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Abstract
We propose a strategy to design infrared emitters with predefined spectral response using aluminum gratings as building blocks. We begin by identifying 3 target spectra with resonances in the 7–15 µm wavelength range. Next, we use FDTD simulations and interpolation to create a reference library of gratings relating their structural parameters to attributes of their infrared spectra. By using a search algorithm based on minimization of errors in spectral attributes, we identify gratings from this library corresponding to peaks in the target spectra. Finally, we discuss an approach for designing hybrid structures from these gratings to generate each of the 3 target spectra. This strategy can be extended to design structures with complex spectral responses.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The ability to design absorptive features in different wavelength ranges is of immense importance for several applications, such as surface enhanced detection [1,2], biosensing [3,4] and solar energy harvesting [5–7]. To this end, various nanostructured designs have been explored to tailor the absorption spectrum [8–12].
In the mid- to long- IR, many designs have exploited metal-insulator-metal (MIM) resonances to achieve enhanced absorption over a desired wavelength range [13–21]. A typical MIM device is comprised of a dielectric layer sandwiched between an array of nanoscale metal elements and a metal back reflector. While this approach offers flexibility in spectral design, its realization requires fabrication of a three-layer structure. For simplicity of fabrication, it is attractive to consider simple metal gratings for tailored absorption. Coupling of light to surface plasmon modes in stainless-steel gratings has been used to design absorption peaks in the visible as well as infrared [22,23]. Since for the majority of materials, absorptivity is equal to emissivity, such design techniques can also be used to create emissivity peaks.
Here we consider aluminum gratings as building blocks for spectral emissivity design. Aluminum is highly abundant in nature and hence inexpensive. This makes Al gratings a potential cost-effective alternative for applications requiring large-area fabrication. In addition, the skin depth of aluminum in the infrared is a few tens of nanometers. This offers the possibility of using Al-coated dielectric or polymer gratings instead of conventional bulk aluminum gratings, allowing structural flexibility while retaining spectral response.
Below, we propose a strategy for designing samples with particular emissivity spectra of interest. In this work, we focus on normal-incidence emissivity. For concreteness, we consider three different target spectra, containing peaks with various amplitudes, widths, and locations. Once we have generated a reference library of aluminum grating designs, we match a desired spectrum by combining several gratings with appropriate area-weighting fractions. We find that for aluminum thicknesses larger than the skin depth, the spectral features obtained from the grating do not depend on the underlying substrate. This approach could therefore offer a useful alternative to MIM structures for spectral design applications.
2. Defining target spectra
In general, a resonance can be characterized using three spectral parameters: the peak location λ0, peak width Δλ0, and peak emissivity ɛmax. For simplicity, we assume that our material system is at thermal equilibrium and hence Kirchoff’s law is valid. This enables us to write emissivity equal to absorptivity. The peak amplitude ɛmax refers to the system emissivity at λ0, while Δλ0 is the full width at half maximum (FWHM). Below we consider how to generate spectra with several resonances.
We begin by considering 3 target spectra, each comprised of 3 resonances at different wavelengths and having different line widths and amplitudes (Fig. 1). Assuming a Lorentzian line shape for the emissivity at each resonance, the total emissivity can be written as:
Fig. 1. (a) Target spectrum 1 with peak locations 8, 11 and 14 µm and amplitudes 0.34, 0.66 and 1 respectively. (b) Target spectrum 2 with peak locations 8, 11 and 14 µm and amplitudes 1, 0.34 and 0.66. (c) Target spectrum 3 with peak locations 10, 11 and 12 µm and amplitudes 0.34, 1 and 0.66.
