1. Introduction

With the ever increasing advances in nanofabrication techniques, many remarkable optical phenomena have been discovered, thus stimulating considerable interest in light scattering at the nanometer length scale. Within the emerging field of nanoplasmonics, metallic nanostructures can be tailored to harness collective optical effects, namely, surface plasmon resonances (SPRs). Localized field enhancement is one of the key underlying physical characteristics in promising new nanotechnological applications. Due to relatively high energy confinement (Q factor) and small mode volume, SPRs have been used in the construction of tightly packed photonic devices [1], and play a major role in such applications [2–9] as, surface-enhanced Raman scattering [2, 3], photoconversion [4, 5], and Metamaterial designs [8, 9]. However, SPRs are extremely sensitive to material loss, which diminishes the Q factor significantly and can limit its benefits for the envisaged applications. Discovery of a new subwavelength field-enhancement mechanism would be greatly beneficial in future nanoscale research.

It is well known that surface plasmons cannot be excited by TM polarization (H is perpendicular to the axis of the rod) [10, 11]. With TE polarization SPRs can be strongly excited at normal incidence, but diminish as the incident angle approaches grazing incidence. Although scattering by a cylindrical object is an old topic [12–16], grazing scattering has a broad applicability [17–19] and is a nontrivial and longstanding problem due to singularities arising at the zero grazing angle, and can be further complicated by the emission of leaky waves from the cylinder [20, 21]. As the grazing angle approaches zero, intrinsic singularities in the scattering solutions reveal possible numerical instabilities. The general solution intended for non-grazing incidence often fails to provide a clear picture of this limiting case due to divergences arising at the zero grazing angle. Even in the macroscopic and long wavelength regime, descriptions of grazing behaviors often lead to contradictions [19].

In this paper, we investigate unexplored and nontrivial grazing phenomena in long nanowires. We undertake the analytic and numerical challenges inherent in grazing scattering at the nanoscale and develop a robust analytic solution to Maxwell’s equations for grazing incidence, from which the essential physics can be extracted. At grazing incidence, scattering and surface wave propagation are interconnected. Physically meaningful concepts like scattering cross section and efficiency Q must therefore be redefined. By taking proper account of the guided surface waves, we generalize the definition of the scattering efficiency. Our calculations reveal a family of non-plasmonic resonances with a much higher Q-bandwidth product and which are more sustainable to material loss than SPRs. It is determined that these resonances, or enhanced fields, originate from the excitation of long-range guided asymmetric surface waves corresponding to the n = 1 azimuthal mode of the nanorods, akin to Sommerfeld waves, traditionally ascribed to the azimuthally symmetric (n = 0) case for TM waves propagating along a conducting cylinder [22]. However, these higher order (n ≠ 0) periodic solutions have been historically overlooked due to the typically high attenuation at long wavelengths [23]. It is remarkable that indeed grazing light can intricately couple to these first order (n = 1) cyclic surface waves, i.e. cyclic Sommerfeld waves (CSWs), and contribute to exotic grazing resonances [24], or cyclic Sommerfeld resonances (CSRs). The cyclic Sommerfeld wave is an azimuthally periodic surface wave and is formed by two azimuthally counter-propagating helical surface waves along the rod. As will be shown, CSRs have a much higher Q-bandwidth product and thus, a much higher Q factor and broader bandwidth than SPRs. CSRs cannot in general be trivially scaled to other spectral regimes due to material dispersion. Contrary to the quasistatic nature [25] of SPRs, these grazing resonances are strictly non-electrostatic. This phenomenon of non-electrostatic resonances in the subwavelength regime was brought to light in a recent work involving the effect of volume plasmons on enhanced fields in ultrasmall structures [26], revealing the richness of resonant scatterings in the nanoscale.

