1. Introduction

Phase resonances (PRs) in compound transmission gratings (CTGs) and compound reflection gratings (CRGs) have extraordinary optical properties that have been attracting significant interest [1–8]. Similar resonance modes have also been studied in commensurate metallic gratings (i.e., long-period gratings containing several identical, deep, subwavelength-sized grooves per period) [9,10]. PRs (also called π resonances) are resonant optical modes composed of coupled p-polarized waveguide cavity modes (WCMs) within multiple dissimilar grooves within the unit cell of CTGs and CRGs. The dissimilarity, or asymmetry of the grooves breaks a particular mirror symmetry in a simple lamellar grating (SLG) (i.e., a grating with identical uniformly spaced grooves) and lifts the restriction of the phase relationship of the fields within the cavities having to adhere to Floquet’s theorem, and in the case of PRs in CTGs with two dissimilar grooves in the unit cell, the phase difference between the fields in the two coupling cavities is approximately π radians. The asymmetry of the grooves can either be different relative sizes of the grooves, different dielectrics filling the grooves, or nonuniform spacings of the grooves in the unit cell. In prior works on PRs, the optical properties of p-polarized PRs have been studied in CRGs [1, 2], CTGs [3–5], in CTGs with multiple layers [6, 7], and it has been also been shown that s-polarized PRs exist in CTGs [8]. Most of these prior works describe in great detail many properties of PRs in structures that support WCMs that have moderate to high Q factors and moderate to small bandwidths (more specifically, the transmission peaks produced by the WCMs are clearly identifiable). The properties described in these prior works include the coupling the fields of the WCMs in the grooves of the unit cell, with the fields having a π radians phase difference, and producing either sharp changes in the efficiency curves of the reflected order in reflection gratings [1, 2], or a narrow bandwidth transmission dip in a wider bandwidth transmission peak in transmission gratings as shown in Fig. 1 (i.e., a ”forbidden channel within a permitted band” [7]) [3–5, 8]. Also, Crouse et. al. has argued that CTGs with two dissimilar grooves within the unit cell can support PRs that produce counter-propagating mutually-canceling circulations of light leading to a large field concentration and amplification but little to no net transmission of light through the grooves at the peak of the PR [8].

 

Fig. 1 (a) A cross-sectional schematic of the two-groove-per-unit-cell CTG that supports PRs. If the grooves are identical and uniformly spaced, then the CTG reverts to the parent simple lamellar grating (SLG) that does not support PRs. Mirror symmetry about Plane A needs to be broken for the structure to support PRs. (b) The simulated transmittance (solid) and re-flectance (dashed) for normal incident light for a CTG with WCMs with moderate Q and with the structure’s dimensions tuned to produce a PR at what was initially (i.e., before a dissimilarity between the grooves is introduced) the very peak of the transmittance. Again, note that before any groove dissimilarity is introduced, the transmission peak produced by the WCMs is a smoothly varying Gaussian shaped peak from 10 GHz to 14 GHz. The PR introduces a complete inversion of the transmissivity/opacity of the film. The structure has a period Λ = 19.05 mm, both grooves have widths of w₁ = w₂ = 1.905 mm, but with Groove 1 having ε₁ = 1 and Groove 2 having ε₂ = 1.5; the height of the grooves is h = 9.525 mm, and the width of the aluminum wire with Groove 1 on the left and Groove 2 on the left is s₁₂ = 2 mm, the substrate and superstrate is vacuum. A RCWA algorithm is used to simulate the optical properties of the structure shown in this figure [16].

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The effect of a narrow bandwidth transmission peak within a larger bandwidth reflection peak in a system composed of coupled oscillators has been receiving significant attention recently, particularly in coupled optical ring resonators [11–14] and in compound split ring resonator (SRR) arrays [15]. Several works have shown that coupled ring resonators are a classical, all-optical analog to electronically induced transparency (EIT) in atomic systems [11–14]. In EIT, destructive interference between in-phase and out-of-phase normal modes results in no power transfer to the system even when excited by some signal [11]. The classical all-optical analog to EIT was experimentally demonstrated in several works on ring resonators [12, 14]. EIT-like behavior can occur in wider range of structures, including SRR arrays, micro-resonator arrays, and multilayer systems where particular optical modes are disallowed due to symmetry-breaking. In the type of CTGs studied in this work, PRs do indeed occur as a result of the coupling of normal optical modes within oscillators (i.e., the two grooves of within the unit cell). However, this coupling creates destructive interference for the transmitted beam, such that all the energy of the excitation beam is reflected or trapped within the CTG. Thus within CTGs, there is electronically induced reflection or absorption as opposed to EIT.

