Optimal Design of Dielectricloaded Surface Plasmon Polariton Waveguide with Genetic Algorithm
 Author: Jung Jaehoon
 Organization: Jung Jaehoon
 Publish: Current Optics and Photonics Volume 14, Issue3, p277~281, 25 Sep 2010

ABSTRACT
We propose a design and optimization method for a dielectricloaded surface plasmon polariton waveguide using a genetic algorithm. This structure consists of a polymer ridge on top of two layers of substrate and gold film. The thickness, width and refractive index of the ridge are designed to optimize the figures of merit including mode confinement and propagation length. The modal analysis combined with the effective index method shows that the designed waveguide exhibits a fundamental propagation mode with high mode confinement while ensuring that the propagation loss remains relatively low.

KEYWORD
Surface plasmon polariton , Waveguide , Genetic algorithm

I. INTRODUCTION
Recently, Surface Plasmon Polaritons (SPPs) have gained much attention due to their ability to confine light at a subwavelength scale [1]. SPPs are electromagnetic waves coupled to free electrons in metal. They propagate tightly bound to interfaces between a dielectric and a metal [2]. Optical waveguides based on SPPs have been extensively studied as modulators [3], filters [4,5], optical attenuators[6], and switches [7]. Among a variety of SPP waveguides proposed to date, the InsulatorMetalInsulator (IMI) [8,9] or the MetalInsulatorMetal (MIM) [10,11] have been the most widely investigated and have become the most fundamental building blocks of nanophotonic integrated circuits. The IMI SPP waveguide exhibits very low loss propagation with poor localization on the order of several micrometers and it is not suitable for highdensity photonic integration. Conversely, the MIM structure can confine light beyond the diffraction limit, allowing rendering of highly integrated optical devices. Its propagation length, however, is extremely short compared to the IMI and to any other conventional dielectric waveguides [11]. Thus there is a tradeoff between good confinement and long propagation length. In order to overcome this problem, DielectricLoaded SPP Waveguides (D
L _{SP}PWs) [12,13] and a symmetric DL _{SP}PW (SDL _{SP}PW) [14] have been proposed and comprehensively analyzed, but the optimal design method has not been presented to date.On the other hand, Genetic Algorithms (GAs) have been employed for optimal design of optical devices such as optical filters [15,16], longperiod fiber gratings [17], and waveguides [18]
In this paper, we propose the use of a Genetic Algorithm (GA) to optimize mode confinement and propagation length of a D
L _{SP}P waveguide at a telecom wavelength (1550 nm). First, we investigate the effects of variations in the thickness, width and refractive index of the dielectric ridge on mode confinement and propagation length. The Effective Index Method (EIM) [19] was employed to obtain the optical properties of the DL _{SP}PW. The design results show that a DL _{SP}PW mode with high confinement and relatively low loss is realizable.II. ANALYSIS OF D
L _{SP}PWA schematic diagram of the D
L _{SP}PW is shown in the inset of Fig. 1 (a). The structure consists of a polymer ridge with thicknesst and widthw on top of a gold film of thickness d. The gold film is on top of a substrate. In the calculations hereafter, the reference parameters are as follows: the refractive indexes of the polymer, substrate, gold and air are 1.65, 1.46 [14], 0.5590 + 9.8100i [20] and 1, respectively. The metal thicknessd =100 nm, and the excitation wavelength is 1550 nm. We use these values if not mentioned otherwise.The electric field can be expressed in the form:
where β is the complex propagation constant β=β'iβ'', which is related to the mode effective index according to β'=
k _{0}n _{eff} and the propagation length is defined asL _{sp}=1/(2 β'') with propagation constantk _{0} in a vacuum.In Fig. 1 (a), we show the effective index
n _{eff} and propagation lengthL _{sp} of the fundamental DL _{SP}PW mode as a function of the ridge widthw for different ridge thicknesses. It is clear that asw increases,n _{eff} increases andL _{sp} decreases, which corresponds well with the general characteristics of general dielectric waveguides. Increasing the ridge width produces a high electric field along the interface between the polymer and metal, and thus, the fraction of power density in the metal rises, implying a higher propagation loss and a corresponding decrease of propagation length. Similarly, it is also shown that increasing thicknesst causes highern _{eff} and shorterL _{sp}.In general, there is a tradeoff between the mode confinement and propagation length of the SPP modes. For a metric to evaluate waveguide performance including both parameters, we define the Figure of Merit (FOM) as (수식처리) where
M _{s} is the area in which the power density is more than1/e ^{2} times its maximum value.In Fig. 1 (b), we show the FOM as a function of the ridge width
w for different ridge thicknesses. All other parameters are the same as those in our reference structure. It can be seen that the FOM is not monotonic with respect to the ridge widthw but exhibits a maximum around 100200 nm. It should also be noted that ast increases, the FOM increases, due to the fact that ast increases,L _{sp} increases sharply butM _{s} does not change significantly.In Fig. 2 (a) we show the effective index
n _{eff} and propagation lengthL _{sp} of the fundamental DL _{SP}PW mode as a function of the ridge thicknesst for different ridge widths. It is clear that for very thin film thickness, the values ofn _{eff} andL _{sp} approach those of SPP propagating along a goldair interface in all cases:n _{eff} →1.0052, andL _{sp} →206.4 μm. It is apparent from the figure that the effective index increases monotonically with respect to both the ridge width and thickness. Decreasing the film thickness, however, raises the fraction of power density in both the polymer and the metal due to the squeezing effect [12], and therefore reduces the propagation length. The propagation length decreases with decreasing polymer thickness until it reaches some point at which it exhibits a minimum. Beyond the thickness, lengthL _{sp} increases rapidly with decreasing polymer thickness.In Fig. 2 (b) we show the FOM as a function of the ridge thickness
