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Abstract
We fabricated a rigid bandpass filter with a broad far-infrared wavelength range of high transmission using a silicon subwavelength structure with a Babinet complementary metamaterial half-mirror pair, despite its apparent light-blocking structure. The rigid one-piece filter was produced by a simple process involving photolithography, dry etching, and deposition, each performed only once. The transmission principle relies on the Fabry-Perot resonance with a metamaterial half-mirror pair that exhibits extraordinary optical transmission due to spoof surface plasmon polaritons. The transmission center wavelength was successfully predicted by the basic equation of Fabry-Perot resonance with an effective medium approximation. In contrast, a narrower bandwidth and a lower minimum transmittance than those predicted from the basic equation were provided by the subwavelength Si structure between the metamaterial mirrors, resulting in enhanced bandpass filter characteristics.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The mid- and far-infrared wavelength ranges, approximately ranging from 2.5 µm to 100 µm [1], is crucial in astronomy and gas sensing due to its suitability for observations of thermal emission and characteristic absorption lines of various molecules [2]. Astronomy benefits from mid- and far-infrared light, which allows for spectroscopy of cosmic dust particles to gain insights into the molecules present, leading to better understanding of planet formation, the interstellar medium, and galaxy evolution [3,4]. In the spectroscopy, optical bandpass filters are essential for accurately isolating radiation between discrete lines [5]. Gas sensing detects, for example, volatile organic compounds and CO2 concentrations, using mid- and far-infrared light. This allows for air quality monitoring and hazard assessment, and assists in breathing diagnostics [6,7]. Optical gas sensors including non-dispersive infrared, photoacoustic spectroscopy, and tunable diode laser absorption spectroscopy use a bandpass filter to selectively detect the absorption lines of the target gas and to reduce the background.
The bandpass filters for these applications require high mechanical and thermal robustness. For spacecraft observations, they must withstand mechanical vibration during launch, and thermal cooling cycles for low thermal background measurements. Currently, multilayer dielectric interference filters and metal mesh filters are the primary types of infrared bandpass filters. A multilayer dielectric filter consists of materials with different refractive indices that can be damaged during thermal cycles because of their different thermal expansion coefficients. A metal mesh filter consists of a woven mesh of metal wires in a specific shape, typically a regular array of square holes or an array of metal squares as its complementary structure [8,9]. The square hole array is called an inductive mesh because it is a metamaterial half-mirror that operates as a low-pass filter, while the square metal array is called a capacitive mesh because it operates as a high-pass filter. Bandpass filters can be created by forming a Fabry-Perot resonator with two metamaterial mirrors [10] or a metal hole array metasurface [11]. A representative hole design with unpolarized transmission properties is the cross-shaped geometry [12,13]. Hollow type metasurface filters, despite its excellent transmission properties, are vibration-sensitive. Metal hole array metasurfaces with tunable characteristics have been reported using a superconductor [14], a phase-change material [15] and a nematic liquid crystal [16]. The design of metasurface bandpass filters is closely tied to materials, fabrication processes, and modulation methods. Achieving narrower bandwidths and more precise resonance frequencies requires a sophisticated design including these factors. Multilayered plasmonic metasurfaces have been designed and fabricated to enable broadband band-pass filters based on couping of electric or magnetic modes [17,18]. A bilayer-metamaterial structure with a square-loop hole periodic array [19] offers a stopband rejection approaching 40 dB while the maximum transmittance remains above 65%. Because the multilayered plasmonic bandpass filters rely on the coupling of plasmonic modes, the relationship between the structure and the optical characteristics is not straightforward. A recently studied, robust bandpass filter for astronomical observations uses two Si substrates, with one having an inductive mesh and the other a capacitive mesh. This filter offers a wide transmission frequency range (200–400 GHz) and high spectral resolution (∼100) [20,21], but requires a precise mirror positioning mechanism.
