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Triple mechanisms were employed to trap mid-infrared (mid-IR) rays within a semi-transparent SiO2 film sandwiched between gold gratings and a gold substrate. Dimensions of four absorbers were explicitly determined using an LC (inductor-capacitor) circuit model considering the role transition of SiO2 film. The film behaves as a capacitance and an inductance when the real part of relative electric permittivity for SiO2 is positive and negative, respectively. At the normal incidence of transverse magnetic waves, every absorptance spectrum of absorbers showed a peak at wavelength λ = 10 μm due to the first mode excitation of magnetic polaritons (MP). At oblique incidence, the Berreman mode led to another peak at λ = 8 μm while its bandwidth was expanded with epsilon near zero mode excited by diffracted waves. The full-width-at-half-maximum of both peaks exceeded 0.6 μm thanks to the SiO2 loss. Other minor absorptance peaks in the mid-IR were caused by variants of the same MP mode.
© 2013 Optical Society of America
1. Introduction
Trapping light or enhancing absorptance (A) has been useful in energy harvesting [1] and thermal imaging [2]. Moreover, a perfect (A = 1) absorber is also an ideal emitter because absorptance equals emittance, as given by Kirchhoff's’ law [3]. Enhancing emittance helps beam the light [4] and recycle the waste heat [5]. Above applications have motivated various ways of trapping light, and the most popular one is to employ periodic structures and metamaterials to excite resonances [6, 7]. Well known resonance types include the magnetic polaritons (MP) [8, 9], Fabry-Perot resonance [10, 11], cavity resonances [12], surface plasmon/phonon polariton [13, 14], and localized surface plasmon polariton [15, 16]. When a mode of resonance is excited, the absorptance spectrum exhibits a peak of narrow bandwidth. The bandwidth can then be expanded using merged peaks from dense modes of the same resonance type [17–19] or concurrent excitation of two types of resonance [20, 21].
In contrast to resonance excitation, trapping light did not readily exploit material intrinsic loss. The usual weakness of material intrinsic loss necessitated a long optical path for the complete absorption of incident waves. Sometimes, an absorptance peak shifted its wavelength [22] and lowered the magnitude [23, 24] due to the material loss. Therefore, an absorber that traps light is much preferred for exciting resonances at wavelengths outside the lossy range of componential materials. However, the absorption through material loss withholds unique advantages. One is the commonness of material loss in insulators and lightly doped semi-conductors. These materials are inexpensive and suit current micro/nano-fabrication techniques very well. In addition, the spectral range of absorption through loss is wide and insensitive to incidence orientation. If advantages on the absorptance enhancement from both resonances and material loss can be realized, trapping light becomes more practical in a broad band and wide angular range.
The objective of this work is thus to trap light using benefits from resonances and material intrinsic loss. The loss here mainly refers to the Berreman mode and epsilon near zero (ENZ) mode. Both modes occur as the real part of relative electric permittivity, Re(ε), for an involved material approaches to zero [25]. Spectra of numerically developed absorbers will demonstrate two nearby and connected peaks at the transverse magnetic (TM) wave incidence. The peak at the wavelength λ = 10 μm is going to utilize the resonance MP and material loss together. The other peak at λ = 8 μm will use the Berreman mode, which can be excited by incidence from free space directly because the mode is within the light cone. The bandwidth of peak is then going to be expanded with the epsilon near zero (ENZ) mode, which can be excited with evanescent waves only. Bandwidths of two peaks will be insensitive to the incidence polar angle (θ) or the parallel component of wavevector if the film is thin enough [25]. Moreover, the peaks are wider than those do not involve material loss. The maximum of two peaks will be designed to be unity (A ≈1). The absorber baseline will be composed of a semi-transparent film sandwiched by metallic gratings and a substrate to support MP as well as the Berreman and ENZ modes. Similar structures were discussed in excellent studies [9, 13, 15, 26–29], but rare attention has been paid to the presence of material loss together with a resonance. The semi-transparent film uses SiO2 for its intrinsic loss at wavelengths from 7.6 μm to 10.6 μm. Its Re(ε) experiences twice sign changes for being able to excite both the Berreman and ENZ modes. Because the region contains wavelengths of the CO2 laser and the human body thermal emission peak, developed absorbers are potentially attractive to laser machining and thermal sensing. The generality of the development process will not be lost if the target wavelengths are switched based on the lossy range of employed materials. The metal uses gold due to its ease of microfabrication and negligible loss.
