1. Introduction

Surface plasmon polaritons (SPPs) are propagating surface waves at metal-dielectric interfaces and originate from a coupling of electromagnetic radiation with the collective response of the electron plasma of the metal. SPPs have found a wide range of applications, such as biochemical sensing using surface-enhanced Raman spectroscopy (SERS) [1], subwavelengthimaging [2] or even high-harmonic generation within nonlinear optics [3–5]. Besides planar plasmonic waveguides, SPPs propagating on metallic nanowires have recently gained strong interest due to promising applications for subwavelength light sources [6, 7], quantum optics [8,9], nanoscale integrated circuits [10, 11] and new types of plasmonic excitations [12–14]. So far, the fundamental properties of such SPPs have been investigated by numerically solving Maxwell’s equations [15–17], which allows calculating the corresponding properties, but fails to give a detailed picture of the ongoing physics. It remains unclear what physical influence the curvature of the wire implies to the propagation properties of the SPPs, i.e. how it affects the waveguide dispersion, even though numerical calculations suggest a substantial influence.

It is well known that curved spaces can affect the propagation properties of quantum particles [18, 19], as observed in e.g. photonic topological crystals [20]. However, only a few groups have focused on the effect of interface curvature on the propagation of planar SPPs [21–23].

In this article we reveal the fundamental properties of SPPs propagating on metallic wires by introducing two curvature-induced geometric momenta which significantly change the velocity of the SPPs. We show how these surface modes are connected to their planar counterparts by a superposition of two counter-rotating trajectories and derive an explicit analytic dispersion equation for all guided SPP modes which is in excellent agreement with the rigorous numerical solutions down to wire radii in the order of the wavelength.

2. Theory

The wave vector of an SPP propagating along the planar interface of two semi-infinite metal-dielectric half spaces is determined by the interface boundary conditions leading to [25]

Here, k₀ = 2π/λ₀ is the magnitude of the vacuum wave vector (λ₀: vacuum wavelength), while ε₁ and ε₂ denote the permittivity of the metal and the dielectric (or vice versa), respectively.

Considering a cylindrical surface, we want to discuss how the propagation properties of a guided SPP change. A metal wire can be interpreted as a curved planar interface with curvature radius a. The wire radius should be much larger than the penetration depth in the metal, allowing to neglect the interaction of electromagnetic field components on opposite sides of the wire, which holds in the visible and near-infrared spectral range if a ≳ 100nm. Since a cylinder and a plane are locally isometric, we can project the wave vector of the planar SPP onto the azimuthal and longitudinal axis:

Here, ${\overrightarrow{e}}_z$ and ${\overrightarrow{e}}_{\theta }$ are the unity vectors in the cylindrical coordinate system and k_z and k_θ are the lengths along those directions. For constant values of k_z and k_θ, Eq. (2) implies that all SPPs propagate along helical trajectories on the wire surface (see Figs. 1(a)–(c)), which is already well known from literature [26–28] Longitudinal invariant field patterns (i.e. guided eigenmodes) are formed by a superposition of two counter-rotating SPPs on trajectories with opposite helicity (±m, number of periods in the k_θ -axis). Unfolding the wire creates a finite planar surface with a width equal to the circumference of the cylinder. The angle between the wire axis (here z-axis) and the direction of the SPP is the corresponding helix angle α (see Fig. 1(d)).

 

Fig. 1 Trajectories of the (a) m = 0, (b) m = ±1 and (c) m = ±2 SPP on a metal wire. (d) Deformation of the planar SPP in momentum space from a circle (yellow) to an ellipse (purple) due to the curvature-induced scaling factors s_z and s_θ. The green arrow refers to the fundamental SPP. Red and blue arrows correspond to the left- and right-handed helices of the m = ±1 and m = ±2 SPPs, respectively. Dotted arrows indicate the original (unscaled) wave vectors, while solid arrows show their stretched counterpart. The angles α₁ and α₂ refer to the helix angles of the corresponding SPP.

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In the classical particle picture, the SPP propagating along the two-dimensional surface cannot distinguish whether the surface is planar or cylindrical, since the intrinsic (Gaussian) curvature of both surfaces equals zero (see Appendix A for details). However, quantum particles and quantum quasi-particles (e.g. SPPs) are associated with wave functions and thus with evanescent fields. As a consequence, they are sensible to the extrinsic (mean) curvature and the three-dimensional space adjacent to the surface (i.e. they “feel” the curvature). In quantum mechanics the extrinsic curvature leads to a modification of the kinetic energy of a particle, resulting in an additional term in the Hamiltonian of the system [18, 19]. This additional term is called geometric potential. It was shown that a similar modification can be derived for the momentum operator, resulting in a geometric momentum depending on the extrinsic curvature of the interface [29, 30].