Here, we demonstrate the fabrication of a robust terahertz bandpass filter by simply depositing Au on the entire surface of subwavelength structures on a single Si substrate [Fig. 1(a)]. The Au deposition enables the spontaneous formation of a Babinet complementary pair [22] of the top metal hole and bottom square metal. Despite completely blocking the light's path perpendicular to the filter, the Babinet complementary metal pair works as a high transmittance far-infrared bandpass filter. The bandpass filter is mechanically and thermally robust due to its integration on a single silicon substrate and the fixed cavity length determined by the subwavelength structure depth. This study shows, for the first time, that a pair of Babinet complementary metamaterial mirrors, placed less than a wavelength apart, offer good transmission properties despite their opaque configuration.
Fig. 1. (a) Schematic drawing of the bandpass filter with period p, side length of an etched square a, and etched depth d. (b) Optical micrograph and SEM image of a fabricated filter. p = 15 µm, a = 12.3 µm, d = 24.4 µm.
2. Experiment
We fabricated a bandpass filter, illustrated in Fig. 1(a), on a 625 ± 15 µm thick, a high-resistivity (>3 kΩ·cm) (100) single-side polished silicon wafer. We formed square holes through photolithography and dry etching using the pseudo-Bosch process [23,24] with SF6 as the etching gas and C4F8 for the passivation. Electron-beam deposition of a 10/100 nm thick Ti/Au film in a direction perpendicular to the square hole array layer produced an Au square grid on the top, corresponding to the inductive grid, and an Au square array on the bottom, corresponding to the capacitive grid. The fabricated filter is shown in Fig. 1(b) with optical microscope and SEM images.
We evaluated the bandpass filter characteristics by measuring the transmission spectrum in the far-infrared region using a Bruker VERTEX 80 v Fourier transform infrared spectrometer (FTIR) with a Globar lamp source under room temperature (25°C) and vacuum (0.5 Pa) conditions. We note that the visibility of the interference in a 625 µm thick Si substrate was approximately 0.2-0.3%, which was hardly observed in the measurement of the bandpass filters. The measured transmission spectra were compared to 3D electromagnetic simulation results using COMSOL Multiphysics finite element method software. The 3D model's structural parameters were obtained from SEM measurements. We assumed a constant refractive index of 3.42 for Si in the wavelength range of 50 µm–150 µm and adopted the frequency-dependent complex refractive index of Au from the literature [25]. We imposed periodic boundary conditions on the x- and y-directions of the unit cell in the calculation model. We inputted light polarized in the y-direction from the air region with refractive index n = 1 above the filter, propagated it in the -z direction, and determined the transmittance of the light passing through to the Si substrate. We applied the transition boundary condition to the Au thin film to reduce the calculation time. This transition boundary condition reduced the calculation time to less than one-tenth compared to calculations without it and did not significantly affect the transmission spectrum. For example, the calculation with the transition boundary condition for a structure with p = 15 µm, a = 12 µm, and d = 20 µm resulted in a 0.6% red shift in the transmission center wavelength, a 0.4% decrease in the maximum transmittance, and an 8% decrease in the full width at half maximum (FWHM), compared to the calculation without it.
3. Results and discussion
3.1 Transmission in a Babinet complementary structure
Figure 2 compares transmission spectra between FTIR measurement and 3D electromagnetic simulation in four filters with different structural parameters. The filters achieved 40% maximum transmission despite being a Babinet complementary structure pair where the bottom square Au fully blocks the light passing through the top square hole. Because the measured total transmittance includes the Fresnel reflection at the substrate's bottom Si/Air interface, the transmittance of the essential filter part of the mirror pair is about 60%. The experiment and simulation demonstrate that the structure operates as a bandpass filter with a redshifted central wavelength as the depth increases or side length decreases.
Fig. 2. FTIR spectra for four fabricated (symbols) and simulated (solid lines) filters with period p = 15 µm, side length a, and depth d.
