1. Introduction

Optical filters that selectively reflect or transmit input light are ubiquitously used in spectroscopic measurements, laser systems, optical communications, displays, hyperspectral imaging and many other practical applications. As optical filters, various resonant diffractive structures are widely used, including thin-film structures [1], subwavelength-period guided-mode resonance gratings (GMRGs) [2–7], and “combined” multilayer structures containing GMRGs embedded into multilayer thin-films [7–10].

The simplest thin-film reflecting filter is the Bragg grating (BG), which corresponds to a periodic quarter-wave layer system. A phase-shifted Bragg grating (PSBG) consists of two symmetric BGs separated by a half-wave spacer (“defect” or “phase-shift” layer) and enables obtaining a narrow transmission passband centered at the design wavelength [1]. In the early works dedicated to the GMRG-based filters, only the filters operating in reflection were proposed [2–4,8]. The utilization of combined multilayer structures (i.e. GMRGs in combination with multilayer thin-film structures) made it possible to simplify the filter configuration and decrease the required number of layers. The first transmission bandpass filters having such configuration were presented in 1995 [9]. More advanced GMRG-based transmission filters include only a few thin-film layers or even a single-layer GMRG [6,7,10].

As a rule, transmission filters based on PSBGs or GMRGs enable obtaining a Lorentzian transmittance peak. At the same time, the Lorentzian line shape does not always fit the design requirements. For many applications, achieving a nearly rectangular transmittance spectrum with flat top and steep slopes is highly desirable [1,11,12]. Currently, flat-top narrow-band filters operating in transmission are mostly based on thin-film Bragg structures containing multiple defects (i.e. phase-shift spacers breaking the periodicity of the structure) [1]. In such structures, several leaky modes localized at the phase-shift regions may exist. The existence of several modes makes the line-shape of the resonance significantly more complex and makes it possible to achieve a flat-top transmittance peak by optimizing the parameters of the phase-shift spacers [1,11]. A similar approach can be applied to the design of GMRG-based transmission filters. In particular, Ref. [12] considers the design of flat-top bandpass filters consisting of several GMRGs separated by phase-shift spacers. Let us note that, to the best of our knowledge, no theoretical description of the formation of a flat-top transmittance peak has been presented in the published works. The parameters of the multilayer thin-film structures and GMRG-based structures are, as a rule, determined using numerical optimization.

In this work, we present a theoretical analysis of the spectra of composite structures consisting of several identical resonant diffractive structures (RDSs) with a Lorentzian transmittance profile. Using the scattering matrix formalism, we analytically demonstrate that composite structures comprising two or four RDSs optically implement the Butterworth filters of the order two or three, respectively, and enable achieving flat-top transmittance spectra with steep slopes and no sidelobes. In addition, we show that composite structures consisting of three or four second-order Butterworth filters approximate the Butterworth filters of the order four or five, respectively.

The obtained theoretical results are confirmed by rigorous numerical simulations. As the resonant building block of the composite structures, a simple three-layer resonant structure is utilized. The presented simulation results of the designed composite structures confirm the formation of flat-top transmittance peaks, the shape of which fully agrees with the derived theoretical description. Moreover, an exceptionally simple mechanism of controlling the transmittance peak width is demonstrated, which consists in changing the thicknesses of the layers of the initial resonant structure.

2. Theoretical analysis of the spectra of composite structures

In this section, we consider composite structures consisting of several identical resonant dielectric structures. As such structures, guided-mode resonance gratings, phase-shifted Bragg gratings or other resonant thin-film structures can be used. For example, Fig. 1(c) shows a composite structure comprising two so-called W-structures (or W-type waveguides) [13–16]. The W-structure is one of the simplest resonant structures supporting leaky modes. W-structures consist of three dielectric layers: a high-index core layer and two low-index cladding layers [Fig. 1(a)]. The structure is supposed to be located in a high-index symmetric dielectric environment. The term “W-structure” is due to the fact that the refractive index profile of the structure resembles the letter “W” [see Fig. 1(b)]. Such a refractive index profile enables resonant tunneling of obliquely incident light through the structure caused by the excitation of a leaky mode localized in the central core layer.

