Analysis of optical waveguides with ultra-thin metal film based on the multidomain pseudospectral frequency-domain method

Po-Jui Chiang; Yen-Chung Chiang; Nai-Hsiang Sun; Shi-Xi Hong

doi:10.1364/OE.19.004324

1. Introduction

The principle of the surface plasmon resonance (SPR) technology has been proposed for decades [1–6], and it was widely used in optical waveguide design [7–15], in which metals (Au, Ag, Al, Cu) are deposited on the dielectric materials. Due to SPR, the power response of optical devices becomes extremely sensitive, i.e., a small amount of optical power of the particular polarization (surface plasmon polariton, SPP) can induce highly intensity surface plasmon (SP) modes [12–15] or hybrid surface plasmon (HSP) modes [16,17]. The field intensity of major SP modes is confined in the metallic surface, while that of the HSP modes is distributed both in the dielectric region and the metallic surface. The property of SPP forms the basis of many biochemical sensors, SPP-based modulators and switches.

The combination of a waveguide mode with the resonant field enhancement of a surface plasmon was described in [8,13,16–18] by deposing finite metallic layers on typical multi- and single-mode waveguides. However, the SP and the HSP modes of the metallic ultra-thin films with the properties of both guided and leaky modes (mixing modes) are difficult to analyze by conventional low-order numerical methods [19–28] without any further improvement. Recently, the pseudospectral method [29–35], which is a high-order method, has been proposed for precisely modeling optical waveguides and devices. Compared with conven- tional methods, there are more compacted points near curved boundaries and it is easier to achieve higher order convergence for the pseudospectral method, thus it is an effective method to solve electromagnetic fields with violent variation at the interfaces. As mentioned previously, the nature of the SP and the HSP modes has field distributions with extremely variation on the adjacent boundary of metallic surfaces and the pseudospectral method is very suitable to calculate such problems.

Multidomain pseudospectral frequency-domain (PSFD) method has been proposed to analyze photonic band gaps (PBGs) and traditional optical waveguides in [33,34], respectively. This method is applying pseudospectral formulation on the Helmholtz equations based on nonuniform Chebyshev-Gauss-Lobatto grid points with suitable boundary conditions (Dirichlet or Neumann type boundary conditions, DBC or NBC) at curved boundaries. Further, in our previous paper [35], the original PSFD method is improved by adopting the modified PML (MPML) for both leaky and guided modes and it can further save CPU time and computer memories. Hence, we will apply the PSFD-MPML developed in [35] to calculate field distributions and complex effective indices n _eff of the SP and the HSP modes in this work. However, the core waveguide structures treated in this paper consist of dielectric-metallic interfaces and it is noticeable that the boundary condition significantly influences the convergence behavior while solving optical waveguide problems by numerical methods. Compared with the dielectric-dielectric interfaces in [35], the dielectric-metallic boundary is more complicate because of the complex index of the metal. In this paper, using Maxwell’s equation we propose the boundary condition between the complex metallic index and the real dielectric index. Using the proposed boundary condition, the PSFD-MPML is stable and has spectrum convergence.

Some novel optical components, such as the polarization/mode selection devices [36,37] and the polarizing beam splitters [37–39], are getting more and more important in optical integrated circuits. With the help of polarization/mode selection devices, different modes at certain wavelengths can be packed into the multi-mode waveguides and thus it can improve the flexibility of the electromagnetic field distributions. And such characteristics can be adopted in the design of the polarizing beam splitters by coating the gold thin film in the optical waveguides and couplers, which is useful to reduce the scale. We will explain the property of HSP modes and apply it to design these two kinds of optical components.

In Subsection 2.1 of this paper, we will firstly investigate the field characteristic of the SP and HSP modes by applying our PSFD-MPML method in the 1-D dielectric-metal waveguide. In Subsection 2.2, to validate our PSFD-MPML method in the SP and HSP modes analysis, we then compare our result of the HSP mode for the multilayer waveguide with that obtained in [8] and it is shown that accurate effective indices can be obtained by using only a small computation window. The mode selection device and polarizing beam splitter are developed in Subsection 2.3. The conclusion is drawn in Section 3.