The experimental and simulation spectra generally agree. An experiment (simulation) using a filter with a period p = 15 µm, side length a = 12.7 µm, and depth d = 18 µm showed a central transmission wavelength of 70.4 µm (70.2 µm), a maximum transmittance of 40% (37%), and a FWHM of 6.4 µm (4.4 µm). The cutoff transmittance was estimated to be 1.5% (0.9%) at 60 µm on the short wavelength side of the center wavelength and 6% (3.5%) at 80 µm on the long wavelength side of the center wavelength. The experimental spectra had a wider FWHM than the simulated spectra because of the distribution of a and d, which was approximately ±0.2 µm. The central transmission wavelength is sensitive to changes in a and d, resulting in a shift of approximately 6 µm and 3 µm, respectively, for every 1 µm change.
3.2 Fabry–Perot interference
To understand the transmission mechanism, we plot the transmission center wavelengths of various filters against the optical path length 2n0d for a single cavity round trip between the top and bottom mirrors [Fig. 3(a)]. n0 refers to the effective refractive index of the Si subwavelength structure with square holes, which is calculated by the zero-order effective medium approximation as ${n_0} = 3.42({1 - {a^2}/{p^2}} )+ {a^2}/{p^2}$ [23,24]. The transmission center wavelength is nearly proportional to the optical path length 2n0d, demonstrating that the main transmission mechanism is the Fabry-Perot interference between the top and bottom mirrors. Figure 3(b) shows that changing the depth d from 16 µm to 24 µm with a fixed value of a = 12 µm tunes the transmission wavelength from 68 µm to 95 µm, while maintaining a transmittance of 40% or higher. The central wavelength is also shifted by varying the side length a, mainly due to a change in the effective refractive index. The details of side length dependence are discussed in section 3.5.
Fig. 3. (a) Path length dependence of the transmission center wavelengths of eight experimental and five simulated filters, as well as those of Fig. 2. The black solid line indicates the line where the transmitted wavelength λ is equal to the optical path length, 2n0d. (b) Simulated depth dependence of the transmission spectrum with p = 15 µm and a = 12 µm.
3.3 Effective refractive index and phase shift
Both the experimental and simulated transmission center wavelengths in Fig. 3 redshift by 5-10 µm from the λ = 2n0d line. This is because of the deviation from the zero-order effective medium approximation of the effective refractive index and the effect of the phase shift at reflection by the metamaterial half-mirrors. A more accurate effective refractive index neff, including wavelength dependence, was obtained by using the anti-reflection condition for the Si subwavelength structures [26]. The effective refractive index neff was obtained from neff = λ/4 dmin, where dmin is the depth that gives the minimum reflectance in the Si subwavelength structure without Au layers. The wavelength-dependent effective refractive index neff almost matched the zero-order effective refractive index n0 at wavelengths above 150 µm, as shown in Fig. 4(a). At shorter wavelengths, neff increased as the wavelength decreased. For example, neff = 1.941 at 81.4 µm, which was the transmission center wavelength of the filter with p = 15 µm, a = 12 µm, and d = 20 µm, was approximately 4% larger than n0 = 1.871.
Fig. 4. (a) Wavelength dependence of the effective refractive index obtained from the anti-reflection condition for p = 15 µm and a = 12 µm. The dashed line represents the effective refractive index n0 calculated by the zeroth-order effective medium approximation. (b) Schematic model structures with the effective refractive index neff for the subwavelength structures, and the phase shifts, δr,top and δr,bottom, for reflection from the top and bottom metamaterial mirrors, respectively. (c) Wavelength dependence of δr,top and δr,bottom. (d) Wavelength dependence of the phase for one round trip in the resonator, shown for three cases: using n0, neff, and neff with the phase shift. The dotted line represents the phase for the first-order resonant condition (2π).