2. Component materials and absorber base
Our numerical model for developing absorbers includes the materials SiO2, Au, and free space. The air is filled in the free space, whose optical constants (refractive index n and extinction coefficient κ) at all wavelengths are approximated to n = 1 and κ = 0. In contrast, the optical constants of SiO2 and Au are wavelength-dependent. Figure 1(a) shows optical constants of SiO2 from [30] after interpolation between neighboring data points. Their spectral range from 0.4 μm to 12 μm is wide enough to contain both lossless and lossy regions of SiO2. The lossless region is where λ ≤ 7.6 μm, as depicted by the negligible κ. At 7.6 μm ≤ λ ≤ 10.6 μm, both optical constants change obviously. They indicate the formation of electric dipoles due to lattice vibration (phonon) as depicted by the Lorentz Oscillator model [31]. The incidence energy is thus absorbed by phonons; moreover, the frequency of absorption maximum matches that of the dipole vibrational mode.
Fig. 1 (a) Optical constants of Au and SiO2; (b) The angle of refraction (θr) and reflectance spectra of a semi-infinite SiO2 substrate at the TE and TM wave incidence of λ = 8.0 μm.
The optical constants of Au can be found in [30] as well, which gives the longest wavelength as 9.92 μm. Instead of extrapolation, optical constants are alternatively from the relative electric permittivity or dielectric function ε = (n + iκ)2, where i is the square root of (−1). The Drude model is employed to obtain ε for Au [32]:
where ε∞ is the high frequency dielectric constant, and ω is the incidence angular frequency. ε∞ is set to unity, the plasma frequency is ωp = 7.25 × 104 cm−1, and the damping frequency is ωτ = 2.16 × 102 cm−1 [32]. Calculated optical constants of Au are also shown in Fig. 1(a). Both optical constants linearly increase with the wavelength, but κ is always larger than n.The occurrence of Berreman mode in SiO2 not only requires Re(ε) ≈0 but the angle of refraction (θr) less than 90°. As an example, Fig. 1(b) plots correlations between θr and θ at λ = 8.0 μm from the equation below [3]:
where nSiO2 = 0.41 and κSiO2 = 0.32, making Re(ε) of SiO2 a positive number. The maximum θr is 82° and no total reflection occurs. That is, refracted waves propagate within the SiO2 film. The figure also shows the directional reflectance spectra at the incidence of TE (transverse electric) and TM waves. The reflectance at normal incidence is larger than 0.2 and the Brewster angle is 22.7°. Other wavelengths within 8 μm ≤ λ ≤ 9 μm are also investigated although results are not shown. For example, λ = 8.5 μm is selected for nSiO2 < κSiO2. Correlations between θ and θr as well as the reflectance spectra are similar to those in Fig. 1(b).Optical responses of a semi-infinite substrate and a film supporting the Berreman mode are different. Figure 2(a) further shows the absorptance from a semi-infinite SiO2 substrate, a free-standing 200-nm-thick SiO2 film, and a 200-nm-thick SiO2 film above an Au substrate at the TM wave incidence of λ = 8 μm. The absorptance is calculated using programs based on thin film optics [33]. The absorptance from substrate is large at small θ because the transmission coefficient reaches its maximum at the Brewster angle. Most energy of the incidence is carried by the refracted wave and is then totally absorbed within the substrate. However, the absorptance from a free-standing film is relatively low. The low absorptance attributes to the little κ and small film thickness. Since the penetration depth δ = λ/4πκ is about 2000 nm, only a large θr can provide a long propagation distance to enhance absorptance. Therefore, the absorptance peak shifts to θ = 74.4°, leading to θr = 81.6°. The arguments are also applicable to the absorptance from a SiO2 film on an Au substrate, which will serve as the base for developed absorbers. The absorptance peak still occurs at large θ (θ = 63.7° and θr = 80.4°). The addition of Au substrate provides a perfect mirror to refracted waves, whose propagating distance doubles before waves reach the film interface with air. Therefore, the absorptance is almost twice that from a free-standing film. The maximum absorptance can even attain 0.995. Briefly speaking, the absorptance enhancement at λ = 8 μm is due to the bouncing of refracted waves supporting the Berreman mode. Though κSiO2 is 0.32 only, waves dissipate their energy fully in a small number of bounces as long as θr is large. At λ = 8.5 μm (nSiO2 = 0.47 and κSiO2 = 0.93), the maximum absorptance is at θ = 76.0° and the above arguments remain valid.
