A wavelength-selective but polarization-insensitive thermophotovoltaic emitter was numerically developed with a binary tungsten grating and its appealing emittance spectra were demonstrated with analysis. Ranges of emitter dimensions were preliminarily confined with the excitation of the surface plasmon polariton, cavity resonance, and Wood’s anomaly at specified wavelengths. Then, a hybrid scheme (the rigorous coupled wave analysis together with a genetic algorithm) was able to finely tailor the grating profile such that emittance could be significantly enhanced in the near infrared region. The peak emittance at the transverse electric and transverse magnetic polarizations was 0.997 and 0.935, respectively. The emittance was actually almost twice that from a plain tungsten plate at short wavelengths but significantly reduced at long wavelengths. Moreover, such spectral emittance is insensitive to the polarization and 5% dimension modification, making the emitter ideal for thermophotovoltaic applications. Patterns of electromagnetic fields and Poynting vectors were able to confirm the excitation of physical mechanisms.
©2012 Optical Society of America
The wide profile diversity, excitation of unconventional physical mechanisms, and advances in micro/nanofabrication have together significantly facilitated tailoring absorptance/emittance of an absorber/emitter with its structural profile . Thus, many absorbers and emitters have been developed with one-, two-, or three-dimensional (1D, 2D, or 3D) micro/nanostructures [1–15]. Their tailored properties can fit applications such as solar cells [5, 9, 15, 16] or thermophotovoltaic (TPV) devices [1, 8, 11–14, 17] by matching band gaps with their operating wavelength region. One of well-studied application is the TPV emitter, which usually owns a complicated profile, i.e., 2D or 3D structures [8, 11–14]. For example, 2D tungsten (W) grating composed of rectangular microcavities experimentally demonstrated a strong emission peak and high thermal stability over 1400 K . A 3D metallic woodpile structure as a thermophotovoltaic emitter was recently fabricated with the efficiency over 32% . In contrast to symmetric structures, a 1D periodic structure as TPV emitter was rarely found due to its orientation-sensitive radiative properties. The only exception might be the complex grating , whose profile was still not simple and emittance spectra are polarization dependent.
However, 1D structures as emitters are always much preferred for their relative fabrication easiness and little cost as long as the performance is acceptable. This study thus is going to numerically propose a polarization-insensitive TPV emitter with a simple omnidimensional periodic structure. Expectations to the proposed emitter include not only its simple profile but also following objectives. First, its emittance should be large at wavelengths between 0.6 μm and 2.0 μm at its operating temperature to match the band gap of common TPV cells. Second, the emittance needs to be significantly suppressed at wavelengths longer than 2.0 μm to diminish useless thermal emission. Third, such wavelength-selective emittance characteristics are desirable to exhibit polarization independence and direction insensitivity. Forth, the profile feasibility is a big concern to micro/nanofabrication such that the aspect ratio should be no more than 3 and reasonable dimension tolerance (5%, for example) should be allowed. Last but not least, the emitter must survive at high temperature with the same appealing emittance spectra.
The realization of multiple objectives above is going to take a two-stage development process. The first stage is to preliminarily narrows down dimension ranges via meeting the requirements to excite several physical mechanisms, which can enhance emittance at target wavelengths and polarization. The second stage is to finely tune each dimension within the range with a systematic way, a hybrid numerical scheme containing the rigorous coupled-wave analysis and a genetic algorithm. Though details of each methodology will not be given in this work, the big picture of the development process will be covered below. The beginning of the next section is going to brief the construction of the employed numerical model and the selection of the material. Next, both stages during the emitter development will be illustrated. An optimized profile will then be proposed and followed with its emittance spectra. Those spectra will be able to confirm the fulfillment of design objectives above. Additionally, patterns of electromagnetic fields will also be plotted to confirm the excitation of physical mechanisms.