The first target spectrum [Fig. 1(a)] is chosen such that a reduction in peak width is accompanied by an increase in peak amplitude. In order to demonstrate that other trends are achievable, we create a second target spectrum [Fig. 1(b)] in which the narrowest peak has the lowest amplitude. Finally, the third target spectrum [Fig. 1(c)] shows the possibility of moving the peaks closer together in wavelength, while changing their relative amplitudes. It is important to note that in principle, there are an infinite number of spectra which can be generated by using metal gratings. However, for simplicity we focus our attention on these three test cases to demonstrate the general spectral design capabilities.
2. Building the grating library
Figure 2 shows a schematic of a single unit cell of an Al grating with period a, slot depth d and slot width w. For simplicity, we simulate the case where the entire structure is made of Al. This structure is known to support a surface plasmon-polariton (SPP) resonance characterized by a surface wave which propagates along a metal-dielectric interface and decays exponentially in the perpendicular direction [24]. For small periodic perturbations at the interface (small d and w) one can relate the resonant wavelength λn for the nth SPP mode to the period a by the following equation:
Fig. 2. Unit cell of a bulk aluminum grating used in simulations. The grating period, slot depth and slot width are denoted by a, d and w respectively.
To create a reference library of gratings, we consider variations in the grating parameters. We start with a set of 8 gratings having the same slot dimensions (d, w) but different periods (a). For the ith grating, the period ai = (6 + i) µm, while the slot depth di and width wi are kept fixed at 0.5 µm for all gratings. Since the slot dimensions are an order of magnitude smaller than the grating period, we expect λ0 to be well approximated by using Eq. (2). Next, for each grating we vary both the slot width and depth from 0.5 to 2.5 µm in steps of 0.5 µm. This creates 25 gratings for each of the 8 periods. For each of these gratings, we compute the normal-incidence emissivity spectra using Lumerical FDTD solutions and determine the peak position λ0, peak width Δλ0, and peak amplitude ɛmax. We assume that emissivity is equal to absorptivity. Absorptivity is calculated as 1 – reflectivity; there is no transmission through the metal grating. We assume the incident radiation to be transverse magnetic (TM) – polarized.
Figure 3 shows the variation of these spectral parameters with slot dimensions for three grating periods. It can be observed that for fixed w and d, λ0 increases with an increase in the grating period a. When w, d << a, the peak is located at a wavelength approximately equal to the grating period. This is consistent with what one would expect from Eq. (2). However, an increase in slot dimensions is accompanied by a red shift in the peak location. These observations can be understood by looking at the peak location colormap for a = 12 µm. For w = d = 0.5 µm, λ0 ≈ 12 µm while for w = d = 2.5 µm, λ0 ≈ 16 µm.
Fig. 3. Variation of resonance peak parameters with slot dimensions for three different grating periods.
The peak width on the other hand, follows a trend opposite to that observed for peak location. An increase in a for fixed values of w and d causes a reduction in Δλ0. However, for a fixed grating period, the peak width can be increased by increasing the slot dimensions. This is evident from the peak width colormap for a = 8 µm. In this case, the peak width can be increased from few tens of nanometers to about 6 µm with a change in d from 0.5 to 2.5 µm, keeping w fixed at 0.5 µm.
For a fixed grating period, a change in slot dimensions can be used to move towards or away from the maximum-amplitude band on the color map. The band can be shifted by changing the grating period. This effect can be seen by observing the upward shift in the yellow region of the peak amplitude colormaps for a = 8, 10 and 12 µm. In general, the radiative and absorptive rates depend on the grating parameters (d and w) in different ways.
Within the framework of coupled-mode theory [25], the maximum amplitude band corresponds to the critical-coupling condition, where the absorptive and radiative loss rates of the grating are equal. For equal rates, lower values correspond to narrower peak widths. In general, the radiative and absorptive rates have different dependencies on the grating parameters. Since these dependencies can be difficult to predict from intuitive arguments alone, we can use the color maps of Fig. 3 to identify spectral peaks with the desired characteristics.