2. General Solution at Oblique Incidence

Consider an infinitely long cylinder of radius a with its rotation axis along the z-direction [use cylindrical coordinates (ρ,ϕ, z)], and with permittivity ε ₁ and permeability µ ₁. The background permittivity ε ₀ and permeability µ ₀ are uniform throughout the space. A plane TM wave is incident onto the cylinder with a grazing angle θ with respect to the z-axis as shown in Fig. 1. Without loss of generality, assume the incident wave resides in the x-z plane with a harmonic time dependence, exp(-iωt). Thus, in cylindrical coordinates, the incident plane wave has components,

where x≡k ₀ ρ sinθ, ψ(z)≡E ₀exp(ik ₀ z cosθ), $k_0\equiv \omega \sqrt{{\epsilon }_0{\mu }_0}$, ${풴}_0\equiv \sqrt{{\epsilon }_0{\mu }_0}⁄{\mu }_0$, and J_n(·) are Bessel functions of the first kind. Due to the non-normal incidence, the TM and TE modes are coupled in the scattered field. The general solution [27] involves an infinite series of Bessel functions. Here we take an alternative approach to derive the general solution. We first solve the Helmholtz equation in cylindrical coordinates for E_z and H_z. The remaining transverse field components are then determined from [28]:

where ⊥ refers to transverse dimensions and β ² _j=ω ² ε_jµ_j-k ² z,(j=0,1). When the grazing angle θ→0, β0→ ₀. Thus, the transverse components have singularities two-orders higher than the longitudinal components. By matching tangential components of the fields at the cylinder surface (ρ=a), i.e., the continuity of E _z,ϕ and H_z,ϕ, the general solution for both inside and outside the cylinder and for arbitrary incidence angles can be cast into the form,

where the index ν corresponds to z or ρ. Thus there are six components in Eq. (3). Inside the cylinder, Eq. (3) refers to the total field. Outside the cylinder, Eq. (3) refers to the scattered field, and the total field is the scattered field plus the incident field, E ⁱ +E ^s and H ⁱ+H ^s. For convenience, we define dimensionless quantities: ε_r≡ε ₁/ε ₀, µ_r≡µ ₁/µ ₀, $\eta \equiv \sqrt{{\epsilon }_r{\mu }_r-{\cos }^2\theta }$, ${\eta }_0\equiv \sqrt{{\epsilon }_r{\mu }_r-1}$, δ≡sinθ, _r≡k ₀ ρ, r ₀≡k ₀ a, x≡k ₀ ρδ, x ₀≡k ₀ aδ, x ₁≡k ₀ aη, y≡k ₀ ρη, and z ₀≡k ₀ aη ₀. The radial dependence of the scattered fields outside the cylinder [Eq. (3) with the superscript t=s] are given by

where H_n(x)≡H ⁽¹⁾ _n(x) is the first kind Hankel function, and

where n=0,1,2, 22EF;, and

Resonant modes mathematically correspond to the poles of the scattering matrix. Physically, resonant modes are related to the excitation of waveguide modes that can be found by determining where the denominators of the coefficients a_n and b_n vanish, i.e. A ² _n+I ^ε _n I ^µ _n=0, which yields,

where k=ω/c and ^c is the velocity of light in vacuum. The ε _0r, ε _1r, µ _0r, and µ _1r are the relative permittivity and permeability in the corresponding medium.

 

Fig. 1. Schematic: A plane wave of TM mode is incident onto a long cylinder. The grazing angle θ is the angle between the k-vector of the incident wave and the symmetry axis of the rod.

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Inside the cylinder the radial dependence of the fields [Eq. (3) with the superscript t=c] are,

where J_n(·) represents either Bessel or modified Bessel functions depending on the material properties of the cylinder. The coefficients are given by,

and b ₀=d ₀=0. The general solution involves an infinite series of Bessel functions, and often poses numerical difficulties for small grazing angles.

