1. Introduction

Coaxial waveguides are used extensively at the microwave frequencies because they support a propagating TEM mode for infinitesimal dimensions. For visible and infrared frequencies, the situation changes and there has been considerable effort to understand the influence of surface plasmons within cylindrical coaxial waveguides and geometries [1–10]. The cylindrical surface plasmon (CSP) can extend the cut-off of the waveguide modes for narrow gaps between the metal sides. The Bessel field profile of the CSP can have rapid decay in metal and free space, with sharp localization at metal boundaries. In this regard, phenomena such as extraordinary optical transmission requires consideration of the localized resonances associated with the CSP, which have shown to play an important role, both theoretically and experimentally [4,11–16]. The properties of CSPs have been of interest to a wide range of applications including; nonlinear optics [4,5], metamaterials [17,18], THz waveguiding [19], subwavelength and near-field optics [3,20–25], and band-pass filters [26,27].

While past works focused on the propagation of light within the aperture structure, the reflection properties at the end-face are critical to determining both the wavelength and quality of Fabry-Perot resonant transmission from the CSP. The phase of reflection associated with the end-faces of a coaxial aperture affect the Fabry-Perot resonances seen at the microwave frequencies [28]. Recently, a theory to account for those resonant shifts for the perfect electric conductor (PEC) case was presented [29]. In the real metal case, the phase and amplitude of reflection are quantitatively different from the PEC case because of plasmonic effects. In this work, we quantify the differences in the phase and amplitude of the reflection coefficient, which clearly demonstrate that the PEC case cannot be used (not even approximately) to model the real metal case. The difference arises from the new physics associated with surface plasmons: in particular, changes in the propagation constant, finite penetration of the field into the metal region, changes in the sign of the normal electric field component at the boundary, and the sharply peaked field profile. Therefore, care should be taken to include the real metal response.

It is common in the literature to first consider the PEC case as simplified theory that accounts for the geometric optical physics, and later to develop a more detailed theory to account for the plasmonic influence. For example, the progression of study on the single slit problem has followed that trend. An early work to account for the phase and amplitude of reflection for a single slit in a metal concentrated on the PEC case, showing significant geometric influence on the phase of reflection [30]. Following that work, the effects associated with the finite conductance of the metal were shown to dominate the resonances in the microwave regime [31]. For real metals in the visible-IR regime, later theories revealed the plasmonic influences on the reflection properties [32] and the ability to generate surface plasmons at the slit [33]. In each of those works, new physics was uncovered when accounting for the real response of the metal.

In this paper, we present an analytic theory that accurately describes the reflection of the radial CSP mode within a coaxial geometry. The theory provides the reflection amplitude and phase of radially polarized surface plasmon waves from the end faces of an annular aperture in a plate made of a real material. Based on this theory, it is demonstrated that transverse resonances produce oscillations in the dependence of reflection amplitude and phase on the aperture geometry, which is of direct relevance to the wavelength and quality of the plasmonic resonances. The theoretical approach is also used to tune apertures to a specific resonant transmission peak, as confirmed by comprehensive finite-difference time-domain (FDTD) simulations.

2. Theory of end-face reflection from an annular aperture in a metal plate

The theoretical approach is based upon the single-mode-matching method, where a single mode within the waveguide region is matched to a continuum of radiation and evanescent modes to obtain the reflection properties. With the reflection coefficient and the mode’s propagation constant, the wavelength and quality of the localized resonances can be obtained using Fabry-Perot theory. This method is accurate for subwavelength systems where the single-mode approximation represents the field distribution well. It has been applied successfully to the reflection from an annulus in a perfect electric conductor [29] and a number of other systems including subwavelength slits [30,32], and surface plasmons at a step-edge [34,35].

Figure 1 shows a schematic of the geometry under consideration. An annular aperture in a metal film is coaxial with the z-axis within the cylindrical coordinate system$(\rho ,\varphi ,z)$, and the end-face of the metal terminates at $z=0^{}$. Considering only the lowest-order mode (CSP) [36], the field at $z=0^{-}$can be expressed as:

 

Fig. 1 Schematic view of an annular aperture in a metal film. The regions 1, 2, 3 and 4 have relative permittivity constants of ε₁, ε₂, ε₃ and ε₄ .