This paper describes new aspects of PRs in the frequency and time domains, and also studies in greater detail some already known properties of PRs. In the frequency domain, a design methodology is briefly described that allows one to design a CTG with PRs that have particular frequencies and momenta. This design methodology introduces a closed-form equation that describes the dispersion of PRs in CTGs with two grooves per unit cell with the two grooves being only slightly different relative to each other. Also described in this work are the dependencies and evolution of the optical properties of the PRs on the amount of dissimilarity of the grooves in the unit cell. It will be shown how PRs and their electromagnetic properties arise either immediately (for lossless structures) or gradually (for structures with optical loss) upon the breaking of a particular symmetry of a SLG. These electromagnetic properties include: the Q of the PR, the electromagnetic field intensities of the PRs, and the relative phase difference of fields within the groove. The second part of this paper studies PRs’ properties in the time domain, including the excitation of a PR by an incident beam, the time-evolution of the PR and the scattered radiation, the concentration and guiding of light, and methods to extract energy from the PRs. Application of PRs to optical filters and corrugated surface antennas are discussed. Even though the effects of PRs are the most pronounced in structures with low optical loss, examples are given in this work of CTGs with PRs in the infrared spectral range. Also discussed is how such structures can potentially be used in electromagnetic filters, antennas and infrared sensing applications.

2. Frequency Domain Analysis of Phase Resonances

We will briefly summarized two methods to design CTGs with two dissimilar grooves in the unit cell (a 2-groove CTG) with PRs with particular frequencies, momenta and Q values. The first method is straightforward and has been described in prior works on PRs [8,16] and is only summarized here. In this first method, one starts by first considering a SLG with one groove per unit cell and adjusting the dimensions of this groove to first tune the wavelength of the WCM to the desired value, as described in [16]. Then two of these grooves are introduced into the unit cell while keeping the period of the unit cell small enough to avoid far-field diffraction. Next, a relative difference or dissimilarity in the grooves is introduced that breaks mirror symmetry about Plane A in Fig. 1. Regardless how small this perturbation is, a PR occurs in lossless structures at some wavelength within the bandwidth of the two WCMs and produces a complete inversion of the transmissivity/opacity of the structure. The smaller the perturbation is, the narrower the bandwidth and higher the Q of the PR will be. Yet too small of a perturbation will produce such a narrow bandwidth PR that it may not be resolved by most electromagnetic modeling programs unless an extraordinarily fine frequency sweep is performed. Next, the bandwidth of the PR can be made larger by increasing the dissimilarity between the grooves. Finally, the separation between the two grooves and the widths of the grooves are adjusted to tune the wavelength of the PR (i.e., λ_pr). As the separation between the two grooves within the unit cell is decreased, there is a general decrease λ_pr and an increase in the bandwidth of the PR.

Such a design process can be performed using commercially available finite element or finite integration algorithms (e.g., HFSS and CST Microwave Studio); indeed, such programs were used in this work to check the results and to provide information on the time domain behavior, and the optical characteristics of finite-sized, nonperiodic versions of the structure. For the initial design of the structure and to generate the full dispersion curves of the phase resonances, a rigorous coupled wave algorithm (RCWA) algorithm was used and is significantly faster and more able to resolve narrow bandwidth features such as phase resonances [16,17]. Additionally, the RCWA algorithm can be easily programmed in Matlab and make use of its genetic algorithm toolbox to optimize a CTG to have particular properties. However, with the RCWA used in this work being able to solve over 10,000 frequency points in less than 30 s on a typical desktop computer, optimization can be done manually.

The second method uses an equation that describes the dispersion relation for a CTG structure with two dissimilar grooves in the unit cell, but with the dissimilarity being small [18]. In other words, the two grooves initially have the same height h, widths w_g, dielectric constants ε_g, and are uniformly spaced apart from each other, but then one of these values is infinitesimally perturbed to break mirror symmetry about Plane A in Fig. 1. The resulting CTGs will support PRs with frequencies (ω_pr) and momenta (k_x) for which the following equation is satisfied [18]:

Equation (1) is only used to adjust the structural and compositional properties such that the structure will support a PR at the desired frequency and then to identify or locate (in frequency) the PR at this frequency with the RCWA or other modeling algorithms. Once located, the frequency of the PR can be tracked as it changes as one adjusts other aspects of the device to tune the Q of the PR. (For CTGs significantly perturbed away from their parent SLG, the frequencies of the PRs in the CTG may be significantly different than what Eq. (1) predicts, as is seen in Fig. 2. The reason why one may want to further modify the structure is because the PRs for which Eq. (1) applies have Q values that are very high for lossless structures. To reduce the Q (and increase the bandwidth), the dissmilarity between the grooves is increased.