3. Grazing Incidence Solution

At near-zero grazing incidence, the general solution in Section 2 becomes singular. A wellbehaved grazing solution can be derived by asymptotic expansion of the general solution at the singularity θ = 0. Although the derivation is tedious and lengthy, this approach has the advantage of a simpler analytical solution (written solely in terms of elementary functions). Moreover, the grazing solution provides a simpler picture in which to extract the essential physical quantities of interest. From the grazing solution it is clear that only the n = 0 and n = 1 azimuthal modes have significant contribution to the EM field at the near-zero grazing angles. For convenience, a set of useful properties and asymptotic expansions of Bessel and Hankel functions with small arguments used here are listed in the Appendix. When the grazing angle θ → 0, δ → 0 and x ₀→0, from the properties in the Appendix, the small-argument expansion of Eq. (6) yields, $A_n\approx \genfrac{}{}{0.1ex}{}{n{H}_n\left(x_0\right)}{r_0{\delta }^2}\left[1-\left(\genfrac{}{}{0.1ex}{}{1}{2}+\genfrac{}{}{0.1ex}{}{1}{{\eta }_0^2}\right){\delta }^2+\left(\genfrac{}{}{0.1ex}{}{1}{{\eta }_0^4}+\genfrac{}{}{0.1ex}{}{1}{2{\eta }_0^2}-\genfrac{}{}{0.1ex}{}{1}{8}\right){\delta }^4\right],$ $P_n\approx -{\left(-i\right)}^n\genfrac{}{}{0.1ex}{}{n{J}_n\left(x_0\right)}{r_0\delta }\left[1-\left(\genfrac{}{}{0.1ex}{}{1}{2}+\genfrac{}{}{0.1ex}{}{1}{{\eta }_0^2}\right){\delta }^2+\left(\genfrac{}{}{0.1ex}{}{1}{{\eta }_0^4}+\genfrac{}{}{0.1ex}{}{1}{2{\eta }_0^2}-\genfrac{}{}{0.1ex}{}{1}{8}\right){\delta }^4\right],$ $Q_n\approx i{\left(-i\right)}^n\genfrac{}{}{0.1ex}{}{{𝒴}_{0}n{J}_n\left(x_0\right)}{r_0\delta }\left\{1-\left[\genfrac{}{}{0.1ex}{}{{\epsilon }_r{B}_{n0}}{{\eta }_0^2}+\genfrac{}{}{0.1ex}{}{r_0^2}{2n\left(n+1\right)}\right]{\delta }^2-\left[\genfrac{}{}{0.1ex}{}{r_0^4}{8n{\left(n+1\right)}^2\left(n+2\right)}+\genfrac{}{}{0.1ex}{}{q_n{\epsilon }_r}{{\eta }_0^2}-\genfrac{}{}{0.1ex}{}{{\epsilon }_r{B}_{n0}}{{\eta }_0^4}\right]{\delta }^4\right\},$ $I_n^{\epsilon }\approx \genfrac{}{}{0.1ex}{}{i{𝒴}_{0}n{H}_n\left(x_0\right)}{r_0{\delta }^2}\left\{1+\left[r_0^2{\alpha }_n^{\left(1\right)}\left(x_0\right)+\genfrac{}{}{0.1ex}{}{{\epsilon }_r{B}_{n0}}{{\eta }_0^2}\right]{\delta }^2+\left[r_0^4{\alpha }_n^{\left(2\right)}\left(x_0\right)+\genfrac{}{}{0.1ex}{}{q_n{\epsilon }_r}{{\eta }_0^2}-\genfrac{}{}{0.1ex}{}{{\epsilon }_r{B}_{n0}}{{\eta }_0^4}\right]{\delta }^4\right\},$ $I_n^{\mu }\approx \genfrac{}{}{0.1ex}{}{in{H}_n\left(x_0\right)}{{𝒴}_0{r}_0{\delta }^2}\left\{1+\left[r_0^2{\alpha }_n^{\left(1\right)}\left(x_0\right)+\genfrac{}{}{0.1ex}{}{{\mu }_r{B}_{n0}}{{\eta }_0^2}\right]{\delta }^2+\left[r_0^4{\alpha }_n^{\left(2\right)}\left(x_0\right)+\genfrac{}{}{0.1ex}{}{q_n{\mu }_r}{{\eta }_0^2}-\genfrac{}{}{0.1ex}{}{{\mu }_r{B}_{n0}}{{\eta }_0^4}\right]{\delta }^4\right\}.$ The reason for maintaining the third term δ ⁴ in the above equations is because the first two terms are canceled in the coefficients of a_n, b_n, and the transverse components of the EM fields. Thus,