t for different ridge widths. We observe that the FOM is not monotonic with respect to the ridge thicknesst but exhibits a minimum around 100200 nm. It is noteworthy to mention that asw increases, the FOM increases.Furthermore,
n _{eff} andL _{sp} are dependent on the refractive index of the ridge polymer. In Fig. 3 (a) we show then _{eff} andL _{sp} values of the fundamental DL _{SP}PW mode as a function of the refractive index n for different ridge thicknesses. Bothn _{eff} andL _{sp} exhibit monotonic variation with respect to the refractive indexn . FOM also decreases monotonically with n unlike the cases of ridge width and thickness variation (as shown in Fig. 3 (b)).In Fig. 4 (a) we show the
n _{eff} andL _{sp} values of the fundamental DL _{SP}PW mode as a function of the refractive index n of the ridge polymer for different ridge widths. It is apparent thatn _{eff} increases andL _{sp} decreases with respect to the refractive indexn . From Fig. 4(b) it can be seen that the FOM, however, does not exhibit monotonic variation with respect to n, unlike the cases of different ridge thicknesses (as shown in Fig. 3 (b)).III. OPTIMIZATION OF D
L _{SP}PW USING GENETIC ALGORITHM3.1 Genetic Algorithm
The Genetic Algorithm (GA) is a powerful and efficient method for optimization problems of a given function. The basic theory of the GA was first proposed by John Holland in 1975 [21], and further developed by Goldberg [22]. It is based on a stochastic technique, and a population of individuals is created randomly at each generation. The GA emulates natural evolution selection and survival of the fittest. At each step, it modifies a population of individual solutions and creates a new generation based on the three operators of selection, crossover and mutation. Figure 5 shows the flow chart for designing an optimal D
L _{SP}PW for a FOM, which is expressed as an objective function in the chart and is also called a fitness function in optimization theories.3.2. Design Variables and Objective Function
GA optimization is applied to the thickness
t , widthw , and refractive indexn of the ridge in order to design a DL _{SP}PW with high mode confinement and low propagation loss. All other parameters are the same as those in our reference structure. Here we define the objective function as the FOM defined earlier. In this form of GA optimization, the search ranges of the thickness, width and refractive index of the ridge (t, w, n ) are chosen as [50 800] (nm), [75 1000] (nm), and [1.5 3.5], respectively. The mode profile of the DL _{SP}PW is not symmetric [12], and the lateral extent is wider than the vertical one at the same width and thickness of the ridge. Since increasing the thickness renders the lateral extent shorter, we apply the following additional constraint to our problem for coupling with other optical waveguides having a symmetric mode:Furthermore, Figs. 2(b) and 3(b) show that a larger thickness is advantageous for our FOM. However since such waveguides operate in multimode fashion, we apply a constraint of singlemode operation to our optimization problem.
IV. OPTIMIZATION RESULTS
In our computations, the following parameters were used: the population size was 100, the crossover rate was 0.8, and hence the mutation rate was 0.2. The number of generations was fixed at 300 runs. In the attempt to obtain the optimal design parameters satisfying the constraints, we implemented the GA with the above setting parameters. The optimal parameters obtained were
t =191.0 nm,w = 106.1 nm,n =2.97. In Fig. 6, we show the optimization results with respect to the number of generations, where it can be seen that the FOMs increase and converge to the optimal value of 254.0 as the number of generations increases. The objective function corresponding to the best design at each generation will converge faster if the population size is larger, which is consistent with general expectations. It was also found that the maximum number of generations, 300, is sufficient for our form of optimization. The optimum design results are summarized in Table 1. When compared with the previous results [12], the FOM value was improved by a factor of 3.2 (300×300 nm2: 79.375) and 1.6 (600×600 nm^{2}:158.75)Fig. 7 shows the electric field distribution (arrows) and power density (colored surface) for the D
L _{SP}PW with this optimal FOM. From the figure, it can be seen that the final DL _{SP}PW confines the light within the ridge successfully while ensuring lowloss propagation. The numerical results also show that the mode profile is nearly square and the dimensions are small enough to implement singlemode operation as a result of introducing our mode shape and singlemode operation constraints.V. CONCLUSION
This paper presents an efficient and powerful approach to optimization of the D
L _{SP}PW for mode confinement and propagation loss. The thickness, width and refractive index of the ridge have been optimized to achieve wellconfined light propagation with low propagation loss. The EIM has been employed to analyze the DL _{SP}PW and obtain the optical parameters. In order to optimize the FOM including mode size and propagation length, the GA has been used. The FOM is as large as 254.0 with the optimal design variables, which is improved by a factor 3.2 and 1.6, respectively when compared with the results previously reported. It is also possible to develop a similar optical waveguide with subwavelength confinement based on SPP using the optimization scheme proposed here.

[FIG. 1.] (a) Effective index and propagation length and (b) FOM of the fundamental DLSPPW mode as a function of the ridge width w for different ridge thicknesses.

[FIG. 2.] (a) Effective index and propagation length and (b) FOM of the fundamental DLSPPW mode as a function of the ridge thickness t for different ridge widths.

[FIG. 3.] (a) Effective index and propagation length and (b) FOM of the fundamental DLSPPW mode as a function of the refractive index n for different ridge thicknesses.

[FIG. 4.] (a) Effective index and propagation length and (b) FOM of the fundamental DLSPPW mode as a function of the refractive index n of the ridge polymer for different ridge widths.

[FIG. 5.] Flow chart of GA optimization.

[FIG. 6.] Best fitness function with respect to number of generations.

[TABLE 1.] Optimum parameters of the GA

[FIG. 7.] Electric field distribution (arrows) and power density (colored surface) for the DLSPPW with this optimal FOM.