 

Fig. 2 The dispersion curve calculated using the RCWA for a CTG with Λ = 19.05 mm, h = 9.525 mm, w₁ = 11.43 mm, w₂ = 5.08 mm, s₁₂ = 0.3175 mm, ε_t = ε_b = ε₁ = ε₂ = 1, and for p-polarized incident light. The dispersion curve shows the transmittance of the structure for p-polarized light, showing that the PR inverts the transmissivity/opacity of the film, has a large Q, and has a negative group velocity. Also shown is the PR dispersion curve predicted by Eq. (1) for a perturbed SLG with Λ = 19.05 mm (Λ is actually twice the period of the SLG), h = 9.525 mm, w_g = 8.255 mm, s₁₂ = 1.27 mm, ε_t = ε_b = ε₁ = ε₂ = 1. The frequencies of the PRs predicted by Eq. (1) differ from what is obtained using the RCWA because the CTG is significantly perturbed away from the parent SLG for which Eq. (1) is most accurate.

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It will be seen later in this work, that if one can have near perfect control of the sizes structural features and composition (i.e., have very tight tolerances), then PRs with arbitrarily high Q factors can be obtained in low loss structures. Yet in practice, there is always some variation in structural feature size and material parameters. With devices that operate in the microwave spectral range, a high degree of control of feature sizes can be obtained using computer numerical control milling machines (CNCs). Whereas for structures that support PRs in the optical spectral regions, microfabrication processes need to be employed that produce much greater variability in structural feature sizes. The effect of these variations in feature size and composition is to broaden out the bandwidth of the PR, as is seen in numerous experimental papers on PRs, including [2, 19].

The motivation for this work was to study whether PRs in CTGs can be configured to serve as add-drop filters (ADFs) and optical antennas, as has been done with photonic crystal structures [20]. However, a couple problematic aspects of the optical properties of PRs had to first be addressed. In a prior work by Crouse et. al. [8], it became evident that the PRs have wavelengths within the bandwidth of the two coupling WCMs, with the exact wavelength being determined by the groove dissimilarity and groove separation. But outside the bandwidth of the two coupling WCMs, the structure was not very transmissive, a property that is not desirable for ADFs. To ameliorate this issue, a well known fact was drawn upon, namely that in the absence of diffraction, the off-band transmissivity (i.e., outside both the WCM and phase resonance bandwidths) for p-polarized light is largely determined by the ratio of the area occupied by air-filled grooves to the total area of the unit cell. Hence the question arose concerning the necessity of having moderate to high Q WCMs in order to support phase resonances, or whether very low Q WCMs that would yield high off-band transmission are sufficient. To address this question, the bandwidths of the two p-polarized WCMs were increased to the maximum extent possible by increasing the widths of the grooves to as large of values as possible (while still having Λ < λ_pr to ensure that no diffraction occurs) so that the structure was highly transmissive for all wavelengths below the onset of diffraction, except of course for λ = λ_pr. The resulting structure is very similar to the CTG shown in Fig. 1, except for having significantly narrower metal wires. In particular, the structure has a period of Λ = 19.05 mm, vacuum for the superstrate and substrate, two vacuum filled grooves per unit cell each with a height of h = 9.525 mm but with one groove (Groove 1) having a width of w₁ = 11.43 mm and the other groove (Groove 2) having a width of w₂ = 5.08 mm, and with the width of the perfect electric conductive (PEC) wire separating Groove 1 and Groove 2 as s₁₂ = 0.3175 mm (thus the width of the second wire within the unit cell is s₂₁ = 2.2225 mm). Also, the aspect ratio of the grooves (i.e., height/width) is at most 2:1, thus a larger dielectric constant material is not needed in the groove that would allow the height to be reduced while keeping λ_pr constant. The full dispersion curve (obtained using RCWA), showing the transmittance of the structure as a function of energy and in-plane momentum (i.e., k_x) is shown in Fig. 2. At normal incidence, the Q of the PR is approximately 2000, yet at a 45° incident angle, the Q for the PR decreases to approximately 87. Figure 2 shows that the structure is predominantly transmissive except at the frequency of the PR, at which point the structure is entirely reflective; hence the transmissivity/opacity of the lossless film is inverted at this frequency due to the excitation of the PR.

