1. Introduction

The design of curved waveguides is of central importance in integrated optics [1] since achieving efficient directional change is necessary in numerous structures such as offsets (e.g.: S-bend), splitters and combiners (e.g.: Y-junction), Mach-Zehnder interferometers (e.g.: back-to-back Y-junctions) and couplers (e.g.: coupled S-bends). Achieving an efficient directional change implies minimising for instance the area or path length required for the curve while minimising its insertion loss. The circular curved waveguide is a suitable structure for achieving directional change and its analysis as a dielectric structure has been the subject of intense research for at least a few decades [2].

Recently a straight thin metal film of finite width surrounded by a dielectric has been characterised theoretically and proposed as the foundation waveguide for a new integrated optics technology [3]. This symmetric “metal stripe” waveguide supports as a fundamental mode, a long-range (low-loss) surface plasmon-polariton (LRSPP) wave, identified as the ${\mathit{\text{ss}}}_b^{\mathit{0}}$ mode [3]. The existence of this mode and its properties have been verified experimentally [4–6]. While numerous elements and devices incorporating curved metal stripes propagating the long-range ${\mathit{\text{ss}}}_b^{\mathit{0}}$ mode have been demonstrated [7–12], no report addressing the modal analysis of such curves has appeared in the open literature. As is well understood, the loss and confinement of the ${\mathit{\text{ss}}}_b^{\mathit{0}}$ mode vanish as the metal vanishes [3], and therefore so does its ability to propagate around curves. Clearly then, it is important to accurately model curved metal stripes in order to determine the onset of significant curvature induced radiation and to optimise structures incorporating curves. This paper deals with the theory and modelling of such curves, and particularly, on the propagation characteristics of the long-range ${\mathit{\text{ss}}}_b^{\mathit{0}}$ mode therein.

2. Method

The curved waveguide of interest is shown schematically in cross-sectional view in Fig. 1(a) and in top view in Fig. 1(b), along with the cylindrical coordinate system (ρ, ϕ, z) used for its analysis. The metal film of thickness t and width w has an equivalent complex permittivity ε ₂ and is embedded in an infinite background dielectric of permittivity ε ₁. The circular curve has a radius of curvature r₀ that is invariant with ϕ so the problem is two-dimensional and the structure is completely characterised via the modes that it supports along the propagation direction which is ϕ. The straight waveguide is obtained in the limit r₀ → ∞.

Compared to the straight waveguide, bending induces: (i) radiation loss due to the finite speed of light in the cladding medium, (ii) modal offset (towards the outside of the curve) and modal distortion leading to a transition loss at the junction between waveguides of different radii of curvature, and (iii) changes in the effective index of the mode. These effects become more pronounced as the radius of curvature decreases, and are generally deleterious, with effects (i) and (ii) leading to higher insertion loss and effect (iii) leading to phase distortion in interferometric structures for example. However, reducing the radius of curvature reduces the size of components, which is generally desirable. Fortunately, all of these effects (i) - (iii) can be accurately quantified via modal analysis of the curve, so appropriate compensation and trade-off can made in the design of a component.

 

Fig. 1. Curved waveguide in (a) cross-sectional and (b) top views.

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The modes supported by the curved waveguide must satisfy the time-harmonic (e ^+jωt) Maxwell’s equations for source-free isotropic media [13]:

with ε_r being a function of ρ and z. For convenience of analysis, an electric Hertzian vector potential ∏_e is introduced and its curl defined on the basis of the null divergence of H and the vector identity ∇ ∙ ∇ × A = 0 (where A is an arbitrary vector) as:

Using this definition and manipulating Maxwell’s equations (1)-(4) in the usual way [13] yields the vector wave equation governing ∏_e :

and the following relationship between E and ∏_e :

where β ₀ = √μ ₀ ε ₀ is the phase constant of free-space. The modes supported by the curved waveguide propagate along the ϕ axis, so a mode exhibits an e ^-jβ_ϕϕ = e ^{-jn_effβ₀r₀ϕ} dependency for propagation along + ϕ, where n_eff is the complex effective refractive index of the mode. The loss of the mode a is given in dB by:

for a curve subtending an angle of θ in radians, with r₀ in m and β₀ in m^-1.

