## Abstract

A three-dimensional method for obtaining the bending losses and field distributions of bent surface plamon-polariton waveguides is presented. The method is based on a so called ‘method of line’, which discretises potential in the direction of the metal-widths, and leads to Airy-equations in the radial direction. From the results obtained. It is confirmed that thiner metal waveguide enable longer-ranging propagation of surface plasmon-polariton mode, but the weakened confinement requires larger bending radii on order to keep radiation loss.

© 2006 Optical Society of America

## 1. Introduction

Surface plasmon-polaritons(SPPs) attract much interest since they are sensitive to interface properties. Due to the relatively small propagation length, SPPs are considered to be somewhat limited in their applications. Recently, long-range surface plasmon-polaritons(LR-SPPs) for the guiding of light along thin metal stripes embedded in dielectric were suggested[1–4]. LR-SPPs have been proposed as the foundation waveguide of a new integrated optic technology for several reasons. First, the attenuation of the symmetrical mode, also called LR-SPP, decreases drastically with the decrease of the film thickness, leading to an increase in the propagation length. At the same time, LR-SPPs field components extend over several micrometers into the cladding via two identical evanescent tails, facilitating the optical excitation, because the mode size is now close to that of the standard SMF[2–4]. Especially, high flexibility of metal waveguides makes LR-SPPs waveguides worth considering as a promising planar technologies for integrated optics.

In recent years, curved LR-SPPs waveguides were analyzed by Berini and Lu using a rigorous vectorial numerical method and an absorbing boundary condition[5]. They concluded that long-range structures are not incompatible with bending and that the effects caused by bending are the same as those encountered in conventional dielectric waveguides. A curvature of metal waveguides in the direction of the thin metal-heights is inevitable for applying LR-SPPs waveguides into flexible optical wiring, for example, in mobile folder phone in future. It is expected that the radiation characteristics with vertical curvature is expected more sensitive to bending radii, but has not been reported yet, as far as the author get informed.

In this paper, we present a three-dimensional numerical method for the rigorous solution of the scalar wave equation in bent surface palsmon-polariton waveguide. We use discetisation in the direction of metal-widths and calculate the potential only along a set of lines in the radius direction. To avoid difficulties for evaluation of complex order and argument, we linearize properly the index profile, thus replacing the Bessel-type equation by an Airy-type equation. Calculation results show that radiation characteristics including the bending loss of the curved metal waveguide.

## 2. Theory

A configuration of the bent surface plasmon-polariton waveguide is shown in Fig. 1, along with cylindrical coordinate system(*r*, *ϕ*, *y*). It cosists of a metal film of thickness *t* and width *w* has an equivalent complex permittivity *ε*
_{2} and supported by a semi-infinite homogeneous dielectric substrate of permittivity *ε*
_{1} and covered by a semi-infinite homogeneous dielectric superstrate of permittivity *ε*
_{3}. For purpose of analysis, the cross-section of the waveguide is subdivided into three regions I, II, and III. The vector wave equations for the surface plasmon-polariton waveguide are approximated by the scalar Helmholz wave equation, and we only analyze the case of transverse magnetic(TM) modes with E_{r}, electric fields in the radial direction here since E_{r} is the main transverse electric field component in most practical structures *w*/*t*≫1. In the present article all the fields are assumed to have harmonic time dependance exp(*jwt*). For TM waves propagating in the *ϕ* direction, one can write the field component E_{r} in the following form

where *β* is the propagation constant along the mean radius R. *E*_{r}
(*r*, *y*), satisfies the following Helmholtz equation,

where *k*
_{0} is the wave number in free space and *n*(*y*) is the refractive index. Replacing
${E}_{r}(r,y)=\frac{\psi (r,y)}{\sqrt{r}}$
, Eq. (2) may be rewritten as

Replacing *r*=*R*+*x* and taking it into consideration that radius *R* is by far greater than metal thickness *t*, we can take only the first order of the Taylor expansion of

Equation (3) thus becomes

To arrive at a solution we discretise the waveguide in the y-direction and claculate the potential only along a set of lines in the radius direction. To relate the potentials between these lines, we replace the partial differential operator
$\frac{{\partial}^{2}}{\partial {y}^{2}}$
with a diffence operator [*D*_{yy}
] of the form of a matrix[6]. Introducing the column vector {*ψ*_{i}
} in each area, we obtain from Eq. (5) a set of equations:

with [${n}_{i}^{2}$]=*diag*[(${n}_{i}^{\left(k\right)}$)^{2}], where ${n}_{i}^{\left(k\right)}$ designates the refractive index on the *k* th line. To decouple the equations we take the sum of the matrices and transform them into the diagonal form by means of the orthogonal transformation:

where the matrix [*T*_{i}
] is the eigenvector and the diagonal matrix [Λ] is the eigenvalue of [*D*_{yy}
]+${k}_{0}^{2}$[${n}_{i}^{2}$]. Substitution of this expression into Eq. (6) yields a equations below.

