1. Introduction

Since their first experimental demonstration [1], photonic crystal fibers (PCFs), optical fibers with holes running along their length, have become a major topic of research. Not only have PCFs allowed technological breakthroughs such as hollow-core guidance or endlessly single mode fibers, with many applications in metrology, sensing, dispersion management, nonlinear optics or particle guidance [2], they have also enabled researchers to unveil new aspects of waveguidance and discover new physical phenomena. PCFs with high-index fluid filled holes [3] or with solid inclusions having a refractive index higher than that of the background material [4] can exhibit guidance which is not due to Bragg reflection from the surrounding photonic crystal. Rather, as for planar anti-resonant reflecting optical waveguides (ARROWs), guidance in these fibers relies on enhanced back-scattering at the anti-resonances of single inclusions. Many of the properties of these fibers, also called ARROW fibers, can be deduced from the properties of single inclusions [5, 6]. From this model and other studies, it appears that many properties of PCFs can be designed by adjusting the resonances of single holes. Resonances of single cylinders, however, are well known and offer only limited flexibility. A natural way of gaining further control over the number, location and nature of resonances is for example by adding a coating layer of dielectric, metal or composite material to the inclusions.

Coated spheres [7] and coated cylinders [8–10] have been shown to be able to widen bandgaps when used instead of their un-coated equivalent in two- or three-dimensional photonic crystals. Very recently, Sazio et al have demonstrated that holes of PCFs can be coated with a whole range of materials through high-pressure chemical vapor deposition over lengths of up to 70 cm , and that they could also be coated locally, on micrometric length-scales, using laser assisted chemical vapor deposition [11]. Sazio et al’s technique promises the development of novel complex all-in-fiber passive as well as active photonic devices. For example, adding a metallic coating to all or some of the holes of a PCF allows exploitation of surface plasmon polariton resonances to modify PCF properties. Although this is expected to increase losses because of the metal’s material absorption, localized metallic coatings over micrometric lengths in the fiber could enable the exploitation of very strong plasmonic resonances with minimal loss. However, the theory of PCFs with coated cylinders and their modal properties have so far to our knowledge not been studied.

Here, we extend the multipole formalism for PCFs [12] to include coated cylinders. Using this formalism, we calculate modes and modal properties for several PCFs with metal and dielectric coated holes. Because of intrinsic geometrical limitations [12], the multipole method is not the most versatile PCF simulation tool; although it is generally considered to be computationally very efficient, other very fast and more versatile mode finders exist [13]. However, because it is rigorous and analytical, the multipole method is certainly the most accurate and because of this has been widely used in calibration of subsequent methods [13–24]. Our aim here, besides demonstrating future interesting possibilities of PCFs with coated holes, is to provide accurate data for several examples of coated hole PCFs which may be used as a reference to verify simulations made using other methods. Indeed, especially for structures coated with metals having large complex dielectric constants, simulations using other methods can be delicate [25, 26], and it is good practice to verify their convergence using a few reference examples. For this purpose, we provide convergence studies on selected examples, with tables of numeric data for easy comparison in Section 3. Finally, we study the guidance properties of two types of PCF containing coated inclusions: First a PCF with high index coated holes, for which we demonstrate that the dominant guidance mechanism is antiresonant scattering at short wavelength and modified total internal reflection at long wavelengths, and second a PCF containing silver coated holes for which we exhibit coupling of the core more to surface plasmon polaritons, associated with strong field localization and very large group velocity dispersion.

The remainder of the paper is organized as follows: In Section 2 we describe the mathematical framework of the multipole method for PCFs including coated cylinders; Section 3 gives numerical examples and tables of convergence; Finally we study guidance properties of two examples in Section 4.

2. Multipole method for coated cylinders

A PCF typically consists of a narrow rod of high index material with holes running along its length (Fig. 1). Holes are usually arranged periodically, following a triangular lattice, around a central core. In the present study, the core will consist of a single missing hole at the center of the fiber.

In the multipole method [12], the longitudinal components of the fields are expanded in Fourier-Bessel series, containing Bessel and Hankel functions of the local coordinates, around each hole. The coefficients for Bessel and Hankel functions are linked through the scattering matrix of each inclusion, whereas coefficients in the series for different holes are linked through Graf’s addition theorem. Graf’s theorem along with the scattering matrices is sufficient to entirely determine the system, and since both are known in closed form for simple circularly symmetric holes, the multipole method is rigorous, semi-analytic, fast and accurate.

