1. Introduction

The unique optical responses, such as enhanced transmission or wavelength-selective emittance, of one-dimensional periodic structures have been intensively studied for their potentials in numerous applications [1–18]. Depending on the wavelength and structure characteristics, these properties are mostly attributed to either one or the interplay of multiple physical mechanisms, including cavity resonance, surface plasmon/phonon resonance, Wood’s anomaly, and effective medium behavior [11,15]. Among all excellent works, quite a few were devoted to correlations between tailored optical responses and their physical mechanisms brought by various slits formed between rectangular lamellae. An example of such slits is Case A shown in Fig. 1 while its profile could vary in opening width (w), slit depth (d), lamella lateral length (l), and structure period (Λ) [2–5,9]. Other types of slits are arrays composed of alternative metallic and dielectric strips, multiple lamellae unevenly aligned in a period, or rectangular metallic strips partially replaced with dielectrics [6,10,12]. The assumptions of exact rectangular cross-sections for slit profile not only benefit numerical and analytical studies by simplifying geometry and boundary conditions but also obtain good enough optical responses compared to those from experiments.

However, the expectation of smooth side walls and sharp corners for periodic structures is not always met during micro/nanofabrication [8,16]. Either photoresist residues or the intrinsic limits of fabrication process may cause imperfection of structure profile and attached features at slit openings. Though very rare studies were conducted, negligible effects of tiny structures seemed to be a plausible but not a confirmed assumption, especially at long wavelengths. Another questionable assumption is the larger transmittance guaranteed by wider slits provided other conditions are the same. Since validations of these assumptions are usually partial or intuitive, it is better to verify them more systematically and look for possibilities of their violation. On the other hand, if optical responses can indeed be modified by intentionally-fabricated irregular features, slits with non-rectangular cross-section may offer an additional freedom for practitioners to manipulate optical responses.

Hence, this work is going to numerically investigate influences of slight slit geometry modification on their optical responses (absorptance, transmittance, and reflectance) and serve several purposes. One is to draw our attention to irregular slit geometry by demonstrating the invalidity of two popular but questionable assumptions mentioned in last paragraph. In order to serve this purpose much easily here, the target physical mechanism aims at the cavity resonance because it is distinct from other mechanisms [14]. The excitation of cavity resonance depends mainly on cavity characteristics but little on the structure period and lamella material. For metallic slit arrays, the cavity resonance can lead to enhanced transmittance of the TEM modes, regardless of groove depths and opening widths [3]. Another objective of this work is to seek tentative explanations to observable discrepancy between numerically- and experimentally-acquired optical responses of slits. Moreover, it can offer an additional flexibility in refining known or developing new devices with desired optical responses by tailoring slit profiles.

 

Fig. 1. Cross-sections of five slit cases with nanoscale geometry modification, where a=20 nm, d=3600 nm, l=750 nm, w=50 nm, and Λ=800 nm represent the square feature size, slit depth, lamella width, slit width, and slit period, respectively. E and H are the electric and magnetic field vector of the incident transverse magnetic wave while k is its wavevector and θ is the angle of incidence or the polar angle. The incidence is marked with red arrowheads and its wavelength ranges from 3 to 15 µm.

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Only a few profile variations will be looked into for simplicity because this study is an initiation while there are obviously numerous feature variations. One way of profile variation here is realized by attaching square features (side length, a in Fig. 1) or rectangular ones at lamella edges, which could be found in real cases [16]. Another way is to comprise two types of slits in a period of structures, which were also proposed recently [12,17]. The wavelength (λ) range is set between 3 and 15 µm because infrared optical responses of metallic slits were rarely studied but attracted much interests recently [16–18]. Besides, once the wavelength is much longer than all dimensions of slits, effects of feature variation can be much emphasized. For the same purpose, the minimum feature in this work is the slit opening 10 nm, which is narrowed by attached features. The tiny feature is within the limit of current technology for nanofabrication, such as the electron beam lithography or the focused ion beam milling [19]. Programs developed with an efficient algorithm, rigorous coupled-wave analysis (RCWA), are employed for obtaining all optical responses, electromagnetic (EM) fields, and Poynting vectors in this study [17].