Equations (5)-(7) are solved numerically for any mode of interest using the Method of Lines (MoL) formulated in cylindrical coordinates [14,15]. An absorbing boundary condition positioned on the radiating side of the curve (right side in Fig. 1(a)) is required and used [16,17]. The formulation of the MoL for curves is vectorial and well suited to this waveguide problem since it makes use of a 1-D discretisation along the ρ axis and analytical forms along z. The analytical forms make it easy to handle either a very large or an infinitesimal metal thickness t, while the width of the metal w and the radius of curvature r₀ set the discretisation. Using the MoL, the complex effective index n_eff of a mode, along with the spatial distribution of its six field components, can be determined.

3. Results

The formulation and software developed [15] were validated by: (i) reproducing the 1-D field distributions and radiation losses computed for dielectric curves using the MoL [14] (the radiation losses reported therein were also validated experimentally [18]); (ii) comparing favourably with the radiation losses computed for dielectric curves using the finite element [20] and finite difference [21] methods (the radiation losses computed therein were also validated experimentally [19]); (iii) reproducing the transition losses computed using the finite-difference method [21] for a straight dielectric waveguide end-coupled to a curved one as a function of r₀ ; (iv) reproducing the phase and attenuation constant versus thickness t (and the field distribution at t = 100 nm) of the ${\mathit{\text{ss}}}_b^{\mathit{0}}$ , ${\mathit{\text{as}}}_b^{\mathit{0}}$ , ${\mathit{\text{aa}}}_b^{\mathit{0}}$ , ${\mathit{\text{sa}}}_b^{\mathit{0}}$ modes for straight 1 μm wide metal stripe waveguides [3] in the limit r₀ → ∞; and (v) making comparisons with experiment for S-bends fabricated using thin narrow Au stripes in SiO₂ and propagating the long-range ${\mathit{\text{ss}}}_b^{\mathit{0}}$ mode at λ₀ = 1550 nm [11].

In order to remain consistent with the results reported in Ref. [3] for the straight metal stripe, a set of curves were analysed at the same wavelength of λ₀ = 633 nm and assuming the same materials: an Ag film having ε_r,2 = - 19 -j0.53 surrounded by a dielectric of relative permittivity ε_r,1 = 4. The radius of curvature r₀ is variable and the cross-sectional dimensions (w, t) of the film considered are: w = 0.5, 1 μm with t = 10, 11, 12, 13, 14, 15 nm. The highest loss, highest confinement waveguide among this set is the w = 1 μm, t = 15 nm structure, for which the attenuation of the long-range ${\mathit{\text{ss}}}_b^{\mathit{0}}$ mode is 26 times lower than that of the surface plasmon supported by the corresponding single interface (compare the ${\mathit{\text{ss}}}_b^{\mathit{0}}$ mode for w = 1 μm, t = 15 nm with the ${\mathit{\text{ss}}}_b^{\mathit{0}}$ mode for t → ∞ in Fig. 2(b) of Ref. [3]). Thus, the ss_b⁰ mode remains long ranging for all (w, t) combinations considered.

 

Fig. 2. (a) Insertion loss, (b) radiation loss and (c) Re{n_eff } of the ${\mathit{\text{ss}}}_b^{\mathit{0}}$ mode versus radius of curvature for w = 1 μm and t = 15 nm. (d) Summary of r_0,opt and IL_min for all (w, t) cases considered.