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The propagation constant β for the fields in this structure can be found via dispersion relation:

For $z=0^{+}$, the electric and magnetic fields are expanded in terms of a continuum of modes with the same symmetry as the CSP:

The transverse electric and magnetic fields are matched at$z=0$, and the mode orthogonality relations are used to determiner. Equation (1) is equated to Eq. (5) and both sides are multiplied by$J_1(k\rho )\rho$ and integrated over ρ from 0 to ∞. Considering the orthogonality of the Bessel functions, this integration gives:

The solution in Eqs. (12) and (13) represent the main result of this work. It captures the amplitude and phase of reflection of the lowest order mode at the interface to free space (or uniform dielectric). This incorporates both the physical effects of mode-shape mismatch and impedance mismatch that lead to the reflection coefficient. The theory is limited to the subwavelength regime. For larger slit widths, where higher order radial modes are allowed to propagate, the theory will give inaccurate results and full numerical simulations will typically be required to capture the scattering of such a system. Furthermore, for large radii, higher order azimuthal modes may be allowed. This does not necessarily mean that the theory presented will not be invalid – so long as the rotational symmetry of the zeroth order mode is preserved, the theory presented here is valid for larger radii. The theory assumes that the plasmon mode is supported, as defined by the existence of solutions to Eq. (3).

3. Behavior of reflection coefficient

Having developed a theoretical expression for reflection, we proceed to evaluate the reflection amplitude and phase of various structures. Figure 2 shows the calculated amplitude and phase at the end-face of an annular aperture in PEC case and real gold. The calculation is done for wavelength of 632.8 nm and a relative permittivity of −11.694 + 1.225i in real case [37], with the dielectric medium having relative permittivity of 1. Two main features can be seen from this figure: that the amplitude and phase experience an offset when changing the slit width of the annulus (b-a), and that there are oscillations with variation in the annulus radius.

 

Fig. 2 Evaluation of the theoretical reflection expression (Eq. (12)) for annular apertures in gold plate at 632.8 nm free-space wavelength. The results are contrasted with PEC case, using dashed lines. (a) The reflectivity and (b) the phase of reflection as function of inner radius, a, for slit widths (b-a) of 20 nm and 50 nm.

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The changes with slit width result from variations in coupling to the free space (or uniform dielectric) modes. Narrower slits have larger reflection amplitudes because of increased impedance mismatch and mode-shape mismatch (between the CSP mode in the slit and the continuum of propagating modes in the uniform dielectric) as the width is decreased. It is well known for gap plasmons that decreasing the distance between two metals can increase the propagation constant of the mode in the gap, and the same is true for the CSP here [1]. Furthermore, the narrower gap confines the electromagnetic energy to a subwavelength region.

The oscillatory behavior with changes in the annulus radius arises from transverse resonances. Such transverse resonances are not present for a single linear slit, but do arise for double slits, where the electromagnetic energy is scattered resonantly between the slits [38]. For the annulus here, the scattering occurs transversely between opposite sides. For a more mathematical description of this phenomenon, it is instructive to consider Eq. (13), for which the integrand diverges when $k^2={\omega }^2{\mu }_0{\epsilon }_4$. The integrand at this value of the wave-vector will play an important role in the value of r, and the oscillation arises from the oscillatory nature of the J Bessel functions with variations in the radius. It should be noted as well that the integrals in denominator have monotonic behavior with changes of radius because of the presence of modified Bessel functions.

Figure 2 shows considerable differences between PEC case and real metal case. This difference increases further if the wavelength is closer to the plasmon resonance. At the wavelength of 500 nm, the reflectivity is noticeably different, as shown in Fig. 3 . The differences in the locations of the extrema in PEC and real metal case are also due to plasmonic effects. These differences increase from less than 20 nm in Fig. 2 to 40 nm in Fig. 3 by change of excitation wavelength from 632.8 nm to 500 nm.

 

Fig. 3 Comparison between reflection for the PEC case and the real metal case, both for annular aperture with slit size of 20 nm illuminated with 500 nm wavelength light. (a) Reflectivity and (b) phase.