 

Fig. 1. Three-layer W-structure (a), refractive index profile of the W-structure (b), and a composite structure consisting of two W-structures (c).

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Let us emphasize that the theoretical results obtained below in this Section are quite general and can be applied not only to the thin-film structures, but also to composite structures consisting of resonant subwavelength diffraction gratings. Some comments on such grating-based structures are given at the end of this section.

Let us suppose that the lossless RDSs constituting the composite structure have a Lorentzian transmittance spectrum. In this case, the complex transmission and reflection coefficients of the RDS considered as functions of the angular frequency $\omega$ of the incident light can be approximated by the following expressions [17,18]:

For the analysis of composite structures, it is convenient to describe the optical properties of an RDS at a fixed incidence angle $\theta$ by the $2 \times 2$ scattering matrix [18–21]

The matrix ${{\mathbf {S}}_1}(\omega )$ relates the complex amplitudes of the plane waves ${i_u}(\omega )$ and ${i_d}(\omega )$, which impinge on the RDS from the superstrate and the substrate regions, respectively, with the complex amplitudes of the reflected and transmitted waves $r(\omega )$ and $t(\omega )$:

Let us consider a composite structure consisting of two identical RDSs described by the scattering matrix of Eq. (2) and separated by a phase-shift layer with the thickness ${l_1}$. In this case, the scattering matrix of the composite structure can be expressed through the matrix ${{\mathbf {S}}_1}(\omega )$ in the form [19,22]

By using the definition of the Redheffer product given by Eq. (5) in Eq. (4), one can obtain the scattering matrix of the composite structure in the following form:

2.1. Butterworth filter of the second order

A more complex form of the coefficients ${r_2}(\omega )$ and ${t_2}(\omega )$ gives an opportunity to control the shapes of the reflectance and transmittance spectra. Let us consider a special case when the thickness of the phase-shift layer ${l_1}$ satisfies the condition

 

Fig. 2. Locations of the poles of the designed filters (crosses) and of the Butterworth filters of orders 1–5 (small circles). Dashed lines show the circles with radii ${\Delta _1} = {\mathop{\rm Im}\nolimits}\ {\omega _p},{\Delta _2},{\Delta _4},{\Delta _6}$, and ${\Delta _8}$.

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2.2. Butterworth filter of the third order

Next, let us consider a composite structure consisting of four RDSs and corresponding to the superposition of two second-order Butterworth filters. The scattering matrix of such structure has the form

It is interesting to note that the obtained third-order Butterworth filter consists of four resonant structures. One could expect that such a structure should support four modes resulting from the coupling of the modes of the four RDSs. However, only three poles are present in Eq. (18). Computations show that the fourth pole actually exists when the distance ${l_2}$ is arbitrary, however, once ${l_2}$ tends to the value defined by Eq. (16), the fourth pole and a corresponding zero both move to infinity and, in the limiting case, cancel each other out, thus “disappearing” from Eq. (17).

2.3. Approximations of the fourth- and fifth-order Butterworth filters

Unfortunately, at $N > 2$, the composite structures comprising N second-order Butterworth filters (or 2N RDSs) do not exactly implement higher-order Butterworth filters. However, the transmission coefficient of the composite structure consisting of three second-order Butterworth filters is a good approximation of the Butterworth filter of the fourth order. Indeed, the scattering matrix of this structure has the form

Similarly, one can consider a composite structure consisting of four second-order Butterworth filters:

Let us note that the theoretical description presented in this section can be applied not only to thin-film structures, one of which will be considered in the next section, but also to composite structures containing diffraction gratings. The latter, however, requires the fulfillment of several additional conditions. Firstly, the grating period should be small enough so that only the 0^th diffraction orders are non-evanescent both outside the structure and in the spacer layers. Secondly, the spacer layers have to be thick enough to make the near-field interaction between the adjacent gratings negligibly small. Note that it can always be achieved by choosing sufficiently large values of the integers ${m_1}$ and ${m_2}$ in Eqs. (10) and (16).