2. Model development and numerical methods

2.1 Geometry

Figure 1 shows cases of selected slit profile, the local coordinate defined for each case, and values of all dimensions, including a=20 nm, d=3600 nm, l=750 nm, w=50 nm, and Λ=800 nm. Each of the presented five slit profile cases owns its uniqueness: no attached features (Case A), symmetric square features at the slit opening facing the incidence (Case B), symmetric rectangular features at the slit opening facing the incidence (Case C), symmetric square features at both slit openings (Case D), and symmetric square features at both slit openings with a little extended slit depth (Case E). The slit profile of Case A does not have any attached feature, such that its optical responses can serve as a reference. Slits of Case B come from those of Case A by simply extending square features into slit inlet opening. While not shown here, slits of Case B’ also come from those of Case A by extending the same square features into slit opening, except at the outlet of slits. The profile of Case C is similar to that of Case B by elongating square feature into a rectangular feature with 2a length. This work does not discuss Case C’ to avoid the redundancy. On the other hand, if a square features is extended at both inlet and outlet openings of slits, the profile is exhibited as Case D. The slit profile of Case E is similar to that of Case D for discussing effects of extended slit depth. Moreover, the cavity bounded between extended features is the same as that of Case A slits such that influences of boundary conditions at slit openings can also be studied.

Only the transverse magnetic (TM) wave incidence is considered here because the transverse electric (TE) wave transmittance through metallic slits is close to zero [15]. Note that the plane of incidence is usually defined by the incidence wave and the surface normal. For slits shown in Fig. 1, the plane of incidence is set the x-z plane even at normal incidence for consistency. The wavevector k specifies the direction of incidence and its magnitude k=2π/λ. The angle between k and the surface normal is θ, the angle of incidence. For the TM wave, its magnetic field H always oscillates perpendicular to the plane of incidence. The angle between the reflected/transmitted wave and the surface normal is the same as θ owing to the same mediums (free space) above and below slits. Another reason is the large ratio of λ/Λ, which makes the zeroth order diffraction the only propagating wave. That is, the reflectance and transmittance are exactly the zeroth order reflected and transmitted diffraction efficiency, respectively. Other orders of diffraction are evanescent waves, which decay exponentially along the vertical direction and can contribute to EM fields only in the near-field. In RCWA calculation, the efficiency of each diffraction order is computed from the time-average Poynting vector in the z-direction normalized to that of incidence wave. The directional-hemispherical reflectance/transmittance is the summation of all reflected/transmitted order efficiency. Once the reflectance (R) and transmittance (T) are obtained, the absorptance (A) can be calculated by A=1–R–T based on the energy balance.

2.2 Optical constants

In contrast to multiple geometry variations in this work, only three mediums, gold, silver, and free space, are considered. For most results, gold is selected as the material of metallic lamellae as well as attached features because of its ideal physical and optical properties. For one, the malleability and ductility of gold minimize the difficulty in nanoscale fabrication, especially for non-rectangular metallic strips. Additionally, the resistance of gold to oxidization maintains structure geometry and high reflectivity at the IR incidence. The slit is free space with the dielectric function ε=(n+iκ)²=1, which is the same as those of regions above and below slit arrays. Optical constants n and κ are the refractive index and extinction coefficient, respectively. On the other hand, the dielectric function ε of gold is modeled by the Drude model and is expressed as [20]:

where ω is the angular frequency, γ is the scattering rate or damping constant, ω_p is the plasma frequency, and ε_∞ is the constant that accounts for high-frequency contributions. The scattering rate and plasma frequency of gold are taken from Ref. 21 as γ=216 cm^-1 and ω_p=7.25×10⁴ cm^-1. In the present study, ε_∞ is set to 1 and ε(ω) varies little with ε_∞ up to 2. Though optical constants of gold below 9.9 µm are tabulated in Ref. 22, they do not cause significant optical response difference in the spectral range of interest [16]. However, the Drude model is somewhat restrictive and caution should be taken once the wavelength is near the interband transition threshold. The dielectric functions of noble metals were found to be different for nanostructures and bulk materials [23]. Since this work mainly aims at influences from the modified structure profile, keeping a simple model for optical constants should be appropriate for the clarity and allow comparisons with prior works [9,16].

Silver is also employed as another lamella material for some results in this work due to following reasons. One is to confirm the generality of conclusions drawn from this work to most metallic slits because optical constants of silver can be described by the Drude model as well [9]. The constants for silver in Eq. (1) are ε_∞=1, γ=1.14×10¹⁴ rad/s, and ω_p=1.29×10¹⁶ rad/s. On the other hand, effects of different optical constants can also be obtained by comparing results between gold and silver slits of the same slit geometry. Furthermore, using silver can validate the capability of developed codes by duplicating results in Ref. 9.