, where the vector {*U*_{i}
}=[*T*_{i}
]^{-1}{*ψ*_{i}
}. We can rewrite Eq. (8) in terms of {*U*_{i}
} and obtain

Solutions to Eq. (9) may be expressed in terms of Airy functions[7]:

In region I, the solution needs to be a standing-wave *Ai*(${Z}_{I}^{\left(k\right)}$) in order for the field to decay as *x* goes to -∞(${Z}_{I}^{\left(k\right)}$→∞). In region II, the solution becomes a combination of *Ai*(*Z*
^{(k)}
_{II}) and *Bi*(${Z}_{\mathit{\text{II}}}^{\left(k\right)}$) to account for the fluctuating standing-waves. In region III, however, the field is to be described in the form of radiative traveling-wave to show the leaky-mode characteristics. Among arguments
${Z}_{\mathit{III}}^{\left(k\right)}{e}^{j\frac{2}{3}\pi}$
and
${Z}_{\mathit{III}}^{\left(k\right)}{e}^{-j\frac{2}{3}\pi}$
, the latter is selected to represent the outward-traveling wave with respect to *x*.

Elimination of the coefficients ${c}_{i}^{\left(k\right)}$ by matching the fields at the interfaces *r*=*R*, *r*=*R*+*t* leads to the following complex matrix equation:

It is known that a homogeneous linear matrix equation shows intrivial solutions only when the determinant of the matrix is eual to zero. Thus the propagation constant is determined by solving the determinant equation

The solution for *β* is complex and its imaginary part is the radiation loss coefficient of the bent surface plasmon-polariton waveguide.

## 3. Results

For given waveguide parameters such as width *w*, wavelength *λ*, curvature radius *r*
_{0} and values of indices *n*
_{1} and *n*
_{2}, evaluation of Eq. (13) gives us the eigenvalue *β* of complex number.

Figure 2(a) and 2(b) show relative contours of the |*E*_{r}
| of the ${\mathit{\text{ss}}}_{b}^{0}$ [1] mode for straight waveguide and bent waveguide with *r*
_{0}=5mm, respectively. Figure 2(c) shows relative distribution of the mode field along a vertical cut immediately above the metal film for *r*
_{0}=5mm. As the figures confirm, the bend effects the mode field to shift toward the outer interface. The leaky field is also observed along the outside of the bend.

The propagation loss is plotted as a function of the radius of curvature *r*
_{0} in Fig. 3 As the figures confirm, thick metal waveguide and abrupt bending cause large attenuation. The result also reveals that the propagation losses converge into those of straight waveguides with corresponding thickness as indicated as arrows.

(*w*=4*µm*, *λ*=1.3*µm*, *n*
_{1}=1.535, *n*
_{2}=0.3859-i7.965)

(*w*=4*µm*, *λ*=1.3*µm*, *n*
_{1}=1.535, *n*
_{2}=0.3859-i7.965)

Figure 4 shows the excess loss, caused by bending, which is calculated from the difference between the propagation losses of bending waveguide and straight waveguide. As expected, a slight change of the curvature radius causes a change of the loss coefficient by many orders of magnitude. Our results reveal numerically that thin metal waveguide enable long-ranging propagation of surface plasmon-polariton mode but that the weakened field confinement requires larger bending radii in order to keep radiation loss low.

## 4. Conclusion

In the present article we have introduced a simple and effective numerical method to investigate the propagation losses of bent surface plasmon-polariton waveguide. We use discetisation in the direction of metal-widths and calculate the potential only along a set of lines in the radius direction. To avoid difficulties for evaluation of complex order and argument, we linearize properly the index profile, thus replacing the Bessel-type equation by an Airy-type equation. Calculation of the propagation loss as a function of the radius of curvature is performed. From the result and propagation loss of a straight waveguide, the excess loss due to bending structure can be evaluated. Our results reveal that thiner metal waveguide enable longer-ranging propagation of surface plasmon-polariton mode, but the weakened confinement leads to larger bending radii in order to keep radiation loss low. It is shown that the method is useful for studying the qualitative dependence of bending losses on several parameters of interest and is sufficiently accurate to design bent slab metal waveguides with a given bend loss.

## References and links

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