2. Structure profile and the employed material
The proposed structure composes of a grating layer atop a substrate as shown in Fig. 1 . The omnidirectional profile can be depicted by its period (Λ), width of ridges (w), and grating thickness (d). The groove width (a = Λ − w) and the filling factor (f = w/Λ) are fixed once Λ and w are determined. Figure 1 actually demonstrates the numerical model for obtaining absorptance at a linearly-polarized plane wave incidence. When the locally thermal equilibrium is reached, the emittance is the same as the absorptance based on the Kirchhoff’s law . Note that the emission orientation is the reverse of the incidence while their polarizations are identical. In other words, this work indirectly acquires the emittance from gratings although patterns of the electromagnetic fields should be the same. The absorptance is the remaining of unity after subtracting the reflectance (R) according to the energy balance for an opaque object . Here, the incidence is a plane wave from free space and its orientation is specified with a polar angle θ between its wavevector k and the surface normal z. The plane of incidence (PoI) is defined by the incidence and the surface normal. Only the incidence on the x-z plane will be discussed hereafter for simplicity. In contrast, the manuscript considers incidence of both linear polarizations, transverse electric (TE) and transverse magnetic (TM) waves. Since those two polarized waves can synthesize emittance of any other polarization, the comprehensiveness of discussion is therefore assured. Note that the calculation in this study is only applied for the classical diffraction as the PoI is perpendicular to the grating grooves.
Tungsten is selected for attractive properties, such as the strong resistivity against corrosion and high temperature. Moreover, its thermal expansion coefficient and temperature-sensitivity to temperature variation are both sufficiently small such that tungsten emitters can perform well at various temperature levels for TPV applications. Both the grating and its supporting substrate are thus made of tungsten. The employment of single material and binary profile clearly lead to fabrication easiness and the reduced cost in contrast to gratings of complicated profiles and materials. In the numerical model, the wavelength-dependent dielectric function ε of tungsten is inputted in programs. It is converted from optical constants by, where n is the refractive index andis the extinction coefficient, is taken from the handbook .
3. Grating profile determination
3.1 Design guidelines and physical mechanisms
The first stage of profile determination is to roughly confine dimensions with the excitation frequency of various physical mechanisms. These frequencies are strongly correlated with grating geometry via simple equations, but their excitations could enhance emittance within the spectral region of interest. On the other hand, those tailored dimensions also set up the cut-off frequency of those mechanisms such that the emittance at longer wavelengths cannot raise due to the same reason. The mechanisms employed here include surface plasmon polariton (SPP), cavity resonance (CR), and Wood’s anomaly (WA) effects [21–27]. Their characteristics, correlating equations, ranges of dimension, and expected excitation frequency are detailed below.
The SPP is a coupled, localized electromagnetic wave that propagates along the interface between two different media due to charge density oscillation. The excitation of SPP requires a magnetic-field component parallel to the grating groove . Only the TM wave incidence satisfies this condition when the PoI is the x-z plane. The excitation condition of SPP at a shallow diffraction grating can be expressed as [21, 27]:Eq. (1), θ is a monotonic function of nd, and so taking the derivation of it is seen that the effectively excitation of SPP generally requires . The appropriate grating period to exciting SPP with both positive and negative diffraction orders can therefore be obtained as 
The CR caused by interference effects in the grooves of gratings can enhance transmission or absorptance. Its wavelength corresponding to the TE wave is defined as :Eq. (3), the maximum λmn can be determined by either l = 0 or m = 0 . Here, we pick l = 1 and m = 0 such that only a dominates the frequency and . Note that the calculated wavelength λmn for optimization is between 0.6 μm and 2 μm.
The WA causes abrupt changes in the reflectance, transmittance, and absorptance spectra because its diffraction order (j) emerges or disappears at the grazing angle. Wood’s anomaly can be predicted with the following equation :
Equation (4) presents that Wood’s anomaly depends mainly on the grating period Λ and is independent from other dimensions. The plus and minus first diffraction orders contribute to Wood’s anomaly when the wavelength is equal to the grating period (Λ = λ) at normal incidence (θ = 0°). They are also considered as the orders with energy dominating others.
Though only the TM wave can excite SPP for metallic gratings, the CR and WA can be excited with both linearly-polarized waves and clearly seen for deep gratings . These mechanisms could modify radiative properties of gratings significantly. In this work, grating dimensions are thus obtained by employing mentioned equations to fit the excitation frequency into the target spectral region. The deep grating having a ratio of the grating thickness d and groove width a, i.e., , is considered in this study. From the ratio d/a together with Eqs. (2), (3), and (4), we can predict a range of designed parameters such as the period (0.3 μm < Λ < 2 μm), the grating thickness (0.3 μm < d < 2 μm), and the filling factor (0 < f < 1).