2. Numerical examples and discussion

2.1 Dielectric-metal waveguide

We firstly investigate the 1D dielectric-metal waveguide as shown in Fig. 1(a) , with magnetic permeability μ₀ and piecewise uniform refractive indices n = n_a and n = n_b for dielectric and metal, respectively, where dielectric n_a is a real number and metallic n_b is a complex number. In the dielectric-metal waveguide, the fields of the TM mode ( $\vec{E} =$ {E_x, 0, E_z}, $\vec{H} =$ {0, H_y, 0}) propagate toward the z-direction and the magnetic field can be expressed as $\vec{H} = {\hat{a}}_{y} H_{y}$ $= {\hat{a}}_{y} H (x) \exp (- j γ z),$ which satisfies

(\nabla^{2} + k_{0}^{2} n^{2}) [H (x) \exp (- j γ z)] = 0,

where γ is the propagation constant,

\nabla^{2}

is the Laplacian operator, and k ₀ = ω /c is the free-space wave number, in which ω is the angular frequency, and c is the velocity of light in vacuum.

Fig. 1 (a) Sketch of a dielectric-metal waveguide. (b) H_y-field distribution of the SP mode along the x direction with wavelengths λ = 0.5 μm, 0.7 μm, 0.9 μm, and 1.55 μm, respectively.

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Equation (1) can be reduced to

(\frac{\partial^{2}}{\partial x^{2}}) H (x) = (γ^{2} - k_{0}^{2} n^{2}) H (x) .

Referring to Fig. 1(a), the field is confined at the vicinity of interface between the dielectric and the metal and is exponential decaying toward x- and z-directions. Accordingly, the solutions of the field in Eq. (2) satisfying such characteristics can be expressed as

H (x) = {\begin{matrix} H_{0} \exp [- j (\sqrt{k_{0}^{2} n_{a}^{2} μ_{0} - γ^{2}}) x] = H_{0} \exp (- j k_{1 x} x) & for x < 0, \\ H_{0} \exp [j (\sqrt{k_{0}^{2} n_{b}^{2} μ_{0} - γ^{2}}) x] = H_{0} \exp (j k_{2 x} x) & for x > 0. \end{matrix}

Due to phase matching, we can obtain γ and the effective index n _eff from Eq. (3) as

n_{eff} = \frac{γ}{k_{0}} = \sqrt{\frac{n_{a}^{2} n_{b}^{2}}{n_{a}^{2} + n_{b}^{2}}},

where n _eff is in general a complex number. Besides of the analytic solution in Eq. (4), we may employ the PSFD-MPML introduced in [35] to analyze such problems as well. The formulas given in [35] are derived under dielectric-dielectric interfaces condition and modification on the Neumann type boundary conditions is required for the dielectric-metallic adjacent region for modeling the structure in Fig. 1(a). Considering the refractive indices of the metallic n_b (a complex number) and the dielectric n_a (a complex number), we derive the following boundary conditions by Maxwell’s curl equations

{\frac{\partial H_{y}^{a}}{\partial x} |}_{x = 0} = \exp^{- j θ} (\frac{n_{a}^{2}}{| n_{b}^{2} |}) {\frac{\partial H_{y}^{b}}{\partial x} |}_{x = 0}

where

H_{y}^{a}

and

H_{y}^{b}

are local fields of the dielectric and the metallic regions, respectively, and θ is equal to

\tan^{- 1} [Im (n_{b}^{2}) / Re (n_{b}^{2})] .

The condition of Eq. (5) for the dielectric-metallic interface is required to guarantee the stability of the PSFD-MPML method and the spectrum convergence characteristic. Similar relations can also be derived for the 2D cases, and we will not address them explicitly here.