Performing an asymptotic expansion of the Bessel functions [31] in the dispersion equation of the metal wire for the fundamental SPP (m = 0), we find that the longitudinal wave vector k_z (propagation constant) corresponds to that of a planar SPP with an additional term depending on the radius of the wire (see Appendix C). The azimuthal wave vector is analyzed in a similar way by considering eigenmodes solely propagating around the circumference of the wire with zero longitudinal momentum (k_z = 0, see Appendix C). While the curvature-induced change of the longitudinal wave vector has been intensively addressed in literature, the azimuthal component has hardly been investigated [23]. We find that both longitudinal and azimuthal wave vectors are rescaled by particular factors k_z → k_z/s_z and k_θ → k_θ/s_θ with

Here, ε₁ and ε₂ indicate the permittivity of the wire and the surrounding cladding, respectively. The parameter V = k₀a(−[ε₁ + ε₂])^1/2 has a positive real part for all cases considered here. The symbol V has been chosen due to a similar mathematical form as the V-parameter used in the dielectric waveguide theory. The different signs in Eqs. (3a) and (3b) correspond to the case of a metal wire embedded in a dielectric (upper sign) and to the situation of a dielectric pin inside a metal (lower sign), respectively. Since we aim to discuss SPPs on metal wires, we focus our discussion to the first case (ε₁ → ε_M, ε₂ → ε_D). The scaling factors are solely curvature-induced and are equal to unity in the limit of a/λ₀ → ∞. Substituting the geometry factors (3a) and (3b) into Eq. (2) and squaring both sides yields an expression which we call the momentum ellipse (ME)

 

Fig. 2 (a) s_z and s_θ as given by Eqs. (3a) and (3b) for different wire radii in the case of a silver wire embedded in silica at λ₀ = 1000nm. (b) Positive quadrant of the momentum ellipse. The horizontal black lines refer to the quantized values of the azimuthal momentum. Circles indicate the intersections with the momentum ellipse, representing guided modes. The vertical dashed-dotted line corresponds to the quasi-cutoff condition $k_z^{qco}=k_0{\epsilon }_D^{1/2}$. (c) Geometric slowdown factor ζ = k_plan/k_z and (d) helix angle as function of azimuthal wave vector. The calculations in (b)–(d) were performed for three different wire radii: a = 1μm (blue solid line), a = 10μm (red dashed line) and a = 100μm (quasi-planar case, green dotted line).

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Guided eigenmodes require that the azimuthal phase of the SPP accumulating during one circulation is a multiple of 2π (a similar condition is used in Bohr’s model of atoms). The result is a quantization of the azimuthal wave vector k_θa = m with m = 0, ±1, ±2, …, where m is the azimuthal mode order of the SPP. Positive (negative) numbers of m indicate left-handed (right-handed) helical trajectories. In general, a translational invariant mode pattern is obtained by superposing two helices of opposite helicity (±m), leading to a vanishing net azimuthal momentum of the eigenmode. A special case of this superposition is the fundamental SPP which reduces to a single (m = 0) trajectory without helicity. Substituting this resonance condition into the Eq. (4), we can identify the guided mode by the intersection points of the horizontal lines (lines of constant azimuthal momentum: k_θ = m/a) with the ME (see Fig. 2(b)), leading to an explicit analytical dispersion equation of SPPs on metal wires:

The two square-root factors in Eq. (5) fundamentally result from the two geometric momenta (i.e. scaling factors s_z and s_θ) and thus only appear due to the propagation of the SPP on the cylindrical surface. Equation (5) can directly be used to calculate the dispersion of guided SPPs of any azimuthal order and consists of three factors: (i) the wave vector of a planar plasmon, (ii) a contribution from the longitudinal geometric momentum which is larger than unity and increases k_z, and (iii) a term taking into account the quantized azimuthal momentum and decreases k_z. In the limit a/λ₀ → ∞ (planar metal-dielectric interface), V approaches infinity and Eq. (5) converges to $k_z^{\left(m\right)}=k_{\mathrm{plan}}$ as expected. We want to emphasize again that Eq. 5 was derived under the assumption of non-absorbing materials. However, as we will see in the next section, our model yields very good results for the real part of the propagation constant even if the ohmic losses of the metal are considered.