where n=1,2,3, ⋯, and

where

The quantities B _n0, q_n, α (1) _n (x ₀), and α ⁽²⁾ _n (x ₀) are given in the Appendix.

Substituting Eq. (10) into Eqs. (5) and (9), we obtain, for n≠0,

where

and

The n=0 terms require a separate treatment. From Eq. (5)

which, as θ→0, yields,

where

Substituting Eq. (17) into Eq. (9), we obtain

where

Substituting Eqs. (13)–(17) into Eq. (4), we obtain the asymptotic solution for the radial dependence of the scattered fields outside the cylinder:

where σ denotes ρ or ϕ, δ≡sinθ, x≡k ₀ ρ δ, x ₀≡k ₀ aδ, and

and

In fact, in Eq. (23) only the D ₁ remains and all the n>1 terms vanish as the grazing angle approaches zero. The other related coefficients in Eq. (21) are succinctly written as

where g ⁽¹⁾ _n and g ⁽²⁾ _n are given by Eq. (15); χ ^(±) _n is given by Eq. (12); and

Here α ⁽¹⁾ _n (x) and α ⁽²⁾ _n (x) in Eqs. (24) and (25) are given in the Appendix [Eq. (37)]. The solution inside the cylinder is simpler, which can be obtained by substituting c_n and d_n in Eqs. (13) and (19) into Eq. (8). While the above solution was derived for TM polarization, the TE-mode solution can be obtained by imposing the property of duality of the fields: E→H, H→E, ε_r→µ_r, µ_r→ε_r, and 𝓨 ₀→-𝓩 ₀, where ${𝒵}_0\equiv \sqrt{{\epsilon }_0{\mu }_0}⁄{\epsilon }_0$.

The waveguide eigenmode relation, Eq. (7), becomes singular in the limiting case (k_z→k ₀, β ₀→0), and can lead to inaccurate mode solutions when solving Eq. (7) directly. Using the asymptotic expansions in the Appendix, the eigenmodes of Eq. (7) for this limiting case reduces to,

where $k_0\equiv \omega \sqrt{{\epsilon }_0{\mu }_0}$ is the background wave number. By taking the minimum of the absolute value of the left hand side of Eq. (26), the surface modes in the limiting case can be easily identified.

The grazing solution in Eq. (21) indicates that the scattered fields outside the nanorod are long-range guided surface waves. Furthermore, only the n=0 and 1 azimuthal modes have significant contribution at the near-zero grazing incidence. The n=0 term corresponds to the radially symmetric Sommerfeld wave. Historically, only the n=0 term is considered, and the n≠0 terms are neglected due to high losses in the radio frequency (RF) region [23]. However, in the optical regime and at grazing incidences, the n=1 term has a great impact on the resonance effects. Although the grazing solution is apparently complicated, it is written solely in terms of elementary functions, thus providing a clearer physical picture. Also, without the infinite series, the overall field expressions are simpler since only the n=0,1 terms need to be considered. Therefore, it is computationally efficient and more stable. To ensure the accuracy of the above asymptotic expansions, several tests have been conducted. Shown in Figs. 2–4 are the EM field coefficients in Eq. (21) for the case of n=0 and n=1 versus the grazing angle along with their counterparts for the exact result. We also studied the coefficients for other cases (n>1). In each case studied, all six components of the EM fields in the grazing solution accurately approached the general exact solution at low grazing angles. Overall, the grazing solution coincides with the exact result for grazing angles θ≲10°. Although the grazing angle can be arbitrarily small from a strictly mathematical perspective, it is clear that the given geometry sets physical limits for the minimum angle of incidence given by θ _min=a/L, where L is the distance between the observation and incidence points; and a is the radius of the rod. Thus in what follows, θ→0 implies θ→θ _min.