2.3 Numerical algorithm

Since originally developed in the 1980s, RCWA has become one of the efficient tools for acquiring optical responses. Taking a binary grating for example, both the dielectric function of materials and the EM fields are expanded in a Fourier series resulting from the structure periodicity. When both Fourier series are substituted into Maxwell’s equations, an infinite series of coupled equations are formed. Enough boundary conditions to those equations are obtained by satisfying the continuity of tangential components in EM fields at region interfaces. Detailed derivations for the TM wave incidence on periodic structures can be found in Ref. 24 and its references thereof. Attention must be paid because only finite equations can be numerically solved with the help of computers such that the accuracy of RCWA strongly depends on the number of attacked equations or corresponding Fourier terms. The sufficiency of Fourier terms in this work will be validated later while brief explanation to RCWA algorithm for slits with attached features is illustrated in the following.

Taking Case B slits in Fig. 1 for example, the TM wave is incident from the free space below the slits (Region I) while the transmitted wave is in free space above the slits (Region III). Region II is composed of mediums gold and free space such that the dielectric function is a periodic function of x. Here, Region II is divided into two slabs with different thickness and the metallic filling ratio is not equal in each slab. Though boundary conditions of RCWA also require continuous tangential EM field components at the slab interface, the additional interfaces between slabs in Region II causes the instability of inverse matrixes during calculation. Such instability can be resolved by the numerically-stable transfer matrix approach in Ref. 25.

The validation of RCWA codes was examined in several ways, including numerical convergence and energy balance check. One way of convergence check is to increase the number of slabs in Region II for the same slit geometry. For example, the reflectance and transmittance of the representative Case A slits can be obtained with one 36000-nm-thick slab or three slabs with total thickness of 3600 nm. Even though the thickness of each slab is varied, the reflectance and transmittance deviation is less than 10^-5 at normal incidence and λ=3 µm. Another way is to increase the Fourier transform terms of periodic dielectric functions in Region II. A total of 361 Fourier transform terms are sufficient for convergence because the relative difference in optical responses of Case D slits is less than 1% as the number of terms increases to 401, and 1001. On the other hand, energy balance is confirmed by the unity summation of R and T if Au is replaced with a dielectric (no absorption).

3. Results and discussion

3.1 Freestanding gold slits at normal incidence

Figure 2 shows the transmittance and absorptance spectra at normal incidence for five cases of slits shown in Fig. 1. The wavelength range is 3 µm ≤ λ ≤ 15 µm and three modes of resonance excitation are exhibited in each spectrum. The resonance is formed by the interference of the forward and backward EM waves within the slit cavity. Such resonances were also known as the organ pipe mode in Ref. 1 or cavity resonances in Ref. 9, Ref. 15, and the current work. The wavelengths of three modes corresponding to slits of Case A and Case E are specifically marked on peaks with arrowheads. Each type of arrowheads represents peaks in the spectrum of distinct slits. Moreover, relative positions of arrowheads emphasize the order of optical responses for slits of five cases in each plot. For example, the transmittance through slits of Case A is the maximum while that of Case E is the minimum at λ=3.4, 5.0, and 10.1 µm as shown in Fig. 2(a). On the contrary, the transmittance through slits of Case E is the maximum while that of Case A is the minimum at λ=3.6, 5.4, and 10.9 µm. Interestingly, the three transmittance peaks of the same spectrum almost remain constant while peaks in other spectra at neighboring wavelengths almost decay from slits of Case A to Case E. The absorptance of five slit cases is plotted similarly in Fig. 2(b). The peak wavelengths marked in the figure and orders among peaks in five spectra at specified wavelength are the same as those of transmittance plots. Furthermore, the three absorptance peaks of the same spectrum also remain relatively constant. The only significant discrepancy between transmittance and absorptance plots is the magnitude order among peaks at neighboring wavelength. In Fig. 2(b), orders of peaks at neighboring wavelengths are not the same to three groups of peak although absorptance by slits of Case E is always a little higher than that of Case A. Detailed discussions of Fig. 2 are in the following.

 

Fig. 2. Optical responses of gold slits at normal incidence: (a) Transmittance; (b) Absorptance. The arrowhead type illustrates the order of optical responses among five slit geometric modification cases at peaks in a spectrum with wavelengths specified by numbers above arrowheads in the unit of micrometer. The relative heights among arrowheads also tell the relative difference of optical responses.