3. Comparison with numerical calculations

3.1. Radius dispersion

As an example we investigate the propagation of SPPs on a silver wire embedded in silica and compare our analytic model to the corresponding numerical solutions of Maxwell’s equations. All calculations were performed at a constant wavelength of λ₀ = 1000nm leading to ε_M = −45.7534 + 2.7213i [32] and ε_D = 2.1037 [33]. Although our model was derived for lossless materials, it is also capable of handling lossy systems. However, as we will discuss later, the imaginary part of k_z cannot be reproduced near the cutoff of all SPPs with m > 1. Therefore, we will restrict the discussion to the real part of k_z. The fundamental SPP (m = 0) of a metal wire corresponds to an azimuthally symmetric mode which is TM-polarized (TM₀) [16]. It is well known that this eigenmode is strongly confined to the interface and does not exhibit a cutoff, i.e. that it will be guided for any wire diameter. In our model, this SPP is described by a non-helical trajectory. The azimuthal term (right-handed square-root term) of Eq. (5) becomes unity and the equation reduces to $k_z^{\left(0\right)}=k_{\mathrm{plan}}{\left(1+1/V\right)}^{1/2}$. Thus, the TM₀ SPP only experiences the longitudinal geometric momentum. Its influence, and correspondingly the propagation constant, strongly increases towards smaller radii in accordance with the exact numerical solution (see Fig. 3). Our model describes the dispersion of the TM₀ SPP down to wire radii in the order of the wavelength. For smaller radii, the assumption κ_ja ≫ 1 (j = M, D, see Eq. (11)) used in the derivation of the longitudinal geometric momentum does not hold anymore and deviations occur. If the longitudinal geometric momentum is neglected (blue dashed lines in the inset of Fig. 3), the dispersion of the TM₀ SPP would become radius-independent, which is obviously incorrect. Therefore, this model reveals that the longitudinal geometric momentum is the physical origin of the waveguide dispersion of the fundamental SPP on a metal wire. Contrary to other models which do not take into account the curvature-induced geometric momentum [26, 27], our model uniquely explains the behavior of the fundamental SPP very well.

 

Fig. 3 Real parts of the normalized propagation constants of our model (Eq. (5), red solid lines) and numerical solutions of Maxwell’s equations (Eq. (8), black dotted lines) for a silver wire embedded in silica at λ₀ = 1000nm. The black dashed line corresponds a planar SPP. Black stars and circles represent the quasi-cutoff and the cutoff of the particular SPPs, respectively. The inset is a close-up of the graph for radii larger than 1μm. Blue dashed lines represent the waveguide dispersion if all geometric momenta are neglected. The integer m indicates the azimuthal mode order. The graph in the top row shows the (logarithmic) relative percentage error between our model and the numerical solutions. Green and red shaded regions indicate errors smaller and larger than one percent, respectively.

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The SPP carrying the smallest amount of azimuthal momentum (m = ±1, k_θ = ±1/a, HE₁-mode) is a hybrid-mode (HE₁) with six non-vanishing field components [16]. Generally, it reveals a long-range SPP behavior [27], exhibiting no cutoff, weak confinement and low losses in the limit of a/λ₀ → 0. Figure 3 shows that the propagation constant of the HE₁ SPP is predicted properly by our model down to wire radii in the wavelength range. Below this region the deviation is clearly visible, which we address to two reasons: (i) the propagation constant of the HE₁ SPP strongly depends on the quality of the approximation of the TM₀ SPP, which gets more crucial towards small wire radii; (ii) the assumption m ≫ 1 used for the asymptotic expansion of the azimuthal momentum does not hold in the case of |m| = 1.

In contrast to planar interfaces or films, metal wires can support higher order SPPs (|m| ≥ 2) whose propagation constants show so-called quasi-cutoffs (intersections with the silica light line, $\Re \{k_z\}=k_0{\epsilon }_2^{1/2}$ and distinct cutoffs (vanishing modal loss, $\mathfrak{J}\left\{k_z\right\}=0$). Exactly at the cut-off, the mode fields show no evanescent decay but rather a purely oscillating behavior outside the wire. The real parts of the propagation constants of the higher-order SPPs are extremely well reproduced by our model with an improving accuracy towards larger values of |m| (see Fig. 3). Especially the dispersion within the positive region of the diagram is reproduced extremely precise, whereas assuming no geometric momenta fully fails to explain the dispersion of the HE_m SPPs (blue dashed lines in the inset of Fig. 3). It is interesting to note that Eq. (5) intuitively suggests that all higher-order SPPs result from the dispersion of the fundamental SPP by a plain multiplication of a factor originating from the additional azimuthal geometric momentum. Figure 3 visualizes the previously discussed fact, that the waveguide dispersion of the fundamental SPP on a wire is described by a single function (first square-root term in Eq. (5)), whereas all higher order modes correspond to a multiplicative superposition of two functions (both square-root terms in Eq. (5)). Our model also predicts the quasi-cutoffs very accurately, which is the key for many phase matching situations [13]. Though, it does not predict the appearance of real mode cutoffs (black circles in Fig. 3), which we attribute to the assumption of real-valued k_z in Eq. (4). However, these cutoffs are located at small values of k_z, which is far away from the typical region of phase matching in many photonic/plasmonic systems [13].

