1. Introduction

Applications such as multi-spectral imaging using mosaic-array cameras [1] require polarization-insensitive narrowband spectral filters with transmission characteristics that do not change significantly over a wide range of incident angles. Subject of special interest are filters for the mid-infrared range operating in the atmospheric transparency windows with wavelengths in the 3µm-5µm and 7µm-14µm spectral intervals.

Narrowband filtering can be achieved using Fabry-Perot cavities with multilayer dielectric mirrors. All-dielectric multilayer structures can certainly be scaled to operate in the mid-infrared range [2]. However, Fabry-Perot filters are not angle-independent as the resonant wavelength gets blue-shifted for increased angles of incidence. To some degree the problem can be mitigated by filling the cavity with high-index materials, but the blue shift as such remains an unavoidable feature of the Fabry-Perot resonator with flat mirrors. Another drawback is technological difficulty of depositing multilayers with a rather large total thickness needed for implementing filters operating at longer wavelengths. The latter issue can be addressed by using metal-dielectric [3,4] instead of all-dielectric multilayers – the metal layers need to be only few dozen nanometer thin. The blue shift of the resonant wavelength remains the issue though.

Optical filtering has also been achieved by utilizing localized surface plasmon resonance to produce angle-independent filters in the UV and visible spectra [5]. Gold nanoparticles can be used to create a local resonant field which achieves angular independence at the cost of a low Q-factor. Another known technique to achieve narrowband filtering is to use guided mode resonances on gratings [6–9]. Periodical thin metal films supporting plasmonic modes also show sharp resonances [10,11]. The filter structure can then be very thin, even in case of a dielectric waveguide-based mid-infrared filter [12], but the spectral resonances are angle-dependent and in many cases polarization-dependent. The physical reason for angular dependence of the resonant wavelength in all of the above cases – from Fabry-Perot resonators excited by oblique incident wave to waveguide gratings to periodical metal films supporting surface modes – is that the resonant field in all these cases is an electromagnetic mode traveling along the lateral direction. Matching the tangential component of the wave vector of the incident wave to the effective wave vector of the traveling mode then sets the link between the incident angle on one side and resonant wavelength on another, and thus the transmission characteristics become angle-dependent.

To break away from the angular dependence of the resonant wavelength, the filter operation should rely upon localized rather than traveling fields. An array of resonators, that are sparse enough to avoid coupling between them, which effectively would lead to formation of a traveling mode that hops from one resonator to the next, and at the same time dense enough to ensure strong resonant transmission, is expected to make an angle-independent filter.

The simplest resonator of this sort, proposed back in 1980 [13], is a narrow slit in a metal film of finite thickness as illustrated in Fig. 1 at the left.

 

Fig. 1. A narrow slit in a metal film of finite thickness (left), and a pair of thin metal screens with narrow slits (right).

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In the direction orthogonal to the plane of the film, light propagates back and forth in a slit as a guided mode in a metal-dielectric-metal waveguide. TM-polarized modes can be supported by infinitely narrow slits, while the slit width W must be at least half-wavelength $({W > \lambda /2} )$ in size to support TE-modes. Reflections from the edges at the input and output apertures of the slit form Fabry-Perot style transmission resonances when the thickness of the film is close to multiple ($m$) integer of half-wavelength for the mode in the slit: $d \approx m\lambda /2{n^\ast }$, and ${n^\ast }$ is the modal index for guided modes in the slits. More precise evaluation of the resonant wavelengths should also account for the phase of reflection at the apertures.

At the resonant wavelengths ${\lambda_m}$, assuming the actual width of the slit is small $W \ll \lambda /\pi $, the effective transmission width of the slit for TM-polarized light waves appears to be $W_{eff}^{ \bot res} = {\lambda_m}/\pi $ regardless of the actual slit width. This result was also verified using generic temporal coupled-mode theory in [14]. In a similar manner, effective transmission area in case of a cylindrical cavity terminated by small circular apertures made of perfect metal is $A_{eff}^{res} = ({3/4\pi } ){\lambda ^2}$ regardless of how small the apertures are [15].