Fig. 2 (a) The absorptance by a semi-infinite SiO2 substrate, a free-standing 200-nm-thick SiO2 film, and a 200-nm-thick SiO2 film on an Au substrate at the TM wave incidence of λ = 8.0 μm; (b) The absorptance of TE and TM waves from the base of absorbers at θ = 0°, 30°, and 45°.
Figure 2(b) shows the absorptance spectra from the base for developed absorbers at the normal (θ = 0°) and oblique (θ = 30° and θ = 45°) incidences of both linearly polarized plane waves. The figure and one of its insets show the absorptance at the TM and TE wave incidence, respectively. Another inset depicts the oblique incidence of the TM wave, whose wavevector is k. E and H symbolize the electric and magnetic field vectors, respectively. The TM wave incidence is emphasized in this figure and hereafter because it supports the MP as well as both the Berreman and ENZ modes. Since none of MP modes is excited, the absorptance spectra exhibit peaks contributed by the SiO2 loss and Berreman mode only. One minor peak is at λ ≈9.5 μm, and its maximum increases from 0.05 to 0.1 as θ shifts from 0° to 45°. The wavelength corresponds to the peak of nSiO2 as well as the largest difference between nSiO2 and κSiO2. Since the peak also displays at the TE wave incidence, it is purely caused by the material loss and irrelevant to the Berreman mode. Spectra at θ = 0° are identical for both polarizations because the base surface profile is isotopic. But the peak at TE wave incidence decreases its maximum with θ although the decrease is trivial.
The other absorptance peak appears only at the oblique incidence of TM waves because the TE wave incidence cannot excite the Berreman mode. The maximum of peak is at λ = 8.0 μm, and it increases significantly with θ. Absorptance spectra of a free-standing 200-nm-thick SiO2 film are investigated to confirm the absorptance peak appearance at the oblique incidence of TM waves. As explained earlier, the peak is simply caused by the Berreman mode excited by refracted waves within SiO2. Specifically, θr is larger than 60° when θ is larger than 28.4°. The large θr makes the refracted waves propagate almost horizontally in a wide range of θ. For the base of absorbers, the high absorptance can be observed in a wide range of θ. But the bandwidth of peak is fixed by κ > 0, and the absorptance maximum is not high at small θ due to the insufficient propagation distance for refracted waves. The absorption from Au substrate is trivial for its high plasma frequency. The substrate is a perfect mirror confining trapped light within SiO2.
While the current base is able to trap mid-IR rays with the Berreman mode and SiO2 loss, the absorptance can be enlarged and its bandwidth expanded by adding Au gratings at the top of base. The left inset in Fig. 3 shows the baseline for developed absorbers. The addition of gratings does not prohibit the incidence into the base because nanoscale gratings are almost transparent to TM wave incidence, specifically in the mid-IR. The transparency can be explained by the Ewald-Oseen extinction theorem [34], which states that movement of induced electrons by incidence is restricted within lamellae. As a result, the re-emitted field is low, and the incidence wave can penetrate easily. The zeroth order of a diffracted wave acts like the refracted wave without gratings. On the other hand, gratings on top bring about two advantages. One is the capability of absorbers to excite MPs, specifically in the lossy region of SiO2. When a mode of MPs is excited, currents having opposite directions are formed at the top and bottom metallic surfaces [27, 35]. The energy is confined and finally absorbed within the SiO2 film that is sandwiched by the grating and substrate. The other advantage is the generation of multiple orders of diffracted waves, whose wavevectors have a large parallel component to support the ENZ mode. Because their energy dissipates in the long run, the absorptance from absorbers with gratings should exceed that from only the base. Specifically at normal incidence, most energy of the zeroth-order diffracted wave reflects. Only other orders of diffracted wave can enhance the absorptance.