3.2. Wavelength dispersion

Figure 4 shows the real part of the normalized propagation constants k_z/k₀ (i.e. effective mode index) in the spectral range from 400 nm (VIS) up to 1500 nm (NIR) for SPPs from m = 0 to m = 9. The dispersion obtained by Eq. (5) accurately resembles the results from the full numerical solutions of Maxwell’s equations with a relative error smaller than 1% for the TM₀ and HE₁ SPP, even at a wavelength of 1500 nm which is in fact three times larger than the wire radius. Thus, in contrast to previously introduced models which do not take into account the curvature-induced geometric momenta [26, 27], our model is capable of approximating the propagation constants of all SPPs extremely well. It needs to be pointed out that the model introduced by Schmidt and Russell [27] gives a reasonable approximation for m ≥ 2, but relies on an unjustified mathematical substitution (m → m − 1) [27], with the origin of this substitution being unclear and fails to describe the TM₀ and HE₁ SPPs. In contrast, our model approximates all guided modes in the systems without any spurious substitutions.

 

Fig. 4 Comparison of the spectral distributions of the real parts of the (k₀-normalized) propagation constants of our model (Eq. (5), red solid lines), the numerical solutions of Maxwell’s equations (Eq. (8), black dotted lines) and results when all geometric momenta are neglected (blue dashed lines) (silver wire, a = 500nm embedded in silica). Black stars and circles represent the quasi-cutoff and the real cutoff of the particular SPPs, respectively. The integer m indicates the azimuthal mode order. The graph in the top row shows the relative percentage error between our model and the numerical solutions. Green and red shaded regions indicate errors smaller and larger than one percent, respectively.

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3.3. Quasi-cutoffs

We also investigated the spectral position of the quasi-cutoffs for different SPP mode orders (m = 2 to m = 5) of silver wires with radii between 100 and 500 nm and compared our findings with the model introduced by Schmidt and Russell (relying on the heuristically reduced mode order) and the full numerical solutions (see Fig. 5(a)). Equation (5) accurately resembles the evolution of the quasi cutoff with increasing accuracy towards higher mode orders. The deviations are smaller than 13 nm for the HE₂ SPP and decrease to below 3 nm for the HE₃ and all higher order SPPs. The corresponding relative percentage error |∆λ_qco/λ_qco| is smaller than 1% for the HE₃ and all higher order SPPs (see Fig. 5(b)).

 

Fig. 5 (a) Quasi-cutoff wavelengths as a function of radius of guided SPPs on a silver wire embedded in silica for different mode orders (red solid lines: quasi-cutoff wavelengths calculated using Eq. (5), black dotted lines: exact solutions, blue dashed lines: Schmidt/Russell model [27]. (b) Corresponding radius dependence of the percentage error of both models. The different mode orders are indicated by the integer numbers.

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4. Limits of the model

The presented model approximates the fundamental SPP using an asymptotic expansion of Bessel’s functions for large arguments (κ_ja ≫ 1), where κ_j (j = M, D) are the transverse wave vectors given by Eq. (11). Inside the metal wire this condition is generally fulfilled for k₀a ≳ 1 in all spectral domains (including the THz regime) since the negative sign of ε_M always increases the radicand in Eq. (11). Inside the cladding, however, κ_Da can reach very small values if $k_z^2\approx k_0^2{\epsilon }_2$ (quasi-cutoff) and, as a consequence, the condition κ_Da ≫ 1 is only fulfilled if k_Da ≫ 1. Although we found that reasonable results can be achieved down to values of κ_Da ≈ 1, the approximation of the dispersion of guided modes using Eq. (5) gets less accurate for κ_Da < 1. In case of the azimuthal wave vector, all higher order modes (m ≥ 1) contain an additional azimuthal momentum, which is approximated by an asymptotic expansion of Bessel’s functions in the limit of large Bessel function orders. This expansion yields good results for mode orders m ≥ 2 again with increasing accuracy towards larger mode orders. Equation (5) qualitatively reproduces the evolution of the imaginary part of the propagation constant with less accuracy than the corresponding real part (see Appendix D for further details). Thus we want to emphasize that Eq. (5) can be used to reveal the fundamental dispersion properties of SPPs on wires and not the modal attenuation (with the exception of the TM₀ SPP, which is in fact nicely reproduced), even though the agreement is substantially better than given by models reported in literature [26, 27]. While the quasi-cutoff of a particular SPP can be found straightforward by our model using the real part of the propagation constant $(\Re \{k_z\}=k_0{\epsilon }_2^{1/2})$, the real cutoff cannot be calculated, since our model fails near the cutoff condition $\mathfrak{J}\left\{k_z\right\}=0$. In the case of non-absorbing materials $(\mathfrak{J}\{{\epsilon }_{1,2}\}=0)$, however, k_z cannot cross the silica line and as a result, the cutoff and the quasi-cutoff are identical. In this particular case, our model allows to calculate the cutoff of any guided SPP.