Strikingly, in case of a perfect conductor, even an infinitely narrow slit would resonantly transmit finite amount of optical power $IW_{eff}^{ \bot res} = I{\lambda_m}/\pi $, where I is the intensity of the incident wave. The effective transmission coefficient $W_{eff}^{ \bot res}/W$ then diverges at $W \to 0$. Effective transmission width $W_{eff}^{ \bot res}$ can also be treated as a width from which light is collected before passing through a much narrower slit. In case of periodical structures with periods comparable or shorter than ${\lambda_m}/\pi $ neighboring resonators will share the area from which they collect incident light and thus be coupled.

Optical filters based on periodical arrays of slits in a metal film have been considered in recent studies [16–19]. These studies often focus on achieving high transmission in the visible spectrum. Slits designed and simulated in one study were determined to achieve angle independence by decoupling the angle-dependent grating resonance from the slit resonance [17]. This was accomplished by reducing the slit width with respect to the periodicity. They also note that mode confinement is enhanced by filling in slits with a high index dielectric medium. Another study fabricated an optical blue filter with polarization insensitivity and angle independence up to 20° using metal disks on a dielectric waveguide [18]. The mode confinement is achieved through a hybridization of surface plasmon modes from the periodicity of the disks and localized Fabry-Perot resonances in the gaps.

In [20], filters for the microwave spectral range have been studied theoretically assuming lossless metals. It has been shown that with proper choice of geometric parameters of the structure, spectrally narrow transmission can be maintained within a rather large range of incident angles up to 30° with respect to the normal to the plane of the filter. Looking forward, our paper deals with realistic materials in the mid-infrared spectral range and demonstrates angular tolerance up to 60^o, with a smaller Q-factor of the resonance though. The results in this paper overall are consistent with the theoretical analysis in [20].

Another structure with similar functionality is a pair of thin metal screens with narrow slits [21] as shown in Fig. 1 at the right. Sharp transmission resonances appear when distance between the screens is multiple integer of half-wavelength. Due to possible excitation of propagating electromagnetic modes in the gap between the conductive planes, the transmission may not be perfect.

Case of TE-polarized waves was recently studied in [22]. For infinitely thin perfectly conducting slotted films the resonant transmission of an optimally focused incident wave reaches unity, even though transmission through a single slit may be infinitely small. This property closely resembles transmission characteristics of a symmetric FP filter with highly reflective mirrors: even with vanishing small transmission through a single mirror, resonant transmission through a pair of mirrors reaches unity. In real metals, the peak transmission drops rather quickly though. For instance, at microwave frequencies in the 10 GHz range the magnitude of the transmission peak for a representative structure, both measured and calculated, was reported to be around 1% [23].

More complicated structures of this sort, with larger number of slotted metal screens, were studied in [24]. Demonstrated filters operate in visible through near infrared spectral range. Additional metal layers help to suppress unwanted transmission peaks and implement a filter with single transmission band in the spectral range of interest.

Reported here are design and experimental verification of narrowband mid-infrared optical filters with transmission characteristics that are practically constant over a wide range of incident angles. We start with a narrow slit configuration as in Fig. 1 at the left, with the slits filled with high-index dielectric such as Ge. We proceed to considering slit-like cavities with input/output apertures smaller than the slit width. This helps to better control reflection at the apertures leading to improved filtering characteristics. Then we show that a two-dimensional array of cylindrical cavities with small circular apertures becomes an angle-independent and polarization-independent narrowband filter.

2. Mid-IR nano-resonator filters

2.1 Geometry of the nano-resonators

Possible implementations of Fabry-Perot style filters in a thin metal-dielectric structure are shown in Fig. 2. In the simplest case, this is a dielectric slab cavity terminated by optically thin semitransparent metal films Fig. 2(a). Coupling to/from the resonator involves light transmission through the metal films, which inevitably leads to some optical losses. Using slotted Fig. 2(b) or perforated Fig. 2(e) metal films provides an alternative way of getting light into the resonator – through the voids in the metal films – which may help reducing overall optical losses. Filters of these types will likely show significant angle dependence.