The extension of the multipole method to the case of coated inclusions is hence straightforward since only the scattering matrices of the inclusions need to be modified. Accordingly, we will not re-derive the multipole method in its entirety (for this, see Ref. 12), but solely detail the derivation of the scattering matrices for coated cylinders, which form the elements of 𝑹 in the mode’s eigenvalue equation (Ref. 12, Eq. (30)).

For coated cylinders with circular symmetry we obtain closed form expressions for the scattering matrices, starting from the scattering matrices for a single interface and using recurrence relations; although the choice of basis functions and notations differ, the approach we follow to extract the scattering matrices is mathematically equivalent to that used in previous studies of scattering by coated cylinders in oblique incidence [27].

We consider two concentric circularly symmetric interfaces, such as depicted in Fig. 2. The structure being infinitely long, all components of the electric and magnetic fields can be obtained [28] from E_z and H_z , and at a given frequency their longitudinal and time dependence can be separated from the transverse dependence:

where V is either E_z or H_z , β is the propagation constant and ω is the light’s angular frequency. The transverse fields V_t then satisfy the Helmholtz equation with propagation constant β and can be expanded in each region in Fourier Bessel series:

where V is either E_z or H_z , l is one of e (exterior), s (shell or coating) or i (interior), k _⊥ = (k ₀ n ^l₂ - β ²)^1/2 and r is in region l. We define the column vectors for each field and region A ^V,l = [$A_{\mathrm{\nu }}^{V,l}$] and B ^V,l = [$B_{\mathrm{\nu }}^{V,l}$], and for each region, the vectors

and similarly B̃^l. The vectors Ã^l and B̃^l of adjacent regions are linked through the boundary conditions at the interface between those regions, and for a single interface one can define reflection and transmission scattering matrices such that

where the + and - superscripts refer to the outside and inside of a single interface respectively.

For circularly symmetric interfaces, the transmission and reflection matrices are known explicitly and given in Ref. [12], Appendix C, whereas for non-circularly symmetric interfaces, they can be computed using a number of numerical techniques, eg differential methods using the fast Fourier factorization method [24, 29].

The overall scattering matrix S̃^e-e of the coated inclusion required for the multipole method’s eigenvalue equation is defined by B̃^e = S̃^e-e Ã^e. When there are no sources in the inclusion B̃ⁱ = 0, so that through elementary matrix manipulations Eqs. (4–5) at the e-s and s-i interface yield

where I is the identity matrix and the subscripts refer to the two interfaces. Solving the eigenvalue equation yields B̃^e for all inclusions, and hence the fields everywhere in the PCF’s background material [12]. The fields inside the inclusions are obtained using Eq. (2) where Ã^l and B̃^l (with l = i, s) are obtained from B̃^e using Eqs. (4–5).

 

Fig. 1. Geometry of a PCF with coated cylinders.

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3. Numerical examples and convergence

Our first example is a coated version of the PCF studied in Ref. 12, to enable direct comparisons. The inclusions are coated by dielectric material of higher refractive index than the background material. The geometry is a single hexagon of inclusions in a background material extending to infinity, with parameters defined in the caption of Table 1. Table 1 shows a convergence test done on the fundamental mode (confined mode with highest real part of the effective index) of that structure, at a wavelength of 1.45 μm, along with the Wijngaard parameters W _E and W _H defined in Ref. 30. W _E and W _H can be seen as a very loose upper bound of the relative error of the fields E and H respectively [31]. Fig. 3 shows the field distributions of the studied mode, for M = 8. Note that the fundamental mode for such a geometry is doubly degenerate, and we show field plots for only one of the two degenerate modes. As can be seen from Table 1, the real and imaginary part of the mode’s effective index converge very quickly with M. Above M = 8, the Wijngaard parameters increase, showing that accuracy is lost in fields, and eventually also in the effective index. This is due to higher order coefficients of the Fourier Bessel expansion becoming so small that they are numerically negligible and ill-defined, and only add numerical noise to the general formulation. Although not documented before in the context of PCFs, this is a general feature of the multipole method, which could be somewhat improved by different normalization of the Fourier Bessel coefficients. However its effect becomes relevant only after the effective indices converge almost to machine precision; when accuracy is paramount, it is nevertheless important to choose the best possible value of M, which varies with geometry, by carrying out a convergence test.

