## Abstract

The feasibility of nanometric practical optical waveguide circuits based on surface plasmon polariton gap waveguides (SPGWs) is investigated in detail through three-dimensional simulations. H-plane planar branching waveguide circuits of subwavelength scale are shown to be possible using SPGWs. The waveguide characteristics of the circuits are found to be highly sensitive to the dimensions of the optical circuit, indicating that highly accurate computer-aided design and simulations are necessary for the construction of practical SPGW-based optical circuits.

©2005 Optical Society of America

## 1. Introduction

The construction of optical circuits on subwavelength or nanometric scales has attracted much attention of many researchers recently. The strong light confinement in structures with high index contrast makes it possible to achieve highly compact optical circuits [1]. Techniques based on photonic crystal have also been used to construct small optical circuits [2]. However, the size and density of optical devices employing conventional dielectric optical waveguide and photonic crystals will in principle be limited by the diffraction limit of light. In contrast, optical waveguides based on surface plasmon polaritons (SPPs) can be miniaturized much further, leading to the possible development of nanometric integrated optical circuits. Thus, optical circuits based on SPPs are promising as future nanoscale optical circuits [3–7]. Although SPPs can travel no more than a few micrometers before extinguishing, such distances are sufficient for nanometric optical integrated circuits. Many interesting experimental and theoretical works treating practical nanometric optical circuits based on SPPs have been reported [3–7], and the present authors recently proposed an SPP gap waveguide (SPGW) as a basic element of nanometric optical circuits [8, 9]. The waveguide mechanism of the SPGW is derives from the low phase velocity exhibited by SPPs in nanometrically narrow gap regions between two parallel metal substrates compared to that in wide gap regions. SPGWs have been demonstrated to guide, divide, and bend optical waves with acceptable losses in nanometric circuits [8, 9].

In this paper, the basic waveguide characteristics of a practical subwavelength SPGW optical circuit are investigated through three-dimensional numerical simulations. A planar device for a complex branching optical circuit is considered, consisting of one right-angle corner or bend and two T-shaped branches in the subwavelength area. The dependence of the waveguidance characteristics on various parameters of the SPGWs are investigated in detail, and it is shown that dense and complex optical circuits on subwavelength scales are possible using SPGWs as a basis.

## 2. Nanometric optical circuit

The operating wavelength considered in this analysis is λ=573 nm, the metal supporting the SPP is assumed to be silver with a relative permittivity of *ε*
_{1}=-12.4-*j*0.85, and the system is assumed to be governed by an exp(+*jωt*) time dependence. A schematic of the H-plane planar optical SPGW circuits considered here is shown in Fig. 1. A square hole of dimensions *C _{x}*×

*C*×

_{y}*C*is bored into the metallic substrate such that the hole is surrounded by four walls and one bottom plate. The walls and bottom plate are assumed to be thicker than the optical skin-depth of the metal, defined as

_{z}*d*in Fig. 1. Inside the square hole, the ridge structure is preserved, forming a four-port asymmetric branching circuit in the

*x*-

*y*plane. A cover plate of thickness

*d*is then placed over the hole, creating a small gap between the ridge and the cover plate. The width of the ridge is given by

*w*and the gap between the ridge and the cover plate is given by

*g*. Since the phase velocity of the SPP in the gap region is smaller than that in the surrounding region in the hole, the gap between the ridge and the cover plate constitutes an SPGW and optical waves are expected to be guided along the ridge structure in the

*x*-

*y*plane, i.e., constituting a four-port branching circuit. From a manufacturing point of view, the structure shown in Fig. 1 is not unrealistic.