 

Fig. 2. Variety of implementations of Fabry-Perot style filters.

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Further sophistication of the filter structure involves breaking the dielectric slab into an array of independent nano-resonators, such as slits Fig. 2(c) or cylindrical holes Fig. 2(f) in an optically thick metal film. With large enough lateral distance between the resonators, the filters or this sort become angle-independent. For better control of light reflection at the input/output apertures of the resonators, the actual apertures with opening smaller than the lateral size of the resonators can be introduced Figs. 2(d) and 2(g). One dimensional arrays Figs. 2(b), 2(c), and 2(d) will likely be polarization-dependent, while two-dimensional arrays Figs. 2(e), 2(f), and 2(g) can be designed to be polarization-independent.

The structure we start with in this study is a one-dimensional array of nano-resonators (Fig. 2(c)) with subwavelength apertures (Fig. 2(d)). The array period is Λ. The structure in general is treated as a three-layer system, with each layer being a slotted metal film. Top and bottom films of thickness t with slit apertures of width w separate the nano-resonators in the middle film from the substrate and cover. The middle film of thickness H has slit-shaped voids with width $W \ge w$ that form the nano-cavities. The resonator dimensions are H×W. The entire structure is held by a substrate assumed to be transparent in the spectral range of interest.

In a particular case, the setting $w = W$ transforms the structure into a single slotted metal film with total thickness of $d = H + 2t$. In this case, it is appropriate to treat the slits as resonators with dimensions d×W. This paper studies angle-independent mid-infrared transmission resonances in a slotted metal film with voids filled with dielectrics with high refractive index n, including the case of actual nano-cavities with $w < W$.

The structure under consideration is schematically shown in Fig. 3. With proper settings for the dimensions and optical constants of the materials, it turns into a narrow slit in a metal film of finite thickness as in Fig. 1 at the left, or into a pair of perforated thin films as in Fig. 1 at the right. Consequently, it is expected that transmission through the narrow apertures will be generally speaking small, except for the wavelengths at which FP-style resonances are excited in the nano-resonators. We find these resonances well suitable for designing angle-independent narrow-band mid-infrared filters. These filters are polarization-dependent though.

 

Fig. 3. Details of the nano-resonator array: 3D view (left) and cross section (right).

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In simulations below the substrate is a hypothetical lossless dielectric with refractive index of n_s = 1.40, close to that of BaF₂ in the λ = 10µm wavelength range. Barium fluoride is transparent to about 12.5µm while simulation in this paper is continued for slightly longer wavelengths. Material filling the cavities and apertures is a hypothetical lossless dielectric resembling Ge (n = 4.0). Spectral dispersion of both the substrate index and the fillings index at the moment is neglected. The metal (M) permittivity follows Drude formula with plasma frequency 7.25·10⁴ cm^–1 and damping rate 1.45·10² cm^–1, close to the parameters of Ag. Unless mentioned otherwise, these material parameters are common for all the numerical examples below.

The incident wave, unless mentioned otherwise, is assumed to be TM-polarized.

2.2 Filtering by high-index-dielectric-filled slotted metal films

Representative transmission spectra for slotted metal film (w = W, t = 0) calculated using the scattering matrix algorithm [25] are shown in Fig. 4. The simulation settings are as follows. The film thickness is d = 1µm, slit width is W = 0.5µm, and the period is Λ = 2µm. Transmission resonances are found for both TE and TM polarized inputs (Fig. 4, left). The TE resonances, as expected, are only visible at shorter wavelengths – the wavelength has to be short enough for TE modes to be guided by the slits. Choice of metal affects the losses in the resonators and thus the maximal transmission at the resonances (Fig. 4, right). Gold and silver show similar performance, as far as applications in the mid infrared range, while aluminum is significantly worse.