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For the wavelengths and geometries considered here, the reflection is larger for the PEC case; but this is not generally true. For instance, in case of a single slit, which can be considered as an annular aperture with the inner radius of infinity, higher reflectivity has been found in PEC compared to real metal case [32]. Close to zero outer radius (for finite inner radius), the PEC case approaches unity reflection because the field is confined to extreme subwavelength dimensions and therefore the mode-shape mismatch with free-space becomes infinite. For the real metal case, however, there is finite extension of the electric field into the metal, even for infinitesimal dimensions, and so the reflection amplitude remains below unity.

The phase of reflection is quantitatively different between the real metal and PEC cases, more than double in the graphs shown in Fig. 2 and 3. In the verbiage of microwave engineering, the increase in the phase of reflection can be thought of as coming from a more inductive termination waveguide. This is consistent with the interpretation that real metals can be thought of as inductive elements in the plasmonic regime [39]. As will be discussed in the next section, the resonant thickness of the metal plates will be considerably less for the real metal case than predicted by the PEC case, even after accounting for the differences in the propagation constant within the aperture.

4. Fabry-Perot resonances

For a finite metal plate of thickness l, the CSP propagating inside the annular aperture will experience reflection at both end-faces. Fabry-Perot resonances arise from multiple reflections between the end-faces, depending on the phase of reflection, ϕ:

Based on the results of Fig. 2 and Eq. (14), we designed several annular aperture geometries to all have a Fabry-Perot resonance at the same wavelength of 632.8 nm. The geometric parameters of each design were then simulated by FDTD (Lumerical Solutions Inc.). The metal plate spanned from $z=0$ to $z=l$ nm, with l the thickness of the plate. A parametric dispersion model was used with gold permittivity set to −11.694 + 1.225i at 632.8 nm [35] (that software uses a proprietary multi-coefficient model for extrapolating the permittivity in different wavelengths fitting the experimental data). A z-polarized broadband dipole source was located at $z=100$ nm. The perfect matched layer (PML) boundary conditions were used for the computational domain, and the symmetric boundaries were adopted at x and y axis. A 1 nm mesh was set at both ends of the annular aperture and the mesh resolution was set to smaller than 2 nm inside the annulus. The convergence was ensured with variations of the simulation region size and simulation time.

Figure 4 shows the results of the FDTD simulations for the various designs. Each case clearly shows a peak in the transmission through the aperture near the specified wavelength of 632.8 nm. The variations in the peak-widths and heights are expected from variations in the reflection amplitude and the propagation loss for the different geometries.

 

Fig. 4 (a) Transmission in arbitrary units for (b-a) value of 20 nm. Each curve relates to a different structure as specified in the legend and the dashed line is at 632.8 nm. In this figure a is the inner radius and l is the thickness of the plate (b) The (b-a) value is changed to 50 nm.

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As it can be seen, by comparing Fig. 4(a) and 4(b), the quality of the frequency profile of the resonance not only changes with a but also with slit size (b-a). This is a direct effect of propagation constant and reflection amplitude being functions of a and b as stated in Eqs. (3), (4), and (12), and shown in Fig. 2. For example, the reflection amplitude for the red curve at inner radius of 50 nm is higher than reflection of green curve at inner radius of 250 nm (Fig. 2(a)), and this correlates with the higher quality resonance seen in Fig. 4(a). It can be inferred from Fig. 4(b) that change of slit size (b-a) causes the relative value of peaks to be influenced. This is because the reflectivity in this case follows 50 nm curve in Fig. 2(a), and therefore the variations in reflection amplitude are more significant.