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For highly reflecting metals, resonance wavelengths of cavities with both ends open should satisfy the mathematical relation k_z,m=mπ/d _eff [20], where k_z is the z-component of the wavevector, the subscript and number m is a non-negative integer, and d _eff is the effective cavity depth. Note that k ²=k ² _x+k ² _y+k ² _z while k_y=0 for discussion in this manuscript. Though the phase of the reflected wave generally shifts at the end of the cavity, such shift is neglected here for simplicity. The physical thickness of cavities d is the same order but not identical to d _eff because cavities do not have physical boundaries or symmetric boundaries. While both phase shift and effective thickness may lead to little discrepancy in quantitatively predicting cavity resonance wavelengths from those employing k_z,m=mπ/d, the frequencies of all modes should still be multiple of the same fundamental frequency once the phase shift is negligible [9]. Note that only resonance along the vertical direction is discussed in this work. In fact, the frequencies of three modes in each spectrum indeed are multiples of a fundamental frequency. The angular frequencies correspond to slits of Case A are ω≈1.87×10¹⁴, 3.76×10¹⁴, and 5.54×10¹⁴ rad/s while those of Case E are ω≈1.73×10¹⁴, 3.49×10¹⁴, and 5.23×10¹⁴ rad/s. That is, the peaks at short, medium, and long wavelengths in all spectra represent the third, second, and first mode resonance within slits, respectively. Further confirmation of the cavity resonance and their corresponding mode will be corresponding EM fields shown later.

Since the resonance wavelength/frequency changes among slits of five cases, nanoscale geometric modifications indeed vary optical responses in the IR region. According to the mathematical relation mentioned above, the wavelength of the same resonance mode increases with the slit depth such that the peak wavelengths of Case E slits are longer than those of other cases. Though depths of other four case slits are the same, attached features shift their resonance wavelengths as well. The shift should come from modifications in both the slit profile and boundary conditions at slit openings; that is, the cavity resonances are not simply determined by physical depth of slits [15]. Hence, caution should be paid when using the mathematical condition to predict optical responses of slits of irregular cross-section at resonance. Predictions of resonance wavelengths associated with irregular slit geometry will be considered in the future work.

The comparison between Figs. 2(a) and 2(b) provides more information about geometric modification effects on optical responses. Firstly, even the cavity resonance wavelength shifts by features, the peak wavelengths of transmittance spectra still agree well with those of absorptance spectra. Secondly, the narrow openings of slits with features obviously block incidence more than wide ones such that transmittance peaks of Case A slits are higher than other peaks. However, the role of features in absorptance is not quite clear here. At long wavelengths, the absorptance peak of Case A slits is lower than other peaks, suggesting attached features enhance absorptance somewhat. This argument is acceptable because only the extinction coefficient of gold is not zero for absorption. However, the same argument becomes invalid when absorptance peaks of Case B slits are the lowest at short wavelengths. The reason is the increased reflection from attached features at inlet openings and reduced incidence funneling through the opening. In other words, attached features at inlets indeed benefits absorptance, but they also obstruct incidence wave into slits for absorption. The net effects are the absorptance reduction and the absorptance peak is lower than others. These contradict roles of features at slit inlets will be confirmed later by showing Poynting vectors around features as well as comparing absorptance spectrum of Case B and Case B’.

3.2 Freestanding silver slits

Figure 3 shows the transmittance and absorptance spectra of silver slits for four selected cases. The slit profiles of Case A, B, and D are the same as those of gold slits shown in Fig. 1. The profile of Case B’ is the same as that of Case B while the incidence is on the side without features. The peak wavelengths in the spectrum of Case A slits are specified with numbers like those in Fig. 2. Insets in Fig. 3 are duplication of Fig. 1 in Ref. 9, serving as a validation of codes developed in this work. Though the wavelength domains in Ref. 9 and this manuscript are not the same, the inset can also confirm the physical mechanism, organ pipe modes or cavity resonance modes, for spectral peaks showing in the absorptance and transmittance spectra here. The interested wavelength domains are different between two works because the height/depth of slits is 400 nm and 3600 nm in Ref. 9 and here, respectively. The wavelength of resonance modes is almost proportional to the slit height/depth provided other conditions remain unchanged. In Fig. 3(a), the transmittance peaks of Case A slits are higher than those of other slits and the corresponding wavelengths are very close to those of gold slits. The transmittance spectrum of Case B and Case B’ slits overlap with each other because they physically are the same. Clearly, most conclusions drawn from gold slits are applicable to silver or other metallic slits because their optical constants from the Drude model are similar. Discrepancy in optical constants may lead to reduction in transmittance at resonance, but its wavelength does not shift much. Attached silver features within slits also reduce the transmittance, regardless at inlets or outlets. The transmittance is further reduced once features are located at both slit openings.