The phase shifts δr,top and δr,bottom during reflection by the top and bottom metamaterial half-mirrors were obtained by simulating the reflection of a plane wave at the metamaterial half-mirrors placed on the top and bottom surfaces of an effective medium with the effective refractive index neff, replacing the Si subwavelength structure, as shown in Fig. 4(b). We note that the phase shift sign was taken to be positive in the direction of increasing phase of the time evolution of the complex electric field exp(+iωt). Figure 4(c) shows that the phase shift of the top mirror δr,top asymptotically approaches π on the long wavelength side, while the phase shift of the bottom mirror δr,bottom asymptotically approaches -π on the short wavelength side. This corresponds to the reflectance of the inductive and capacitive grids approaching unity on the long and short wavelength sides, respectively [8,9]. The Babinet complementary structure cancels δr,top and δr,bottom, resulting in a relatively small total phase shift δr,top+δr,bottom. Using the total phase shift and the effective refractive index neff obtained from the anti-reflection condition, the cavity round trip phase is expressed as 2kneffd-(δr,top+δr,bottom). A Fabry-Perot resonator maximizes the transmittance when the cavity round-trip phase equals an integer (m) multiple of 2π, where m is the interference order. The transmission center wavelength at m = 1 is determined by the intersection of the cavity round trip phase and 2π. Figure 4(d) shows that the intersection of the cavity round-trip phase including the total phase shift and neff provides a center transmission wavelength of 81.7 µm. This value is nearly the same as the 81.4 µm obtained in the 3D simulation. Thus, the interference condition of the Fabry-Perot resonator with the effective medium approximation, including the phase shift of the metamaterial mirrors, reproduces the transmission center wavelength.
3.4 Transmission path
Figure 5 shows the electric field intensity and Poynting vector distribution in the first-order Fabry-Perot resonance at the 81.4 µm transmission center wavelength. The z direction electric field distribution shows a standing wave with nodes at the top and bottom mirrors and an antinode at the center. A similar electric field distribution was also confirmed by an eigenmode analysis. The electric field intensity in the Si subwavelength structure depends on the orientation of the Si side-wall relative to the polarization direction of the incident light. According to the boundary conditions of Maxwell's equations for dielectrics, the electric field intensity in the Si side-wall parallel to the y-polarized light is large, as shown in the xz cross-section, whereas the intensity in the Si side-wall perpendicular to the polarization is small, as shown in the yz cross-section. The Poynting vectors indicate the light transmission path. The y-polarized light transmits through the top square hole toward the Si side-wall parallel to the light polarization [Fig. 5(a)]. The xz cross-section shows no Poynting vector in the bottom square Au array, while the yz cross-section [Fig. 5(b)] and bird's-eye view [Fig. 5(c)] show large Poynting vectors in the gap between the Au squares with sides perpendicular to the polarization of light. This means that light traveling in or near the Si side-wall parallel to the polarization does not pass straight through the gaps between the bottom Au squares with sides parallel to the polarization, but bends toward other gaps between the Au squares with sides perpendicular to the polarization and then transmits.
Fig. 5. Color plots of electric field intensity |E| at the 81.4 µm transmission center wavelength in (a) xz and (b) yz cross-sections at the filter center with p = 15 µm period, a = 12 µm side length, and d = 20 µm depth. The black arrow indicates the Poynting vector. (c) Bird's-eye view of Poynting vectors, with the Au regions highlighted in yellow.
The side length of the upper square hole a = 12 µm, through which light passes, is about 15% of the wavelength (81.4 µm), and the width of the gap between the lower Au square holes is about 4% of the wavelength. The light transmission through these holes and gaps, which are much smaller than the wavelength, is due to the extraordinary optical transmission caused by the spoof surface plasmon polaritons (SPPs) in the metal periodic structures [27–31]. The transmission of light primarily through the gaps with sides perpendicular to the light polarization rather than those parallel to it, is attributed to the spoof SPP being excited only by the polarization perpendicular to the gap sides. This phenomenon is similar to the anomalous transmission of slit-shaped metal holes, where light with polarization perpendicular to the slit predominantly transmits [32]. Thus, the far-infrared light follows a “twisted” transmission path due to the dielectric boundary conditions in the Maxwell equations and the polarization selection rule for the spoof SPP excitation.