Now, we assume that the index of the dielectric material is n_a = 1.52, the metal is gold, and it is operated at wavelengths λ = 0.532 μm to λ = 1.55 μm. The index n_b of gold at different wavelengths is referred to [40]. According to Eqs. (3) and (4), the magnetic field profiles of the SP mode at wavelengths $λ = 0.5 μm,$ 0.7 μm, 0.9 μm, and 1.55 μm are shown as in Fig. 1(b). It is obvious that there is a longer field tail extended into the dielectric at longer wavelengths, but, oppositely, a shorter field tail in the metal at longer wavelengths as shown in the inset of Fig. 1(b). Furthermore, it can be observed from Fig. 1(b) that the field tails will decrease to a negligible level when the metallic thickness is larger than 0.25 μm. In applying numerical methods, we may employ a large computational window to precisely model the long field tails in the dielectric. The large computational window, however, corresponds to longer computing time and larger memories. Furthermore, the change of the electromagnetic fields become very sharp at the boundary between the dielectric and metal, and the phases k ₁ _x and k ₂ _x of Eq. (3) are complex numbers, i.e., there are propagating and evanescent components in the x direction. Hence, our PSFD-MPML is an appropriate method for analyzing such a structure.

To validate the MPML for analyzing this problem, we modify the original problem in Fig. 1(a) as that shown in Fig. 2(a) in which the MPML with thickness d ₃ = 0.1 μm is imposed and the thicknesses of metallic and dielectric layers are d ₁ = d ₂ = 0.3 μm, respectively. Figure 2(b) shows the calculated magnetic fields at λ = 1.55 μm and it is observed that the results by adopting our PSFD-MPML agree with the exact solutions very well. Table 1 lists the calculated effective indices (n _eff) of the SP mode by using Dirichlet, PML, and the MPML boundary conditions in the same computational window size as shown if Fig. 2(a). It can be seen that the results of using MPML agree with the exact n _eff up to 9 significant digits as the degree N is larger than 22, while the results by using both Dirichlet and PML can even not converge. Hence, the MPML has been demonstrated the capability of modeling unbound areas in such cases and it will be adopted in the subsequent studies.Furthermore, to demonstrate the superiority of the PSFD-MPML for solving the SP and HSP modes over other methods, we reconsider the structure shown in Fig. 2(a) and adopt the thicknesses of gold and dielectric layers as d ₁ = 0.5μm and d ₂ = 1.42μm, respectively, with λ = 0.5μm. We solve this problem by using our PSFD-MPML method and the improved finite-difference mode solver (FDMS) proposed in [21] for comparison. For the FDMS, we employ both formulas with convergent orders O(h ²) and O(h ⁴), respectively, where h is the grid size and corresponds to (d ₁ + d ₂)/h total unknowns. And for the PSFD-MPML, only two sub- domains are adopted with the same degree N, corresponding to $(N + 1) \times 2$ total unknowns. Figure 3(a) shows the relative errors in the calculated effective indices (n _eff) with respect to the number of unknowns for using O(h ²)-FDMS, O(h ⁴)-FDMS, and PSFD-MPML, respectively. The relative error in n _eff is defined as $| n_{eff; calculated} - n_{eff; exact} | / | n_{eff; exact} |$ . It is seen that using the PSFD-MPML can achieve very high accuracy and maintain the spectrum convergence characteristic. The computer memory required to reach a certain level of relative error in n _eff for each method is also shown in Fig. 3(b). As indicated in [21,22], the operator matrix for the O(h ²)-FDMS and O(h ⁴)-FDMS is very sparse. However, the memory required for the PSFD-MPML to obtain the same accuracy is dramatically smaller than the improved FDMS because of the spectrum convergence of the PSFD-MPML. Accordingly, we again demonstrate that the PSFD-MPML is very suitable to model the dielectric-metallic structures supporting the SP and/or HSP modes. Note that the structures shown in Figs. 1(a) or 2(a) are simple exercises with the SP mode and then the subsequent practical components with ultra-thin metal films are complicated structures with both the SP and HSP modes, which is very difficult to accurately analyze their n _eff using the uniform-grid and low-order numerical methods.