In contrast, the absorptance modifications by placing attached features are different at inlets and outlets as shown in Fig. 3(b). When features are attached at the inlets (Case B), their role of obstructing incidence prevails over benefiting absorptance. On the other hand, the features at slit outlets (Case B’) not only help absorptance but also keep the incidence wave trapped within slits. As a result, the absorptance peak of Case A slits is slightly higher than that of Case B slits but a little lower than that of Case B’ slits. Indeed, Case B and Case B’ are the same structures such that their resonances locations can be quite close. Since the only difference is the backward or forward incidence, spectral optical responses can be somewhat different. When features are attached at both slit ends, features play dual roles at the same time and the absorptance peak of Case D slits is between those of Case B and B’ slits. Since the slit profile vary among Case A, B, and D, the resonance wavelengths are different while those of Case B and B’ slits are the same. In fact, the absorptance spectrum of Case B and B’ slits are very close to each other, except their peaks at resonances. Though wavelengths of absorptance peaks are a little different from those of transmittance peaks, their agreement can still be considered pretty well.

 

Fig. 3. Optical responses of silver slits at normal incidence: (a) Transmittance; (b) Absorptance. The geometry of four slit cases is the same as that of gold slits in Fig. 1, except the incidence of Case B and B’ is on opposite sides of slits. Insets of two contour plots are duplications of Fig. 1 in Ref. 9 as a validation of codes employed in this work.

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3.3 Resonance confirmation and angular dependence

The cavity resonance mode and roles of attached features can be further investigated by exhibiting the Poynting vector S and EM fields around slits. The gold Case D slits at normal incidence is taken as an example and λ=3.58 µm is the third resonance mode. Figures 4(a) and 4(b) show the time-averaged Poynting vectors and the magnitude square of the y-component magnetic field in the logarithmic scale (i.e., log₁₀|H_y|²), respectively. The Poynting vector is defined as [20]:

where Re represents the real part of the complex quantity. E and H* are the electric field and conjugated numbers of the magnetic field, respectively. The subscripts y is for the y-component because the plane of incidence is the x-z plane. In Fig. 4(a), arrowheads tell directions of Poynting vectors such that the path of net energy flow is clear. The lengths of arrowheads represent magnitudes of Poynting vectors while the scale has been enlarged such that penetration depths of arrowheads are not the actual representation. Note that the penetration depth (λ/4πκ) of gold is about 11 nm at λ=3.58 µm. The penetration depth is on the same order of feature size in this case. The transmittance (T) and absorptance (A) are specified as 0.383 and 0.477, respectively. Since the Poynting vectors around attached features are critical, those in the middle of slits are omitted and not shown in the figure.

 

Fig. 4. Poynting vectors and the magnitude square of complex magnetic field in the logarithmic scale for gold slits of Case D at θ=0° and λ=3.58 µm: (a) Poynting vectors; (b) the magnitude square of complex magnetic field. The transmittance (T) and absorptance (A) in this case are listed in the figure as T=0.383 and A=0.477, respectively. Lamellae boundary in Fig. 4(b) is marked with grey lines.

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Poynting vectors in Fig. 4(a) clearly confirm discussions about dual roles of attached features. For both features at inlets and outlets, Poynting vectors are somewhat guided by those features into lamellae such that absorptance is enlarged. That is, attached features within slits act like antennas attracting the energy flow. On the other hand, features serve as obstacles to energy funneling through slits because the incidence wave mostly reflects from metallic parts. Accordingly, the Poynting vectors below metallic lamellae and features are smaller than those below slits. Furthermore, since partial energy is also absorbed by slit walls, Poynting vectors become shorter within slits even between attached features at openings. It is reasonable that no obvious Poynting vectors exist inside gold lamella because most transmittance should funnel through slits [17]. Figure 4(b) shows the complete magnetic fields around two slit periods to confirm the resonance mode. Three anti-nodes (the maximum of the magnetic field) are observed within each slit and vertical standing waves in the cavity are clearly seen. The magnetic fields correspond to other peaks in the same spectrum and peaks in other spectra are also plotted but not shown here. Those plots verify the resonance mode specified previously and they also suggest the cut-off wavelength of slits shown in Fig. 1 is less than 11 µm. Here, the cut-off wavelength is the longest wavelength can excite cavity resonance along the vertical direction and such resonance mode is the basic mode. Moreover, since the first and last nodes (the minimum of the magnetic field) are not exactly at z=0 and z=3600 nm, physical slit depth is not the only criteria determining the cavity resonance. Geometric modification certainly can shift the frequency by changing slit characteristics as well as boundary conditions.