In order to generate a more exhaustive set of resonances in the 7–15 µm wavelength range, we interpolate the peak parameters for intermediate values of grating dimensions. We define a three-dimensional grid of grating parameters with a ∈ [7,14] µm, d ∈ [0.5, 2.5] µm, w ∈ [0.5, 2.5] µm and a grid point spacing of 0.1 µm along all three dimensions. Using MATLAB’s cubic interpolation function with previously calculated values of peak parameters, we determine λ0, Δλ0 and ɛmax for all points on the grid. This approach creates a database of resonant grating structures from which design parameters can be selected to generate resonances with specific spectral characteristics.
3. Results for spectral matching
We return to the problem of designing hybrid grating structures to generate the emission spectra presented in Fig. 1. We choose peaks with Δλ0 ∈ {0.25, 0.5, 0.75} µm and λ0 ∈ {8, 10, 11, 12, 14} µm to form our basis set. In order to find resonant gratings corresponding to each of these peaks, we write a search algorithm based on maximizing the peak amplitude while ensuring that errors in peak location and width are within acceptable tolerance levels, chosen to be 0.05 µm and 0.03 µm respectively. We test the correctness of our search algorithm by simulating gratings with design parameters determined in the previous step. The resulting emission spectra are plotted in Fig. 4. All peaks are within 0.08 µm of their desired locations and 0.06 µm of their desired widths. Resonances with higher accuracy in peak location and width can be obtained by minimizing the sum of squared errors in λ0 and Δλ0 instead of maximizing the amplitude. However, this would result in peaks with lower amplitudes. From comparing Figs. 5(a) through 5(c), it can also be observed that for any given peak wavelength, the peak amplitude tends to decrease with peak width.
Fig. 4. Emission spectra for gratings derived from the search algorithm to generate peaks at 8, 10, 11, 12 and 14 µm with widths: (a) 0.25 µm, (b) 0.50 µm, and (c) 0.75 µm. The desired peak locations are indicated by dashed lines.
Fig. 5. Structural parameters and area fractions for gratings used in the construction of hybrid gratings corresponding to the 3 target spectra. The schematic on the top summarizes the relation between various design parameters and the grating structure and that between the normalized and unnormalized grating spectrum. The table lists the values of these parameters.
To match the target spectra of Fig. 1, we use 3 gratings with different resonant wavelengths for each case. Consider a substrate patterned with n gratings having emissivities ɛ1(λ), ɛ2(λ) … ɛn(λ). We assume that the ith grating occupies a fraction αi of the total substrate area A. A schematic illustration is shown in Fig. 5 for the case of n = 3. We further assume that the emission is observed in the far-field, such that the emissivities of the individual gratings simply add. The total, effective emissivity of the hybrid grating can then be written as:
We further assume that the lateral extent of each grating is much larger than the grating period, so that the emissivity of each grating, ɛi, is given by the calculations of the periodic structure shown above (Figs. 3 and 4).Consider the emission spectra shown in Fig. 1. Each of these spectra are normalized (have maximum emissivity equal to 1) and have 3 emissivity peaks, with known values of λ0, Δλ0 and ɛmax. Using the values of λ0 and Δλ0, we identify 3 gratings from our basis set corresponding to each emission spectrum. We use Eq. (3) with n = 3. We set ɛeff equal to the value of the target emissivity times a scaling factor s at the three wavelengths corresponding to the peak locations. This yields a set of 3 linearly independent equations. These can be used to determine the area fractions αi and the scaling factor s under the constraint that the area fractions add to 1. Physically, the scaling factor s is a measure of the signal strength from the gratings. For small s, the grating spectrum has a low overall amplitude compared to the target spectrum. For s = 1, the amplitudes match.
Figure 5 lists the design parameters used in the construction of hybrid gratings corresponding to each of the target spectra. The scaling factors s for all three grating spectra are significantly greater than 0, implying that a direct measurement on these gratings will yield sizable emissivities. Moreover, it can be observed that the area fractions for all of the gratings are neither close to 0 nor 1, ensuring that the fabrication of such structures is feasible.
