1. Introduction

Solid silica fibers are well established low loss waveguides in the visible and near infrared (IR) range. In the mid-IR, being highly relevant for emerging areas such as environmental and life-science [1] or metrology [2], such fibers are not employed due to exceedingly large material absorption [3]. Other glass systems solve that issue (e.g., chalcogenides, fluorides or tellurites [4, 5]) but are difficult to process, show low damage thresholds and susceptibility to radiation. Furthermore, applications such as gas spectroscopy or biomedical sensing demand strong interaction of the light with the medium under investigation, which is challenging in solid fibers.

Improvements in fabrication technology have recently led to disruptive developments in the field of hollow core fiber (HCFs) – fiber with air cores – solving the aforementioned issues and opening up new application areas for optical fibers. In this case, the guided power is dominantly concentrated in the ‘empty’ core with only a small field overlap with the host glass, strongly reducing the effect of material absorption. Since the average refractive index (RI) of the cladding is larger than that of core, the modes of HCFs are intrinsically leaky with the light guidance relying on photonic band gaps [6], omnidirectional reflection [7, 8], low-density of states [9], anti-resonant guidance [10, 11] and metamaterials [12–14]. The modes in strongly confining HCFs are comparable to those of hollow metal waveguides, and thus reducing the losses demands increasing core diameters. It is commonly accepted that losses scale with 1/R³ (R: core radius) [14–19], however, no analytic expression exists which clarifies this dependency.

Here, we introduce an approximate model describing the complex effective index of various HCFs with comparably large air core diameters by analytic expressions, emphasizing the 1/R³ dependence (Fig 1(a)). The model reduces the complexity of the cylindrical HCF to a single reflection at a planar interface. The interface properties are included in two dimensionless reflection parameters which are independent of the core diameter and can either be calculated analytically or, in the case of more complicated cladding geometries, obtained numerically. The resulting analytical expressions unambiguously indicate that the 1/R³ dependence is a universal feature of HCFs. As our model substantially reduces the simulation complexity, it enables large scale parameter sweeps, since simulations of the full HCF geometry are unnecessary. We apply our model to the case of a HCF with a highly anisotropic cladding consisting of a ring of longitudinally invariant metallic nanowires, which at high metal filling fractions resembles an indefinite metamaterial (iMM), and define straightforward design rules allowing for modal discrimination engineering.

 

Fig. 1 Physical background of the single-interface reflection model for approximating the complex effective index dispersion of large core hollow-core fibers. (a) Schematic of hollow core fiber highlighting the microstructured cladding. (b) Reflection at the corresponding interface assuming gracing incidence.

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This article is structured as follows. In Sec. 2 we derive approximate analytical expressions for the complex effective index. In Sec. 3 our model is applied to an example system, which is a hollow core fiber with iMM cladding. We show that the model accurately reproduces the complex effective index evolution obtained from finite element modelling. We also define design rules for the optimization of modal discrimination and show the optimal structural parameters for a practically relevant geometry.

2. Derivation of the analytic equations for the complex effective index

Our reflection model assumes that the modal propagation inside HCFs can be approximated by grazing-incidence rays at the air-cladding interface (Fig. 1). The modal attenuation of HCFs thus originates from the attenuation of the individual reflections, i.e., from the non-unity reflection coefficient of the corresponding planar interface (|r|² < 1, r: amplitude reflection coefficient). Our model correlates this reflection with the various leaky modes propagating inside the air core. The derivation starts by analyzing the reflection of a single interface under grazing incidence. The resulting boundary conditions, in combination with the fields of the cylindrical geometry, finally yields analytic expressions for the complex effective index for the various modes, which are entirely characterized by two dimensionless and polarization-dependent reflection parameters. The following sections present the main steps of the calculation, with more details in Appendix A.