4. Grazing Resonance

The quality factor of a resonator is usually linked to the photon life time [29, 30]. In the scattering cases, the scattering efficiency Q is related to the scattered field [16, 27]. At grazing incidence, both the scattered waves and excited surface waves are interconnected. To incorporate both effects, it is convenient to define a θ-dependent total Q_θ factor,

where

 

Fig. 2. Comparison of the exact and grazing results: plots of the electric field coefficients versus the grazing angle for the n=0 azimuthal mode. Blue: Exact solution [Eq. (4)]. Green: Grazing solution [Eq. (21)]. Top: Real part. Bottom: Imaginary part. Clearly, the grazing solutions asymptotically match the exact solutions as the grazing angle approaches zero.

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Fig. 3. Electric field coefficients as in Fig. 2, except now for the n=1 azimuthal mode.

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Here $I_0=\genfrac{}{}{0.1ex}{}{1}{2}{𝒴}_0{\mid E_0\mid }^2$ is the input intensity (𝓨 ₀ is the background admittance) and R is the radius of the integration circle around the rod. The Poynting vector components S^sρ and S^s_z are along the ρ̂ and ẑ directions respectively, and are calculated from the scattered waves. The quantity Q ^s _ρ describes resonant scattering and the Q^s_z describes guided surface-wave excitations. The Qθ factor is a useful metric to characterize the resonant scattering and the efficiency of channeling of the EM field through the nanowire. The Qθ appropriately reduces to the traditional Q when θ=90° (normal incidence). Our studies have shown that Q^s_ρ is independent of R for all points outside of the cylinder, as is the case for the traditional Q. Since the radius of the rod is on the order of the skin depth of the metal, the EM field can fully penetrate the nanowire. To describe the effective power of the surface wave inside the nanowire, similar to the definition of the Q^sz, we define,

where S^c_z is the z-component Poynting vector inside the cylinder. Thus, the factor Q^c_z represents the efficiency of the guided wave propagating inside the nanorod. We also define a dimensionless resonance bandwidth:

where λ ₁ and λ ₂ are the wavelengths corresponding to the half-maximum value of the Qθ of the resonance; and a is the radius of the rod. Thus, the dimensionless Q-bandwidth product (QBW) is given by QBW≡Q_θ×BW. Although the bandwidth of the resonances is in the several tens of THz, due to the subwavelength nature of the problem, i.e. a≪λ, the dimensionless BW defined by Eq. (30) can be very small.

 

Fig. 4. Comparison of the exact and grazing results: Magnetic field coefficients in Eq. (21) for the n=1 azimuthal mode. Blue: Exact solution. Green: Grazing solution. Top: Real part. Bottom: Imaginary part.

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Fig. 5. Comparison of SPR and CSR in a silver nanorod. Left: TE mode at normal incidence and the excitation of SPR at the wavelength about 309 nm. Right: TE mode incident at the grazing angle of 0.01° and the corresponding resonance (CSR) at about 380 nm. Solid Blue: without material loss. The Q of CSR is about 10 times higher than that of SPR. Dashed Green: with material loss. The Q of CSR is then about 23 times higher than that of SPR. Moreover, due to the higher QBW of CSR, CSR has a broader bandwidth than SPR. Without the material loss, QBW≈0.04 for SPR and QBW≈0.7 for CSR.