Fig. 2 (a) Scheme of the MPML in a dielectric-metal waveguide. (b) Field profiles of a dielectric-metal waveguide at the wavelength λ = 1.55 μm.

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Table 1. Convergence behavior of the SP mode effective index of the dielectric-metal waveguide with the degree N at the wavelength λ = 1.55 μm

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Fig. 3 (a) Relative errors in the calculated n _eff versus the number of unknowns for the SP mode of the dielectric-metal waveguide. (b) Necessary computer memory versus relative errors in n _eff for the SP mode of the dielectric-metal waveguide.

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2.2 Mode selective waveguide

We then consider the HSP mode in a multilayer waveguide structure [8] which is depicted as in Fig. 4(a) . Referring to Fig. 4(a), the structure consists of a glass substrate with n _sub = 1.5195, a glass waveguide core with n _core = 1.5595 and d_c = 2 μm, a SiO_x buffer layer with n _buf = 1.520 and d_b = 0.5μm, a gold film with n _Au = 0.463 + j2.4 and $d_{g}$ = 0.053μm, and a water subphase with n = 1.333. The operation wavelength was chosen to be λ = 0.532 μm. In order to compare the attenuation behavior of different modes, the imaginary part of effective indices Im(n _eff) (which is denoted as κ in [8]) is plotted versus the gold film thickness for the TM HSP modes as shown in Fig. 4(b). The circles are the results obtained by using our PSFD-MPML based on Chebyshev polynomials of degree N = 12. The solid lines are the results from the commercially available program TRAMAX [8] which uses a transfer matrix formalism based on the Fresnel equations. It is seen that the two results agree with each other quite well and the resonant coupling thickness of the gold appear at around 0.04 μm. To examine convergence behavior of our PSFD-MPML method in the HSP mode case, we list the calculated n _eff for theTM₁ HSP mode and the corresponding numbers of degrees N in Table 2 . It is seen that a relative error smaller than the order of 10⁻¹² can be achieved by using degree N = 18 in gold thickness $d_{g}$ = 0.01 μm case.

Fig. 4 (a) Sketch of the multilayer waveguide structure with the gold thin film. (b) Imaginary part of the effective refractive index Im(n _eff) of the TM₀, TM₁, and TM₂ HSP modes versus the gold thickness.

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Table 2. Convergence of the calculated effective index of the TM₁ HSP mode with respect to different degrees N for gold thickness d_g = 0.01 μm at the wavelength λ = 0.532 μm

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For a further investigation on the relation between the geometry and the waveguide characteristic, we fix the gold thickness $d_{g}$ being 0.04 μm and buffer layer d_b = 1.9 μm, and let the core thickness d_c varying from 0.9 μm to 1.6 μm. Figures 5(a) and 5(b) show the calculated Re(n _eff) and Im(n _eff) for the TM₀ and the TM₁ HSP modes, respectively, versus the core thickness d_c. It can be observed that the gap of Im(n _eff) between the TM₀ and the TM₁ HSP modes becomes larger as d_c decreases. Thus the TM₁ HSP mode can be facilely eliminated with d_c = 0.9 μm in the waveguide while the TM₀ HSP mode sustains a relative smaller attenuation.

Fig. 5 (a) Real part and (b) imaginary part of the effective index n _eff of the TM₀ and the TM₁ HSP modes, respectively, of the waveguide versus core thicknesses d_c with the wavelength λ = 0.532 μm.