3. Profile identification and absorptance spectra
To expand the bandwidth of absorption enhancement, the MPs need to be excited at the lossy region of SiO2. Because absorptance is surely enhanced at λ ≈8 mm, the long-wavelength limit of the lossy region (λ = 10 μm) is set as the target wavelength for defining absorber profiles using an LC circuit model [9, 27, 36]. Figure 3 shows a unit cell of the absorber baseline and its equivalent circuits in the LC circuit model. Circuits in the left panel is valid for nSiO2 ≥ κSiO2, but circuits in the right panel works for nSiO2 < κSiO2. The frequency leading to zero total impedance of the circuit is the excitation frequency of the first MP mode (MP1). In the left circuits, the spacer and free space in a slit are approximated by the capacitances (Cm) and the capacitance (Ce), respectively [27]. Each side wall is replaced with electric and magnetic resonance inductances (Le and Lm) connected in a series. Detailed expressions of those elements are specified in the figure also. The total impedance (Ztot) of the circuit can be expressed in the following form [27]:
On the other hand, the Ztot for nSiO2 < κSiO2 is different from that above because the role of SiO2 film changes from the capacitance (Cm) to an inductance Such a role transition originates from the negative [37]. As Cm is replaced with the total impedance becomes the form below.where the new inductance is defined with a fitting coefficient Note that the LC circuit model is an approximation, we thus simply consider the dominant role. But when κ is not zero, the role of SiO2 film can be dual (inductor and capacitance). Actually, the omission of SiO2 minor role may lead to the mismatch of MP1 wavelength from LC circuit model and from RCWA modeling results later on. is similarly defined as and the difference between them is explained below. The w/d1 is replaced by d2/w because of the inductance orientation. The plasma frequency of Au (ωp) is substituted by the incidence angular frequency (ω), which associates with the sharp change of optical constants due to the phonon absorption. Since the new inductance originates from and the transition should be smooth, the fitting coefficient should account for both c1 and The minus sign fixes the flaw of negative As a result, the coefficient is wavelength-dependent. The original fitting coefficient γ is less than unity. It is related to not only the incidence penetration depth but the unevenness of charge distribution [38]. The effective cross section of metals is smaller than their physical section area, for example, the fitting value 2/3 originates from [27]. Equation (4) considers only without the imaginary part based on the LC circuit model employed in [27] for lossy metal W. The error caused by this omission should be trivial because κSiO2 is not large.The zeroing of two total impedances actually gets the same cubic equation below:
where all symbols and constants are listed with their applicable units in the figure. Besides setting d2 = 200 nm, identifying the profile of an absorber requires the slit width b, grating height d1, and ridge width w. Equation (5) depicts w as the fitting constant. In fact, w is also the critical dimension in fabricating an absorber. Four sets of other two dimensions are intended to generalize the development process. For each, two values are alternatively employed such that (b, d1) for the four sample absorbers are (100, 60), (100, 80), (150, 60), and (150, 80) in the unit of nanometer. Table 1 lists the unique real root of w from Eq. (5) when populating b, d1, and other constants. The other two roots of w are conjugate complex numbers. The grating period Λ is also listed for reference. Using d1 for Le of metallic gratings looks intuitive because it is the physical thickness. Since no other dimensions are referable for the substrate, d1 is borrowed to serve as a first-order approximation. The success of substitution can be explained with the penetration depth (δ ≈12 nm) of incidence, but itself is not enough as the thickness using in the LC circuit model. Only the thickness larger than three times of penetration depth is thick enough to block incidence. The employed thickness (d1 = 60 nm or 80 nm) here is sufficiently thick and not too thick. Moreover, the constant d1 is able to serve in a wideband well because the extinction coefficient of Au increases linearly with the wavelength. As a result, the thickness of 60 nm or 80 nm is indeed suitable for Le of both Au gratings and substrate.
Table 1. Dimensions of Developed Absorbers from the LC Circuit Model
Their absorptance spectra and EM field patterns of developed absorbers are acquired from programs using the rigorous coupled-wave analysis algorithm (RCWA) [39]. Figure 4(a) shows absorptance spectra of developed absorbers at normal incidence. As expected, MP1 is excited at λ ≈10 μm, thereby bringing about a peak in every spectrum. The values of maximum absorptance peak, max[Aλ], of absorber II is 0.851. But that of other three peaks exceed 0.996. All of them are the highest absorptance in a spectrum aided by SiO2 loss. Their corresponding wavelengths are somewhat larger than 10 μm because the LC model is an approximation only. For each spectrum, the full-width-at-half-maximum (FWHM) is wider than 0.6 μm. The spectral region influenced by MP1 is actually larger because the two peaks at λ ≈6.5 μm and λ ≈8.5 μm are also correlated with MP1. These two peaks also cannot be found in the spectrum from a base at normal incidence as shown in Fig. 2(b). Maxima of the two peaks are lower than the one at λ ≈10.0 μm for their corresponding wavelengths are at the margin or even beyond the SiO2 lossy region. Because they are weaker and not expected by the LC circuit model, we consider them variants of MP1. More evidence for their mechanisms will be offered later, based on electromagnetic (EM) field patterns and absorptance contour plots.