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In the simulations, the permittivity of the silver nanorods was obtained from a polynomial fit to experimental data [31], appropriate for the visible spectrum of interest:

where λ is in units of µm. For the results below, we take R=3a with a=15 nm for all the figures. All field quantities are normalized with respect to the input field. For a silver nanorod, the SPR at λ≈309 nm can be excited by a TE wave at normal incidence, as shown in the left plot in Fig. 5. The frequency of the SPR is insensitive to the rod diameter [25], however the Q factor is sensitive to material loss. At low grazing angles, this picture changes dramatically: a series of non-plasmonic resonances emerge as shown in the narrow peak in the Q_θ factor. Although Q_θ demonstrates the resonant peaks at both normal and grazing incidences, the underlying physical mechanism of these resonances is different. The QBW of these grazing resonances is much higher than that of SPRs (0.7 vs. 0.04). Hence, the grazing resonances can have a much higher Q factor and broader bandwidth than the SPRs, as clearly indicated in Fig. 5. Moreover, by comparing the blue and green curves in Fig. 5, it can be seen that these grazing resonances are much less susceptible to material loss than the usual SPR. As the incident angle transitions from normal to grazing, the SPR continuously transforms into a new resonant mode, i.e. cyclic Sommerfeld resonance (CSR), with a corresponding redshift of the resonant frequency. This result is consistent with the waveguide modes calculated from Eq. (26) for the nanorods, which are shown in the right plot of Fig. 6. As k_z/k ₀ tends to unity, small changes in the grazing angle result in significant shifts in the resonant wavelength. This explains the features found in the left plot of Fig. 6, where the resonant peak continuously shifts to longer wavelengths when approaching the minimum angle. As indicated in Fig. 6, the grazing resonances appear near to the cutoff wavelength of n=1 azimuthal modes of the circular waveguide. The field enhancement near to the cutoff wavelength was also demonstrated in other geometry [32] and appear to be a universal feature of subwavelength structures. Due to the material dispersion, the grazing resonance cannot in general be trivially scaled to other spectral regimes. The grazing resonances reported here are consistent with a Q factor enhancement reported in previous work [15, 16].

 

Fig. 6. CSRs and corresponding waveguide modes in a silver nanorod at the low grazing angles. The resonant peaks are associated with the excitation of a family of guided cyclic periodic (n=1) surface waves, i.e. cyclic Sommerfeld waves. Left plot: the peaks from low to high correspond to θ=2°, θ=1°, θ=0.5°, θ=0.1°, θ=0.05°, θ=0.02°, and θ=0.01°. The Q_θ value and the wavelength of grazing resonances increase continuously as the grazing angle approaches the minimum angle. Right plot: the n=1 waveguide modes calculated from Eq. (26) with the same kz values of the CSRs for the silver nanorod. These modes are related to the first-order CSRs shown on the left plot.

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In Fig. 7, we illustrate the robust nature of CSRs for a dielectric and metallic nanorod with |ε _1r|=4 and µ _1r=1. These grazing resonances again correspond to the excitation of first-order (n=1) cyclic Sommerfeld waves. The upper left panel demonstrates how Qθ and the wavelength of grazing resonances increases continuously as the grazing angle approaches θ _min. Both the dielectric and metallic cases have resonances with similar magnitudes but with a clear shift in the wavelength. The lower-left panel shows the corresponding n=1 waveguide modes calculated from Eq. (26) and with the same kz values of the CSRs. Again, for both cases, we see that as k_z/k ₀ approaches unity, the resonant wavelength is highly sensitive to small changes in the grazing angle. The upper-right panel is the calculated Q bandwidth product of the corresponding CSRs as a function of grazing angle. There is a monotonic decline that levels off at larger impact angles. In contrast, the true bandwidth, ΔF, of the corresponding CSRs increases with grazing angle (lower right panel).