2.1. Field in the core

The fields of cylindrical waveguides with longitudinal invariance are given by the solutions to the Helmholtz equation [20, 21]:

Here, J_m and J^′_m are the Bessel function of m-th order and its first derivative, m is the azimuthal mode index, β and κ are the axial and radial wave vector components (Fig. 1(b)), k₀ is the vacuum wave number with $k_0^2={\beta }^2+{\kappa }^2$, and $Z_0=\sqrt{{\mu }_0/{ϵ}_0}$ is the vacuum impedance. In the following we will derive approximate solutions for the lowest order modes in case of large core diameters (R ≫ λ), resulting in small radial wave vectors (κ ≪ k₀), which corresponds to the case of grazing incidence ψ = arcsin κ/k ≪ 1 (ψ: complementary angle of incidence, Fig. 1(b)).

2.2. Planar model

In our model, we separate the calculation of the modal parameters (κ and β) from the modeling of the cladding. This is achieved by defining two dimensionless reflection parameters ν_TE and ν_TM characterizing the reflection properties of the cladding. As a first step we consider the field in a small volume element of the size Δr × rΔϕ × Δz at the core-cladding boundary. For Δr ≪ R and Δϕ ≪ 1 we can neglect the curvature of the boundary and approximate the total field as a superposition of incident and reflected plane waves (Fig. 1(b)). This model requires that the reflection at the core-cladding interface solely occurs within the mentioned volume element. For the situation of a simple capillary this condition is always fulfilled, whereas for a HCF with microstructured cladding the thickness t of the reflecting layer must be small compared to the core radius (t ≪ R). The coordinates and field components of the planar model are related to the quantities of the cylindrical waveguide by x + R → r, y → rϕ, E_y → E_ϕ etc. In the case of grazing incidence the reflection coefficients for TE and TM waves are close to −1 and their dependency on ψ can be approximated by a first order series expansion

For a homogeneous dielectric cladding (i.e., a capillary) Eqs. (7) and (8) have analytic forms and can be derived from the Fresnel Equations (see Appendix A). In the case of more complicated claddings such as the iMM considered here, ν_TE and ν_TM depend on material properties, wavelength λ and geometrical parameters, and must be obtained numerically. For the fields at the boundary we can derive the following relations between the longitudinal and the transversal field components (see Appendix A for details). For TE waves we obtain

In the following, Eqs. (9) and (10) will be used as boundary conditions for the mode fields at r = R with the substitutions E_y → E_ϕ, and H_y → H_ϕ.

2.3. TE modes

TE modes are represented by Eqs. (4) and (5) with A₁ = 0 and m = 0. In the limit of a perfectly reflecting boundary (r_TE = −1) the transverse field component is zero at the boundary since perfect conductors have zero penetration depth. Equation (5) leads to $E_{\varphi }\left(R\right)\propto {J^{\prime }}_0\left(\kappa R\right)=0$, and the values of the radial wave number for perfect reflection κ₀ are then defined by

In the more general case (r_TE ≈ −1) the boundary condition in Eq. (9) results in

Using a perturbation ansatz κ = κ₀ + κ₁ with |κ₁| ≪ κ₀ and a first order series expansion with respect to κ₁ we obtain

Using ${\beta }^2=k_0^2-{\kappa }^2$, the complex effective index n_eff = β/k₀ can be calculated assuming |κ₁| ≪ κ₀ ≪ k₀

2.4. TM modes

The derivation of the TM-case is similar as for the TE-waves using the correct field distribution and the boundary condition (Eq. 10, see Appendix A.2 for details), resulting in the following expressions for the TM_0n modes:

2.5. HE/EH modes

For a non-zero azimuthal order (m ≠ 0), the modes have six field components (so-called hybrid modes [22]), leading to a more elaborate derivation. Starting again from the case of a perfectly reflecting interface (r_TE = r_TM = −1) the transverse field components at the boundary are zero (E_ϕ (R) = 0, H_ϕ (R) = 0). Using Eqs. (5) and (6) we obtain a homogeneous set of equations for the coefficients A₁ and A₂ requiring the determinant to vanish.

Since R ≫ λ we can set β ≈ k₀ resulting in

Using Eq. (36) we obtain

To be consistent with the mode terminology of step index fibers [23] we label the modes related to j_m−1 (i.e., s = −1) as HE and those related to j_m+1 (i.e., s = +1) as EH.