3. Numerical investigation of composite W-structures

Let us illustrate the theoretical results obtained in the previous section by the results of rigorous numerical simulations of the considered composite structures. As the initial RDS, let us use the W-structure described above [Fig. 1(a)]. Let us remind that this structure consists of three layers: a core layer with the refractive index ${n_{\textrm{core}}}$ and thickness ${h_{\textrm{core}}}$ and two cladding layers with the refractive index ${n_{\textrm{cl}}} < {n_{\textrm{core}}}$ and thickness ${h_{\textrm{cl}}}$. The structure is located in a symmetric dielectric environment with the refractive index ${n_{\textrm{out}}}$. If ${n_{\textrm{out}}} > {n_{\textrm{cl}}}$, the W-structure supports leaky eigenmodes described by the dispersion relation given in Ref. [13]. For a leaky mode, the eigenfrequency ${\omega _p}$ is complex. For the considered structure, the quantity ${\Delta _1} = |{{\mathop{\rm Im}\nolimits}\ {\omega_p}} |$, which determines the width of the resonance, is an exponentially decaying function of the ratio $\gamma = {{{h_{\textrm{cl}}}} \mathord{\left/ {\vphantom {{{h_{\textrm{cl}}}} {{h_{\textrm{core}}}}}} \right.} {{h_{\textrm{core}}}}}$ [13]. At a large cladding thickness ${h_{\textrm{cl}}}$ (at $\gamma \gg 1$), the modes of the W-structure can be approximately described by the dispersion relation of a slab waveguide with the same core layer as in the W-structure and semi-infinite claddings with the refractive index ${n_{\textrm{cl}}}$.

As an example, Fig. 3 shows the reflectance and transmittance spectra of the W-structure. The spectra in Fig. 3, as well as all the simulation results presented below, were calculated using a numerically stable implementation of the enhanced transmittance matrix approach [24] for an obliquely incident TE-polarized plane wave. The following values of the material and geometrical parameters were used: ${n_{\textrm{core}}} = 2.2698\;({\textrm{Ti}{\textrm{O}_2}} )$, ${n_{\textrm{cl}}} = 1.457\;({\textrm{Si}{\textrm{O}_2}} )$, ${n_{\sup }} = 1.7786\;({\textrm{SF11}} )$, ${h_{\textrm{core}}} = 40\,\textrm{nm}$. The simulations were performed for three different thicknesses of the cladding layers: ${h_{\textrm{cl},1}} = 700\,\textrm{nm}$, ${h_{\textrm{cl},2}} = 800\,\textrm{nm}$, and ${h_{\textrm{cl},3}} = 1000\,\textrm{nm}$. The obtained spectra have a Lorentzian line shape and are well described by Eq. (1). Let us mention that the spectra in Fig. 3 corresponding to different cladding thicknesses were calculated at different incidence angles: ${\theta _1} = 60.707^\circ$ (${h_{\textrm{cl},1}} = 700\,\textrm{nm}$), ${\theta _2} = 60.713^\circ$ (${h_{\textrm{cl},2}} = 800\,\textrm{nm}$), and ${\theta _3} = 60.715^\circ$(${h_{\textrm{cl},3}} = 1000\,\textrm{nm}$). At the angles ${\theta _i}$, $i = 1,2,3$, the reflection coefficients of the W-structures with the cladding layer thicknesses ${h_{\textrm{cl},i}}$, $i = 1,2,3$ vanish at the same frequency $\omega = {\omega _0} = 2.811 \cdot {10^{15}}\,{\textrm{s}^{ - 1}}$ (free-space wavelength ${\lambda _0} = 670\,\textrm{nm}$).

