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Abstract
We develop the theory of all-dielectric absorbers based on temporal coupled mode theory (TCMT), with parameters extracted from eigenfrequency simulations. An infinite square array of cylindrical resonators embedded in air is investigated, and we find that it supports two eigenmodes of opposite symmetry that are each responsible for half of the total absorption. The even and odd eigenmodes are found to be the hybrid electric (EH111) and hybrid magnetic (HE111) waveguide modes of a dielectric wire of circular cross section, respectively. The geometry of the cylindrical array is shown to be useful for individual tuning of the radiative loss rates of the eigenmodes, thus permitting frequency degeneracy. Further, by specifying the resonators’ loss tangent, the material loss rate can be made to equal the radiative loss rate, thus achieving a state of degenerate critical coupling and perfect absorption. Our results are supported by S-parameter simulations, and agree well with waveguide theory.
© 2017 Optical Society of America
1. Introduction
Electromagnetic wave absorbers based on metamaterials have attracted much interest over the past decade. The most extensively explored design is the metal-dielectric-metal (MDM) three layered structure due to its ease of fabrication, frequency scalability, wide-angle absorption, tunability, and application in diverse areas as: thermal emitters, sensors, detectors, and spatial light modulators [1–7]. Although there is continued interest in the basic physics and applications of metal-based absorbers, they posses various shortcomings, which may ultimately limit their usefulness. For example, the high operational temperatures (T>1000 °C) required for energy harvesting using thermal emitters in thermophotovoltaics (TPVs) is above the melting point of commonly used metals [8]. Further, the performance of metallic metamaterials is inextricably interwoven with both the electrical conductivity and thermal conductivity, as prescribed by the Wiedmann-Franz law [9]. Thus high performance metal-based metamaterials are constrained to also be good thermal conductors.
An alternative approach using all-dielectric materials to form metasurface absorbers has recently been proposed and demonstrated [9–11]. Experimental verification of high absorption at both 1 THz and 600 GHz was demonstrated, and a test system consisting of an uncooled terahertz imager was also shown. It has further been proposed that dielectric metasurfaces may be useful as high temperature emitters for energy harvesting applications [9]. Metasurface absorbers are fashioned from subwavelength dielectric particles [12,13], which can be achieved utilizing arrays of various geometrical shapes with a specific amount of material loss. In [10], Shadrivov et. al. show that the high absorption occurs due to the overlap of electric dipole (ED) and magnetic dipole (MD) resonances supported by a square array of dielectric cylinders. In [9] and [11], Padilla et.al. experimentally verified and confirmed that the high absorptive state is due to dipole resonances – in particular the ED and MD resonances were shown to be the hybrid electric (EH111) and hybrid magnetic (HE111) modes, respectively, of a dielectric cylindrical waveguide.
Here we investigate the mechanism underlying high absorption in all-dielectric absorbers using temporal coupled mode theory (TCMT) [14–17]. We find that all-dielectric metamaterial absorbers achieve degenerate coupling of the EH111 and HE111 modes [16], with each providing half of the absorption. Further, the two hybrid modes possess opposite symmetry and each achieves a state where the radiation loss rate (γ) is equal to the material loss rate (δ), i.e. critical coupling γ = δ. For simplicity, we consider an array of sub-wavelength free-standing lossy dielectric disks in air at normal incidence. The all-dielectric metasurface absorber is thus a mirror-symmetric 2-port resonator with the symmetry plane lying in the center of the disk perpendicular to the cylindrical axis. It is important to note that the all-dielectric absorber may be coupled to with one or two inputs [16]. Thus it is distinct from MDM absorbers where often a continuous ground plane only permits coupling with a single port [18, 19], and different from coherent perfect absorbers [20] where two ports must be simultaneously excited with specific inputs. In what follows, we will separate contributions of the even EH111 and odd HE111 modes to absorption, and demonstrate that they are degenerate, uncoupled to each other, with each independently achieving nearly critical coupling and thus high absorption.