 

Fig. 2. Geometry and field expansion coefficients for a single coated inclusion.

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The fields of Fig. 3 show that the mode is predominantly confined in the core (the region between the inclusions). However, a non negligible fraction of the field resides in the high-index coating. This is to be compared with the same mode without coating in Ref. 12, Fig. 4, where the fraction of the field inside the low index inclusions is negligible. Also, the effective index of the mode is raised compared to that of the same mode without high-index coating, and the propagation loss is higher by almost two orders of magnitude. We will see in Section 4 that this is linked to coupling to leaky modes of the coatings, and is highly wavelength dependent.

Table 1. Real and imaginary parts of n _eff and Wijngaard parameters W _E and W _H as a function of truncation parameter M of the Fourier Bessel series. Single hexagon with pitch Λ = 6.75 μm of inclusions having ρ _e = 2.5 μm, ρ _i = 1.5 μm, n _i = 1, n _s = 1.7, n _e = 1.45, λ = 1.45 μm.

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Table 2 shows a convergence test for the fundamental mode of a structure containing 50nm metallic (silver) coatings; the geometry and parameters of the structure are described in the caption. Again, convergence is very rapid with increasing M, although now M = 9 is now required when M = 7 was sufficient in the dielectric case. Losses are significantly higher, due to the high absorption of silver at the chosen wavelength. Fig. 4 shows field distributions of the fundamental mode. Note how these differ significantly from those of similar geometries with non-coated or dielectric coated cylinders. Fig. 4(d) demonstrates that a significant fraction of the power carried along the fiber is concentrated at the dielectric/silver interface. We will show in Section 4 that plasmon resonances indeed strongly influence guidance properties of this type of PCF.

 

Fig. 3. Selected field distributions of the mode studied in Table 1: (a) ∣E_z ∣, (b) ∣H_z ∣, (c) Re(S_z ). (d) color scale. All fields are computed with M = 8. The color scale for this and all subsequent figures is linear.

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Fig. 4. Selected field distributions of the mode studied in Table 2: (a) ∣E_z ∣, (b) ∣H_z ∣, (c) ∣S_z ∣, (d) detail of ∣S_z ∣ around the rightmost cylinder, showing the strong field concentration on the silver surface. All fields computed with M = 9. The color scale is the same as in Fig. 3.

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4. Properties of coated PCFs

4.1. High-index dielectric coated inclusions

Resonances of coated cylinders can differ substantially from those of uncoated cylinders. This has been used by Stone et al [33], who have recently exploited the fact that higher order resonances of annular high-index regions are frequency shifted compared to solid high index rods to design all-solid photonic bandgap fibers with reduced bend losses. Here, we illustrate the diversity of phenomena accessible with coated PCFs using the example of a solid core PCF similar to that in Fig. 1, but with three rings of identical inclusions. Each inclusion has a hollow core (air, n _i = 1) and a high index coating n _s = 1.6, with ρ _e = 1 μm and ρ _i = 0.8 μm. The background material has refractive index n _e = 1.45. The average refractive index of the cladding is hence lower than that of the core, and one could naively expect the PCF to guide light in the core through “index guidance” or “modified total internal reflection.” [34] Figures 5 and 6 respectively show the real and imaginary part of the effective index of the fundamental mode. For comparison, Figs. 5 and 6 also show the real and imaginary part of the effective index of a PCF with equivalent homogenized holes, ie with three rings of homogeneous inclusions of radius 1 μm and refractive index (${\mathit{\text{fn}}}_{\mathrm{i}}^2$ + (1 - f)$n_{\mathrm{s}}^2$)^1/2 ≃ 1.24964. This value is derived from the mean of dielectric constants (f = ${\rho }_{\mathrm{i}}^2$/${\rho }_{\mathrm{e}}^2$ is the filling fraction of air in the cylinder), and corresponds to the homogenized refractive index parallel to the fiber axis [35]; using this homogenization for a single inclusion does not rely on any rigorous approach, and other choices (eg Maxwell-Garnett type formulas) could also be argued for [35]. Note that the scale on the horizontal axis for both figures is reciprocal, so that frequencies are evenly spaced: the x-axis on top of the figures indicates the normalized frequency V, which we define relative to the coating parameters:

Table 2. Real and imaginary parts of n _eff and Wijngaard parameters W _E and W _H as a function of truncation parameter M of the Fourier Bessel series. Single hexagon with pitch Λ = 1.5 μm of inclusions having ρ _e = 0.4 mm, ρ _i = 0.35 mm, n _i = 1, n _s = 4.33480043837×10^-1+i8.70529497278,λ= 1.45μm and n _e = 1.45. Note that the value used for n _s results from an interpolation of measured data for silver [32]. The value given is the one used in our simulation, however it should be noted that the number of significant digits given here bears no relation to the accuracy with which the refractive index of silver is known at that wavelength.