 

Fig. 5. Contour plots of optical responses for gold slits of Case D at various angles of incidence and wavelengths: (a) Transmittance; (b) Absorptance.

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Figure 5 shows the transmittance and absorptance contour plots of Case D slits at different angles of incidence. While not showing here, contour plots of other case slits are quite similar and the only difference among plots is the vertical bright band shift in wavelengths due to attached features. Other effects of attached features may be trivial because the cavity resonance is little dependant on the angle of incidence. Figure 5 clearly shows that the enhanced transmittance and absorptance resulting from cavity resonance are at the same wavelengths, regardless of angle of incidence. However, the transmittance peak gets wider in the spectrum and becomes higher at larger angles of incidence. When the direction of incidence is close to the grazing angle (θ>80°), the transmittance enhancement is in a wide band and irrelevant to resonance. Such broad transmittance enhancement comes from the effective medium behavior of metallic slits, which is obvious at wavelengths much longer than structure dimensions [15]. The role of the extinction coefficient becomes insignificant and the structures act like a virtual dielectric medium with a large refractive index. Most incidence wave can funnel through slits because the reflectance is close to zero at large angles, which are close to the Brewster angle of the virtual dielectric. On the other hand, the absorptance peaks diminish at large angles of incidence because the absorption reduces with small extinction coefficient.

3.4 Slit density effects

The normal transmittance, absorptance, and reflectance for freestanding gold slits are shown in Figs. 6(a), 6(b), and 6(c), respectively. Two square features are attached at both inlets and outlets such that the slit profile is the same as that of Case D shown in Fig. 1. Four spectra in each figure represent different slit densities achieved by varying the metallic strip length (l) and the structure period (Λ). The slit density is defined as w/Λ, the ratio of slit width with respect to the slit period. Note that the solid line in blue color symbols exactly the spectrum of Case D slits with Λ=800 nm and the slit density is about 0.063. The periods of other three spectra are Λ=400, 1200, and 1600 nm such that the slit density ranges from 0.031 to 0.125.

 

Fig. 6. Optical responses for gold slits of Case D with selected slit densities (w/Λ=0.031, 0.042, 0.063, and 0.125) at normal incidence: (a) Transmittance; (b) Absorptance; (c) Reflectance. The legend in Fig. 6(a) specifies the slit period, lamellae width, and slit density of structures discussed here while the inset in Fig. 6(b) correlates the transmittance/absorptance with slit density at different spectral regions. In the regions marked with lines with arrowheads, both the transmittance and absorptance decrease with reducing w/Λ ratio. In contrast, the transmittance decreases but the absorptance increases with reducing w/Λ ratio in the spectral region marked with green line sections without arrowheads.

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In Fig. 6(a), four transmittance spectra show three peaks at the same wavelengths because the cavity resonances are independent of the strip length (l). The transmittance through slits of higher density is larger than that through lower density slits within the whole spectral range. However, the transmittance increment among spectra is not proportional to the slit density and the increasing amount changes with the wavelength. Within the spectral range, the transmittance through slits is mostly larger than the slit density, specifically at resonance wavelengths. The minimum of the spectrum is 0.08 with the slit density 0.125 while the minimum of other spectra is less than 0.04. One interesting characteristic of the transmittance peaks is their full-width-at-half-maximum (FWHM) gets wider with higher slit density. The same characteristic is also demonstrated in the absorptance spectra shown in Fig. 6(b).

Except the four absorptance spectra, Fig. 6(b) also clearly exhibits effects of slit density on the transmittance and absorptance spectrum. At three wavelengths of resonance, the low slit density guarantees low transmittance while the absorptance peaks show almost a reverse order. The lowest absorptance peak is for slits with the highest slit density and those peaks are almost identical for slit density of 0.031 and 0.042. As specified in the inserted table, two types of green line symbol two kinds of spectral region for four absorptance spectra. In the regions marked with lines with arrowheads, both the transmittance and absorptance decrease with reducing w/Λ ratio. In contrast, the transmittance decreases but the absorptance increases with reducing w/Λ ratio in the spectral region marked with green line sections without arrowheads. The shown absorptance peaks can be considered as the maximum allowable absorptance by slits because further reduction in slit density makes structures close to a thick film. Note that the absorptance of a gold film calculated from thin-film optics is less than 0.1. Since the absorptance at the highest slit density also approaches to zero under full transmittance, the existence of absorptance maximum for slits at resonance is understandable. On the contrary, both transmittance and absorptance reduce with low slit density when the wavelength is outside the range of resonance. Though the FWHM of absorptance peaks are not the same as those of transmittance peaks, it still gets wider with higher slit density.