 

Fig. 7. CSRs for a dielectric nanorod (solid curves) with the relative permittivity ε _1r=4 and permeability µ _1r=1 and for a metallic rod (dashed curves) with the relative permittivity ε _1r=-4 and permeability µ _1r=1: These grazing resonances correspond to the excitation of the first-order (n=1) cyclic Sommerfeld waves. Upper-left panel: the peaks from low to high correspond to θ=2°, θ=1°, θ=0.5°, θ=0.1°, θ=0.05°, θ=0.02°, and θ=0.01°. The Qθ value and the wavelength of grazing resonances increases continuously as the grazing angle approaches the minimum angle. Lower-left panel: the corresponding n=1 waveguide modes calculated from Eq. (26) with the same k_z values of the CSRs. These modes are related to the first-order CSRs shown on the Upper-left panel. Upper-right panel: the QBW of the corresponding CSRs vs. grazing angle. Lower-right panel: the true bandwidth ΔF of the corresponding CSRs vs. grazing angle.

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The correlation between the grazing resonances and the n=1 cyclic periodic surface waves are clearly shown in the upper-left panel of Fig. 8, demonstrating the consistency of the two curves identifying the resonant wavelengths. Thus, these grazing resonances result from the excitation of two first-order azimuthally counter-propagating guided waves, which travel with a spiraling trajectory along the nanowire and constructive interference significantly enhances the resonant scatterings. The enhancement of the Q factor at near-zero grazing angles is due to guided asymmetric surface waves and a coherent superposition of scattering states, which reinforces the resonance effect. The CSR also complies with the general rule that an increase of the Q factor implies a decrease of the bandwidth. As illustrated in the lower panels of Fig. 8, the increase of the Qθ (lower-left plot) is accompanied by the decrease of the bandwidth ΔF (lower-right plot) in the grazing region.

Another interesting phenomenon is that while the excited surface wave propagates forward outside the nanorod, the power flows backward inside the rod due to the boundary conditions and negative permittivity of silver. This backward propagation is expressed as the negative value of the quantity Q^c_z since it describes the axial power flow inside the rod, as shown in the upper-left plot of Fig. 9. When the grazing angle approaches zero, more energy in the incident wave is transformed into the guided surface waves, and thus the absolute values of both Q^c_z and Q^s_z increase as shown in the upper panels of Fig. 9. The ratio Q^c_z/Q^s_z in the lower-left plot of Fig. 9 describes the ratio of the axial power inside and outside the nanorod. As the grazing angle decreases, the scattering in the radial direction becomes weaker and wave guiding along the rod becomes stronger. Since the power inside the rod arises from the radial scattering, the value |Q^c_z/Q^s_z | decreases as θ→θ _min. Shown in the lower-right plot of Fig. 9 is the ratio Q^s_z/Q_θ versus the grazing angle. This value increases as approaching the minimum angle and the quantity Q^s_z becomes dominant in the total quality factor Q_θ due to the increase of the guiding effect with the decrease of the grazing angle. To better visualize the spatial characteristics of the resonant fields, we show in Fig. 10 the Poynting vector from several perspectives at near-zero grazing incidence. As shown in the left panel, the power (S_ρ,S_x,S_y) in the x-y plane is toward the center of the rod and has nearly perfect circular symmetry. The degree of symmetry depends on how small the angle is. However, once the energy penetrates the wire, it travels asymmetrically backward along the silver nanowire, as shown in the upper-right panel. Figure 11 shows the intensity of the total electric field close to the surface ρ=a inside and outside the silver rod. It reveals a two-dimensional standing wave pattern in the azimuthal and propagation directions, as the result of phase-matched spiraling propagation along the nanorod. The geometric influence on the grazing resonances is shown in Fig. 12 for a lossless silver nanowire. In the upper-left panel, both the resonant wavelength and the corresponding QBW increase with the increase of the radius of the nanorod. The lower-left panel shows the total quality factor Qθ and corresponding bandwidth of CSRs versus the radius of the rod. In the lossless nanorod, the total quality factor Qθ decreases with the increase of the radius, indicating this peculiar high Q grazing resonance is a nano-phenonmenon or a phenomenon more pronounced in the nano-region. The smaller the radius, the stronger the guiding of the surface wave along the nanorod. Due to the finite skin depth of silver, when the radius of the nanorod increases, less EM power penetrates the rod, and thus the absolute value of the quantity Q^c_z reduces, as shown in the upper-right panel of Fig. 12. The lower-right panel shows the ratio Q^c_z/Q^s_z versus the radius of the rod, showing again the backward propagation and the decrease of the power inside the silver nanorod as the radius increases.