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Figure 5 also shows the imaginary part of the effective index of the fundamental mode of a PCF with homogeneous inclusions having refractive index 1.6 (ARROW fiber), which guides purely by antiresonant effects [5]. In Fig. 5 the green vertical lines mark the cutoff wavelength of modes of a single coated inclusion, while in Fig. 6 we also plot the real part of the effective index of leaky and guided modes of single coated inclusions.

At long wavelengths, losses of the coated PCF asymptotically follow those of the equivalent fiber with homogenized inclusions, demonstrating all characteristics of index guidance. On the contrary, the ARROW fiber losses diverge rapidly with increasing wavelength, as expected from the ARROW model [36]. At shorter wavelengths however, guidance in the coated PCF exhibits all characteristics of ARROW guidance: losses peak at cutoffs of the modes of individual inclusions, with low loss bands in between cutoffs. Note that while the definition of a normalized frequency is standard for the ARROW fiber, there is no unique way of defining the normalized frequency for the coated holes, so that we can’t directly compare frequency values between the two cases. Propagation losses for the PCF with coated inclusions at their minima in higher order bands are substantially lower than those of the equivalent PCF with homogenized inclusions, and are also lower than those of a PCF with air holes of radius ρ _e or ρ _i (data not shown). From Fig. 6 it appears that loss peaks are due to avoided crossings of the core-guided mode with leaky modes of the individual coated inclusions near cutoff, a characteristic feature of ARROW guidance [6,37,38].

 

Fig. 5. Imaginary part of n _eff as a function of wavelength for a PCF with high index coated holes, for an equivalent PCF with homogenized inclusions and for a PCF with inclusions having the refractive index of the coating (ARROW fiber). The three structures are detailed in the text. Green vertical lines denotes cutoffs of the modes of a single coated inclusion. The imaginary part of n _eff for the ARROW fiber for V > 5 are too small to be computed with the multipole method.

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The PCF with high-index coated holes hence exhibits two distinct guidance mechanisms, depending on wavelength. At long wavelengths, when the transverse wavelength becomes large enough compared to the size of the inclusion for homogenization arguments to hold, the PCF is index-guiding. As soon as the wavelength becomes small enough for resonances of the coating to appear, the ARROW mechanism becomes the dominant waveguiding mechanism.

4.2. Metallic coated inclusions

By coating PCF holes with thin metallic layers, PCF guidance can be combined with surface plasmon resonant effects, with the prospects of strong field localization and sharp resonances. Here we demonstrate excitation of a surface plasmon in a PCF with holes coated by a thin layer of silver. The structure is that of Fig. 1, with only the second ring of holes being coated. Fig. 7 shows the real and imaginary parts of the effective index of the fundamental mode as a function of wavelength. Around λ = 1.72μm an avoided crossing with another mode occurs. Fig. 8 shows the field distributions around the avoided crossing. It appears clearly that the mode causing the avoided crossing is a surface plasmonic resonance of the ring of silver coated holes: the fields are strongly localized at the surface of the coated inclusions. Fig. 9 shows a detail of the electric field across a coated inclusion. The fields are evanescent in the dielectric matrix as well as in the air hole, and strongly decaying in the silver region, characteristic of a surface plasmon resonance.

 

Fig. 6. Real part of n _eff as a function of wavelength for the fundamental mode of a PCF with high index coated holes (red) and of an equivalent PCF with homogenized inclusions (blue). The real part of n _eff of modes of a single coated inclusion are also shown; the latter are leaky for Re(n _eff) < n _e = 1.45 and guided when Re(n _eff) > n _e = 1.45. The structures are detailed in the text.