 

Fig. 3. Reflectance (а) and transmittance (b) spectra of the W-structure at different thicknesses of the cladding layers: ${h_{\textrm{cl},1}} = 700\,\textrm{nm}$ (dashed blue line), ${h_{\textrm{cl},2}} = 800\,\textrm{nm}$ (dash-dotted red line), and ${h_{\textrm{cl},3}} = 1000\,\textrm{nm}$ (solid yellow line).

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As noted above, an increase in the cladding layer thickness ${h_{\textrm{cl}}}$ at a fixed value of ${h_{\textrm{core}}}$ leads to a decrease in the resonance width ${\Delta _1} = |{{\mathop{\rm Im}\nolimits}\ {\omega_p}} |$. This effect is clearly visible in Fig. 3. In particular, at the thickness ${h_{\textrm{cl}}} = {h_{\textrm{cl,3}}} = 1000\,\textrm{nm}$, the resonance width is almost 20 times less than at ${h_{\textrm{cl}}} = {h_{\textrm{cl,1}}} = 700\,\textrm{nm}$.

As the building block for constructing the composite structures, we will use the W-structure with the cladding thickness ${h_{\textrm{cl}}} = {h_{\textrm{cl,1}}} = 700\,\textrm{nm}$. For this structure, we rigorously calculated the complex frequency of the leaky eigenmode ${\omega _p} = 2.811 \cdot {10^{15}} - 4.2 \cdot {10^{12}}\textrm{i}\,\ {\textrm{s}^{ - 1}}$ using the method presented in Ref. [25]. According to Eq. (1), the reflection coefficient vanishes at the frequency $\omega = {\omega _0} = {\mathop{\rm Re}\nolimits}\ {\omega _p} = 2.811 \cdot {10^{15}}\,{\textrm{s}^{ - 1}}$. The half-width of the resonant transmittance peak (reflectance dip) at the 0.5 level amounts to ${\Delta _1} = |{{\mathop{\rm Im}\nolimits}\ {\omega_p}} |= 4.2 \cdot {10^{12}}\textrm{i}\,\ {\textrm{s}^{ - 1}}$ (∼1 nm-wide wavelength interval).

Figure 4 shows the rigorously calculated reflectance and transmittance spectra of the composite structures consisting of N stacked W-structures for several N values (solid lines). For comparison, dotted lines show the spectra of the initial W-structure. Figures 4(а) and 4(b) show the spectra of the composite structures consisting of two and four W-structures, respectively. The thicknesses of the spacer layers between the W-structures are determined by Eqs. (10) and (16). In this case, according to the results obtained in Section 2, the complex transmission coefficients of the composite W-structures coincide with the transfer functions of the Butterworth filter of the second and third orders. The transmittance spectra ${T_N}(\omega )= {|{{t_N}(\omega )} |^2}$ are described by Eqs. (14) (at $N = 2$) and (19) (at $N = 4$). The presented simulation results [Figs. 4(a) and (b)] are in full accordance with the theoretical description presented in Section 2. Indeed, the squared moduli of the transfer function of the Butterworth filters of Eqs. (14) and (19), which are shown with dashed lines in Fig. 3, almost completely coincide with the calculated transmission spectra. Let us remind that the poles of the designed filters, as it is shown in Fig. 2, coincide with the poles of the corresponding Butterworth filters. Note that the widths of the resonant features (transmittance peaks and reflectance dips) of the composite structures are less than the resonance width of the initial W-structure by $\sqrt 2$ [Fig. 4(a)] or by 2 [Fig. 4(b)] times.

 

Fig. 4. Reflectance (red) and transmittance (black) of the composite W-structures with (solid lines) at $N = 2$ (а), $N = 4$ (b), $N = 6$ (c), and $N = 8$ (d). Dotted lines show the spectra of the initial W-structures. Dashed green lines show the squared moduli of the transfer functions of the corresponding Butterworth filters.