2. Degenerate critical coupling of EH111 and HE111 modes
In Fig. 1(a) we show a schematic of one unit cell of the all-dielectric absorber disk array embedded in air, with r the radius, h the height, and p the periodicity of the square array. The mirror plane is shown in the z=0 plane in Fig. 1. One typically excites metamaterial absorbers with electromagnetic radiation incident from a single side, i.e. a 1-port excitation. However, due to mirror symmetry, the all-dielectric absorber possesses two identical input facets. Thus we may decompose the 1-port excitation into a combination of even and odd eigenexcitations from the two opposing identical ports [16], each with half of the total power as depicted in Fig. 1. The two excitations – one from each port – are equal in electric field amplitude (E0/2), but possess opposite symmetry with respect to the mirror plane. The net result of the above procedure is to decompose a single excited two port resonator into two 2-port resonators, each excited with half of the original power. Further, the decomposition specifies a boundary condition for the mirror symmetry plane, i.e. a perfect magnetic conductor (PMC) for even eigenexcitation, and perfect electric conductor (PEC) for odd eigenexcitation. Therefore the even and odd eigenexcitations only couple to either the EH111 (even) or HE111 (odd) modes of the same symmetry, respectively [16, 21].
Fig. 1 (a) Schematic of one unit cell of the all-dielectric absorber with a single input, where r is the cylindrical radius, h is the height, and p is the period of the square array. The mirror plane (gray area) is at z=0. The single input from one port can be represented as a combination of even (b) and odd (c) eigenexcitations. Each eigenexcitation contains half of the power compared to the single input case, and two equal amplitude waves at each port, which are symmetric and anti-symmetric (with respect to the mirror plane) for the even and odd eigenexcitation, respectively.
If the EH111 and HE111 modes possess resonant frequencies that are close to each other – but far from other higher order modes – their behavior can be described with TCMT [14–17]. We may thus express the total absorption as a linear sum of absorption due to even and odd eigenexcitations. If the incident field values are not too large, such that the system responds linearly, the loss from each mode remains independent and thus the total absorption can be expressed as (see Appendix A) a sum of two Lorentz terms,
where subscripted “even” and “odd” stand for the even (EH111) and odd (HE111) hybrid waveguide modes, respectively. In Eq. (1), ω0 is the center frequency of each mode, γ and δ are the radiation loss rate and material loss rate, respectively, for each mode. We note that the Lorentz parameters given in Eq. (1) are governed by the geometry of the resonator, filling fraction of the array, and the resonator’s complex dielectric function [18, 19]. As can be seen from Eq. (1) both terms can achieve a maximum absorption of 50% at ω = ω0 when the radiation loss is equal to the dissipation loss, i.e. the so-called critically damped state γ = δ. Moreover, if we additionally have the condition ω0,1 = ω0,2, we achieve a state of degeneracy, and thus perfect absorption at ω = ω0. Relaxing the condition for critical coupling, we note that it is still possible to achieve significant absorption in each mode if the radiative rate and dissipation rate are close, since Eq. (1) for each mode gives, which is valid for both the even and odd modes independently. Thus if both eigenmodes occur at ω0 and achieve the criteria specified in Eq. 2, the total absorption is at least A(ω0) = Aeven + Aodd = 98%.The structure investigated here is similar to that in [11], where the disk material consists of boron doped silicon with relative permittivity described by the Drude model [22] as , ϵ1r and ϵ2r are the real and imaginary parts, respectively, of the relative permittivity ϵr, and tanδ = ϵ2r/ϵ1r is the loss tangent. Here we use ϵ∞ = 11.9, ωp = 2π × 1.27 × 1012Hz and ωs = 2π × 0.64 × 1012Hz. In order to reduce the existence of – and potential coupling to – higher order modes (evidenced by the occurrence of Fano lineshapes [23]) – the height h and radius r of disks are chosen to be close to the cut-off values of the EH111 mode [9, 11, 21], determined by and r = J1,1 λ0/2π(ϵ1r −1)1/2, where J1,1 is the first positive solution of the Bessel function of the first kind, and λ0 = 2πc/ω0, where c is the speed of light. We further require that the period p of the array should be smaller than λ0 in order to avoid any diffractive effects. For a target frequency of ω0 = 2π × 1.0 THz, we find h=45.8µm, r=58.6µm from the cut-off conditions of the EH111 mode, and use a periodicity of p <300µm. In our study, we investigate the absorptive properties of the all-dielectric metamaterial using the optimized parameters of h=50µm, r=60µm, and p=210µm, unless specified otherwise.