Fig. 4 The absorptance from four identified absorbers at the TM wave incidence of: (a) θ = 0°; (b) θ = 30°; (c) θ = 45°.
Figure 4(b) and 4(c) show absorptance spectra at θ = 30° and θ = 45°, respectively. The absorptance peak jointly caused by the MP1 and SiO2 loss remains at λ ≈10 μm in all eight spectra. The max[Aλ] of each peak is close to unity and remains the highest absorptance in a spectrum. Such absorptance peak is indeed enhanced regardless of θ for developed absorbers. The other absorptance peak at λ = 8.0 μm is significantly raised at oblique incidence compared to normal incidence. Its maximum is higher than 0.6 and 0.8 for θ = 30° and θ = 45°, respectively, while it is about 0.2 for θ = 0°. In addition, the peak shifts its wavelength from λ = 8.5 μm (θ = 0°) to λ = 8.0 μm (θ = 30° and 45°). These changes associated with the responsible mechanisms. As previously mentioned, one MP variant and SiO2 loss jointly enlarge the absorptance at normal incidence. But they are not the major contributors to the sharp peak at oblique incidence. Both the Berreman and ENZ modes cause the absorptance peak and the wavelength shift. Clearly, four examples successfully trap mid-IR rays in the expected spectral range. Other absorptance peaks in Fig. 4 are relatively low and often have a very narrow bandwidth for not having benefited from the SiO2 loss. Their physical mechanisms will also be investigated with EM field patterns later. Although the investigation was conducted on all four examples to assure generality, only the patterns of absorber I are presented, to avoid redundancy.
4. Electromagnetic field patterns
Figure 5(a) shows EM field patterns as the MP1 is excited at λ = 10.36 μm. The contour background displays a strong magnetic field enhancement that resembles an oval within the SiO2 film. Electric field vectors depict a counterclockwise circular pattern, which is emphasized with curved arrows at the right part of the figure. The left part also contains a unit of absorber I, but for clarity, curved arrows are not shown. Electric field patterns also demonstrate that generated currents propagate in an opposite directions at the top and bottom metal surfaces. These patterns are indeed characteristic of MP1. The EM fields can also obtain the Poynting vector S = E × H [34]. Figure 5(b) shows the time-averaged Poynting vectors <S>, which are essentially the energy flow in space. Poynting vectors cannot penetrate the metallic lamellae because the heights of metallic lamellae are thicker than the penetration depth. But those vectors can funnel through the nanoscale slits and then decay quickly in the SiO2 film.
Fig. 5 (a) The EM field patterns and (b) Poynting vectors of the absorber I when the MP1 is excited (λ = 10.36 μm) at θ = 0°. Arrowheads in (a) represent the normalized electric field vectors.
EM field patterns of the other designed absorptance peaks are shown in Fig. 6. Figure 6(a) shows arrows pointing upwards in both the free space and the film. Those in the film align vertically and have almost the same magnitude. Their orientations symbolize the capacitance role for the SiO2 sandwiched by Au. The uniformity in magnitude results from the transparency of Au gratings at the TM wave incidence of the long wavelength. Arrows in the free space are tilted and relatively short due to the weak and oblique reflectance. The background contour does not show any magnetic field enhancement because no resonances are excited. Only refracted waves propagate and then dissipate their energy within the SiO2 film. Figure 6(b) shows Poynting vectors, which further supports above arguments. Horizontal Poynting vectors in the SiO2 film depict the energy propagation direction as it dissipates simultaneously. The absorptance peak is thus confirms as the Berreman mode within the lossy SiO2 film of . The sharp contrast between Fig. 5 and Fig. 6 conveys the uniqueness of the dual mechanisms aided by the loss.
Fig. 6 (a) The EM field patterns and (b) Poynting vectors of the absorber I as the wave guide mode (λ = 8.0 μm) occurs at θ = 45°.