3.5 Hole size dependence and effective medium approximation
We investigate here the effect of the characteristic transmission path on the transmission spectrum by comparing the simulated spectra with those obtained by the basic Fabry-Perot resonator equation [33] with the effective medium approximation. Figure 6(a) shows how the transmission spectrum depends on side length a, with a fixed p = 15 µm and d = 20 µm. The transmittance reaches its maximum and the FWHM becomes minimum at a = 12 µm. To understand the dependence, we simulated the reflectance (Rtop, Rbottom) and absorption (Atop, Abottom) for the top and bottom mirrors individually using the effective medium approximation for the Si subwavelength structure, as shown in Fig. 4(b). Figure 6(b) illustrates the values at the transmission center wavelengths. Rtop decreases as side length a increases, while Rbottom increases. The a dependence of the reflectance is attributed to the decrease of neff and the resulting shift of the transmission center wavelength toward shorter wavelengths as a increases. The wavelength dependence of the reflectance corresponds to the reflectance of inductive and capacitive grids approaching 1 at longer and shorter wavelengths, respectively. The top and bottom mirrors exhibit low absorption; Atop is less than 5% and Abottom is less than 3%, for all the side lengths. Thus, for simplicity, we will ignore the absorption in mirrors in the following discussion.
Fig. 6. (a) Transmission spectra dependence on side length a, with fixed period p = 15 µm and depth d = 20 µm. (b) Reflectance (Rtop, Rbottom) and absorption (Atop, Abottom) of the top and bottom single mirrors at the corresponding transmission center wavelengths. (c) Maximum transmittance (Tmax) dependence on side length a. Triangular symbols represent Tmax obtained from the simulated spectrum (a). The solid line represents Tmax obtained from Eq. (1). (d) FWHM dependence on side length a. Square symbols represent the FWHM obtained from spectrum (a), and the solid line represents that obtained from Eq. (2).
The maximum transmittance Tmax, or the transmittance at the transmission center wavelength, in an asymmetric Fabry-Perot resonator, composed of a uniform dielectric structure with two no-loss half-mirrors with Rtop and Rbottom is expressed as:
The FWHM Δ is expressed using the effective refractive index between the two mirrors neff and the cavity length d:
Figures 6(c) and (d) compare Tmax and Δ obtained from the simulated spectra [Fig. 6(a)] with those calculated by Eqs. (1) and (2). Qualitatively, the side length dependence is reproduced by Eqs. (1) and (2). Quantitatively, Tmax and Δ are approximately 50-70% and 40-50% of their predicted values from Eqs. (1) and (2), respectively. The deviation is attributed to the Si subwavelength structure and the characteristic transmission path. The transmission path shows that some of the light in the air region with n = 1 at the upper part of the square hole propagates in the Si sidewall, parallel to the polarization of the light. In addition, some of the light in the lower part of the square hole at n = 1 passes through the Au gap whose longitudinal direction is perpendicular to the polarization, and propagates in the Si sidewall, which has a refractive index of n = 3.42. These transmission paths involve local Fresnel reflections. The position-dependent local Fresnel reflections and the twisted transmission paths can cause a deviation from the basic equation that assumes a plane wave passing through a uniform dielectric.