3.2 Finely tuning grating profile with a hybrid method
The second stage of determining grating dimensions is to optimize the emittance spectra with a hybrid numerical method. The method is composed of the genetic algorithm (GA) [19, 32] and the modal method or rigorous coupled-wave analysis (RCWA) method [33–37], which serve for searching the designed parameters and calculating the spectral absorptance A, respectively. In the calculation, the GA commences by establishing an initial population of candidate solutions (i.e. the grating period, the filling factor, and the grating thickness). The RCWA method is then applied to calculate the corresponding TE and TM emittance spectra for each candidate solution at discrete and representative wavelength between 0.6 μm and 4 μm. Values of emittance are then substituted into a fitness function to determine the best dimension combination. If the difference between the ideal emittance and the calculated one in the fitness function is little enough, the optimization program is terminated and the corresponding dimensions are extracted. Otherwise, the GA applies its operators (selection, crossover, and mutation), and the TE and TM emittance spectra are recalculated. The ideal emittance spectra is set to unity at wavelengths of interest (0.6 μm ≤ λ ≤ 2 μm) and zero at wavelengths (2 μm ≤ λ ≤ 4 μm) while the calculated emittance spectra (A) is indirectly obtained from the reflectance R, i.e., A = 1 - R. Detail calculation of the algorithms can be found in  and its references.
In contrast to the RCWA used to solve Maxwell’s equations in the frequency domain, the finite different time-domain (FDTD) method solves problems in the time domain, which can easily observe a wide variety of EM fields and Poynting vectors distributions. For the EM fields, the FDTD method (OptiFDTD 4.0) is computed with a mesh size about 2 nm and the time step of 0.0045 fs to ensure the stability . In addition to that, the boundary condition in the z direction is selected as the perfectly matched layer while the periodic boundary condition is used for the x direction. For the Poynting vector, it is defined as , S = 0.5Re(E x H*), where Re represents the real part of the complex quantity. E and H* are the electric field and the conjugated numbers of the magnetic field, respectively. The Poynting vector is normalized to that of incidence. Note that the RCWA code program with necessary modifications for fast convergence is taken in consideration as  and integrated with the GA for efficiently obtaining optical responses. On the other hand, the software (OptiFDTD 4.0) based on the FDTD method can provide the electromagnetic field patterns to confirm the physical mechanism and was also validated with Ref [38, 39].
4. Performance demonstration and physical mechanism confirmation
4.1 Emittance spectra
The proposed structure is depicted with the optimized parameters by the GA, i.e., Λ = 0. 650 μm, d = 0.6 μm, and f = 0.28. These dimensions are feasible to the current fabrication technique with an aspect ratio of the grating thickness and the ridge width around 3. Figure 2(a) shows the emittance spectra of this grating structure at and of the TM wave. A high emittance peak about 0.935 can be seen at the wavelength of 0.890 μm for and another is about 0.98 at the wavelength of 0.905 μm for . However, the bandwidth of the emittance spectra becomes narrower and the peak shifts a little when the incidence angles increases at . For the tungsten plain, the emittance obtained is about 0.5 at . Similarly, for the TE wave shown in Fig. 2(b) the emittance peaks are obtained perfectly of 0.997 at the wavelength of 0.890 nm for both angles of 0° and 20°, although the emittance spectrum at is a little narrower. From the results, it is clear that the energy of the incidence wave is efficiently absorbed by the structure without the reflectance at wavelengths from 0.6 to 2.0 μm. Notice that the substrate is thick enough in the modeling, and so the transmittance here is considered as negligible. Overall, the device achieves the high emittance above 0.90 for the TM and TE waves at the angles from 0° to 20°.
4.2 Patterns of electromagnetic fields and Poynting vectors
Figure 3 shows the spatial distributions of magnetic (H) and electric (E) fields, as well as of Poynting vectors (Sx, Sz)  for the TM and TE polarizations at the lowest and peak wavelengths of 0.645 μm and 0.890 μm, respectively. As described in the previous section, the predicted resonance wavelength’s value of Wood’s anomaly is equal to that of Λ. The wavelength of propagation modes λmn is 2a. After optimizing dimensions and substituting them in Eq. (3) and (4), λmn = 0.940 μm and Λ = λ = 0.650 μm reasonably agree with the peak wavelengths as shown in Fig. 2. In addition to that, the relation of the optimized grating period and the resonance wavelengths as described in Eqs. 2(a) and 2(b) is also found to be good agreement, i.e., with λlowest = 0.645 μm while j > 0; and with λpeak = 0.890 μm while j < 0.