5. Conclusions

In conclusion, we have shown that the propagation of planar plasmons on cylindrical metallic wires is fundamentally affected by two geometric momenta, extending the current knowledge of propagating plasmons. These momenta are solely induced by the curved wire surface and result in a significant change of the phase velocity and helix angle of the SPPs. Guided SPPs on such wires are formed by a superposition of two planar SPPs propagating on helical trajectories with opposite helicity, leading to modes with defined mode numbers. We obtained a straightforward-to-use analytic equation to describe the real part of the propagation constant of the guided SPPs, showing excellent agreement with corresponding numerical solutions down to wire radii in the order of the operation wavelength. We showed that the dispersion of the fundamental SPP on a wire solely originates from the longitudinal momentum, whereas all higher order modes are additionally influenced by the azimuthal momentum. We have thus revealed the underlying physics of SPP propagation on metal wires by an intuitive model and anticipate application of our results in all fields related to plasmonics on curved surfaces.

Appendix A: Gaussian and mean curvature

The Gaussian curvature K and the mean curvature H of a two-dimensional surface are defined as [34]

(6)$$K=k_1\cdot k_2,$$

(7)$$H=\frac{1}{2}(k_1+k_2).$$

where k_i = 1/a_i (i = 1, 2) are the principle curvatures. The Gaussian curvature is an intrinsic measure of curvature, characterized by the metric of the surface and not by the three-dimensional space in which the surface is embedded in. Contrary, the mean curvature is an extrinsic measure of curvature that locally describes the curvature of a surface embedded in an ambient space such as Euclidean space. Table 1 shows a comparison of the Gaussian and mean curvatures of different surfaces. For example, a plane and a cylinder have the same Gaussian curvature, i.e. they are (locally) isometric. However, a sphere has a non-vanishing (positive) Gaussian curvature. This results in the effect that the sum of the angles of a triangle that is drawn on the surface of the sphere is larger than 180°.

Table 1. Gaussian and mean curvatures of different 2D-surfaces

View Table

Appendix B: Solution of Maxwell’s equations for an infinite cylinder

The solution of Maxwell’s equations for a cylindrical waveguide is well known from the literature and given by

(8)$$({\mathrm{\Psi }}_1-{\mathrm{\Psi }}_2)({\epsilon }_1{\mathrm{\Psi }}_1-{\epsilon }_2{\mathrm{\Psi }}_2)=m^2{k}_z^2{({\epsilon }_1-{\epsilon }_2)}^2$$

with

(9)$${\mathrm{\Psi }}_1=\frac{{\kappa }_2^2}{k_0}\left(m+{\kappa }_{1}a\frac{I_{m+1}({\kappa }_{1}a)}{I_m({\kappa }_{1}a)}\right)$$

(10)$${\mathrm{\Psi }}_2=\frac{{\kappa }_1^2}{k_0}\left(m-{\kappa }_{2}a\frac{K_{m+1}({\kappa }_{2}a)}{K_m({\kappa }_{2}a)}\right),$$

where k₀ = 2π/λ₀ is the magnitude of the vacuum wave vector, λ₀ is the vacuum wavelength and

(11)$${\kappa }_j=\sqrt{k_z^2-k_0^2{\epsilon }_j},(j=1,2)$$

are the transverse wave vectors inside the wire/pin and the cladding, and m = 0, ±1, ±2, … is the azimuthal mode order. The parameter a represents the radius of the wire, while ε₁ and ε₂ correspond to the permittivity of the wire/pin and the cladding, respectively. The dispersion equation (8) can also be written using the Bessel function J_m(x) and the Hankel function $H_m^{\left(1\right)}\left(x\right)$ by modified definitions of κ₁ and κ₂.

Appendix C: Derivation of geometry factors

I. Longitudinal propagation

For simplicity we assume a lossless metal. For the subsequent expansion of Bessel’s functions, their arguments need to be real-valued. In the case of a metal wire in a dielectric, the results from Eq. (8) show that $k_z\ge k_0\sqrt{{\epsilon }_2}$. Therfore, in order to obtain real-valued arguments, the electromagnetic fields inside the wire and the cladding need to be described by the modified Bessel functions I_m(z) and K_m(z), respectively. In the case of a dielectric pin inside a metallic cladding the choice of the Bessel function becomes more crucial. Inside the cladding the Bessel function K_m(z) still yields real-valued arguments. However, the results of Eq. (8) for this case show that $k_z>k_0\sqrt{{\epsilon }_1}$ for large pin radii and $k_z<k_0\sqrt{{\epsilon }_1}$ for small pin radii. Since the pin radius needs to be larger than several wavelengths in order to avoid an interaction of the electromagnetic fields on opposing sides of the pin, we can assume that $k_z>k_0\sqrt{{\epsilon }_1}$ in this domain and chose I_m(z) as the appropriate Bessel function to describe the fields inside the pin.