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Figures 4(c) and 4(d) show the spectra of the composite structures consisting of six and eight W-structures and described by Eqs. (20) and (24), respectively. The transmittance spectra of these structures are also in good agreement with the theoretical spectra of Eq. (13) at $M = 4$, $M = 5$ and $\Delta = 0.42{\Delta _1}$. The latter are shown in Figs. 4(c) and (d) with dashed lines. However, the difference between the calculated and theoretical spectra is slightly larger in Figs. 4(c) and (d) than in Figs. 4(a) and (b). In particular, the transmittance peak in Fig. 4(d) is not entirely flat-top and has slight variations with the amplitude of about 0.015. This is due to the fact that the composite structures of Eqs. (20) and (24) only approximately correspond to the Butterworth filters of the fourth and fifth orders. This is supported by Fig. 3, which shows that the poles of the designed filters are slightly shifted with respect to the corresponding poles of the “exact” Butterworth filters. Despite these differences, the composite structures with $N = 6$ [Fig. 4(c)] and $N = 8$ [Fig. 4(d)] still enable high-quality almost flat-top filtering profiles with steep slopes and low sidebands.

The width of the transmittance peak of the composite structure is determined by the width of the resonance of the initial W-structure. In the example considered above (${h_{\textrm{cl}}} = {h_{\textrm{cl},1}} = 700\,\textrm{nm}$), the half-width of the transmittance peak of the initial W-structure amounts to ${\Delta _1} = |{{\mathop{\rm Im}\nolimits}\ {\omega_p}} |= 4.2 \cdot {10^{12}}\textrm{i}\,\ {\textrm{s}^{ - 1}}$, which corresponds to an approximately 1 nm-wide wavelength range. For the considered composite structures, the half-width of the peak changes from ${{{\Delta _1}} \mathord{\left/ {\vphantom {{{\Delta _1}} {\sqrt 2 }}} \right.} {\sqrt 2 }}$ (0.71 nm-wide wavelength range) at $N = 2$ [Fig. 4(a)] to $0.46{\Delta _1}$ (0.46 nm-wide wavelength range) at $N = 8$ [Fig. 4(d)]. For the initial W-structure, the resonance width ${\Delta _1} = |{{\mathop{\rm Im}\nolimits}\ {\omega_p}} |$ is an exponentially decaying function of the cladding thickness ${h_{\textrm{cl}}}$ [13]. Therefore, by changing this thickness, one can obtain an arbitrarily narrow transmittance peak (Fig. 3). Consequently, using a composite structure consisting of several identical RDSs, it is possible to achieve a flat-top peak with a required small width. As an example, Fig. 5 shows the rigorously calculated transmittance spectra of the composite structures with $N = 4$ and $N = 6$ consisting of W-structures with different thicknesses of the cladding layers ${h_{\textrm{cl},1}} = 700\,\textrm{nm}$, ${h_{\textrm{cl},2}} = 800\,\textrm{nm}$ and ${h_{\textrm{cl},3}} = 1000\,\textrm{nm}$. Let us note that the transmittance spectra at ${h_{\textrm{cl},1}} = 700\,\textrm{nm}$ shown in Fig. 5 coincide with the transmittance spectra in Figs. 4(b) and (c), where a wider frequency range is shown. From Fig. 5, it is evident that a decrease in the resonance width caused by an increase in the ${h_{cl}}$ value is in accordance with Fig. 3. In particular, the width of the resonance of the W-structure in Fig. 3 at ${h_{\textrm{cl}}} = {h_{\textrm{cl},3}} = 1000\,\,\textrm{nm}$ amounts to ${\Delta _1} = 2.15 \cdot {10^{12}}\textrm{i}\,\ {\textrm{s}^{ - 1}}$ (0.051 nm), which is approximately 20 times less than at ${h_{\textrm{cl}}} = {h_{\textrm{cl},1}} = 700\,\textrm{nm}$. Similarly, the half-widths of the transmittance peaks of the composite structures at ${h_{\textrm{cl}}} = {h_{\textrm{cl},3}} = 1000\,\,\textrm{nm}$ are also approximately 20 times less than at ${h_{\textrm{cl}}} = {h_{\textrm{cl},1}} = 700\,\textrm{nm}$ and equal $1.08 \cdot {10^{11}}\textrm{i}\,\ {\textrm{s}^{ - 1}} $(0.026 nm-wide wavelength range) at $N = 4$ and $9.96 \cdot {10^{10}}\textrm{i}\,\ {\textrm{s}^{ - 1}} $(0.024 nm-wide wavelength range) at $N = 6$.