3. Computational simulations
3.1. Scattering parameter
We perform S-parameter simulations of a single unit cell of the cylindrical resonator, with periodic boundary conditions at the perimeter of the unit cell (±xz −plane and yz − plane), and port boundaries (with perfectly matched layers (PMLs)) on the ±xy −planes placed equidistant from the mirror symmetry plane – see Fig. 1. Thus we may characterize even-eigenexcitations (Fig. 1(b)) or odd-eigenexcitations (Fig. 1(c)) of the metamaterial by driving the ports in-phase or out-of-phase, respectively. Absorptivity due to even or odd eigenexcitations we denote as Aeven or Aodd, respectively. The power supplied to each port – for both the even and odd excitations – is half of the power supplied in the single port simulation shown in Fig. 1(a). The S-parameter configuration described above, and depicted in Fig. 1, further allows us to investigate the sum of even and odd absorption, i.e. AΣ = Aeven + Aodd, as well as the single port absorption A. It is important to note that the 1-port excitation (Fig. 1(a)) drives both even and odd modes simultaneously, in contrast to Fig. 1(b) and Fig. 1(c), where only the even-eigenexcitations or odd-eigenexcitations are driven.
The absorption spectra for both 1-port excitation (red curves) and 2-port excitation of the even (black curve) and odd (gray curve) eigenmodes are shown in Fig. 2. The absorption spectra is calculated as from S-parameter simulations for cylindrical radii of r=45µm, 60µm and 70µm, all for a constant height of h =50µm and period p =210µm. For the parameters studied here, a radius of r =60µm is optimal and gives us a maximum absorption of A = 99.4% at ω0 = 2π × 1.048THz. For radii smaller than optimal we observe that both even and odd modes shift to higher frequencies, and are no longer degenerate. In contrast, for a larger radius of r =70µm, the odd eigenexcitation is nearly unshifted from the optimal case, but the even eigenexcitation shifts to lower frequencies. For the optimal case shown in Fig. 2(b) both the even and odd modes achieve values close to A =50%. The open blue circles in Fig. 2 are the sum of even and odd absorption, i.e. AΣ, and we find excellent agreement between A and AΣ.
Fig. 2 Absorption for 1-port excitation (red curve), even eigenexcitation (black curve) and odd eigenexcitation (solid gray curve) for cylindrical resonators with radii of r=45µm (a), r=60µm (b) and r=70µm (c). The total absorptivity due to both even and odd modes, AΣ, is plotted as open blue circles. The dash horizontal line indicates 50% absorption.
In order to gain insight into the symmetry of each eigenmode, we plot the electric and magnetic fields of the even and odd modes in Fig. 3, on the vertical cut-plane of the disk (y=0 plane), parallel to the incident electric field. As can be observed, the x-component of the electric field (black arrows) is symmetric to the mirror plane for even eigenexcitation (Fig. 3(a)), while for the odd eigenexcitation (Fig. 3(b)), it is anti-symmetric. Also plotted in Fig. 3 is the y-component of the magnetic field (Hy), shown as the colormap. We note that the fields are mainly confined in the disk and their spatial dependence is consistent with the EH111 and HE111 waveguide modes [21,23,24]. In Fig. 3(c) we show a plot of the summed electric and magnetic fields from Fig. 3(a) and Fig. 3(b), as well as the fields resulting from a 1-port excitation in Fig. 3(d). As can be observed, the resulting electric field is asymmetric for both for the summed and 1-port simulations. We also find asymmetry in the magnetic field plotted in Fig. 3(c) and Fig. 3(d), which is not obvious since its amplitude in the HE111 mode (Fig. 3(b)) dominates over that in the EH111 mode (Fig. 3(a)) [10,11,13].
Fig. 3 Electric field (black arrows) and transverse magnetic field Hy (colormap) in the vertical middle cut plane (y=0) of a cylinder for (a) even eigenexcitation, (b) odd eigenexcitation, (c) sum of fields from (a) and (b), and (d) one input excitation, all at ω=1.048THz.
3.2. Eigenanalysis
The S-parameter study presented in Fig. 2 and Fig. 3 verify that the all-dielectric resonator supports two modes of opposite symmetry, but is unable to independently determine the value and significance of the radiative and material loss rates. Thus we next turn toward an eigenanalysis in order to calculate the Lorentz parameters in Eq. (1), i.e. ω0, γ and δ. Here, only the top half of the resonator is simulated and we use a boundary condition in place of the mirror symmetry plane, i.e. a perfect magnetic conductor (PMC) for the EH111 mode and perfect electric conductor (PEC) for HE111 mode [21]. We describe the resulting complex eigenfrequency as , where ω2 = γ + δ. We may also repeat the analysis with material loss removed, i.e. δ = 0, thereby determining the individual radiative and material contributions to resonator loss [16, 18]. Before carrying out eigenfrequency numerical modeling, we first determine the approximate resonant frequencies for the EH111 and HE111 modes from waveguide theory (Eqs. (9)–(12) in Appendix B) and find values of 1.086THz and 1.053THz, respectively [21,24,25]. The Lorentz parameters determined from the waveguide equations and eigenfrequency simulations are shown in Table 1.