Since the transmittance spectra do not interfere but absorptance spectra intersect with one another at resonances, it is better to plot the reflectance spectra to fully understand optical responses. Figure 6(c) shows the reflectance (R) spectra for slits of four densities and R=1- T- A following the energy balance. Obviously, the high slit density suggests the low reflectance in the whole spectral range, especially around the resonance wavelengths. The minimum in those reflectance spectrum can be close to zero while the maximum is higher than 0.85. Such large reflectance difference within the IR range is very attractive for wavelength-selective emitters or sensors. Furthermore, the wide-band low reflectance are little dependant on the angle of incidence, making it more appealing to mentioned applications. Development of devices composed of slits with tailored features can be part of the next work.

3.5 Complex slits

After uniform slits of several features have been looked into, the remaining discussion will focus on non-uniform slits within a period. These periodic structures contain more than one lamella in a period and show promising application potentials [12,26]. Since the grating synthesized from two or more binary gratings is called complex gratings [26], multiple slits with different features in an array here called the “complex slits” and hereafter. In contrast to them, arrays of slit with uniform geometry are referred to “simple slits” during comparison later. As an initiation study to optical responses of complex slits, only slits of two types in a period are discussed here. Moreover, these slit profiles are from Case A, B, C, and D as shown in Fig. 1.

Figure 7 shows the normal transmittance and absorptance spectra of four selected slits combinations. Dimensions of four slits and their sketches are illustrated in the inset of figure. Periods of all complex slits are set to 1600 nm and strip lengths are about 750 nm such that the slit density is the same as most simple slits discussed before. The two spectra plotted with short dashed line (in black) and solid line (in blue) represent optical responses of complex slits composed of slits without any attached features but different slit width. For the dashed line, the slit widths are 30 and 50 nm while those for the solid line are 40 and 50 nm. The dashed line spectrum shows six peaks within the spectral range in both transmittance and absorptance. In Fig. 7(a), three higher transmittance peaks come from the resonance within the 50-nm-wide slits, and one can match wavelengths of peaks as those in Fig. 2(a). In other words, the other three lower transmittance peaks result from resonances within the 30-nm-wide slits. Another support to the argument above is that wide slits without features can usually allow high transmittance through at resonance. Since the resonance wavelengths of two slit types are far apart and the FWHM is not wide enough, six transmittance peaks are distinctly shown. If the slit widths are sufficiently close, such as 40 and 50 nm, the six peaks are partially overlapped as the solid line shown in the figure, specifically at long wavelengths. The area below the merged peaks in the spectrum is larger than that of the dashed line because the transmittance through slits can be considered as the summation from two slits independently. In other words, the transmittance spectrum is a kind of synthesis from spectra of two slit types like the way slit profile comes from. Note that slit density of the two original spectra is half of those in Fig. 1 because a period of 1600 nm has only one slit of each type. As a result, the maximum transmittance of partially-overlapped peaks is still lower than that shown in Fig. 2(a).

 

Fig. 7. Optical responses of gold complex slits at normal incidence while two slits in a period have different profiles: (a) Transmittance; (b) Absorptance. The inset in Fig. 7(a) serves as the legend showing all dimensions of discussed complex slits.

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The center line (in red) plots optical responses of slits composed of Case A and B while the long dashed line (in green) plots optical responses of slits composed of Case A and C. In Fig. 7(a), six transmittance peaks plotting with the center line are not quite clear and they look like three wide peaks. The resonance wavelengths of two slit types are sufficiently close such that two peaks almost merge into one although six apexes can still be observed. When slits of Case B are replaced by those of Case C, the six peaks become more distinct as shown by the long dashed line. It is interesting that the FWHM of partially-overlapped peaks among plots are alike while the areas below peaks are not the same. Note that the FWHM of peaks is not the same at different wavelengths; however, they are pretty similar if they are plotted in wavenumber ν(cm^-1).