 

Fig. 4. Representative normal incidence transmission spectra of high-index-dielectric-filled slotted metal films: TM and TE transmission spectra of Ag/Ge structure (left), and TM transmission spectra of Ag/Ge, Au/Ge, and Al/Ge structures.

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At normal incidence, the transmission spectrum of Ag/Ge structure shows fundamental Fabry-Perot resonance at λ₁ = 8.6µm. Transmission peak reaches 84.5% while transmission between the resonances drops down to about 2.5%. Full width at half maximum for the fundamental transmission resonance is about Δλ₁ = 0.9µm. The resonant wavelength is slightly longer than a simple estimation 2nd = 8.0µm based on the index of the material in the slits and thickness of the film. This is partly due to higher modal index n* for light waves propagating in the slits and partly due to phases of the guided modes reflected at the interfaces with the substrate and cover.

Second, third, and fourth resonances are at wavelengths close to a half, one third, and one quarter of the fundamental resonance wavelength. As wavelengths of the high-order resonances ${\lambda_m} \approx {\lambda_1}/m$, m = 1, 2, 3 … becomes close the structure period Λ = 2.0µm or n_sΛ = 2.8µm, the FP-resonances may become coupled with the resonance due to excitation of surface plasmon-polaritons propagating at metal-cover or metal-substrate interface. The latter are angle-dependent and thus fall beyond the scope of interest of this study. Also, at shorter wavelengths assumptions on negligible losses and negligible dispersion of the refractive index in the dielectric fillings no longer match properties of realistic materials.

It is interesting to observe that maximal transmission for the second order resonance is comparable with that for the fundamental resonance, while higher order resonances are much weaker. This correlates well with estimation for the effective transmission width of a single slit. When effective width ${\lambda_m}/\pi $ is getting smaller than the period Λ of the slits, the resonant transmission is roughly estimated at ${\lambda_m}/\pi \Lambda $ and thus is smaller for resonances at shorter wavelengths, at least in the case of a perfect conductor. The resonances at wavelengths longer than πΛ ≈ 6.3µm are not expected to follow this pattern as effective cross sections of the individual slits overlap.

To further verify the Fabry-Perot nature of the resonances, we traced how the resonant wavelengths change with the film thickness (Fig. 5, left).

 

Fig. 5. Resonant wavelengths – film thickness map (left), and modal index and phase change due to reflections at the apertures versus wavelength (right).

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Fabry-Perot resonances are observed at wavelengths $\lambda $ such that the total round trip phase delay is multiple integer of $2\pi $: $2d{n^\ast }(\lambda )\cdot 2\pi /\lambda + {\psi_1}(\lambda )+ {\psi_2}(\lambda )= 2\pi m$, where ${\psi_1}$ and ${\psi_2}$ indicate phase changes associated with guided mode reflections at the top and bottom apertures of the slit. Solving a pair of such equations with resonant orders m₁ and m₂ with respect to ${n^\ast }(\lambda )$ and ${\psi_1}(\lambda )+ {\psi_2}(\lambda )$ allows to find the modal index and combined reflection phase as functions of wavelength (Fig. 5, right). Estimations based on different pairs of resonant orders m₁ and m₂ are rather consistent. As expected for the fundamental TM mode in the slit, the modal index is slightly larger than the index of dielectric in the slit.

Variations of the transmission spectra for the fundamental resonance associated with other than normal incident angles are shown in Fig. 6, left. Transmission spectra for incident angles from 0^o to 30^o are practically indistinguishable. At larger incident angles the transmission peak is slightly taller and wider, but the resonant wavelength remains the same. In practical sense, the filter retains angle-independent characteristics for incident angles up to about 45^o. In relation to applications in imaging devices, F# 1.0 corresponds to light cone size of approximately $atan({1/2} )\approx {27^{\circ}}$ as measured with respect optical axis.

 

Fig. 6. Main transmission peak for different vales of incident angles (left), structure periods (center), and slit width (right).