In the more general case r_TE ≈ −1 and r_TM ≈ −1 the transverse field components must fulfill both boundary conditions (Eqs. 9 and 10). Inserting Eqs. (3)–(6) into Eqs. (9) and (10) yields a homogeneous set of equations for A₁ and A₂ whose determinant must vanish:

Similar to the derivation of the TE modes, we use the ansatz κ = κ₀ + κ₁ and a first order series expansion of Δ. The resulting expression contains first and second derivatives of the Bessel functions which can be eliminated using Eqs. (38) and (39). Finally, the expression obtained for κ₁ is simplified by a second order series expansion with respect to 1/R, leading to

Inserting κ₁ into Eq. (16) results in the following expression for the complex effective index of the HE_mn (s = −1) and EH_mn (s = +1) modes, respectively:

Together with Eqs. (17) and (19), Eq. (26) clearly shows the scaling of the modal losses with 1/R³ which is reported in numerous publications [14–19] and the influence of the reflection parameters. Particularly for hybrid modes (m ≠ 0), the loss is determined by a superposition of the two reflection parameters, showing that both TE and TM reflections contribute to the modal attenuation. This interesting feature directly emerges from Eq. (26) and results from the fact that all HE and EH modes are principally composed of skew rays [22]. This is the key difference to TE/TM modes which only include one reflection parameter in Eqs. (17) and (19) and are generally described by meridional rays, i.e., rays crossing the fiber axis between two reflections. Due to the different values of the b-coefficients the modal discrimination can be engineered by varying the cladding properties and analyzing the reflection parameters. In the case of lossy cladding materials (e.g., metals) or close to interference resonances, ν_TE and ν_TM can be complex-valued. In this case the real parts determine the losses whereas the imaginary parts give a small contribution to the real part of n_eff. For a homogeneous dielectric cladding (i.e., a capillary) ν_TE and ν_TM can be taken from Eqs. (32) and (35), resulting in full analytic expressions that correspond to those reported by E. A. J. Marcatili and R. A. Schmeltzer [15] and are used for instance to analyze modal dispersions of Kagome HCFs [24]. Our model also yields analytic expressions for homogenous metallic cladding structures. Again the key advantage of our model is that the cladding properties of HCFs can be optimized by analyzing the reflection of a corresponding planar interface only, since the modal properties originate from the reflection parameters ν_TE and ν_TM defined in Eqs. (17), (19) and (26).

3. Example: indefinite metamaterial cladding hollow core fiber

3.1. Introduction to the structure

To demonstrate the capabilities of our reflection model we consider a highly anisotropic HCF with iMM cladding and large core diameter and analyze its properties at two important IR wavelengths (3 µm and 10.6 µm). We recently analyzed this kind of HCF in the case of deep subwavelength wires using an effective medium approach [25]. In contrast to that work, the model presented in this contribution is not restricted to the condition of having deep subwavelength wire diameters and is applicable to a much greater variety of fiber geometries, examples of which include photonic band gap fibers or anti-resonant HCFs. The motivation for choosing a nanowire-based HCF here is to reveal the validity of our model on such an extreme multiscale example containing realistic geometries and materials, including losses. The iMM cladding is composed of a single ring of longitudinally invariant gold wires embedded in silica concentrically surrounding an empty core (Fig. 2(a)). In addition to wire diameter d and inter-wire spacing (pitch) Λ, one key parameter to consider is the gap size between wire and the glass wall g (Fig. 2(b)). If this distance falls within the range of the operation wavelength, interference effect can emerge, strongly modifying the reflection parameters and thus the modal properties. Due to the complexity of the cladding structure we determine the two reflection parameters from numerical calculations and apply them to our model to analyze the modal discrimination within iMM-HCFs.

 

Fig. 2 (a) Schematic of the indefinite metamaterial hollow-core fiber analyzed by using the reflection model. (b) Details of the core-cladding boundary with the corresponding definitions of the relevant parameters.