Table 1. Analytical and Simulated Lorentz Parameters
3.3. Comparison of TCMT with numerical results
We next compare the absorption calculated from Eq. (1) using the eigenfrequencies shown in Table 1, to that of the absorption computed from a 1-port S-parameter simulation. In Fig. 4 we plot the S-parameter A(ω) (open blue circles) and A good (ω) from Eq. (1) as the red curve. We find good agreement between the calculated eigenfrequency A(ω) and the S-parameter simulated absorption near ω0, which gradually worsens away from the resonance frequency. The poor agreement at higher frequencies is due to the occurrence of higher order modes [15, 26, 27], not accounted for in our analysis. Also shown in Fig. 4 are the simulated even (black curve) and odd (gray curve) eigenmode absorptivities. Notably, we find our reflectivity is relatively low across the frequency range investigated (not shown – see [9]), which may be understood by noting that we achieve a critically coupled state γ = δ for each of the two modes and, importantly, all loss rates are close to each other in value. Thus the all-dielectric absorber realizes conditions similar to that of the all-pass filter [17].
Fig. 4 Comparison of absorptivity calculated from Eq. (1) and eigenfrequency simulation (red curve) and by S-parameter simulation (open blue circles). We also plot from Eq. (1)Aeven (black curve) and Aodd (gray curve). The dash horizontal line indicates 50% absorptivity.
4. Dependence of loss rates on metasurface geometry
We next detail the dependence of absorptivity A(ω) and resonant frequency ω0 on the metasurface geometrical parameters of height, radius, and periodicity. In Fig. 5 we show the resonant frequency of both the even (open black circles) and odd (open gray triangles) modes as determined from eigenfrequency simulation. It can be observed that ω0 follows the absorptivity peaks determined by S-parameter simulation, shown as the colormap which is similar to Fig. 1 in [11], for both the even and odd eigenexcitations. The solid curves shown in Fig. 5 are the resonant frequencies for the even (black) and odd (gray) modes, calculated from Eqs. 9–12. We note that the analytical solutions yield good approximate estimates of the geometrical values for design and, in particular, the HE111 mode (solid gray curve) agrees quite well with simulation (gray triangles) for all radii values shown, and we find relatively good agreement with the height. In contrast, the analytically calculated EH111 mode (solid black curve) resonant frequency deviates from the eigenfrequency (black circles) for both r and h, and we believe this is due to the PMC boundary condition assumption in Eq. (9) (see Appendix B).
Fig. 5 Effect of (a) height, (b) radius and (c) period on total absorption A(ω), plotted as a colormap. The circles and triangles are the resonant frequencies of the EH111 and HE111 modes, respectively, determined by eigenfrequency simulations. The solid black and gray lines are the analytical resonant frequencies of EH111 and HE111 modes, respectively. The vertical dash black lines indicate the ideal tan δ value for critical coupling.
In Fig. 6 we show the dependence of the radiative loss rate (solid curves) and material loss rate (dashed curves) on the geometry of the metasurface for both the even EH111 (red) and odd HE111 (blue) eigenfrequencies. The optimal geometrical parameters of h =50µm, r =60µm, and p =210µm for peak absorptivity are shown as the dashed vertical black lines. The waveguide cutoff condition for EH111 is shown as the gray shaded area, and we only consider heights and radii outside of this region. Generally we observe that the material loss rates δ for both modes are roughly independent of the geometrical parameters, whereas γ values vary widely – especially as a function of periodicity where we use an expanded vertical scale.
Fig. 6 Effect of (a) height, (b) radius and (c) period on γ (solid curves) and δ (dashed curves) of the EH111 mode (red curves) and HE111 modes (blue curves). The vertical dashed lines denote (a) h = 50µm, (b) r = 60µm, and (c) p = 210µm. The shaded gray area shows the waveguide cutoff region for the EH111 mode.