A plane wave is assumed to be incident from the negative *x* direction on an entrance hole with a cross-section shown in Fig. 1, exciting the SPPs in the SPGW inside the circuit. The entrance hole is also a SPGW, and the SPP excited within is confined in the narrow-gap region in the hole. The SPP excited in the entrance hole is incident on the port (I) of the branching SPGW circuit (Fig. 1). The output ports (II), (III) and (IV) of the branching SPGW are closed off by metallic walls. To ensure that the field in the entrance hole is the guided mode only, the incident vector is set to be rotated by π/4 with respect to the positive *y* axis to achieve oblique incidence to the entrance hole.

## 3. Volume integral equation

The scattering problem for the metallic structure shown in Fig. 1 is solved using a volume integral equation, as given by [10–12]

where *k*
_{0}=*ω*/*c* (*c* is light velocity in free space, *ω* is angular frequency), ** D**(

**x**) is the total electric flux,

*E*^{i}(

**x**) is the incident electric field, and

*(*

**A****) is the vector potential, which is expressed by the following volume integral.**

*x*Here, **g**(**x**|**x**’) is a free-space Green’s function given by

The volume integral region *V* in Eq. (2) represents the entire space, and *ε*
_{r}(**x**) represents the distribution of relative permittivity, where *ε*
_{r}(**x**)=*ε*
_{1} in silver and *ε*
_{r}(**x**)=*ε*
_{0}=1 in the void space of the circuits. Since the region in which [*ε*
_{r}(**x**’)-*ε*
_{0}] is nonzero is finite and the integral region *V* has finite volume, it is possible to solve Eq. (1) numerically by the well-established method of moments. To obtain the solution, the entire region of the circuit is divided into small discretized cubes of size *δ _{x}*×

*δ*×

_{y}*δ*, and Eq. (1) is discretized by the method of moments using roof-top functions as basis and testing functions (Galerkin method). The resultant system of linear equations is then solved by iteration using the generalized minimized residual method (GMRES) with fast Fourier transformation (FFT) [12, 13]. The validity of the code was checked by confirming that the code gives a reasonably accurate solution compared to the rigorous solution for a dielectric sphere. As the numerical evaluation is long and extraneous, and it can be found in the literature [12], the details are omitted in this paper.

_{z}The parameters used in the branching circuit shown in Fig. 1 are given as follows.

Circuit size: *k*
_{0}
*C _{x}*=5.5 (0.88λ),

*k*

_{0}

*C*=5.4 (0.86λ),

_{y}*k*

_{0}

*C*=1.2 (0.19λ),

_{z}*k*

_{0}

*d*=1.0 (0.16λ),

*k*

_{0}

*l*=2.8 (0.45λ),

_{x}*k*

_{0}

*l*=3.7 (0.59λ),

_{y}*k*

_{0}

*b*=0.8 (0.13λ),

_{1}*k*

_{0}

*b*

*=2.0 (0.32λ).*

_{2}Cross-section of SPGWs: *k*
_{0}
*w*=0.2 (0.032λ), *k*
_{0}
*g*=0.2 (0.032λ).

Cross-section of entrance hole: *k*
_{0}
*a _{y}*=

*k*

_{0}

*a*=0.2 (0.032λ),

_{z}*k*

_{0}

*b*=1.8 (0.29λ),

_{y}*k*

_{0}

*b*=0.4 (0.064λ).

_{z}Discretized cube: *k*
_{0}
*δ _{x}*=

*k*

_{0}

*δ*=

_{y}*k*

_{0}

*δ*=0.05 (0.008λ).

_{z}The four surrounding walls are not essential for the construction of the branching circuit shown in Fig. 1. However, the metallic bottom and cover plates must be included for the construction of optical circuits. The overall effective dimensions of the optical circuits are therefore approximately 0.88λ×0.86λ×0.51λ. The dimensions of the entrance hole, i.e., *a _{y}*,

*a*,

_{z}*b*,

_{y}*b*and

_{z}*d*are kept constant throughout the following numerical examples.