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The reflectivity parameters ν_TE and ν_TM are obtained by simulating the reflection properties of a single rectangular unit cell incorporating one gold wire (Fig. 3) using finite element modelling (COMSOL). To enforce TE- and TM-polarization (corresponding to s- and p-polarization) we use either perfect electric (PEC) or perfect magnetic conductors (PMC) boundary conditions at the sides of the unit cell (with respect to the y-direction). The simulation includes an excitation and monitor port at x = 0 and a perfectly matched layer (PML) terminating the structure. The simulations were performed using a 2D model (Fig. 3) with a large out-off-plane wave vector component that accounts for grazing incidence (ψ ≤ 0.001) and a fixed and experimentally accessible wire diameter of d = 1µm [26, 27]. The optical properties of gold and silica were taken from [28] and [29] (ϵ_silica(3 µm) = 2.012, ϵ_gold(3 µm) = −361.9+58.7i, ϵ_silica (10.6 µm) = 4.740+0.387i, ϵ_gold(10.6 µm) = −3867+1712i). The reflection coefficients are obtained from simulations, and the reflection parameters are given as the respective derivatives with respect to ψ. It is important to note that the two different wavelengths lead to fundamentally different guidance regimes, as the imaginary part of the silica at 10.6 µm is orders of magnitude larger than at 3 µm, and approaches the value of the real part.

 

Fig. 3 Sketch of the unit cell and the associated boundaries used for the finite element simulations to obtain the reflection parameters. The blue and green arrows indicate the incident and the reflected light. The purple lines indicate the locations of the boundary conditions (PEC and PMC) that are used to define the two polarization states.

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3.2. Verification of model on example structure

To validate our reflection model for the case of an iMM-HCF we compare the relative complex effective indices (Δn_eff = Re(1 − n_eff) and Im(n_eff)) of the lowest order modes (TE₀₁, TM₀₁, and HE₁₁) determined by Eqs. (17), (19), and (26) with full numerical calculations of the entire iMM-HCF structure (g = 1 µm, d = 1 µm, f = 0.57). The full simulations of the TE₀₁ and TM₀₁ modes only require circular sectors including one wire with PEC or PMC as boundary conditions. For the HE₁₁ mode we used one quadrant with a PEC and a PMC boundary condition. The reflection parameter retrieval procedure presented in the previous section yields ν_TE = 0.883, ν_TM = 0.193, for λ = 3.0 µm, and ν_TE = 0.989, ν_TM = 0.378, for λ = 10.6 µm. This comparison (Figs. 4(a) and (b)) reveals that our model perfectly reproduces both real and imaginary parts of the effective mode index even at core radii as small 50 µm, which are diameters of typically used silica-based HCFs and is substantially smaller compared to commercially available fibers operating at 10.6 µm (e.g., Omniguide [8]). It should be noted here that the expression for the real part results from the field at an ideally reflecting boundary, i.e. the zeroth-order approximation with respect to κ₁, whereas the imaginary part requires the first order approximation. As a result, Eqs. (17), (19), and (26) precisely describe the dispersions of a large variety of HCFs and show that the physical properties of HCF-modes can be understood on the basis of single interface reflection. It is interesting to note that for the geometry discussed here, the TM₀₁-mode at 3 µm shows the lowest loss of all modes in the system – a situation which is difficult to achieve in dielectric fibers, emphasizing that iMM-HCFs have important polarization properties beyond that of dielectric HCFs.

 

Fig. 4 Comparison of the dependence of the relative complex effective indices on air core radius of the TE₀₁, TM₀₁, and HE₁₁ modes calculated by our reflection model (lines) and full numerical simulations (symbols) at two wavelengths ((a): 3 µm, (b): 10.6 µm). The solid lines refer to the real parts of 1 − n_eff, the dashed lines to the imaginary parts of n_eff.

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In addition, we calculate the dependency of Im(n_eff) on filling factor for a constant radius of R = 100 µm for the two wavelengths using d = 1 µm and g = 1 µm (Figs. 5(a) and (b)). Three different guidance regimes can be identified: For large fill factors (f > 0.5) the nanowire layer acts as an effective medium which reflectivity gradually changes with f resulting in a gradual variation of n_eff. If the filling factor is decreased further the pitch becomes sufficiently large to cause higher diffraction orders into the cladding causing the oscillations in Im(n_eff). In the limit of low filling factors (f < 0.1) the effect of the nanowires vanishes and the properties of the fiber are similar to those of a capillary (f = 0). For any value of f, we observe good agreement between the analytical expressions and the FEM simulations, particular away from the regime of diffraction.