 

Fig. 8. Resonant wavelength, QBW, Qθ, and bandwidth at grazing incidences: Upper-left: Wavelength of CSR vs. grazing angle calculated from the maximum Q_θ (Solid Blue) and from the eigenmode Eq. (26) with n=1 (Dashed Green) for the silver nanorod. The two curves coincide with each other. Upper-right: The QBW of the CSRs vs. grazing angle. Lower-left: Total Qθ factor of the CSRs vs. grazing angle. Lower-right: Bandwidth of the CSRs vs. grazing angle. The bandwidth decreases as the Qθ increases in the grazing region.

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Fig. 9. Various Q factors of the CSRs at near-zero grazing angles for a silver nanowire: Upper-left: The Q^c_z vs. grazing angle. Negative value indicates the backward propagation inside the silver nanowire. Upper-right: The Q^s_z vs. grazing angle. Lower-left: Ratio of the Q^c_z and the Q^s_z. Lower-right: Ratio of the Q^s_z and the Qθ.

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Fig. 10. Radially symmetric scattering and backward propagation in the silver nanorod of radius a=15 nm: θ≈0.001° at the resonance λ=406 nm. Upper-left: Poynting vector (S_ρ) in the ρ-direction. The S_ρ is nearly uniform in the x-y plane. Lower-left: Vector plot of the Poynting vector (S_x,S_y). Power in the x-y plane flows radially towards the center of the rod. Upper-right: The z-direction Poynting (S_z) inside and outside the rod. The S_z is asymmetric and flows backward inside the rod. Lower-right: Contour plot of the S_z further showing the asymmetric distribution in the x-y plane.

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Fig. 11. Intensity of the total electric field at the surface inside and outside the silver nanorod of radius a=15 nm: θ≈0.001° at the resonance λ=406 nm. Showing a 2D standing-wave pattern in the azimuthal and propagation directions. Upper plot: Just outside the surface ρ=a ⁺. Lower plot: Just inside the surface ρ=a ^-. Vertical axis: arc length along the circumference of the rod. Horizontal axis: distance along the axis of the rod.

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Fig. 12. Influence of the radius of a lossless silver nanorod on CSRs at the grazing angle θ=0.01°. Upper-left: CSR wavelength and the corresponding QBW vs. the radius of the rod. Left y-axis for the solid-blue curve; Right y-axis for the dashed-green curve. Upper-right: The Q^c_z of CSRs vs. the radius of the rod. Negative value indicates the backward propagation inside the nanowire. Lower-left: The total Q_θ factor and the corresponding bandwidth ΔF of CSRs vs. the radius of the rod. Left y-axis for the solid-blue curve; Right y-axis for the dashed-green curve. A decrease of the Q_θ factor implies an increase of the bandwidth of CSRs. Lower-right: The ratio Q^c_z/Q^s_z of the CSRs vs. the radius of the rod.

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5. Summary

In summary, we have investigated resonant scatterings of nanowires at near-zero grazing incidences through a newly developed grazing solution. We also interpreted grazing resonances in terms of the excitation of the first-order cyclic Sommerfeld waves and pointed out the non-plasmonic nature of cyclic Sommerfeld resonances. The extraordinary high Q value of these grazing resonances is due to the strong guiding of asymmetric long-range surface waves along the nanorods. This result enriches our fundamental understanding of scattering on the nanoscale and its relevance to many areas within nanotechnology. Since CSRs are associated with the natural modes of the nanorods, they may also be excited by other means. The merit of high Q, broadband, and low loss may render CSRs as an attractive alternative mechanism for enhanced-field applications in the nano-regime.