## 4. Numerical examples

The first case considered is a branching circuit consisting solely of straight SPGWs, as shown by the gray region in Fig. 1. The optical scattering problem is solved for the complex structure using Eq. (1) to obtain the optical field inside the circuit. Circuits with two-port S-shaped, three-port branching and four-port branching configurations were also simulated, as shown in Figs. 2(a)–(c), and the differences in waveguide characteristics were evaluated. Figure 2 shows the distribution of total optical intensity |** E**|

^{2}on the plane parallel to the

*x*-

*y*plane and located at a distance

*k*

_{0}ξ=0.025 from the ridge in the gap region. Standing waves can be clearly seen to arise along the ridge structures in the circuit. Notice that the intensity scale normalized by the incident intensity of these results is 0.0–20.0. The optical field inside the circuits is enhanced by the surface plasmon. The optical fields can also be seen to be vaguely extended to the left of the ridge structure in the circuit shown in Fig. 2(a).

This field can be considered to be radiated from discontinuities of the SPGWs and is undesirable (referred to as radiated fields hereafter). It is seen that the optical fields are guided along the ridge structure as designed. However, it can also be see from in Fig. 2 (c) that the intensity along the waveguide of port (III) is small in the four-port circuit.

The case with round structures in place of the right-angled corners and branches in Fig. 2 was then examined, corresponding to the guides indicated by dotted curves in Fig. 1. The optical intensities are shown in Figs. 3(a)–(c). All bends and branches consisted of round structure with an average radius r given by *k*
_{0}
*r*=1.0. The radiated field shown in Fig. 2(a) is lower in Fig. 3(a), and the small optical fields in the waveguide of port (III) in Fig. 2(c) are increased, as shown in Fig. 3(c). Thus, it is possible to improve the waveguide characteristics by replacing right-angled structures with rounded structures.

The two-dimensional optical intensity distribution on a plane parallel to the *y*-*z* plane indicated by a dotted line in Fig. 3(c) is shown in Fig. 4(a), and the one-dimensional distribution is shown in Fig. 4(b). The optical fields are confined in the gap region between the ridge and the cover plate, and are guided separately to along the three ports (II), (III) and (IV). The approximate full-width at half-maximum (FWHM) values of the optical intensity shown in Fig. 4(b) are given by *k*
_{0}×FWHM≈0.29.

From these numerical examples, the optical fields can be understood to be confined in the gap between the ridge and the cover plate and to be guided along the ridge structure. The subwavelength optical circuit design shown in Fig. 1 can therefore fulfill the desired function. The four-port branching waveguide shown in Fig. 3 (c) is considered exclusively hereafter.

## 5. Dependence of waveguide characteristics on ridge height

The dependence of the waveguide characteristics on the ridge height *h*=*C*
_{z}-*g* in Fig. 1 is shown in Fig. 5 for the condition *k*
_{0}
*g*=0.2 and *k*
_{0}
*w*=0.2. The optical intensities on the plane parallel to the *x*-*y* plane at a distance of *k*
_{0}ξ=0.025 from the ridge inside the gap region are shown in the figure for *k*
_{0}
*h*=1.0, 0.6 and 0.2. The propagation constant *k*
_{w} of the SPGW used is the fundamental constant that characterizes the circuit. From the distance between the standing wave maximums (SWMs) in Figs. 5 (a)–(c), the approximate values of Re(*k*
_{w}/*k*
_{0}) are calculated to be Re(*k*
_{w}/*k*
_{0})≈2.2, 2.0 and 1.6 for *k*
_{0}
*h*=1.0, 0.6 and 0.2, respectively. Thus, the smaller the ridge height, the smaller the Re(*k*
_{w}/*k*
_{0}) values (larger wavelength) of the guided SPPs become. Since the propagation constants *k*
_{w} for *k*
_{0}
*h*=0.2 are very close to those in the region outside the circuit, the radiated field observed in Fig. 5 (c) is considered reasonable.