When looking at the dispersion curve, it is evident that this band of PRs, except for at k_x = 0, has a negative group velocity. Thus, an off-normal incident beam of in-plane vector component k_x will excite a PR of the same k_x and frequency, but with the direction of the in-plane vector component of the group velocity and Poynting vector being opposite those of the incident beam. This property by itself is not unusual, since they occur with Wood’s anomalies, certain surface plasmon modes, diffraction modes, and many modes within photonic crystals. Yet the PR’s unique and highly tunable optical properties (tunable via structure dimensions and composition) make these modes compelling for use in optical and microwave devices that guide and filter radiation, as seen in a later section of this work in which a time-domain analysis of the phase resonances is performed.

Examples of the field intensity profiles and Poynting vector plots for normal incident light are given in many prior works on PRs [1–8], but Fig. 3 show interesting aspects of the optical properties of PRS when light is incident on the structure at off-normal incidence angles, in this case at 45° relative to the y-axis. (Skigin et. al., in [7] shows a field distribution for light incident upon a 2-groove CTG at a 30° incident angle, and Depine et. al., in [1] shows field distributions for PRs in CRGs under illumination from several off-normal incident angles, yet neither show the Poynting vectors for PRs.) In particular, Fig. 3 shows the simulated intensity plot of |H_z|² for an incident beam (incident from the top of the structure) of intensity |H_z,incident| = 1 and a 45° angle of incidence, showing that the intensity of the fields in the two grooves within the unit cell are highly amplified relative to the incident beam by a factor of 40 in Groove 1 and 80 in Groove 2. Also, the Poynting vector shows that the group velocity of the phase resonance is negative, i.e., the flow of energy along the surface due to the phase resonance is in the opposite lateral direction compared with the incident beam that has excited or produced the phase resonance. The aspect of the π radians phase difference between the fields within the two grooves is shown later in this work.

 

Fig. 3 (a) A cross-section of one period of the CTG described in Fig. 2 showing |H_z|² for the p-polarized phase resonance (point (+) in Fig. 2 (ν = 8.46 GHz, k_x/K = 0.38)) for a p-polarized incident beam |H_z,incident| = 1 and 45° incident angle. The intensity amplification (i.e., |H_z,max| // |H_z,incident|²) is only ∼ 87 for this structure and at this angle of incidence. (b) The Poynting vector showing that energy is propagating in the −x̂ direction along the surfaces of the grating even though the Poynting vector for the incident beam has a positive k_x value. This difference in the direction of the flow of energy of the incident beam and the PR is in agreement with the negative group velocity shown in Fig. 2.

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3. Electromagnetic Fields Evolution upon Symmetry Breaking

In this section, a study is undertaken concerning the relative strengths of the electromagnetic field components in the cavities, superstrate and substrate as PRs are excited in CTGs with and without optical loss. To facilitate this discussion, the ẑ component of the magnetic field in the structure, as expressed in the RCWA algorithm that uses a surface impedance boundary condition (SIBC) that includes optical loss in the metals and dielectric materials [7, 16, 17], is given below:

Superstrate:

Substrate:

Grooves

As for the fields in the grooves (g = 1 for Groove 1 and g = 2 for Groove 2), the fields are expressed as follows:

As was noted in a prior work describing PRs within CTGs [8], two of the most striking aspects about PRs are the extraordinarily high Q values of the resonances and the ability of the structures to support PRs even with the smallest possible perturbation in relative groove composition or structural feature size. The Q is determined either by calculating the by ω_pr/FWHM where ω_pr is the frequency of the reflection maximum (transmission minimum) associated with the PR, and the FWHM is the full width half maximum of this reflection peak (or transmission dip). Alternatively, the Q of this mode can be obtained by assessing the maximum intensity amplification produced by the PR, i.e., |H_z,max|²/|H_z,incident|²; both methods of calculating Q give almost identical values. The important thing to note however, is that the Q of these PRs can be made arbitrarily high, limited only by the optical loss within the materials of the structure and structural feature size variations. For structures with large groove asymmetry, or dissimilarity, the Q is small, but as the asymmetry is reduced (i.e., as the grooves become identically composed and equally spaced), the Q increases and goes to infinity. For example, for lossless structures even a minuscule relative difference in the dielectric constant of the two grooves (i.e., ε₁ and ε₂) or the widths of adjacent grooves (i.e., w₁ and w₂), the initially large transmission at the peak of WCMs will go entirely to zero over an exceptionally small bandwidth; the smaller the difference, the smaller the bandwidth and the larger the intensity amplification [8]. The Q and intensity amplification of incident light of the PR as a function of an ”asymmetry factor” f is shown in Fig. 4. The factor f introduces a difference in the grooves by changing ε₂ relative to ε₁ by an amount ε₂ = f · ε₁. The starting structure (i.e., when f = 1) in this Q analysis is a SLG of period Λ = 19.05 mm, with two identically composed and identically spaced vacuum filled grooves of width w = 8.89 mm and height h = 9.525 mm, with vacuum as the superstrate and substrate. To be accurate, the true period for the structure when f = 1 is reduced by half to Λ/2 and the unit cell includes only one of the identical grooves instead of two as would be the case when f ≠ 1 when the period is Λ.