 

Fig. 5. Transmittance spectra of the composite W-structures with $N = 4$ (a) and $N = 6$ (b) calculated for different thicknesses of the cladding layers: ${h_{\textrm{cl},1}} = 700\,\textrm{nm}$ (dashed blue line), ${h_{\textrm{cl},2}} = 800\,\textrm{nm}$ (dash-dotted red line), and ${h_{\textrm{cl},3}} = 1000\,\textrm{nm}$ (solid yellow line).

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It is worth mentioning that due to the fact that the refractive index of the medium surrounding the W-structure ${n_{\textrm{out}}}$ is greater than unity, the practical implementation of the filters based on composite W-structures requires the utilization of two high-index prisms for incoupling and outcoupling of the incident optical beam. This, in general, complicates the integration of the composite W-structures. In our opinion, properly designed diffraction gratings concentrating the incident radiation in a single diffraction order could also be used as couplers for the W-structures. This question might be addressed in a future work.

A promising alternative to the “conventional” thin-film W-structures are the W-structures implemented in the integrated geometry and operating not with the free-space radiation, but with guided waves. Indeed, an efficient on-chip analogue of the W-structure integrated into a slab waveguide was proposed in a recent work by the present authors [16]. In the on-chip structure, the W-shaped refractive index profile of the resonant structure is encoded by the effective refractive index of the guided modes by means of modulating the thickness of the waveguide. In this planar configuration, the W-structure consists of just two grooves on the surface of the waveguide and does not require any dedicated coupling elements [16]. Therefore, we believe that the composite W-structures considered above can efficiently be implemented in the planar geometry.

4. Conclusion

In the present work, we theoretically investigated resonant properties of composite structures consisting of several identical resonant diffractive structures (RDSs) with a Lorentzian transmittance spectrum. Using the scattering matrix formalism, we have analytically demonstrated that the complex transmission coefficients of the composite structures consisting of two and four RDSs coincide with the transfer functions of the Butterworth filters of the second and third orders, respectively. In addition, it has been shown that the composite structures consisting of three and four second-order Butterworth filters approximate fourth- and fifth-order Butterworth filters. The obtained theoretical results are fully confirmed by the results of rigorous numerical simulations of composite W-structures. The simulation results demonstrate the formation of flat-top transmission spectra, the shape of which is in good agreement with the presented theoretical description. Moreover, it is possible to control the width of the flat-top transmission peak by changing the thickness of the cladding layers of the initial W-structure. The presented examples demonstrate the possibility of obtaining flat-top transmission peaks with a significantly subnanometer size. A promising direction for the future research is the design and investigation of on-chip composite W-structures integrated into a slab waveguide. In this case, a simple two-groove structure proposed in [16] could be used as the building block for the composite on-chip filters.

It is important to note that the obtained theoretical results can be applied not only to the design of thin-film composite structures considered in the present work, but also to the design of composite filters consisting of resonant subwavelength diffraction gratings. The investigation of such structures will be the subject of a separate work.

In addition to the applications in spectral filtering, the proposed composite structures can be used for performing the transformations of optical signals. In particular, the complex reflection coefficients of the composite structures consisting of two and four RDSs have real-valued zeros of the order two and three, respectively. This makes it possible to utilize such structures for the optical computation of second- and third-order derivatives of the envelope of the incident optical pulses and beams in the reflection regime [26].