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According to the above discussion, we design an asymmetrical multilayer waveguide by introducing a gold film as shown in Fig. 6 , in which the core width d_c = 3.6 μm, the gold thickness $d_{g}$ = 5 nm, the core-index n ₁ = 1.5595, and the cladding index n ₂ = 1.5195. There are eight modes (four TM modes and four TE modes) in the multimode waveguide for absence of the gold layer at wavelength λ = 0.532 μm. The gold film can be set at the location where the amplitude of the field profile for a certain mode is close to zero and thus minimize Im(n _eff) of this mode. Accordingly, the gold film is chosen to be located between 0.68 μm and 0.685 μm for minimizing Im(n _eff) of the TE₂ mode. Table 3 lists the results of the corresponding n _eff for seven modes of this waveguide. According to Table 3, significant changes in the imaginary part of n _eff can be investigated as the gold film is included. The TE₂ mode has the lowest leaky power than other modes and its real part of n _eff is close to 1.5473738447 which is n _eff of the original waveguide without the gold film. Although the TM₂ mode also has a small amplitude between 0.68 μm and 0.685 μm, it can induce the HSP mode and its Im(n _eff) is in the order close to other modes. Accordingly, we may connect waveguides with and without the metal film and the TE₂ mode will be passed while other modes will be attenuated in the section with the metal film. By properly setting the location of the metal thin film, different modes can be selected through such devices. Note that n _eff of the TM₀ mode is absent in Table 3, because the TM₀ mode will induce the SP mode and its Im(n _eff) is the largest one among these modes.

Fig. 6 Scheme of the mode selective waveguide.

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Table 3. Calculated effective index of the first 7 modes of the multimode waveguide coated with the gold film with thickness d_g = 5 nm between 0.68 μm and 0.685 μm

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2.3 Mode selective coupler and polarizing beam splitter

According to above discussion, modes can be extracted by properly adding thin metal films. We therefore consider the 2-D cross-section of the optical fiber coupler as shown in Fig. 7(a) , in which two D-shape fiber waveguides are glued together with an ultra-thin gold film coated in-between. Referring to Fig. 7(a), the fiber core is consisted by the material with the refractive index n _core = 1.46 and its radius is r = 2 μm, which is surrounded by the cladding with index n ₂ = 1.456 and the aspect ratio d / r equals 5. Due to the geometrical symmetry, we only take a quarter of the original area, which is $8 μm \times 8 μm$ in size, in computing as shown in Fig. 7(b). We chose the operation wavelength λ = 0.6 μm and calculated n _eff of the first four even and odd modes with respect to different thickness $d_{g}$ of the thin metal film. Figures 8(a) –8(d) show field distributions of the dominant components of $H_{y}^{even},$ $H_{y}^{odd},$ $H_{x}^{even},$ and $H_{x}^{odd}$ modes, respectively, under $d_{g}$ = 6 nm.

Fig. 7 (a) Cross-section of coupling mode switching and polarizing beam splitter. (b) Mesh and domain division profile for the optical component.

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Fig. 8 Magnetic field patterns of the dominant components of the first four x- and y- polarized modes of the mode selective coupler and polarizing beam splitter. (a) $H_{y}^{even}$ mode. (b) $H_{y}^{odd}$ mode. (c) $H_{x}^{even}$ mode. (d) $H_{x}^{odd}$ mode.

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Figures 9(a) and 9(b) show the corresponding curves of Re(n _eff) and Im(n _eff), respectively, versus $d_{g}$ with degrees N = 14. Figure 10 shows the Im(n _eff) curves of these modes with respect to λ under d_g = 6 nm. Because the $H_{x}^{odd}$ mode cannot induce the HSP mode and has low field amplitude near the thin metal film as shown in Fig. 8(d), its value of Im(n _eff) is lowest among these modes as shown in Figs. 9(b) and 10. Hence, if we properly design the geometry of the coupler, other modes may be attenuated to a negligible level and left only the $H_{x}^{odd}$ mode and such optical device can be used as a mode selective coupler. Table 4 lists the calculated effective indices of the $H_{y}^{even}$ mode for different degrees N. Obviously, numerical results with five digits of accuracy can be obtained by using only a degree N = 10 and convergence up to the order of 10⁻⁸ can be achieved by using high-order degrees. Note that there is a peak in the curves of the $H_{y}^{even}$ mode as shown in Figs. 9 and 10, which has relation with the plasma resonant frequency and will be discussed in another paper.