 

Fig. 4 The Q of the phase resonance for p-polarized, normal incidence light. The starting structure (when f = 1) structure has the dimensions: Λ = 19.05 mm (actually the true period for the structure when f = 1 is Λ/2), h = 9.525 mm, w₁ = w₂ = 8.89 mm, s₁₂ = s₂₁ = 0.635 mm, ε₁ = 1, and ε₂ = f · ε₁ with f being the asymmetry factor that introduces a dissimilarity between the grooves. It is seen that as f → 1, the Q of the structure becomes infinitely large.

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As is seen in Fig. 4, for lossless structures as f → 1, the Q and field amplification go to infinity, the bandwidth of the PR goes to zero (the bandwidth is inversely proportional to the Q). For the case when f = 1, namely when the grooves are identically composed and uniformly space and the CTG becomes a SLG, PRs do not occur; Floquet’s theorem does not allow a solution that has a normal incident field and the fields within adjacent grooves to have a π radians phase difference. Thus for lossless structures, there is no continuous evolution of the fields as f approaches and then assumes the value of 1; the field intensities diverge to infinity for f approaching 1 but then abruptly become finite and small (relative to the field intensities of the PRs) when f = 1 and PRs are not excited.

The situation is more complex when optical loss is present in the system, as shown in Fig. 5. This structure studied in these figures is the same CTG structure as was modeled in Fig. 2, but with all the structural dimensions having been scaled down in size by a factor of 3175, and with the metal as aluminum (with the complex optical permittivity taken from [21]). Additionally, for this groove dissimilarity analysis the grooves have the same widths and ε₂ is varied relative to ε₁. More explicitly, Λ = 6 μm, w₁ = w₂ = 2.8 μm, 3 μm, and with ε₁ = 1 and ε₂ = f with f = 1 → 2. The frequencies of the PRs in this scaled down device are expected to be scaled up by a factor of 3175 to approximately 31.75 THz (or λ = 9.45 μm), however the aluminum in the structure has much more optical loss, dampening the PRs, broadening their bandwidth, and shifting their energies to lower values. Yet even with significantly larger optical loss, Fig. 5 shows that PRs are still strongly excited. However, because of this optical loss in the structure, the fields do not diverge to infinity as f → 1 as they did in lossless structures, but the PR become over damped such that the magnitude of the fields associated with PRs peak at a particular value of f, but then gradually diminish as f → 1 until they are not cannot be excited at all when f = 1, as seen in Fig. 5. Additionally, when the dissimilarity between the grooves become large, the intensities of the fields in the grooves and ±1 order Floquet modes that compose the PR decrease relative to the exciting incident beam and the phase difference between the fields in grooves becomes lower than π (Fig. 5). Both of these aspects indicate that the PR is becoming weaker, with a lower Q and larger bandwidth as f increases. Thus, it is seen that for structures with lossy materials, such as devices that operate in the optical spectral range, there is an optimal groove dissimilarity f so that the PR is maximally excited. This optimal groove dissimilarity however, depends in complex way on the structural geometry and materials properties.

 

Fig. 5 Top: The transmittance and absorption for a scaled-down version of the device that operates in the infrared spectral region (Λ = 6 μm, w₁ = w₂ = 2.8 μm, h = 3 μm, s₁₂ = s₂₁ = 0.2 μm, ε₁ = 1, ε₂ = f with f = 1 → 2 in steps of 0.06, and aluminum wires with the optical parameters (n and k) obtained from [21]). Even though the PRs have a broader bandwidth and are dampened, their effects on the transmittance are still strong, especially when the asymmetry factor f is greater than 1.4. For structures with optical loss, there is an optimal value for f such that the phase resonances are strong but not overdamped (as occurs in this structure for f < 1.52). Bottom: The intensities of the ±1 order Floquet modes in the superstrate (|R_±1|²), the 0^th order Floquet mode (|R₀|² or specular reflection), and the 0^th order cavity mode |a₀|². As f becomes greater than 1.76, the magnitudes of |R_±1|² and |a₀|² relative to |R₀|² decrease, indicating that the phase resonance is becoming weaker as f increases beyond 1.76.