We next explore the impact of the material loss tangent (tan δ = ϵ2r/ϵ1r) on the reflectivity (R), transmissivity (T), absorptivity, and loss rates. Figure 7(a) shows R, T, and A as a function of tan δ. Here we use a constant ϵ1r = 10.83, thus tan δ ∝ ϵ2r. For zero loss we find that the metasurface achieves a good transmissivity (T ≈ 50%), with only 1×10−3% of the power absorbed. However as loss increases, T and R drop rapidly while A peaks at tan δ = 0.06. In Fig. 7(a) we also plot A(ω = ω0,1 = ω0,2) as the solid red curve from TCMT, i.e. Eq. (2), as a function of the loss ratio, defined as δ/γ, on a separate horizontal scale (top axis). In Fig. 7(b) we show γ and δ for both eigenmodes. We find that γ for both modes does not depend on loss tangent, and that δ is proportional to the materials dielectric loss, i.e. δ ∝ tan δ.
Fig. 7 (a) Dependence of R (green triangles), T (blue squares), and A (red circles) on loss tangent (bottom axis). The solid red curve shows the total absorptivity from Eq. (2) at ω = ω0,1 = ω0,2 as a function of the loss ratio = δ/γ (top axis). (b) Dependence of γ (solid curves) and δ (dashed curves) for both EH111 (red curves) and HE111 (blue curves) modes as a function of loss tangent. The vertical dashed black line indicates the nearly degenerate critical coupling.
5. Discussion
We find that the relative difference between A and AΣ is within numerical precision, thus indicating each mode is orthogonal and provides independent absorption [16]. Further, when we excite only the even (or odd) eigenexcitation – as shown in Fig. 2(a) and Fig. 2(c) – we do not find any characteristic absorption features from the opposite symmetry mode. We note that indeed the sum of the fields for the degenerate critically coupled case, shown in Fig. 3(c), are identical to that shown in Fig. 3(d), and explain the overall asymmetry of the 1-port excited case. That is, when light is incident on the metasurface absorber from one side, the electric fields from the even and odd modes point in opposite directions on the incident side of the mirror plane, and in the same direction on the exit side of the mirror plane. This is the underlying reason for the asymmetric power loss density plots shown in [9] and [11].
The resonate frequencies and loss rates of both eigenmodes are determined from eigenfrequency simulations, which are then used in TCMT to calculate A(ω), and we find excellent agreement with the A(ω) calculated from S-parameter simulations. We also find good agreement between our analytical solution of the waveguide equations for the HE111 mode – given by Eqs. (11) and (12) – and moderate agreement for the EH111 mode determined by Eqs. (9) and (10). Although we have not explored the reason for the discrepancy in the even mode, we believe this is due to the form of Eqs. (9) and (10), which are only approximate.
Although the absorptivity plotted in Fig. 5(c) appears to be a weak function of periodicity, we note that our eigenfrequency analysis shows that the even and odd modes have a strong dependence on p, as shown in Fig. 6(c). Thus neighbor interactions are significant and the relatively weak dependence of the absorptivity on periodicity may be understood through examination of Eq. (2), where it can be seen that the conditions for strong absorptivity from each mode are relatively robust against mismatch of the radiative and material loss rates. More generally, we note that the radiative loss rates for both the EH111 and HE111 modes are relatively strong functions of all geometrical parameters of a cylindrical square array, in comparison to the material loss rates, which realize only weak dependence, as can be observed from Fig. 6. Not surprisingly, Fig. 7(b) shows that, in contrast, radiative loss rates are independent of loss tangent, whereas the material loss rate is linearly proportional, i.e. δ ∝ tan δ ∝ ϵ2r (for ϵ1r constant).
As a general design rule for the construction of all-dielectric absorbers, we first use Eqs. (7) and (8) from [9] to make our cylinder just slightly larger than cutoff for the EH mode. Next we select a periodicity that is large enough to minimize neighbor interaction, but still smaller than the operational wavelength. It’s important to note that we have demonstrated here that – for a square array of cylinders – the geometrical parameters largely determine the radiative loss rate γ. Thus with our geometry now set, we select a loss tangent that give us a material loss rate approximately equal to our radiative loss rate, i.e. δ = γ, as shown in Fig. 7. The later requirement is straightforward when using a semiconductor as the base waveguide material, as one can typically choose a doping in order to provide a prescribed amount of loss.
Lastly we clarify that we have only considered a free-standing disk array. Use of a substrate for structural support [11] introduces asymmetry which has not been addressed. However, if the refractive index of the supporting substrate is low loss – with real values near that of free space [11] – it may be treated as a symmetric structure. Thus the theoretical treatment presented here may be used for initial analysis.