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The configuration studied here is somewhat reminiscent of the one used in dispersion compensating PCFs [39]: both use PCFs with an annular defect surrounding the core. In the case of dispersion compensating PCFs, the second, third or fourth ring of holes around the core is modified typically by reducing the hole-size, so that the ring can support guided modes. Avoided crossings of the fundamental core mode with modes of the ring defect lead to strong group velocity dispersion values. The silver coated PCF studied here is very similar, except that the ring mode is now a surface plasmon polariton. One could therefore expect a sharper resonance, and hence even larger values of the group velocity dispersion. Fig. 10 shows the chromatic dispersion parameter D of the mode of the lower branch of Fig. 7; values of D for the upper branch are similar, but of opposite sign. The unoptimized silver coated PCF achieves values of D comparable to the highest published values for non-coated designs [40], over a similar wavelength range. It is expected that optimization would lead to values of D up to an order of magnitude larger, with comparable bandwidth. However, it must be noted that because of absorption in the silver coating, losses are extremely high. The imaginary part shown in Fig. 7 corresponds to prohibitive losses of the core mode of 10dB/mm off-resonance (λ = 14μm, lower branch), of which 9 dB/mm is due to absorption and the remaining 1 dB/mm is confinement losses [41], and 136dB/mm at resonance (λ = 1.72μm, upper branch), of which 110dB/mm are due to absorption. While confinement loss can be reduced by increasing the size of the inclusions or adding rings of holes, the plasmonic nature of the resonance requires surface currents leading, in all but theoretical metals, to high absorption losses which can hardly be reduced.

 

Fig. 7. Avoided crossing of the fundamental core mode and a plasmonic resonance in a PCF containing silver coated holes. The figure shows the real (upper curves) and imaginary (lower curves) of the two modes involved in the avoided crossing, as a function of wavelength. The structure is that of Fig. 1, with the second ring of holes coated with 20nm of silver. Λ = 1.5μm, ρ _e = 262.5nm, ρ _i = 242.5nm, n ^e = 1.45, n _i = 1.0 and n _s is wavelength dependent, with values resulting from interpolation of published data for silver [32]. Letters (a-d) indicate the location of field plots in Fig 8.

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Fig. 10 also shows the chromatic dispersion for the same PCF but with a coating thickness of 30nm. Compared to the PCF with a coating thickness of 20nm, the resonant coupling to the plasmonic mode has shifted considerably, and the strength of the resonance has also been largely modified. This proves how sensitive the plasmonic resonance is to the actual structure, a fact commonly used eg in sensing to measure minute changes in refractive indices in the immediate surroundings of a thin metallic film or particle [42]. This suggests that even with a range of propagation limited by absorption loss to a few tens of micrometers, a PCF with holes locally coated at its tip could be used for all-optical sensing. Previous approaches to surface plasmon polariton optical fiber sensors required post processing of the fibers (such as tapering, etching or polishing) to be able to create a metal thin film near the core [42]. PCFs “naturally” provide holes near the core without any post-processing, limiting the required post-processing of the fiber to the deposition of a metallic thin film, potentially cutting costs and improving repeatability.

 

Fig. 8. Field distribution ∣E∣ of the mode on the upper and lower branch of Fig. 7 just before and just after the avoided crossing of the core mode with a plasmonic resonance: (a) lower branch, l = 1.7mm; (b) upper branch, λ = 1.7μm; (c) lower branch, ρ = 1.74μm; (d) upper branch, ρ = 1.74μm. The color scale is the same as in Fig. 3. The green segment in (c) represents the coordinates along which the field is plotted in Fig. 9.

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Fig. 9. Norm of the electric field across the upper central coated cylinder in Fig. 8(c).

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Fig. 10. Group velocity dispersion parameter D as a function of wavelength for the structure used in Fig. 7, with silver coatings in the second ring of thickness 20 nm and 30 nm.

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5. Conclusion

The recently demonstrated ability of coating the holes of PCFs opens up a new dimension accessible to PCF designers. We have given here a framework based on the multipole method to explore the physics of these new PCFs further, along with tables for numerical comparisons and calibration of other methods. Our preliminary study of some properties of PCFs with coated inclusions in Section 4 already exhibited two novel phenomena not accessible to non-coated PCFs, namely the ability to have both index and ARROW guidance in a single PCF, and surface plasmon polariton resonances. Although the latter have very high absorptive losses, they may prove useful with short distances of coating at the tip of a fiber for localized sensing applications.