 

Fig. 5 Comparison of the imaginary parts of the effective indices as function of the circumferential filling factor f for the TE₀₁, TM₀₁, and HE₁₁ modes calculated by our reflection model (lines) and full numerical simulations (symbols) at two wavelengths ((a): 3 µm, (b): 10.6 µm). The arrows on the top of both plots indicate the different guidance regimes (cap.: capillary, diff.: diffraction, and indefinite metamaterial).

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3.3. Modal discrimination of the three lowest order modes

The key feature of our model is that the simulation of the HCF-modes is reduced to the analysis of a single reflection at the core-cladding interface, allowing for large-scale parameter sweeps, and in particular to analyze modal discrimination. Equations (17), (19), and (26) suggests that losses increase with increasing mode indices m and n, revealing that the TE₀₁, TM₀₁, and HE₁₁ modes have the lowest losses and are thus the most relevant. Our equations allow specifying design rules for desirably tuning the modal attenuation, i.e., choosing the lowest loss mode in the system. Defining the critical discrimination parameter $C=2j_{11}^2/j_{01}^2-1\approx 4$, we find the discrimination conditions shown in Tab. 1 for the three lowest order modes.

Table 1. Conditions for the critical discrimination parameter for the three lowest order HCF modes.

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To illustrate the capabilities of our model for a modal discrimination analysis, we discuss the variation of the reflection parameters (which determine modal discrimination) of the iMM-HCF at the two wavelengths considered thus far, as function of gap size g and circumferential metal fill factor f = d/Λ, assuming a constant wire diameter (d = 1 µm).

For a wavelength of 3 µm (Fig. 6(a) and (b)), both ν_TE and ν_TM have low values for large fill factors f indicating that low modal attenuation generally appears in the case where the metal wires occupy a large fraction of the unit cell (Fig. 3). This can be explained by the strong reflection of gold at mid-IR wavelengths, resulting from the small penetration of the electromagnetic field into the metal (skin depth < 30 nm). At small filling fractions, however, the reflection of the metal wires becomes less relevant and leakage loss (i.e., transmission through the single interface) dominates. At this wavelength, the small value of Im ϵ_silica(3 µm)< 10⁻⁵ suggests a plane wave absorption length of L_abs = 0.16 m, imposing constructive and destructive interference between the silica wall and the metal wire, which is visible in Figs. 6(a,b) via the dependence of the reflection parameters on gap size (for a fixed circumferential fill factor f). As a result, modal discrimination requires a precise analysis of the individual modes, which as we have shown can be simply achieved by investigating the reflection parameters. One striking feature of the iMM cladding is associated with the large parameter domains in which the TM₀₁ mode has the lowest loss – a situation which is difficult to achieve in pure dielectric HCFs and can lead to new applications, since radially polarized modes offer properties beyond that of HE₁₁ modes (e.g., focusing to extremely small spot sizes [30]). To demonstrate the superior properties of the iMM cladding, it is helpful to compare its reflection parameters with those of a capillary with homogenous silica cladding (ν_TE = 1.98 and ν_TM = 3.98, obtained from Eqs. (32) and (35). The values of our iMM-cladding can be up to two orders of magnitude lower particular away from the interferences, revealing the advantage of using wire based iMM-claddings for achieving low modal attenuation in the mid-IR.

 

Fig. 6 Reflection parameters ν_TE (a) and ν_TM (b) as function of the circumferential fill factor f and gap g at a wavelength of 3 µm. The scale bars (in logarithmic scale) on the top of the plots refer to the value of the individual reflection parameter. The different gray highlighted regions indicate the mode with the lowest loss for this particular combination of g and f.