3.6 Effect of subwavelength structure in multi-order Fabry-Perot resonance
Figure 7(a) shows multi-order Fabry-Perot resonance in a filter with a long resonator length or a deep depth of d = 100 µm, and a subwavelength structure of Si with period p = 15 µm and side length a = 12 µm. The resonance peaks are plotted against wavenumber in units of cm−1 to show that the peaks are separated by approximately the same free spectral range, except for the effect of the phase shifts at the mirrors. The plotted resonance peaks correspond to m = 3 to m = 7. The m = 5 peak near 79 µm or 126 cm−1 exhibits the maximum transmittance among the resonance peaks, as Rtop and Rbottom balance around it. Figure 7(b) shows resonance peaks from m = 4 to m = 8 in a geometrically identical structure but without Si, where the refractive index is n = 1 throughout, except for the Au region. In the structure, the wavenumber at which the transmittance is at a maximum, i.e., Rtop = Rbottom, is located between the peaks of m = 5 and m = 6.
Fig. 7. Transmission spectrum, Rtop and Rbottom of the p = 15 µm, a = 12 µm, d = 100 µm filter with (a) Si subwavelength structure and (b) without Si using n = 1 for all dielectric. (c) FWHM and extinction ratio γ calculated from the spectrum in (a) and from Eq. (2). (d) The FWHM and γ calculated from the spectrum in (b) and formula (3).
The characteristics of bandpass filters with and without Si were investigated using the FWHM and extinction ratio (γ = Tmin /Tmax). The FWHM and extinction ratio were obtained by a 3D simulation and the basic equations of the Fabry-Perot resonator using the effective medium approximation. The Fabry-Perot resonator with Si with the effective medium approximation consists of air (n = 1)/metamaterial mirror/Si effective medium/metamaterial mirror/Si substrate [Fig. 4(b)], while the resonator without Si consists of metamaterial mirror pairs in n = 1 medium. Tmin in the simulation was obtained by averaging the transmittances at the midpoints of adjacent resonance wavenumbers. The extinction ratio γ from the basic equation with no-loss mirrors is expressed as:
A relatively constant FWHM and extinction ratio over a wide wavelength range result from the mirror pair with a Babinet complementary structure, because the Babinet complementary pair of inductive-capacitive grids combines low-pass and high-pass filters [12,13].
The FWHMs and extinction ratios from Eqs. (2) and (3) with Si are compared to those without Si. The FWHMs and extinction ratios with Si are one-fifth to one-fourth of those without. The reduction is attributed to neff in the resonator and increased Rtop and Rbottom by Fresnel reflection at the interfaces between air, Si effective medium, and Si substrate. Smaller FWHMs and extinction ratios on the shorter wavelength side result from the wavelength dependence of neff, as shown in Fig. 4(a). The FWHMs and extinction ratios obtained from the 3D simulation are compared to those from the basic equations. Without Si, the FWHM and extinction ratio from the simulation agree with those of the basic equations [Fig. 7(d)]. On the other hand, the FWHM and extinction ratio obtained from the simulation are about half of the values predicted by the basic equations [Fig. 7(c)], indicating further improvement of the bandpass characteristics. The improvement is attributed to the Si subwavelength structure inducing local Fresnel reflections and twisted transmission paths.
4. Conclusions
We fabricated a robust far-infrared bandpass filter with a Babinet complementary integrated structure by depositing Au onto a Si subwavelength structure, achieving a center wavelength of 60-95 µm and a bandwidth of 6-8 µm. The Babinet metamaterial mirror pair achieved a high transmittance of approximately 60%, despite completely blocking the direct path of light perpendicular to the filter, due to the extraordinary optical transmission resulting from spoof SPPs. The use of a Babinet mirror pair provides a relatively constant FWHM and extinction ratio over a wide wavelength range. The basic equations with the effective medium approximation reproduce the central wavelength of transmission. In contrast, light flow through the non-uniform Si subwavelength structure exceeds the effective medium approximation, resulting in an improved FWHM and extinction ratio to about half of those predicted by the effective medium approximation. The filter's simple and robust integrated structure makes it suitable for stacking, with the potential for further improvements in the FWHM and extinction ratio.
Funding
Budget for Basic R&D on onboard equipment for future space science missions by Advisory Committee for Space Science.
Acknowledgments
The authors would like to thank S. Iwamoto and T. Suzuki for valuable discussions.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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