6. Conclusion
We find that an all-dielectric metamaterial fashioned from a square array of cylinders embedded in air can support two hybrid waveguide modes of opposite symmetry. The geometrical parameters of height, radius, and periodicity may be used to overlap these eigenmodes in frequency, thereby achieving degeneracy. Equally important, we find that the geometry specifies a particular radiative loss rate. Thus by controlling the imaginary portion of the cylinder’s dielectric constant, the material loss rate may be made equal to radiative losses, thus achieving critical damping and perfect absorption. Temporal coupled mode theory and eigenfrequency simulations accurately describe the frequency dependent absorptivity of each eigenmode, as well as their sum – equal to the 1-port excitation. Eigenmode simulations elucidate the radiative and material loss rates, and their dependence on geometry and loss tangent. S-parameter simulations detail the relation of R, T, and A, on the loss tangent, and match well the form of (A tan δ) predicted by TCMT. Other geometrical shapes and alternative systems may be studied with the methods we show here, including bound states in the continuum and coherent perfect absorbers.
Appendix A: Derivation of absorption from TCMT
The behavior of the EH111 and HE111 modes of the mirror-symmetric cylindrical resonator can be described with TCMT [14–17] as,
where a is a vector with an assumed time dependence of exp (−iωt) which describes the mode amplitude, with the stored energy for each mode given by |aj|2. The center frequencies of each modes are given by the real diagonal matrix Ω0; Γ and Δ are two real diagonal matrices describing the radiation loss rate and dissipation loss rate of the modes, respectively. Coupling between the two modes is given by an anti-diagonal matrix K; sin is a vector that represents inputs from each port with |sin,j|2 equal to the input power, with a similar term for the outputs sout; the matrix D describes the coupling between modes and inputs, where D∗D = 2Γ; the matrix C represents the background scattering between ports.We consider the case with input from only a single port and thus write the input sin = [s0, 0]T as a decomposition of even and odd eigenexcitations, as sin,even = [s0/2, s0/2]T and sin,odd = [s0/2, −s0/2]T. Due to the different symmetry of even and odd modes [15,28], they are orthogonal and uncoupled, i.e. K = 0. Thus for an even eigenexcitation of our mirror symmetric resonator, the radiation rates at the two ports is given by γ1,1 = γ2,1 = γ1/2 due to symmetry, and , the complex mode amplitude for the EH111 mode is
and the dissipation loss power is thus the absorption resulting from even eigenexcitation is (normalized to the power of one input case)Similar process is applicable to the absorption resulting from odd eigenexicitation, and total absorption due to independent absorption of the two modes is thus
Appendix B: Analytical resonant frequencies of EH111 and HE111 modes
The resonant frequencies of EH111 and HE111 modes can be analytically estimated from those of single lossless cylinder dielectric resonator antennas in air with some approximation of the boundary conditions from waveguide theory [21,24,25].
For EH111 mode, perfect magnetic wall boundary is set for the side-wall of the cylinder while non-perfect magnetic walls for the top and bottom flat walls, which yield
where kr is the radial wave vector component in the cylinder, kz is the z component of wave vector in the cylinder, while kz0 describes the z component of wave vector in air, they satisfy and , where k0 = ω0/c is the wavenumber in air and ϵ1r is the real part of permittivity of the cylinder. In Eq. (9) the first non-trivial solution is used.Meanwhile, for the HE111 mode, non-perfect magnetic wall boundary is set for the side-wall of the cylinder while perfect magnetic wall boundary for the top and bottom flat walls, which yield,
where u = krr, v = kr0r, kr0 is the radial wave vector in air, J1(u) is the first order Bessel function of the first kind, and K1(u) is the first order modified Hankel function. kr, kr0 and kz satisfy and .By numerically solving Eqs. (9)–(12) with a graphical method, we obtain an analytical estimate of the resonant frequencies of EH111 and HE111 modes. Solutions to Eqs. (9)–(12) are plotted as the solid lines in Fig. 5(a) and Fig. 5(b). Because the approximations for HE111 mode are closer to the physical situation than those of EH111 mode, they yield analytical resonant frequencies closer to the accurate ones.
Funding
Department of Energy (DOE) (DE-SC0014372); National Key Foundation for Exploring Scientific Instrument of China (2012YQ0901670602); China Scholarship Council (CSC) (201606210317).
Acknowledgments
WJP and XL acknowledge support from the Department of Energy (DOE) (DE-SC0014372). LS and XM acknowledge support from the National Key Foundation for Exploring Scientific Instrument of China (2012YQ0901670602). XM is partly supported by China Scholarship Council (CSC) (201606210317). We acknowledge Kebin Fan, Ilya Shadrivov, Andrew Cardin, and David Powell for useful discussions.
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