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4. Time Domain Excitation of the Phase Resonance

In this section the optical response in the time domain of a finite-sized CTG that supports a PR is studied using a finite integration technique (FIT) algorithm (CST MWS). In this time domain analysis, the evolution of a time signal is monitored as it passes through and interacts with the structure, first as it excites the PR, and then as the PR slowly decays as it radiates away energy (Fig. 6).

 

Fig. 6 The change in the field energy contained within structure as the incident pulse passes through the system. The initial large increase in energy is the incident beam passing through the structure while the slow decay of field energy is caused by the slow release of energy trapped within the structure by the PR. This decay rate is inversely proportional to the Q of the PR.

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The CTG that is studied in this section is the same as studied previously in this work, with Λ = 19.05 mm, h = 9.525 mm, w₁ = 11.43 mm, w₂ = 5.08 mm, ε₁ = ε₂ = 1 = ε_t = ε_b = 1, and the metal wires are PEC. While an infinitely periodic structure is not necessary to support PRs [22–24], we observed that PRs were less strongly excited near the edges of a finite sized grating. Thus a large structure with 86 repetitions of the unit cell was used, enough to support PRs in most areas of the structure except for the very ends of the structure (i.e., within 5 unit cells of each end). Because we are modeling the performance of a finite structure, the grating is terminated in open (radiating) perfectly matched layer (PML) that allows for energy to escape the system without causing artificial internal reflection. The PML boundaries truncate the su-perstrate and substrate at a distance of 5 mm above and below y = 0 respectively. The grating was also modeled to be infinitely long in the ±z direction using a Perfect Magnetic Boundary which allows for mirroring of the fields at this boundary, creating periodicity of the CTG in the ±z direction, and in effect creating infinitely long grooves, wires and metal sheets. The time domain simulation is carried out using a pulse (Gaussian shaped in the time domain and having a spatial dependence of a plane wave) that has a frequency range of 9.6 GHz to 10.2 GHz centered about 9.9 GHz, which matches the frequency of the PR (as computed using both RCWA and CST).

Three probe monitors are used in this analysis with each one recording the electromagnetic field strength or energy as a function of time at a different point within the system; the locations of probe monitors are within the CTG (Fig. 6), 1.5 m above the CTG (Fig. 7), and 1 m below the CTG (Fig. 8). For the latter two probes, the large distances away from the structure disallows for the possibility of them recording any near field effects, i.e., the large, trapped fields of PRs within or near the structure. Additionally, for the probe in the superstrate (input side), the large distance away from the structure yields a separation (in time) of its recording of the incident beam and its recording of the radiation emitted by the PR. The time simulation is carried over a duration of 150 ns, time enough for remnants of the incident pulse to leave the neighborhood of the structure and the PR to have decayed such that the total energy in the system is −50 dB relative to the maximum of the incident pulse, as shown in Fig. 6.

 

Fig. 7 (a) The time signal response as captured by a probe above the grating structure. The initial pulse is the Gaussian time signal used for excitation, while the waveform afterwards is the energy released by the slowly decaying PR. (b) The frequency response of the reflected waveform. The maximum of the peak occurs at the same frequency as the frequency of the PR

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Fig. 8 (a) The time signal response as captured by a probe below the grating structure. The initial pulse is the input Gaussian time signal minus a small bandwidth of frequency components that couple to the PR. The second, longer duration waveform is the slow decay of energy released by PR. (b) The frequency response of the transmitted signal shows which frequencies pass through the grating largely unimpeded and which frequencies are reflected or scattered due to the PR.

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For the probe within the structure, three separate stages in time can be identified in Fig. 6 where different optical effects are occurring. The first stage involves the rapid increase in energy as the incident pulse enters the structure and excites the phase resonance. The second stage tracks the rapid decrease in energy in the structure from 0 dB to −10 dB that occurs as the remnants of the incident pulse leave the structure (i.e., the full incident beam minus the energy of the frequency component that has gone into the PR). The third stage follows the slow decay of the PR over 135 ns as the energy trapped within the PR slowly leaks out into the superstrate and substrate (in equal amounts). It is this property of the PR that allows for the grating to behave like an optical resonator by storing and then slowly re-emitting energy. Naturally, the higher the Q of the system, the longer it will take for the grating to fully give up the energy trapped within it.

For the probe on the input side of the system, two separate signals can be identified in Fig. 7, one being the incident pulse passing through the probe location at times from 0 ns to 9 ns, and a second long duration signal occurring between 12 ns at 150 ns. For the first part of this second signal between 12 ns and 16.6 ns, the probe at this location records a rapid increase in radiation that is due to the PR re-radiating energy as it is being excited by the incident pulse. After 16.6 ns, this probe records a decrease in energy as the energy trapped within the PR slowly leaks into the substrate and superstrate. This decaying time signal from 16.6 ns to 150 ns is due to the release of energy trapped within the structure and is a familiar characteristic of any optical resonator. Fourier transforming this recorded, slowly decaying portion of the reflected time signal into the frequency domain shows that this portion of the reflected signal has, as expected, a maximum at the frequency of the PR (Fig. 7).