Figure 7(b) shows the absorptance spectra of four selected complex slits. Absorptance peaks of neighboring wavelengths in the same spectrum indeed merge into a wider and higher one, especially at long wavelengths. Similar to the transmittance, each absorptance spectrum can also be considered as a synthesis from two absorptance spectra of distinct slit types. However, some discrepancy also exists between the transmittance and absorptance spectra. For one, the FWHMs of absorptance peak merged from two closely-nearby peaks are narrower than those of transmittance, such as the center line for Case A and B slits. In contrast to them, the FWHMs of absorptance peaks in solid and long dashed lines are similar to each other and close to those of transmittance peaks. For the short dashed line, each peak has two apexes and a valley and the FWHM of peaks is the widest one among four absorptance spectra. Another discrepancy between transmittance and absorptance spectrum is their peak value compared to that of simple slits. The absorptance, unlike transmittance, does not significantly reduce in spectra of complex slits. Such discrepancy may come from different roles of metallic strips played in the absorptance and transmittance spectrum. The strips defy the cavity characteristics for transmittance enhancement at resonance such that each slit type contribute enhancement distinctively at its resonance excitation. Once the density of each slit type reduces, the transmittance shrinks. On the other hand, the metallic strips are the only part absorbs energy regardless of resonance. The absorptance usually enlarges with the transmittance because energy is more absorbed during incidence wave funneling through slits. As a result, absorptance peaks are strongly correlate to transmittance peaks but not very sensitive to the slit density. Furthermore, the attached features also benefit absorptance such that the valley between two apexes in a peak is somewhat averaged out.

Figure 8 shows the transmittance and absorptance spectra of slits with various slit density, and the two types of slit come from those of Case A and D. For the transmittance, the overlapping of resonance peaks is partial at long wavelengths while peaks at short wavelengths do not overlap at all. Six apexes in a spectrum are obvious and their wavelengths are the same as those of Case A and D slits shown in Fig. 2(a). Furthermore, these peak wavelengths do not shift with the slit density because the cavity characteristics strictly determine the resonance wavelength. On the other hand, the increased slit density raises the transmittance spectrum and the ratio is approximately proportional to the slit density. In contrast, increment in slit density does not assist absorptance significantly. Absorptance spectra of four slit densities are similar and they intersect with each other in the spectral region several times. The FWHM of absorptance peaks is narrower than that of the transmittance peak, especially at highest slit density drawing with a center line (in black).

 

Fig. 8. Optical responses of gold complex slits at normal incidence: (a) Transmittance; (b) Absorptance. The two slit profile in a period come from those of Case A and D while the slit period and lamella width are different for each spectrum. The inset in Fig. 8b is the solid line of absorptance spectrum redrawn in wavenumber ν (cm^-1), but every number above arrowheads is the peak wavelength λ marked in micrometer.

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Additional information can be obtained and some previous discussions can be confirmed by comparing Fig. 8 with Fig. 7. For one, two peaks from basic modes at longest wavelengths can merge into a wide one easier than those of other modes. However, the FWHM of the merged peak is not the widest if the spectrum is plotted in wavenumber as shown in the Fig. 8(b) inset. In fact, they can merge better because their apexes representing the resonance frequency are close enough. Another one is that the left apex is higher than the right one in the transmittance spectra while both values of apexes are close in the absorptance spectra. The significant difference can be explained intuitively. When more features are attached, such as slits of Case D, transmittance peaks definitely become lower than those of Case A slits at resonances. However, the dual roles of attached features on absorptance may compromise each other such that the difference in absorptance apexes is not obvious.

4. Conclusions

We initiated a numerical study on IR optical responses of metallic slit arrays with slight geometric modification in their cross-sections. Even though a simple model of optical constants is employed, the resonance wavelengths can shift significantly by tiny attached features or nanoscale profile variation. Accordingly, all optical responses are strongly modified, and they depend on the slit width, attached feature geometry and size, and positions of attached features. Attached features within slits may reduce the transmittance peak but they play dual roles in absorptance. They in a way benefit absorptance by guiding Poynting vectors into the metallic lamellae while their existence increases the reflectance in another way. Silver slits of several profiles have been employed for codes validation and argument generality confirmation. Poynting vectors and the magnetic fields within slits demonstrate the cavity resonance excitation and its corresponding mode. The transmittance and absorptance enhancement dependence on the angle of incidence is similar but significant discrepancy shows up at large angles due to the effective medium behavior. The idea of complex slits has been introduced in this work by including two or more types of slit profile in a structure period. A wide-band transmittance or absorptance enhancement can be fulfilled with carefully-designed complex slits. Another freedom in designing IR wavelength-selective devices has been obtained with the in-depth understanding to nanoscale geometric modification on optical responses of slits. The next step of future work should be the experimental validation with accurately-fabricated nanostructures.