Figure 7 shows EM field patterns at four wavelengths (λ = 8.1, 8.5, 9.1 and 9.5 μm) in addition to those at λ = 8.0 μm in Fig. 6(a) for demonstrating the role transition of SiO2 film. For n = 0.411 and κ = 0.323 at λ = 8.0 μm, vertical arrowheads representing electric fields confirm the role of capacitance. The magnetic field within film is stronger than that in metals above and below. At λ = 8.1 μm in Fig. 7(a), n = 0.398 is smaller than κ = 0.504. The length of arrowheads in the film is not the same. The magnetic field inside film is stronger than elsewhere. In Fig. 7(b), κ becomes almost twice of n (n = 0.472 and κ = 0.929) at λ = 8.5 μm. The length of arrowheads keeps shrinking, but the magnetic field magnitude within the film is about the same as that in air and metallic gratings. In Fig. 7(c), both n and κ are larger than unity (n = 1.105 and κ = 2.559) at λ = 9.1 μm such that SiO2 film acts like a metallic inductor. Short arrowheads within the film symbolize the weak electric field, while their opposite directions result from the oblique incidence. The magnetic field within the film gets weak and its magnitude approaches to that in the metal. Similar characteristics can also be found in EM field patterns at λ = 9.5 μm (n = 2.705 and κ = 1.716) in Fig. 7(d). But the electric field in the film becomes much weaker than that in air because n is larger than κ now. The SiO2 acts like a semitransparent rather than a metal film again to trap more energy inside than at λ = 9.1 μm.
Fig. 7 The EM field patterns of the absorber I at θ = 45° for the incidence of wavelengths: (a) λ = 8.1 μm (n = 0.398 and κ = 0.504); (b) λ = 8.5 μm (n = 0.472 and κ = 0.929); (c) λ = 9.1 μm (n = 1.105 and κ = 2.559); (d) λ = 9.5 μm (n = 2.705 and κ = 1.716).
Figure 8 shows EM field patterns for absorptance peaks at the normal incidence of wavelength λ = 6.53 μm and λ = 2.17 μm, which are respectively associated with the excitation of an MP1 variant and the MP3. In Fig. 8(a), an oval is formed and sandwiched between the top and bottom metal surfaces. The magnetic field is enhanced, and arrows depicting the electric field are oriented counterclockwise. The only distinct pattern here from that in Fig. 5(a) is the black half ring above the slit opening. Figure 8(b) displays EM field patterns at the excitation of MP3, which leads to a sharp absorptance peak in the spectrum. The resonance is attributed to MP3 for following reasons. First, the resonance wavelength (λ = 2.17 μm) is one-third that of the MP1 variant (λ = 6.53 μm). Second, three ovals appear in the magnetic field patterns, although only two are within the SiO2 film. The other oval is between the side walls of nearby Au ridges. Third, the electric field exhibits circular flowing patterns in each oval. Specifically, the black shadow above the slit becomes a half-elliptic dome, not a circular ring as in Fig. 8(a). The dome is larger with thinner shells to allow for an oval. No EM field patterns of even MP modes are obtainable because they cannot be excited at normal incidence [27].
Fig. 8 At θ = 0°, the EM field patterns of the absorber I at the excitation of: (a) MP1 variant (λ = 6.53 μm); (b) MP3 (λ = 2.17 μm).
For studying even MP modes, MP1 together with its variants, and impacts of incidence orientation on resonances, EM field patterns at θ = 45° are studied for three MP modes. Figure 9(a) shows EM field patterns at the excitation of MP1 (λ = 10.29 μm). Although the incidence is oblique here, patterns are very similar to those at normal incidence, as shown in Fig. 5(a). As a result, the maxima of two absorptance peaks are close. The insensitivity of absorptance peaks to θ is also a characteristic of MP excitation. Figure 9(b) shows EM field patterns at the excitation of one MP1 variant (λ = 5.96 μm). Though a black half-ring exhibits above the slit opening, EM field patterns in Fig. 9(b) are similar to those in Fig. 9(a) such that they both correlate to MP1. That is, MP1 and its variants can be excited with the same absorber at different wavelengths. The relations among MP1 and its variants can be explained using Eq. (5). Initially, the LC circuit model was employed by substituting the target angular frequency ω into the equation to obtain the ridge width w. But Eq. (5) is also a quadratic equation of ω2 if other variables are fixed, including w. Four roots of the equation imply that an absorber of given dimensions is able to excite the MP1 at four wavelengths. But if is fixed to a constant like other variables, two roots of the equation are conjugate complex numbers. The other two are an identical positive number, which is the only one reasonable for the ridge width. As a result, previous works did not show a bent curve from the LC circuit model. The bent curve exhibited in this work is additionally contributed from the wavelength-dependent dielectric function. Precisely speaking, the dually bent curve results from the transition of Beyond the spectral region, multiple roots cannot be observed. The angular frequency corresponding to the absorptance peak at λ = 6.53 μm is confirmed to be a root when the dimensions of absorber I are substituted into Eq. (5).
