter that is intentionally wedged in one direction, providing a position-dependent dispersive optical element. Typically, LVF corresponds to a Fabry-Pérot etalon with varying thickness between two Bragg reflectors [21][22][23][24]. On the basis of the proposed TGPF, a waveguide-integrated LVF can be created, in which the spectral shift of the transmittance peak can be achieved by changing one of the parameters of the structure, e.g. the ridge width wg h . Figure 4 shows the possibility to control the location of the peak by changing the wg h value. In the considered example, the peak is shifted by approximately 7 nm if the wg h value is changed by 2.5 nm. The given examples demonstrate that the proposed TGPF enables obtaining a narrow transmission peak with sub-nm spectral width and Lorentzian lineshape. In practical applications, achieving a nearly rectangular transmittance peak with a flat transmission band and steep slopes is highly desirable [11]. In PSBG-based filters, this problem is solved by introducing several properly chosen phase-shift regions. A similar approach can be applied for the considered TGPF. As a simplest example, let us consider a filter composed of two concatenated TGPFs with 100 nm wg h  and 600 nm cl h  separated by an additional phase-shift ridge with the width 198 nm (inset in Fig. 5). Following the approach used in [25] for the analysis of reflectance spectra of stacked PSBGs, the width of the phase-shift ridge was chosen so that the reflectance spectrum of the structure possesses a so-called second-order zero. In terms of transmittance spectrum, this leads to a shape of the resonant peak that is significantly closer to a rectangle (Fig. 5).
The calculations [20] show that the concatenated structure supports two closely located modes with the complex wavelengths , which provides the so-called flat-top resonance line shape [26]. The quality factors of the modes of the concatenated structure are twice the quality factor of the mode of the initial structure ( ). As the result, FWHM value for the concatenated structure decreases and in our example amounts to 0.22 nm instead of 0.3 nm for the initial structure. A further improvement in the peak shape can be achieved by increasing the number of concatenated TGPFs similarly to the approach used for PSBG-based filters [11].

CONCLUSION
In summary, in the present work we proposed a simple narrowband transmission filter consisting of two grooves on the surface of a slab waveguide and intended for spectral filtering of optical radiation propagating in the waveguide. The spectral filtering is performed due to resonant excitation of an eigenmode of the structure localized at the cavity corresponding to the ridge between the grooves. On the basis of rigorous numerical simulations, the possibility to control the spectral width and location of the transmittance peak by changing the widths of the grooves and the ridge between them was demonstrated. Moreover, for the considered parameters, there is no unwanted parasitic scattering and polarization conversion in the filter. The proposed TGPF can be used for the creation of planar analogues of linear variable filters for waveguide-integrated spectrometers.