For the probe below the CTG in the substrate, the time signal is again composed of two distinct waveforms (Fig. 8). The first part of the waveform that spans from time 0 ns to 13.2 ns are the remnants of the incident pulse after it has passed through the structure. Because a majority of the input signal has components with frequencies different than the frequency of the PR, most of the signal passes through the structure largely unimpeded and maintains a temporal Gaussian profile. However the amplitude and energy of the time signal has decreased slightly relative to the incoming signal due to the excitation of the PR. Additionally, it is well-known that for p-polarized incident light, there is a relatively constant baseline reflection for all frequency components of the incident signal as determined by the ratio of the area occupied by the wires to the total area of the unit cell. Yet for the geometries of the structure studied in this work, with narrow metal wires, the baseline reflection is small.

Concerning the excitation of the phase resonance, as the signal passes through the grating there is a transfer of energy from the incoming plane wave to the PR. Initially, the intensity of this PR increases linearly with time as it is being excited by the incident beam, but the rate of increase slows as the rate of energy being radiated from the PR increases (with the radiation rate being proportional to the intensity of the PR). In less than 10 ns after the incident beam encounters the structure, a steady-state situation is reached where the PR excitation rate equals the PR radiation rate. When the incident beam has passed through the structure and is no longer exciting the PR, the PR continues to radiate energy until the PR is fully extinguished, as seen in Fig. 6 starting at 13 ns until the end of the time simulation at 150 ns. Figure 8 shows the frequency response of the time signal recorded by probe on the exit side (i.e., substrate) and shows that a majority of the frequencies contained within the signal passes through the structure largely unimpeded, minus only the component of the incident signal with a frequency matching the frequency of the PR. This result is expected as the frequency components associated with the PR are no longer part of the output signal and are instead maintained within the structure in the form of a PR. Normalizing this output frequency response to the frequency response of the input signal (Fig. 9) shows that structure has promise as a drop filter, with a narrow bandwidth transmission dip and high off-band transmissivity.

 

Fig. 9 The ratio of the output beam to input beam for the device studied in this work. The portions of the incident beam that do not excite the phase resonance will pass through the structure largely unimpeded, whereas the component of the incident beam that has a frequency that matches the phase resonance will excite the phase resonance, with most of the this component being reflected and some of this component going into intensifying the fields of the phase resonance.

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To summarize this portion of the time analysis study, it has been demonstrated that a CTG can be designed with which a very narrow bandwidth sampling of an incident signal can be performed. In other words, this device allows all components of the incident signal to pass through largely unimpeded except for frequencies within a very narrow bandwidth that match the frequencies within the bandwidth of the PR, at these frequencies none of the signal is transmitted but is partially reflected and partially used to excite the PR. Thus one can envision vertically stacking layers of CTGs with each layer supporting a PR of a different frequency. A signal incident on such a device will drop different frequency components at the different layers of the device. Once the structure has collected the energy of incident beam in a PR of a certain frequency, the next issue concerns the extraction of the energy from the PR. Namely, how does one read out the signal? To address this question, the time dependent flow of energy of PRs within the structure needs to be studied, as is done in a preliminary way in the next section.

5. Time Domain Analysis of Energy Flow

Pertaining to the extraction of energy from PRs, PRs in infinitely periodic lossless CTGs under a steady-state illumination have to give up energy in some way, and it does so in the form of re-flected light as the large amount of energy built up in the system decouples from the PRs to the input side of the structure (i.e., superstrate). Yet if either the periodicity, structure or composition of the grating is perturbed in some portion of the structure, the energy contained within the PR can be released in more useful and controlled ways other than simply as a reflected beam. In this section, methods are described in which the energy of the PR is guided and extracted from the structure.

The starting structure used in this section is the finite-sized CTG described in the previous section, but modified by the introduction of one or several structural or compositional perturbations. The perturbations include the tuning the dielectric constant of the grooves in one portion of the grating and/or introducing a scatterer in close proximity to the structure; both methods will perturb the PR leading to a release of the energy contained within the PR. More specifi-cally, the structure is first split into two portions, with the first portion (see Fig. 10) being the control portion where incident energy is free to interact with the unperturbed, unmodified grating while the second portion has a dielectric material in the grooves with ε = 1.05, which is a 5% perturbation of the dielectric constant. Additionally, a metal sheet placed a distance of 30 mm above the modified portion of the grating so as to shield this portion of the structure from the incident beam and prevent the direct creation of a PR in this region.