Fig. 9 At θ = 45°, the EM field patterns of the absorber I at the excitation of: (a) MP1 (λ = 10.29 μm); (b) MP1 variant (λ = 5.96 μm); (c) MP2 (λ = 3.16 μm); (d) MP3 (λ = 1.75 μm).
Figure 9(c) shows EM field patterns at the excitation of MP2 (λ = 3.16 μm). The patterns within a period (Λ) are not symmetric in the air because the incidence is not normal. The magnetic field displays two ovals in the SiO2 film below edges of a grating ridge. But two nearby ovals merge into a bigger one. In each oval, the electric field exhibits a circular flowing pattern. Although these patterns seem indicative of MP2, the excitation wavelength is half that of the MP1 variant (λ = 5.96 μm), not MP1 (λ = 10.29 μm). Note that the excitation of the MP1 variant also leads to an absorptance peak as shown in Fig. 4(c). Figure 9(d) shows EM field patterns at the excitation of MP3. Three ovals within a grating period appear in the magnetic field contour plot. Arrows representing electric fields form a circular pattern in each oval. The wavelength corresponding to the peak (λ = 1.75 μm) deviates little from one-third that of the MP1 variant, possibly due to the oblique incidence.
5. Absorptance from absorbers of various ridge width
Figure 10 plots the absorptance contour from baseline absorbers having various ridge widths (w). The thickness of SiO2 is d2 = 200 nm, while dimensions b and d1 are identical to those of absorber I. That is, (b, d1) is (100, 60) in the unit of nanometer. Figure 10(a) displays the absorptance at the normal incidence, while their corresponding absorptance at θ = 45° is exhibited in Fig. 10(b). The flipped curves with triangle markers are MP1 wavelengths predicted from the LC circuit model. Each is actually a continuous line, and its two turning points in the unit of micrometer are (w, λ) = (1.42, 9.77) and (w, λ) = (5.57, 9.13). The line appears repeatedly in Fig. 10(b) because θ is not involved in Eq. (5). In contrast to the curve, the resonance modes specified with markers are not from the LC circuit model. They are identified form RCWA modeling results and confirmed with EM field patterns. Little deviation exists between the MP1 wavelengths from the LC circuit model and RCWA modeling results. One reason is the intrinsic flaw of LC circuit model, which does not account for θ. Note that optical responses should vary with θ in general. Besides, the fitting constants c1 and γ are not optimized for proposed absorbers. The resonance wavelength discrepancy can be tailored with fitting constants [22]. Last but not least, dimensions of inductances and capacitances may be regarded shorter at oblique incidence than at normal incidence. For example, b and w are viewed as bcosθ and wcosθ, respectively. The reduction in dimensions can thus lead to significant difference in MP wavelengths from the model and from optical responses. Similar arguments can also be found in [22] when the slit depth gets very shallow.
Fig. 10 The absorptance contours from structures of various w (from 0 to 5 μm) at (a) θ = 0° and (b) θ = 45°. The vertical blue line specifies the absorptance spectrum from absorber I. The flipped curves with triangle markers are MP1 resonance wavelengths predicted from the LC circuit model. Three modes of MP (MP1, MP2, and MP3) are identified with modeling results and are designated in figures by circles, diamonds, and squares, respectively. They are not like flipped curves from the LC circuit model.
In Fig. 10(a), the predicted MP1 wavelengths from the LC circuit model fall inside or approach the bright zones of the absorptance contour. When w is smaller than 1.42 μm, no variants of MP1 are excited. The absorptance peak caused by MP1 excitation is low and in the near-IR region. Wavelengths of the peak increase monotonically but not linearly with w. Other resonances are excited at shorter wavelengths, and the bright zone produced by MP3 excitation can be clearly identified. Bandwidths of other bright zones are narrower, but their wavelengths increase linearly with w. When w is between 1.42 μm and 5.57 μm, MP1 and its two variants can be excited with the same absorber. Its absorptance spectrum thus displays three peaks. If any of them is within the region of SiO2 loss, the resonance is enhanced with the Berreman mode. The peak bandwidth is wider than those from other resonances due to the ENZ mode. Moreover, the peak maximum comes close to unity. Wavelengths of the two MP1 variants seem unaffected by w because both the Berreman mode and ENZ mode are associated with material optical constants only. Conversely, wavelengths of MP1 and other resonances increase with w.