The good agreement of the resonances seen in the comprehensive electromagnetic simulations with the specified wavelength (632.8 nm) shows the predictive capability of the theory presented, which will be useful in future designs of apertures in metal films. Table 1 shows the wavelength at which the peaks arise from several additional simulations, all very close to 632.8 nm. Also, it is further evident from this table that the propagation constant inside the annulus obtained from finite-difference mode solver simulations match the calculated values of Eq. (3). This is because the effective refractive index$(\beta \lambda /2\pi )$ obtained from the finite difference method agrees to that found by the calculation within 0.004 on average, showing that the simulations accurately capture the analytic CSP dispersion.It is important to note that the geometric values presented in Table 1 are not the result of any optimization, but rather, they are found directly from the analytic theory presented in Section 2. With knowledge of a and b, l is derived directly from the theory. To obtain sharper resonances, the reflectivity should be increased (for example, by choosing maxima in Fig. 2(a)), and the material loss should be minimized. Therefore, it is worthwhile to consider the relative contribution of the reflection loss and the propagation loss on the Fabry-Perot resonances. The reflection loss for one round-trip can be written as ${(1-|r{|}^2)}^2$, which describes how efficiently the energy escapes from the cavity. The propagation loss is given by $1-\exp (-4\mathrm{Im}\{\beta \}l)$, where Im is the imaginary part. The propagation loss describes how the energy is lost to the metal absorption. As an example, for the case of the 20 nm slit width red curve in Fig. 4(a), the round trip reflection loss is 0.005 and the propagation loss is 0.28. For the green curve in Fig. 4(a), the round trip reflection loss is 0.28 and the propagation loss is 0.23. Therefore, in the first example, the propagation loss dominates, but for the second example, where the reflection is reduced, the loss from imperfect reflection is larger. Clearly, the proper design of such apertures can have a large impact on the relative contributions to resonant energy transmission and to the loss through material absorption.

Table 1. The Effective Refractive Indexes and Peak Wavelength for Geometries in Fig. 4, for Annular Aperture Structures in Gold Designed to Have Peak at 632.8 nm

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Figure 5 illustrates the changes of resonance wavelength with change of thickness. As can be seen, the resonance wavelength for the same order increases with thickness, and once again, good agreement is seen between our analytic method and FDTD simulations.

 

Fig. 5 Variations of resonance wavelength with plate thickness l. The annular aperture inner radius is kept fixed at a = 50 nm.

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As the effect of inner and outer radius of the annular aperture was fully captured in reflection phase and amplitude, the effect of plate thickness l is important in tuning the plasmonic resonances in such an aperture as well. This is attainable both with the Eq. (12) and (14) and with the FDTD approach. The analytic approach, however, can directly give the resonance thickness simply by substitution of phase in Eq. (14). Unlike FDTD in which the thickness of the structure needs to be known prior to the simulation, the analytic approach can predict the thickness with a choice of a wavelength. This can serve as a prefabrication design guide for an intended wavelength of resonance. Care should be taken when extrapolating the curves in Fig. 5 to thinner plate thicknesses, and to shorter wavelengths. For shorter wavelengths, the CSP mode may not be supported (as prescribed by the existence of solutions to Eq. (3). Furthermore, if the plate is too thin, transmission by higher order modes may start to contribute significantly, and the theory presented here will no longer be valid, as described at the end of Section 2. Here, we have considered only gold, however, the theory may be equally applied to other metals for wavelengths that support the CSP.

Figure 6 shows the distribution of the intensity of the electric-field in cross section of the annular aperture, when on resonance and when off resonance. For the on-resonance case, there is a nearly symmetric field distribution in the aperture (as expected from the Fabry-Perot resonance) and significant electric field intensity transmits through the aperture to the other side of the gold plate. For the off-resonance case, the electromagnetic energy does not build up within the aperture and significantly less transmission is observed.

 

Fig. 6 2D electric field intensity plot in logarithmic scale (base 10) calculated by FDTD showing cross section of the annular aperture with a = 50 nm, b = 100 nm and l = 124 nm. (a) On resonance at the wavelength 632.8 nm. (b) Off resonance at 500 nm wavelength.

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5. Conclusion

We presented a theory for the end-face reflection of an annular aperture in a real metal. This theory is useful for determining the localized surface plasmon resonances associated with an annular aperture in a metal plate. Comparison with comprehensive FDTD simulations showed the good predictive design capability of the analytical approach and the results were contrasted with PEC case. The results presented in this work are relevant to annular apertures for use in metamaterials applications [17,18], near-field optics [21], sensors [22–25], and band-pass filters [26,27]. In particular, metamaterials based on annular structures rely on the plasmonic metal response to obtain interesting new physics, such as negative refractive index [17,18]. However, in those works the important influence of the reflection remains elusive. This work provides a better understanding of the reflection properties in real metals for the annular geometry. For near-field optics applications, annular aperture resonances can boost the performance, for example, as found in a recent work involving near-field hyperspectral Raman imaging [40]. Knowledge of the influence of real metals is necessary to design near-field probes with the desired resonance wavelengths.