Figure 6 presents a comparison between the target spectra and the normalized emission spectra generated by hybrid grating structures. It can be observed that the hybrid grating spectra match well with the target spectra. However, the grating spectra have higher emissivity than the targets at wavelengths away from the peaks. This discrepancy arises as a consequence of our choice of Lorentzian line shapes for resonances while defining the target spectra. In general, SPP resonances do not have a Lorentzian line shape. Despite this discrepancy, the above approach can easily be extended to design hybrid grating structures with a variety of complex spectral responses.
Fig. 6. Comparison between target spectrum and hybrid grating spectrum for targets 1, 2 and 3 (a, b and c).
4. Discussion
One advantage of using aluminum grating structures is the ability to generate similar spectral features on a variety of substrates. Figure 7(a) shows a grating formed by a coating of aluminum on top of a semi-infinite dielectric substrate. The coating thickness of 100 nm is chosen to be larger than the skin depth of aluminum in the wavelength range studied. Figure 7(b) shows the calculated emission spectrum for two different substrates: silicon and glass (SiO2). The main feature is a spectral peak centered close to 11µm, arising from the surface plasmon resonance at the aluminum-air interface. A nearly identical spectral response is obtained for the two dielectric substrates. This is to be expected from the fact that for aluminum thicknesses much larger than the skin depth, the electromagnetic field profile of the surface plasmon mode has minimal overlap with the substrate. The insensitivity to substrate composition suggests significant flexibility in the fabrication approach. For example, other dielectric substrates, such as flexible polymeric materials, could be used provided sufficient aluminum thickness.
Fig. 7. (a) Schematic of an Al surface grating on top of a semi-infinite dielectric substrate. (b) Calculated emission spectra for surface gratings fabricated on silicon and glass.
Throughout this work, we have chosen gratings based on their total, normal-incidence absorptivity, as calculated from full-wave FDTD simulations. For some applications, observation using a limited detector size may introduce additional constraints. In this case, it may be desirable to exclude gratings that emit into diffraction orders, as a portion of the emission will not be collected by the detector. For example, for normal-incidence observation and limited angular acceptance at the detector, one could include only gratings with a period smaller than the minimum wavelength monitored to generate desired spectra.
Depending on application, it may also be desirable to either minimize or maximize the angular dependence of the emissivity. It is known from previous work in the literature [12] that metal gratings support both relatively localized, slot modes and extended, surface plasmon modes. When grating dimensions are tuned to reduce hybridization between the two mode types, the slot modes become angle independent. In future work, the general procedure used here, involving interpolation over a reference library, might further be constrained to include only those gratings with reduced angular dependence in the generation of target spectra.
4. Conclusion
In conclusion, we have designed infrared absorbers with predefined spectral responses using aluminum gratings as building blocks. We started by creating a reference library of gratings to investigate the relationship between their design parameters and spectral properties of their surface plasmon resonances. We expanded the library using interpolation to generate an exhaustive set of resonances in the 7–15 µm wavelength range. By developing an algorithm based on minimization of squared errors, we selected gratings corresponding to resonances in the target spectra. Finally, we proposed a strategy to design hybrid structures using these gratings to generate each of the three target spectra.
In contrast to methods such as inverse design, the approach used here is particularly suitable when one wishes to generate a large number of target spectra. In this case, the great majority of the computational cost is incurred up front, in the calculation of the reference library via electromagnetic simulations. Matching an additional spectrum involves only the very minimal computation required to search over the library.
While we have focused our discussion to the infrared region of the spectrum, a similar approach can be extended to design absorbers for operation in other wavelength ranges by choosing suitable design parameters. The possibility of fabricating these structures on polymers makes them useful for applications benefitting from flexible substrates.
Funding
The Aerospace Corporation.
Disclosures
The authors declare no conflicts of interest.
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