Even modes can be excited at oblique incidence, as shown in Fig. 10(b). Additional bright zones appear in the contour plots, for example, a tilt zone due to the excitation of MP2 at the near-IR region. Peak wavelengths associated with even MP modes increase linearly with w. But the bright zone associated with MP2 diminishes when it meets that of an MP1 variant at λ ≈7.5 μm. In fact, bright zones of two MP1 variants split from each other at oblique incidence. The absorptance peak of one variant remains at λ ≈8.5 μm, but the zone merges with that at λ ≈8.0 μm. On the other hand, the bright zone caused by MP1 excitation is about the same at normal and oblique incidence. The peak wavelength increases with w, and its bandwidth is broader than others in the contour. Peak wavelengths of other odd MP modes linearly augment with w, but they are somewhat blue-shifted at oblique incidence in comparison to normal incidence, such as the peak caused by MP3 excitation. In Fig. 9(b), the success of trapping mid-IR rays is clearly demonstrated by two bright zones. One is caused by MP1 at λ ≈10.0 μm. Its wavelength matches that from the expectation from LC circuit model when defining dimensions. Because the peak wavelengths of the zone follow the LC circuit model as shown, the target wavelengths for trapping can extend to other spectral regions. The wide FWHM (> 0.6 μm) is attributed to the SiO2 loss. The other bright zone is always at λ = 8.0 μm resulting from the Berreman mode. The FWHM of this peak is larger than 0.6 μm because it also benefited from the ENZ mode. As w exceeds 4.5 μm, the bright zone fades, and another one of narrow bandwidth appears at λ > 10.5 μm. The bright band comes from MP2 resonance based on EM field patterns corresponding to points in the band. This MP2 splits from the MP2 at short wavelengths because of the role transition of SiO2 film. Note that MP are supposed to be assisted with a dielectric spacer SiO2. But its Re(ε) experiences twice sign changes within 7.6 μm ≤ λ ≤ 10.6 μm and cannot excite MP2 within the spectral region. The bright zone of MP2 is thus separated.
The absorptance contour from absorbers of b = 150 nm and d1 = 80 nm is also investigated. Results are almost identical to previous ones regardless of the variation in dimensions. The similarity originates from the identical target MP1 wavelength, which is the objective in determining dimensions of absorbers. When the dimensions of the four absorbers were tailored to excite MP1 at the same wavelength, wavelengths of other resonances were accordingly modified. The absorptance dependence on w was also bundled for all four absorbers due to the same baseline structure. As a result, four spectra in any sub-figure of Fig. 4 are alike and contour plots are similar. A precaution to practicians is given here during the simulataneous utilization of MP, Berreman mode, and ENZ mode for trapping light. Though they all lead to absorptance enhancement, their excitation requirements and associated EM field patterns conflict with each other. Therefore, the absorptance at concurrences of them is weakened rather than being further enhanced.
6. Conclusions
Trapping mid-IR rays was realized with aids of the Berreman mode, ENZ mode, MPs in combination with SiO2 intrinsic loss. Absorbers composed of Au gratings, SiO2 film, and an Au substrate were developed using a comprehensive LC circuit model. Even when the role of SiO2 switched between a capacitance and an inductance, the model was able to explicitly identify dimensions of absorbers to support MP1 excitation at λ ≈10.0 μm. Every absorptance spectrum thus showed a peak, and its bandwidth was expanded by SiO2 loss. Absorptance peak also occurred at λ ≈8.0 μm at the oblique incidence and was mainly attributed to the Berreman mode. Its enlarged bandwidth was due to the ENZ mode supported by diffracted waves. The excitation of mechanisms unequivocally produced unique EM field patterns and Poynting vectors. Four examples and the absorptance contour plots demonstrated the generality and of development strategy. The absorptance contour plots also indicated a generous fabrication tolerance to the critical dimension w for practicians.
Acknowledgments
Financial support is highly appreciated from the National Science Council of Taiwan under grant No. NSC-101-2628-E-006-014-MY3 and No. NSC-101-2120-M-006-004-CC1.
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