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For the same reason that resonant wavelengths in the slits are angle-independent, they should not change much with the structure period. Indeed, the angle- and period dependence for transmission resonances on films with gratings is derived from the phase matching condition for excitation of surface modes. In the structures under consideration, each slit is a resonator on its own and periodicity is only an indication of how dense the array of slits is. Thus, it primarily affects the magnitude of the transmission peaks but not the resonant wavelengths. Much longer periods with ${\Lambda} > {\lambda_1}/\pi $ will eventually result in substantially reduced peak transmission as effective transmission width becomes smaller than the period.

Resonant transmission peaks for structures with shorter periods are shown in Fig. 6, center. As effective transmission cross sections of individual slits overlap, the resonances in neighboring slits are coupled resulting in wider transmission peaks. Besides broadening the resonance, the coupling leads to formation of a traveling mode that hops from one resonator to another, which eventually entails dependence of the resonant wavelengths on the incident angle.

Effect of slit width variations on the resonant transmission peaks is illustrated in Fig. 6 at the right. As modal index n^* in the narrower gap gets larger, Fabry-Perot resonances in smaller slits are shifted to longer wavelengths. With the slit width decreasing, the resonance becomes sharper, and the suppression of the out-off-band transmission is getting stronger. Peak transmission is lower though. Interestingly, the slit width of 125nm makes only tiny fraction 6.25% of the period while the peak transmission reaches 33% corresponding to effective per-slit transmission coefficient of 5.3. Also notable is that in this case the slit width is only 1.3% of the wavelength. The resonant interactions indeed facilitate rather efficient channeling of electromagnetic energy through the deep subwavelength apertures.

2.3 Filtering by high-index-dielectric-filled nano-cavities

Structures with nano-cavities are formed when top and bottom metal films, as shown schematically in Fig. 3, have apertures with width w < W smaller than width of the cavity W. A representative transmission spectrum for a structure with H × W = 500nm × 500nm resonators and w = 200nm apertures in t = 50nm resonator terminating films is shown in Fig. 7.

 

Fig. 7. Transmission resonances of an array of nano-resonators (left) at different incident angles (right). Incident wave is TM-polarized.

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Compared to the case of a slit in a film of the same thickness, the fundamental resonant wavelength in the nano-resonator is red-shifted. Resonance is narrower, though maximal transmission is smaller. Free spectral range is wider. Similar to the case of slits, resonant wavelength remains practically the same, peak transmission and spectral width slightly increase with incident angle increasing.

2.4 Filtering by 2D array of high-index-dielectric-filled cylindrical nano-cavities

Filters described above are angle-independent, but not polarization-independent. It has been shown that polarization-independence can be achieved using structures that are axially symmetric [26]. Labeke et al. design a symmetric metal-dielectric cavity structure to achieve angle- and polarization-independent transmission filtering in the visible and NIR spectra. However, the quality factor remains low and they note a shift in the resonance between TE and TM modes. Some polarization dependence may still be present at oblique incidence – even a perfect flat surface shows polarization-dependent reflection and transmission. However, as the resonators as such are isotropic, the filtering characteristics are expected to be largely polarization-independent.

Transmission spectra of two-dimensional arrays of nano-resonators have been simulated using COMSOL (Fig. 8) [27].

 

Fig. 8. Cell structure in the simulation (left), and transmission spectra for TM- (center) and TE- polarized (right) input.

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The resonators were assumed to be placed in square lattice with period $2\mu m \times 2\mu m$. Each resonant cavity is a cylinder with diameter $1.5\mu m$, height $1.2\mu m$, filled with high-index dielectric (Ge). From the top side the cavity is terminated by a $0.1\mu m$ thick metal film with $1.4\mu m$ coupling aperture filled with the same high-index dielectric. From the substrate side, there is another a $0.1\mu m$ thick metal film with a smaller, $1.15\mu m$, coupling aperture. The substrate, as before, is BaF₂.

Field distribution at the resonant wavelength is shown in Fig. 9. In the horizontal plane in the middle of the resonator, the field is that of the HE₁₁ mode of a hollow metal pipe waveguide (turns into TE₁₁ mode in case of a perfect conductor). Along the vertical direction, one can recognize the field distribution characteristic to the fundamental mode of a Fabry-Perot resonator, with maximum in the middle and reduced field strength (nodes) at the resonator apertures.