The dispersion equation of an infinite metal wire with an azimuthal mode order of m = 0 is derived from Eq. (8) and given by

(12)$$-\frac{{\kappa }_1}{{\epsilon }_1}\frac{I_0({\kappa }_{1}a)}{I_1({\kappa }_{1}a)}=\frac{{\kappa }_2}{{\epsilon }_2}\frac{K_0({\kappa }_{2}a)}{K_1({\kappa }_{2}a)}.$$

The first step of the derivation of the geometric momentum relies on performing an asymptotic expansion of the modified Bessel functions I_m(z) and K_m(z) (they are necessary since their argument has positive real values which is needed for the asymptotic expansion). This expansion is given by [31]

(13)$$I_m(z)=\frac{1}{\sqrt{2\pi z}}{e}^z{\displaystyle \sum _{k=0}^{\infty }{(-1)}^k}\frac{a_k(m)}{z^k},$$

(14)$$K_m(z)=\sqrt{\frac{\pi }{2z}}{e}^{-z}{\displaystyle \sum _{k=0}^{\infty }\frac{a_k(m)}{z^k}},$$

with a₀(m) = 1 and

(15)$$a_k(m)=\frac{(4m^2-1^2)(4m^2-3^2)\dots (4m^2-{(2k-1)}^2)}{k!8^k}$$

Truncating the asymptotic expansion after the first order term we are able to write Eq. (12) as

(16)$$-\frac{{\kappa }_1}{{\epsilon }_1}\frac{1+\frac{1}{8{\kappa }_{1}a}}{1-\frac{3}{8{\kappa }_{1}a}}\underset{\approx }{{(k_{0}a)}^2\gg 1}\frac{{\kappa }_2}{{\epsilon }_2}\frac{1-\frac{1}{8{\kappa }_{2}a}}{1+\frac{3}{8{\kappa }_{2}a}}.$$

We have assumed that all terms with (k₀a)⁻² or higher were neglected since k₀a ≫ 1. This assumption significantly reduces the complexity of the resulting equation and holds down to wire radii in the order of the operation wavelength. However, this may lead to inaccuracies near the cutoff when ${\left(n_{\mathrm{eff}}^2-{\epsilon }_2\right)}^{1/2}$ becomes very small. By expanding the left and right hand side of Eq. (16) with (1+3=8κ₁a) and (1 − 3=8κ₂a), respectively, Eq. (16) can be further simplified to

(17)$$-\frac{{\kappa }_1}{{\epsilon }_1}\left(1+\frac{1}{2{\kappa }_{1}a}\right)\underset{\approx }{{(k_{0}a)}^2\gg 1}\frac{{\kappa }_2}{{\epsilon }_2}\left(1-\frac{1}{2{\kappa }_{2}a}\right).$$

We found that 1 − 1/(1 − 2κ₂a) is a much better approximation than 1 − 1/(2κ₂a) for the ratio K₀(κ₂a)/K₁(κ₂a), especially for small values of κ₂a. However, this ansatz leads to a much more complicated mathematical formalism. Squaring both sides of Eq. (17) and using ${\kappa }_1^2={\kappa }_2^2+k_0^2({\epsilon }_2-{\epsilon }_1)$ leads to the quadratic equation

(18)$${\kappa }_2^2-\frac{4a{\epsilon }_1}{4a^2({\epsilon }_1+{\epsilon }_2)}{\kappa }_2+\frac{4k_0^2{a}^2{\epsilon }_2^2+({\epsilon }_1-{\epsilon }_2)}{4a^2({\epsilon }_1+{\epsilon }_2)}=0.$$

In order to simplify the equations we introduce (ε₁ + ε₂), having a similar form as the V-parameter used in fiber waveguide theory. Solving Eq. (18) for k_z by inserting the definition of κ₂ yields

(19)$$k_z^2=\underset{k_{\mathrm{plan}}^2}{\underbrace{k_0^2\frac{{\epsilon }_1{\epsilon }_2}{{\epsilon }_1+{\epsilon }_2}}}\left(1-\frac{1}{4V^2}\left[\frac{{\epsilon }_1}{{\epsilon }_2}+\frac{{\epsilon }_2}{{\epsilon }_1}\pm 2\sqrt{1+4V^2}\right]\right)$$

The positive or negative sign in Eq. (19) corresponds to different directions of the surface curvature (plus: metallic wire in dielectric cladding; minus: dielectric hole in metal cladding). We again neglect the (k₀a)⁻² terms (contained in V) and assume that V² ≫ 1/4 in order to extract the root on the right-hand side of Eq. (19). This leads to the final and very simple equation for a surface plasmon propagating along the axis of the wire