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Figure 2 shows in Part (a) the total insertion loss (IL) of the ${\mathit{\text{ss}}}_b^{\mathit{0}}$ mode for θ= 90°, w = 1 μm and t = 15 nm as a function of r₀ . The total insertion loss is comprised of the propagation and radiation losses. One immediately notes the existence of an optimum radius of curvature r_0,opt where the insertion loss is minimised (IL_min ), occurring in this case at r_0,opt ~ 130 μm. An optimum radius exists because the waveguide includes an absorbing medium (the metal). For r₀ > r_0,opt the insertion loss is dominated by the propagation loss, which increases with r₀ due to the increasing arc length of a fixed angle curve (say 90°), while for r₀ < r_0,opt radiation loss dominates and increases with decreasing r₀ . Part (b) gives the radiation loss component, which is obtained by repeating the computations with Im{ε_r,2 } = 0. It is noted that some radiation loss indeed occurs at the optimum radius of r_0,opt ~ 130 μm. Part (c) plots the Re{n_eff } of the ${\mathit{\text{ss}}}_b^{\mathit{0}}$ mode of the curve as a function of r₀ , showing that Re{n_eff } converges to the value of the corresponding straight case as r₀ → ∞, and diverges as r₀ → 0. This divergence stems from the definition of n_eff for the curve. Consider that n_effβ₀ gives the phase constant in rad/m of the mode propagating along the arc r₀ϕ. Since the mode peak shifts outward as r₀ → 0, then n_effβ₀ must increase if the phase along r₀ϕ is to be the same as the phase of the mode peak, leading to the observed divergence as r₀ → 0. The difference between the effective indices of the curved and straight metal stripes seems to follow an $r_{\mathit{0}}^{\mathit{-}\mathit{2}}$ dependence as in dielectric waveguides [22]. Part (d) summarizes r_0,opt and IL_min for all combinations (w, t) considered and shows a clear trend: the higher confinement, higher attenuation structures (e.g.: w = 1 μm, t = 15 nm) provide a lower IL_min and a smaller r_0,opt .

 

Fig. 3. Normalised contours of Re{E_z } of the ${\mathit{\text{ss}}}_b^{\mathit{0}}$ mode for w = 1 μm, t = 15 nm, (a) r₀ = 1 m and (b) r₀ = r_0,opt . (c) Normalized distributions of Re{E_z } along a horizontal cut immediately above the metal for w = 1 μm, t = 15 nm and r₀ = 1 m, 100 μm and 50 μm. (d) Summary of the transition loss at r₀ = r_0,opt for all (w, t) cases considered.

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Parts (a) and (b) of Fig. 3 show normalised contours of the Re{E_z } of the ${\mathit{\text{ss}}}_b^{\mathit{0}}$ mode (w = 1 μm, t = 15 nm) for r₀ = 1 m and r₀ = r_0,opt respectively, the 1 m radius being used to model the waveguide as straight (r₀ → ∞). Part (c) shows normalized distributions of the Re{E_z } along a horizontal cut immediately above the metal film (w = 1 μm, t = 15 nm) for three radii of curvature: r₀ = 1 m, 100 μm and 50 μm. Two principal effects are noted from these plots as r₀ decreases: (i) the mode peak shifts from the center towards the outside of the bend, and (ii) the field becomes oscillatory along the outside of the bend. The first effect causes a transition (coupling) loss to occur at the junction between waveguides of differing radii of curvature, say curved and straight, as can be appreciated by comparing the contours in Part (a) with those of Part (b). This transition loss can be computed via an overlap integral on the participating normalised mode fields, bearing in mind that the curved mode is a radiation mode [11]. Part (d) summarizes the transition loss for curves designed at r_0,opt and joined to their corresponding straight waveguide (r₀ → ∞) for all combinations (w, t) considered. A clear trend is apparent from Part (d): the lower confinement, lower attenuation structures (e.g.: w = 0.5 μm, t = 10 nm) provide a lower transition loss. This trend leads away from the trend observed Fig. 2(d), however, transition losses can be substantially eliminated by simply laterally offsetting the curved and straight waveguide sections.

4. Conclusions

The main conclusions drawn from this work regarding the long-range ${\mathit{\text{ss}}}_b^{\mathit{0}}$ mode propagating along curves in metal stripe waveguides are that: (i) long-range structures are not incompatible with bending, (ii) the higher attenuation, higher confinement structures considered have a lower minimum insertion loss at a smaller optimal radius of curvature, (iii) reasonably small radii of curvature can be used, and (iv) the effects caused by bending are the same as those encountered in conventional dielectric waveguides. The analysis technique discussed in this paper could also be used to study bends in other surface plasmon waveguide structures [23,24].

Note added in proof: A paper was recently published on the modelling of surface plasmon waveguides, including bend structures, using a 3-D electromagnetic field solver [25].