 

Fig. 9. Resonant electric field strength distribution in a horizontal plane in the middle of the resonator (left) and in the sagittal plane (right).

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The resonant transmission is to a great degree both angle- and polarization-independent. To quantify the angle- and polarization dependence of the spectra, we traced the angular dependence of peak wavelength, full width at half maximum, and peak transmission for both polarizations (Fig. 10). In practical sense, significant difference in spectra is found only at the incident angles about 60^o and larger.

 

Fig. 10. Peak wavelength (left), full width at half maximum (center), and peak transmission (right) versus the incident angle for both polarizations.

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The apertures terminating the resonator cavity help to control the quality factor of the resonance (Fig. 11). The definition of a narrowband filter can be somewhat ambiguous, but certainly a higher Q-factor demonstrate a narrower bandwidth of the filter design. Optical filters with high Q-factors generally come at a cost. Filters using guided mode resonance or cavity-resonator-integrated guided-mode resonance filters (CRIGFs) can achieve a Q-factor over 600 in the visible spectrum [28,29]. However, these approaches utilize resonant coupling to a grating mode, which is angle-dependent. Angular acceptance can be mitigated, but only to a small extent, less than 5° in one case [28]. In our design, improved quality factor is accompanied by smaller peak transmission – as light wave is kept in the resonator for larger number of round trips, it accumulates more losses due to interaction with imperfectly reflecting metallic walls of the resonator.

 

Fig. 11. Resonant transmission peaks for various sizes of the apertures (left), and relation between the full width at half maximum and peak reflection (right).

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The aperture at the substrate side is smaller to symmetrize the reflection coefficients in the Fabry-Perot resonator, which is verified by observing close to zero reflection by the filter structure at the resonance – the incident wave is 100% coupled to the field in the resonators, and then its power is split between the transmission and absorption

Smaller coupling apertures result in weaker transmission, but the resonance becomes sharper. Q-factor around 100 is feasible in the LWIR range if ${T_{max}}\; \sim \; 0.1$ is acceptable. The scaling rule appears to be $\textrm{FWHM}\; \sim \; T_{max}^{3/2}$. The quality factor in the resonators is limited by absorption in metal walls. Filters based on similar principles are expected to show sharper transmission lines at longer wavelengths in the far infrared and THz spectral range.

3. Experimental verification

Transmission filters were fabricated to match the simulated structures of the one dimensional array design. One advantage of this design is the thin film requirements and the fabrication simplicity – the structure only requires a single patterned layer. The substrate used was barium fluoride. Germanium was first deposited by electron-beam evaporation with a thickness of 1µm. A grating structure with period Λ = 2µm and 0.5µm line width was patterned using contactless stepper photolithography to achieve the small critical dimensions. Trenches were then etched into the germanium using an anisotropic reactive ion etch process.

Silver was deposited onto the device prior to the removal of the sacrificial layer used in etching the trenches. Removing the sacrificial layer afterwards leaves Ag in the trenches between the Ge pillars but not on top of them. Upon inspection of a completed device, gaps were observed between the silver and germanium due to shadowing effects from metal depositing on the resist sidewalls, which can be seen in Fig. 12. Various deposition methods were tested in order to mitigate this effect. For the sample shown in Fig. 12, 700 nm silver was deposited by e-beam evaporation, followed by a 300 nm deposition by conformal sputtering. By this method metal partially fills the gaps and reduces their size, while a second layer of metal forms a flat (although rough) top surface. Tilted metal evaporations were also tested, although data is not presented here for those structures.

 

Fig. 12. (Left) Schematic diagram of the fabricated filter. This design corresponds with Fig. 2(c). (Right) Cross section of filter structure imaged by scanning electron microscope. Silver is deposited by e-beam evaporation followed by sputtering, resulting in the metal appearing to have two distinct layers. The gaps between the metal and dielectric are on the order of 30-60 nm, although not uniform.