(20)$$k_z=k_{\mathrm{plan}}\underset{s_z}{\underbrace{\sqrt{1\pm \frac{1}{V}}}}.$$

II. Azimuthal propagation

For a wave that is solely propagating around the circumference of a wire or a pin (k_z = 0), the dispersion equation reads

(21)$$-\frac{1}{\sqrt{-{\epsilon }_1}}\frac{{I^{\prime }}_{\nu }(k_0\sqrt{-{\epsilon }_1}a)}{I_{\nu }(k_0\sqrt{-{\epsilon }_1}a)}=\frac{1}{\sqrt{{\epsilon }_2}}\frac{{H^{\prime }}_{\nu }^{(1)}(k_0\sqrt{{\epsilon }_2}a)}{H_{\nu }^{(1)}(k_0\sqrt{{\epsilon }_2}a)}.$$

and

(22)$$-\frac{1}{\sqrt{{\epsilon }_1}}\frac{{J^{\prime }}_{\nu }(k_0\sqrt{{\epsilon }_1}a)}{J_{\nu }(k_0\sqrt{{\epsilon }_1}a)}=\frac{1}{\sqrt{-{\epsilon }_2}}\frac{{K^{\prime }}_{\nu }^{(1)}(k_0\sqrt{-{\epsilon }_2}a)}{K_{\nu }^{(1)}(k_0\sqrt{-{\epsilon }_2}a)},$$

respectively. Here we used the different Bessel functions in order to obtain real-valued arguments which are necessary for the asymptotic expansion. In contrast to Eq. (12) we used ν = k_θa instead of m to show that the order is in general not integer. For ν ≫ 1 the Bessel functions I_ν(z), K_ν(z), J_ν(z) as well as the Hankel function $H_{\nu }^{(1)}(z)$ can be written in the form of an asymptotic expansion [31]:

(23)$$I_{\nu }(z)\propto \frac{\exp (\sqrt{{\nu }^2+z^2}-\nu \ln [\frac{\nu +\sqrt{{\nu }^2+z^2}}{z}])}{{({\nu }^2+z^2)}^{1/4}}+\dots$$

(24)$$K_{\nu }(z)\propto \frac{\exp (-\sqrt{{\nu }^2+z^2}+\nu \ln [\frac{\nu +\sqrt{{\nu }^2+z^2}}{z}])}{{({\nu }^2+z^2)}^{1/4}}+\dots$$

(25)$$J_{\nu }(z)\propto \frac{\exp (\sqrt{{\nu }^2-z^2}-\nu \ln [i\frac{\nu +\sqrt{{\nu }^2-z^2}}{z}])}{{({\nu }^2-z^2)}^{1/4}}+\dots$$

(26)$$H_{\nu }^{(1)}(z)\propto \frac{\exp (-\sqrt{{\nu }^2-z^2}+\nu \ln [i\frac{\nu +\sqrt{{\nu }^2-z^2}}{z}])}{{({\nu }^2-z^2)}^{1/4}}+\dots .$$

The derivatives with respect to their arguments can be calculated straightforward and yield

(27a)$$\frac{{I^{\prime }}_{\nu }(z)}{I_{\nu }(z)}\approx \frac{2{({\nu }^2+z^2)}^{3/2}-z^2}{2z({\nu }^2+z^2)},$$

(27b)$$\frac{{K^{\prime }}_{\nu }(z)}{K_{\nu }(z)}\approx -\frac{2{({\nu }^2+z^2)}^{3/2}+z^2}{2z({\nu }^2+z^2)},$$

(27c)$$\frac{{J^{\prime }}_{\nu }(z)}{J_{\nu }(z)}\approx \frac{2{({\nu }^2-z^2)}^{3/2}+z^2}{2z({\nu }^2-z^2)},$$

(27d)$$\frac{{H^{\prime }}_{\nu }^{(1)}(z)}{H_{\nu }^{(1)}(z)}\approx -\frac{2{({\nu }^2-z^2)}^{3/2}-z^2}{2z({\nu }^2-z^2)}.$$

The condition ν = k_θa ≫ 1 holds better for higher order surface plasmon polaritons but turns out to give good results even for lower orders as seen in the paper. Substituting Eqs. (27a) and (27d) in (21) yields the approximated dispersion equation for an azimuthal propagating wave on a metal wire

(28)$$-\frac{{\tilde{\kappa }}_1}{{\epsilon }_1}\left(1+\frac{k_0^2{\epsilon }_1}{2{\tilde{\kappa }}_1^{3}a}\right)\approx \frac{{\tilde{\kappa }}_2}{{\epsilon }_2}\left(1-\frac{k_0^2{\epsilon }_2}{2{\tilde{\kappa }}_2^{3}a}\right),$$

where ${\tilde{\kappa }}_j=\sqrt{k_{\theta }^2-k_0^2{\epsilon }_j}$ with j = 1, 2 are the transverse wave vectors for this case. We now search for azimuthal wave vectors of the form