## 6. Dependence of waveguide characteristics on gap width

The dependence of the waveguide characteristics on gap width *g* in Fig. 1 is shown in Fig. 6 for conditions of *k*
_{0}
*h*=1.0 and *k*
_{0}
*w*=0.2. The approximate values of Re(*k*
_{w}/*k*
_{0}) calculated from the distances between SWMs are Re(*k*
_{w}/*k*
_{0})≈1.95, 2.2 and 3.0 for *k*
_{0}
*g*=0.3, 0.2 and 0.1, respectively. Thus, the larger the gap width, the smaller the value of Re(*k*
_{w}/*k*
_{0}) (larger wavelength) for the guided SPPs. The waveguide characteristics are very sensitive to the gap width: while the SPP is not excited effectively along the branching structure for the case of *k*
_{0}
*g*=0.3 (Fig. 4 (a)), the optical wave is strongly guided along the branching circuit for the case of *k*
_{0}
*g*=0.1 (Fig. 4 (c)) and the radiated field is small.

## 7. Dependence of waveguide characteristics on waveguide width

The dependence of the waveguide characteristics on the ridge width *w* in Fig. 1 is shown in Fig. 7 for the conditions *k*
_{0}
*h*=1.0 and *k*
_{0}
*g*=0.2. The approximate values of Re(*k*
_{w}/*k*
_{0}) calculated from the distances between SWMs are Re(*k*
_{w}/*k*
_{0})≈2.9, 2.2, and 2.0 for *k*
_{0}
*w*=0.1, 0.2 and 0.4, respectively. Thus, the smaller the ridge width, the larger the value of Re(*k*
_{w}/*k*
_{0}) (smaller wavelength) for the guided SPPs.

## 8. Other characteristics

Other characteristics that may prove useful in the design of practical nanometric optical circuits based on SPGWs were also investigated. First, the ridge structure of the SPGW connecting to port (II) was cut as shown in Fig. 8(a), where the length of the straight region *l* is given by *k*
_{0}
*l*=0.9 (0.14λ) and the other parameters are *k*
_{0}
*h*=1.0, *k*
_{0}
*g*=0.2 and *k*
_{0}
*w*=0.2. The optical field leading to port (II) is stopped by cutting the ridge structure (Fig. 8(b)). This is interesting in that the SPP appears to be controlled by a space much smaller than the wavelength.

A seven-port branching circuit is shown in Fig. 9 as an example of a more complex and dense circuit. The parameters for the circuit are *k*
_{0}
*h*=1.0, *k*
_{0}
*g*=0.1 and *k*
_{0}
*w*=0.1, with intervals *s* between the six output SPGWs given by *k*
_{0}
*s*=0.5 (0.08λ). The two-dimensional optical intensity is shown in Fig. 9(b) using a normalized intensity scale of 0.0–2.0. The optical waves are guided along all output six branches separately with a small radiated field. The approximate values of Re(*k*
_{w}/*k*
_{0}) calculated from the distances between SWMs is Re(*k*
_{w}/*k*
_{0})≈3.7. The one-dimensional distribution of optical intensities along the broken line in Fig. 9(b) is shown in Fig. 10. The FWHM of the optical intensity shown in Fig. 10 is given approximately by *k*
_{0}×FWHM≈0.14. These examples demonstrate the possibility of constructing dense optical circuits much smaller than the wavelength through the use of SPGWs.

## 9. Conclusions

The feasibility of constructing nanometric optical circuits through the use of surface plasmon polariton gap waveguides (SPGWs) was investigated by three-dimensional simulations using a volume integral equation. H-plane planar branching waveguide circuits were considered. The waveguide characteristics of the nanometric SPGW-based optical circuits proposed in this paper are very sensitive to the parameters of the circuit, demonstrating the need to develop accurate computer-aided design and simulation tools for the design of practical circuits. The matching conditions between the entrance hole and the SPGWs inside the circuits and between terminal ends of the output SPGWs were not discussed in this paper and will need to be investigated in the future.

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