 

Fig. 10 A snapshot in time of the electromagnetic field profile for a 8.17 GHz incident beam with an angle of incidence of −45°. This beam excites a PR that channels the energy along the grating in a direction opposite to the in-plane wave vector component of the incident beam. Upon entering the region of the grating that has its structure perturbed such that it does not support the PR, the energy within the PR is released into the substrate at an angle the matches the angle of incidence of the incident beam.

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As previously mentioned, PRs in 2-groove CTGs have negative group velocities, thus a −45° incident beam of light of frequency 8.17 GHz on the unshielded control portion of the structure will excite a PR with a Q of 80 that produces a flow of energy in a direction of +90° with respect to the normal vector (ŷ) of this structure, namely, flowing in the x̂ direction (Fig. 10). The sign difference between the angle of the incident beam and the flow of energy of the PR is again representative of the fact that this PR can be viewed as having a negative index of refraction. Thus the PR will produce a net flow of energy that propagates along the grating and underneath the metal sheet. The metal sheet, along with the 5% dielectric change in the grooves in this portion of the grating perturb the environment experienced by the PR thus upsetting the mode and causing it to release its energy. The greater the perturbation is, the greater the rate at which the PR gives up its energy. In the structure shown in Fig. 10 that has a long distance of the grating underneath the metal sheet, the release of energy goes into forming what is in effect a transmitted beam, but a transmitted beam that is laterally shifted relative to the incident beam. Again, note that this transmitted beam has been shifted laterally to what would occur if the beam had been directly transmitted through the structure, and the extent of this lateral shift is controlled by the placement of metal plate or other perturbation of the structure.

While the above structure demonstrated the small but negative value of the structure’s index of refraction and the release of the energy contained within the PR, one may want to release the energy of the PR in a more controlled way, say into a waveguide. To do so, the perturbation in ε in the shielded portion was removed (i.e., ε = 1 for all the grooves) and the shielded portion of the grating was truncated to allow for the the PR to last long enough to be able to encounter the end of the grating (Fig. 11). Additionally, a second PEC metal sheet was placed 30 mm below the grating, and together with the sheet above the grating, formed a parallel plate waveguide. Upon reaching the end of the grating between the two metal sheets, the energy of the PR is launched into the parallel plate waveguide yielding a jet of energy propagating laterally away from, and eventually exiting the structure. From the waveguide, the signal can be detected and processed by conventional network analyzers. This structure, with its corrugated structure supporting surface waves that are either being fed into or by a waveguide is similar in many ways to corrugated surface antennas (CSAs) and leaky wave antennas [25–27]. In typical CSAs, a periodic corrugation in a metal surface can support TM polarized waves (i.e., the p-polarized WCM described in this work). When used to emit radiation, the WCMs in CSAs can be excited by, or fed by, a parallel plate transmission line or flared transmission line as described in [27]. Essentially, the CTGs studied in this work are a type of ”leaky wave”, but composed not of individual WCMs, but coupled WCMs with different frequencies and momenta as given by Eq. (1) that have significantly different values and dependencies on structural and material parameters. Such different dependencies and properties of PRs relative to WCMs may provide advantages over conventional CSAs.

 

Fig. 11 A snapshot in time of the electromagnetic field profile for a 8.17 GHz incident beam with an angle of incidence of −45°. Again, this beam excites a PR that channels the energy along the grating in a direction opposite to the in-plane wave vector component of the incident beam. The PR then propagates into the waveguide and releases its energy into the waveguide at the end of the grating. The color scale is the same as in Fig. 10.

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6. Conclusions

In this work, phase resonances in compound gratings with two dissimilar grooves in the unit cell were described in the frequency and time domains. A design methodology was described that can be used to design CTGs with PRs of a particular frequency and momentum. Introduced in this work was a closed-form equation that describes the dispersion of PRs in a 2-groove CTG where the two grooves in the unit cell have only a slight dissimilarity. Several structures were studied, including larger structures that support PRs in the microwave spectral region and smaller structures that support PRs in the infrared spectral region. The study of the evolution of the electromagnetic fields associated with PRs as the a simple lamellar grating is perturbed into a compound grating was performed. In the time domain, the excitation of PRs via an incident beam at normal and off-normal incident angles was studied. The resulting flow of energy of the PR, and the release of that energy into free space or a waveguide was described. Applications of PRs to optical filters and antennas, including a new type of corrugated surface antennas were described.