(29)$$k_{\theta }=A+B/(k_{0}a),$$

with A and B being unknown. Substituting Eq. (29) into the transverse wave vectors ${\tilde{\kappa }}_M$ and ${\tilde{\kappa }}_D$ we find

(30a)$${\tilde{\kappa }}_j^2={\widehat{\kappa }}_j^2+\frac{2AB}{k_{0}a},$$

(30b)$${\tilde{\kappa }}_j={\widehat{\kappa }}_j+\frac{AB}{k_{0}a{\widehat{\kappa }}_j},$$

where ${\widehat{\kappa }}_j=\sqrt{A^2-k_0^2{\epsilon }_j}$. Here we have neglected all terms proportional to (k₀a)⁻² and used the approximation $\sqrt{1+z}\approx 1+z/2$ for small values of z. By using (30a) and (30b) we can rewrite Eq. (28) and compare terms of different order of k₀a separately, yielding

(31a)$$A=k_0\sqrt{\frac{{\epsilon }_1{\epsilon }_2}{{\epsilon }_1+{\epsilon }_2}}=k_{\mathrm{plan}},$$

(31b)$$B=\frac{k_0^3}{2A}\left(\frac{1}{{\widehat{\kappa }}_2^2}-\frac{1}{{\widehat{\kappa }}_1^2}\right){\left(\frac{1}{{\widehat{\kappa }}_2{\epsilon }_2}+\frac{1}{{\widehat{\kappa }}_1{\epsilon }_1}\right)}^{-1}.$$

Using the relation $-{\widehat{\kappa }}_1{\epsilon }_2={\widehat{\kappa }}_2{\epsilon }_1$ we can eliminate ${\widehat{\kappa }}_1$ from the numerator and denominator in Eq. (31b).

A similar treatment of the dielectric pin in a metallic cladding can be performed by substituting Eqs. (27b) and (27b) into the dispersion Equation (22). We find

(32)$$k_{\theta }=k_{\mathrm{plan}}\underset{s_{\theta }}{\underbrace{\sqrt{1\mp \frac{{\epsilon }_1^2\mp {\epsilon }_2^2}{{\epsilon }_1^2-{\epsilon }_2^2}\frac{{({\epsilon }_1+{\epsilon }_2)}^2}{{\epsilon }_1\cdot {\epsilon }_2}\frac{1}{V}}}},$$

containing the earlier introduced variable V. The upper and lower signs in Eq. (32) correspond to a metal wire embedded in a dielectric and a dielectric pin in a metallic cladding, respectively. Please note that we again have used the approximation $1+z/2\approx \sqrt{1+z}$ to bring Eq. (32) in a mathematical form similar to (20).

Appendix D: Imaginary part of the propagation constant

For the sake of completeness, we discuss the imaginary part of the propagation constant $\mathfrak{J}\left\{k_z\right\}$, which is related to the attenuation of the different SPPs. Figures 6(a) and 6(b) show the corresponding imaginary parts of the simulations used to obtain Figs. 3 and 4, respectively. The imaginary parts of the TM₀ SPP, as a function of the radius as well as the wavelength, are reproduced very well by the geometric momentum model. However, the overall correspondence of $\mathfrak{J}\left\{k_z\right\}$ between our model and numerical solutions of the exact dispersion Equation (8) for higher order SPPs is less than for the real part, especially near their cutoffs. In our opinion, this inaccuracy originates from the assumption of non-absorbing materials for the derivation of Eq. (4) as well as the quantization k_θa = m, which implies that k_θ is real-valued. In fact, the azimuthal wave vector is generally complex valued [35], even if no material absorption is considered. This is caused by radiation, which is the dominant loss mechanism for azimuthally guided SPPs. However, for $\Re \{k_z\}>k_0{\epsilon }_2^{1/2}$ the relative error of the imaginary parts of the propagation constants is usually below 1%, which is sufficient for many relevant cases.

 

Fig. 6 Comparison of the imaginary part of the propagation constants between our model and exact numerical solutions. The bottom row shows the imaginary part of the normalized propagation constants k_z as (a) a function of the wire radius for a silver wire embedded in silica at λ₀ = 1000nm and (b) as a function of the wavelength for a silver wire with a radius of a = 500nm. Red lines refer to our model, while black dotted lines correspond to the solution of the exact dispersion relation. The top row shows the corresponding relative percentage error of the model and exact solutions. The integer m indicates the azimuthal mode order. Green and red shaded regions indicate errors smaller and larger than one percent, respectively.

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Acknowledgments

The authors thank Adrian Lorenz, Tino Elsmann and Tobias Tiess for the fruitful discussions, as well as Maria Perel for the help concerning the mathematical derivation of the geometric momenta.

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