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At λ = 10.6 µm, our model reveals a similar magnitude of the reflection parameters but only one interference-induced oscillation (Fig. 7(a) and (b)), which is less pronounced due to the large damping of the wave within the silica (L_abs = 19 µm). The HE₁₁ mode dominates the discrimination behavior over a wide range of parameters, with the TM₀₁ and TE₀₁ modes being only relevant within small intervals. At very large filling factors, the TE₀₁-mode shows the lowest attenuation – a characteristic behavior of metallic tube waveguides and nanowire-enhanced fibers [31, 32]. Higher values of the reflection parameters are observed for smaller filling fractions, indicating the emergence of leakage loss. Compared to a homogenous silica capillary the reflection parameters are about ten to one hundred times smaller (as ν_TE = 1.03, and ν_TM = 4.88), confirming the improved guidance of iMM-HCFs. It is interesting to note that the reflection parameters suggest that the lowest loss capillary mode is the TE₀₁ mode, revealing that a capillary with high material attenuation behaves similar to a metallic tube. For example, to operate the iMM-HCF within the HE₁₁ mode at 10.6 µm with experimentally realistic parameters (d = 1 µm, g = 1µm, f = 0.7) [26,27], we obtain ν_TE ≈ ν_TM ≈ 0.3, leading to Im n_eff = 3.5 · 10⁻³ (λ/R)³. This corresponds to an attenuation of 0.3 dB/m for R = 400 µm, which is less than observed in commercially available Omniguide fibers [8].

 

Fig. 7 Reflection parameters ν_TE (a) and ν_TM (b) as function of the circumferential fill factor f and gap g at a wavelength of 10.6 µm. The scale bars (in logarithmic scale) on the top of the plots refer to the value of the individual reflection parameter. The different gray highlighted regions indicate the mode with the lowest loss for this particular combination of g and f.

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4. Conclusions and outlook

We introduce a mathematical model that allows to interpret the dispersion and attenuation of modes in HCFs with large air cores on the basis of single interface reflections, giving rise to analytic and semi-analytic expressions for the complex effective indices when the core diameter is sufficiently large and the guiding is based on reflection by a sufficiently thin layer. We have identified two dimensionless and core-size independent reflection parameters as the key quantities, and show that the commonly-accepted core size dependence of the modal loss (∝ 1/R³) is a universal feature of HCFs. Our model significantly reduces the simulation complexity and enables large scale parameter sweeps, as we show on the example of a HCF with a highly anisotropic indefinite metamaterial cladding. We reveal design rules that allow engineering the modal discrimination and show for this multiscale example structure that the model is applicable in cases the core size is comparably large. The investigated metamaterial geometry shows principal losses lower than that of a corresponding dielectric capillary in any desired mode for mid-IR wavelength (e.g., at 10.6 µm losses < 1 dB/m can be achieved). Our calculation approach substantially reduces simulation and computational effort as only the reflection at a single interface needs to be analyzed. The application of our model is not restricted to metamaterial fibers, and can be applied to a wide variety of HCFs, provided that the core sizes are large enough so that grazing incidence can be considered, i.e. the incident angle is sufficiently small to allow for linearly approximating the single interface reflection coefficient. Future work will target Bragg- and Omniguide fibers, potentially reaching full analytic equations for the complex effective indices. Due to its simplicity, we expect that the model will be also useful to analyze the guidance properties of advanced HCFs and to optimize their performance, in particular within the scope of recent developments (e.g. antiresonant fibers with junction nodes [10], negative curvature fiber [33], Kagome fibers [34], and metamaterial fibers [12–14]), since our model can be easily applied to other shapes by adapting the geometry (Fig. 3) in the simulation of the reflection parameters. We also believe that the basic idea to relate the fiber losses to the reflection coefficients of the core-cladding interface might be generalized to noncircular fibers. However, in this case the reflection coefficient will no longer be constant along the boundary requiring a more elaborate calculation.