Figure 3(a) shows the EM fields and Poynting vectors distributions at the wavelength of 0.645 μm, at which the emittance spectra are found to be 0.7 and 0.82 for the TM and TE waves, respectively. It can be seen that the pattern of the magnetic field H at the wavelength of 0.645 μm is not completely confined within in the cavity. Thus, the magnitude of H field at the excitation of cavity resonance is weaker as compared to that at the wavelength of 0.890μm For the spatial distribution of the electric field E at the same wavelength, the enhanced field is fully in the center of the cavity grating but negligible along the interface due to weak surface waves. As a result, the emittance peak of the TE wave is higher than that of the TM wave. Furthermore, the Poynting vector distributions (Sx and Sz) reveal that energy of the incidence is almost absorbed by the grating structure as marked with different colors.
Figure 3(b) shows the EM fields and Poynting vectors distributions at λ = 0.890 μm. The emittance peak at this wavelength is higher than that at λ = 0.645 μm, for example, the highest peak obtained 0.997. As shown, the H and E fields are concentrated much more inside the cavity than those at the wavelength of 0.654 μm. Consequently, the enhanced absorptance is larger. For example, for the TM wave the H field is inside the grooves much more than that at λ = 0.645 μm as well as an occurrence of surface plasmon polarition at the interface between tungsten and air. As can be seen in the Poynting vectors plots, the energy in the Sx plot flows up around the surface of the grating and goes through the grating and substrate as shown in the Sz plot. For the TE wave, the E field completely squeezes in the grooves which results in enhancing almost the absorptance as illustrated with orange in the Sz plot. In general, at the wavelength of 0.890 μm the excitation of the CR as calculated in Eq. (3) occurred and it also showed that the magnitude of H field are concentrated strongly in the grating cavitites, which results in enhancing the spectral emittance.
Figure 4 shows the far field region of the H fields at λ = 0.645 μm and 0.890 μm for θ = 0°. From Fig. 4 together with Eq. (4), it is proved that the excitation of the WA occurs at λ = 0.645 μm. Meanwhile, the magnitude of the H field at the far field is weaker at λ = 0.890 μm. It is because the plus and minus first diffraction order are evanescent waves propagating in the near field while they disappear in the far filed as shown in Fig. 4 of the H field distribution at λ = 0. 645 μm. However, at λ = 0.890 μm, the magnetic field H is partially inside the grating grooves while another part concentrates at the tungsten surface, which results in enhancing the absorptance much more than that at λ = 0.645 μm.
4.3 Fabrication tolerance
To ensure the emittance spectra unchanged due to the possible imperfection during real micro/nano fabrication, we conducted some simulations to determine the sensitivity of the emittance peaks to fabrication tolerance, as plotted in Fig. 5 . The emittance spectra are calculated by varying the parameters of the grating such as the grating width (w) and the grating thickness (d) with individually and combined. It can be seen that all cases the emittance was obtained as same as that of the optimal design, for example, about 0.935 for the TM wave and 0.997 for the TE wave. For the simplicity, the data of the combined tolerance were not shown here although the emittance attained was not changed much with the designed structure. However, in the case of the TM polarization, the emittance peaks are shifted a little when increasing the grating thickness but they are constant in the case of TE polarization, i.e., 0.997.
5. Conclusions and discussion
An emitter of a simple profile but appealing emittance spectra was numerically developed with a two-stage process. Dimensions of the emitters were firstly confined with the excitation wavelengths of selected physical mechanisms. Finely tuning dimensions were realized with a hybrid numerical method such that the emittance is largely enhanced at short wavelengths but reduced sharply at long wavelengths. Emittance spectra clearly demonstrated wavelength-selective characteristics regardless of incidence polarization within polar angles 0° ≤ θ ≤ 20°. Choosing tungsten of superior properties as the only material further made the designed structure a promising candidate for TPV emitters. The structure was considered feasible for its profile simplicity and fabrication tolerance. Patterns of electromagnetic fields were able to show physical mechanisms attributed to emittance enhancement and proved the success of the development process. It is concluded that the success of the development way here can definitely expand to the design of other devices of wavelength-selective optical responses for energy harvesting and other applications.
The discussion about conical diffraction is an interesting but complicated subject with GA for an optimum design because another degree of freedom (azimuthal angle, ϕ) needs to be taken into consideration. Actually, each emittance spectrum is not necessarily identical if either of θ, ϕ, or polarization is different. Narrowing the focus of this work benefits the clarity of idea demonstration while a comprehensive investigation on impacts of each variable will be pursued in the future work.
Authors appreciate the financial support by the National Science Council of Taiwan under NSC 98-2221-E-006-053-MY3 and NSC99-2628-E-006-009. Supports from National Synchrotron Radiation Research Center in Hsinchu and National Cheng Kung University are also acknowledged.
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