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Transmission spectra for the filters were measured using a Bruker FTIR with a reflecting microscope objective to narrow the incident beam to a small section of the patterned material. Consequently, the incident light on the sample is not parallel but has a light cone of about 12° - 24° which excludes normal incidence. The sample is tilted with respect to normal incidence of this light cone, and for simplicity the sample tilt angles are reported. Transmittance is reported into the BaF₂ substrate; therefore a bare BaF₂ substrate is used as the reference material.

Figure 13 presents experimental transmission data for both TE and TM polarizations, as well as transmission for various angles of incidence. As predicted by simulations and shown in Fig. 4, different resonant wavelengths are observed for TM and TE polarizations. For each polarization, a fundamental peak is observed with maximum transmission, while lower-order modes are also observed, indicative of a Fabry-Perot resonator and predicted by simulation. For TM polarization, the fundamental resonance is observed at 6.1µm with 84.6% transmission. Transmission drops to 10.8% minimum between the first and second order modes. The full width half maximum for the fundamental mode is Δλ₁ = 1.16µm, slightly wider than the predicted value of 0.9µm. The fundamental transmission peak for TM-polarized light shows little variance as the angle of incidence is increased from 0° up to 60°. The resonant wavelength and bandwidth remain constant, while the transmitted intensity varies between 85-90% over the range of angles measured.

 

Fig. 13. Experimental transmission spectra for fabricated devices based on 1-D grating design. (Left) Transmission at normal incidence for TM and TE polarizations, compared with the simulated transmission shown in Fig. 4. (Right) Transmission spectra as a function of angle of incidence for TM-polarized light.

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While the peak transmission and angular insensitivity match the simulated data, the resonant wavelength deviates significantly from the predicted value of 8.6µm. Additionally, two smaller resonances are observed to be overlapping with the fundamental, one at 6.6µm, which appears as a shoulder on the fundamental peak, and another broad feature centered on 7.3µm, although with only a maximum transmittance of about 30%. The change in resonant wavelength, as well as the additional peaks, are both likely a result of insufficient filling of the gaps between the germanium pillars, as seen in the cross sectional images from Fig. 12. Simulations have shown that small gaps in between the germanium and metal can have a large effect on the resonant wavelength, as shown in Fig. 14. A 10nm rectangular air gap was introduced on both sides of the metal, which shifted the resonant wavelength from 8.6 µm down to 7.3µm. Increasing this gap to 50nm shifts the wavelength down to 5.5µm. When a gap is present, the dielectric chosen to fill the cavity is no longer simply germanium, but rather an effective medium of germanium and air, which has a reduced permittivity. This changes the mode confinement conditions and shifts the resonance. In these cases the peak transmission remains high, the bandwidth narrow and separated from the next harmonic mode.

 

Fig. 14. Simulated transmission spectra for transmission filters at normal incidence for TM-polarized light. The black curve is identical to the transmission from Fig. 4 and represents a cavity completely filled with germanium. The red and blue curves include small gaps in between the metal and dielectric.

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The experimental data presented in Fig. 13 represent an averaging over a large area of the fabricated device. Non uniformities across multiple resonators could result in varying air gap sizes, probably between 10-40 nm based on simulations. These non-uniformities create the multiple peaks observed in the experimental data for the fundamental resonant mode. Increasing fabrication uniformity across a large region would eliminate the multiple resonances and decrease the apparent width of the resonant peak. Nevertheless, the experimental results confirm the predictions provided by the simulations of high transmittance and angular insensitivity up to 60° incidence.

4. Conclusions

In conclusion, we have demonstrated here feasibility of narrowband angle-independent filtering in the mid-infrared spectral range, using arrays of nano-resonators. Experimental data showing high transmittance and angular insensitivity validates the results from simulated structures. Cavities with coupling apertures, compared to slits with straight walls, provide additional degree of freedom tailoring the transmission spectra. Filters based on two-dimensional arrays of axially symmetric cavities, in addition to being angle-independent, are also polarization-independent.