Appendix

A. Relations of the field components at a planar boundary

A.1. TE waves

The fields at a boundary between air and the cladding medium (Fig. 1(b)) are given by an incident and a reflected plane wave $E_y^{(i)}(x,z)=E^{(i)}\exp [i(\kappa x+\beta z)]$ and $E_y^{(r)}(x,z)=E^{(r)}\exp [i(-\kappa x+\beta z)]$, respectively. The reflection properties of that boundary are fully defined by the amplitude reflection coefficient r_TE = E^(r)/E⁽ⁱ⁾ without making any assumptions on the cladding. The longitudinal component of the magnetic field is obtained from the Maxwell equations as

(28)$$H_z=\frac{1}{i\omega {\mu }_0}\frac{\partial E_y}{\partial x}.$$

The total fields are superpositions of incident and reflected waves at the boundary (x = 0) leading to

(29)$$E_y(0,z)=E^{(i)}(1+r_{TE}){\mathrm{e}}^{i\beta z},$$

(30)$$H_z(0,z)=E^{(i)}\frac{\kappa }{\omega {\mu }_0}(1-r_{TE}){\mathrm{e}}^{i\beta z}.$$

and resulting in the following ratio of the forward and transversal field components:

(31)$$\frac{H_z}{E_y}=\frac{\kappa }{\omega {\mu }_0}\frac{1-r_{TE}}{1+r_{TE}}.$$

If the cladding is a homogeneous dielectric medium with dielectric permittivity ϵ the reflection coefficient can be obtained from the Fresnel equations. In the case of grazing incidence (ψ ≪ 1) we obtain the following linear approximation of the Fresnel reflection coefficient

(32)$$r_{TE}=\frac{E^{(r)}}{E^{(i)}}=-1+\frac{2}{\sqrt{ϵ-1}}\psi .$$

For other cladding structures such as the wire-based iMM, we use the linear ansatz for the reflection coefficient given by Eq. (7). The parameter ν_TE depends on the permittivities of gold and silica and the geometric parameters of the metamaterial, and can be obtained by numerically simulating the reflection of plane waves at the interface (Fig. 3). From Eqs. (31) and (7) and κ = k₀ sin ψ ≈ k₀ψ we obtain the relation between the tangential field components and the reflection parameter given in Eq. (9).

A.2. TM waves

In the TM case the relations between the field components are similar and can be obtained from the TE relations by substituting E_y → H_y, H_z → E_z, Z₀ → −1/Z₀ µ₀ → −ϵ₀, ν_TE → ν_TM. In particular, we obtain

(33)$$H_y(0,z)=H^{(i)}(1+r_{TM}){\mathrm{e}}^{i\beta z},$$

(34)$$r_{TM}=\frac{H^{(r)}}{H^{(i)}}=-1+{\nu }_{TM}\psi ,$$

and Eq. (10). In the case of a homogeneous cladding medium with permittivity ϵ the Fresnel equations result in

(35)$$r_{TM}=-1+\frac{2ϵ}{\sqrt{ϵ-1}}\psi .$$

We have defined the reflection coefficient here as the ratio of the transverse magnetic field components in order to get the same mathematical form of expression as for the TE case.

B. Relations of the Bessel functions

The first derivative of the Bessel function fulfills the following relation (given in [35] for s = −1 and s = 1 separately).

(36)$$J_m^{\prime }(x)=-s{J}_{m+s}(x)+\frac{sm}{x}{J}_m(x),s=\pm 1.$$

From the Bessel differential equation [35] and Eq. (36) we obtain

(37)$$J_m^{″}(x)=\frac{m^2-sm-x^2}{x^2}{J}_m(x)+\frac{s}{x}{J}_{m+s}(x).$$

If the argument is a zero of J_m+s we obtain

(38)$$J_m^{\prime }(j_{m+s,n})=\frac{sm}{j_{m+s,n}}{J}_m(j_{m+s,n}),$$

and

(39)$$J_m^{″}(j_{m+s,n})=\frac{m^2-sm-j_{m+s,n}^2}{j_{m+s,n}^2}{J}_m(j_{m+s,n}).$$

Some values of j_mn a given in Tab. 2.

Table 2. Numerical values of j_mn (n-th zeros of the Bessel function J_m).

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C. Numerical values of the coefficients

Table 3 shows numerical values of the a and b coefficients to be used in Eqs. (17), (19), and (26).

Table 3. Numerical values of the coefficients in the equations for the effective index.

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Acknowledgments

The authors acknowledge funding from the German Science Foundation (DFG) via the grant SCHM 2655/8-1. A.T. acknowledges support from the Alexander Von Humboldt Foundation.

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