Fig. 9 (a) Real and (b) imaginary part of the effective index of the first four x- and y- polarized modes versus the gold thickness of the mode selective coupler and polarizing beam splitter with the wavelength λ = 0.6 μm.

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Fig. 10 Imaginary part of the effective index of the first four x- and y- polarized modes versus the wavelength of the mode selective coupler and polarizing beam splitter with the gold thickness d_g = 6 nm.

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Table 4. Convergence of the calculated effective index of the $H_{y}^{even}$ HSP mode with respect to different degrees for gold thickness dg = 6 nm at the wavelength λ = 0.6 μm

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Moreover, another application of this structure is the polarizing beam splitter and its conceptual diagram is as shown in Fig. 11 . We consider the coupling coefficient of i polarized modes, C_i (i = x or y), which is defined as $C_{i} = (β_{i}^{even} - β_{i}^{odd}) / 2,$ where the $β_{i}^{even}$ and $β_{i}^{odd}$ are the propagation constants of i polarization modes. Figure 12 shows C_x and C_y versus wavelengths λ at metal film thickness $d_{g}$ = 0 nm, 6 nm, and 8 nm, respectively. It can be seen that C_x and C_y can be effectively separated with the help of the inserted gold film. Therefore, when beams of both x- and y- polarization magnetic fields are injected into left facet of the structure, only y-polarization magnetic fields can be transferred and propagated into the other side of the waveguide due to the factor of C_y >> C_x. Thus it functions as a polarizing beam splitter. It is as a result of the y-polarization magnetic fields strongly interacting by inducing the surface plasmon. The efficiency of this polarizing beam splitter will be discussed in detail by our future work.

Fig. 11 Scheme of the polarizing beam splitter.

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Fig. 12 Coupling coefficient of i polarized modes, C_i (i = x or y) (a) versus the wavelength λ with the gold thickness d_g = 0 nm, 6 nm, and 8 nm, respectively, and (b) versus the gold thickness d_g with the wavelength λ = 0.6 μm.

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3. Conclusion

We have investigated the surface plasmon (SP) polaritons mode and the hybrid-surface plasmon (HSP) mode, of the dielectric-metal waveguide and the multilayer waveguide coating the ultra-thin gold film [8], respectively. And we have verified that our proposed PSFD-MPML method is especially suitable for analyzing such kind of problems. Therefore, we apply our method in the analyses of the multilayer waveguide structure and the optical fiber coupler both coated with the ultra-thin gold film. Form the simulated results, we can conclude that the PSFD-MPML method is easily generalized for treating the surface plasmon waveguide problems and useful optical components, such as mode selective waveguides and the polarizing beam splitter in optical integrated circuits, can be easily designed by introducing ultra-thin metal films.

Acknowledgments

This work was supported in part by the Ministry of Education, Taiwan, R.O.C. under the ATU plan and by the National Science Council of the Republic of China under Grant NSC97-2221-E-151-055-MY2 and NSC99-2221-E-214-035. The authors would like to thank Prof. H.-C. Chang with National Taiwan University, Taipei, Taiwan for his useful advisory.

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n _eff
Degree N	MPML	PML	Dirichlet
4	$1.53374449563 - j 1.01414068 \times 10^{- 3}$	$1.531 - j 1.9 \times 10^{- 3}$	$1.278 - j 5.5 \times 10^{- 3}$
8	$1.53345862709 - j 1.28117983 \times 10^{- 3}$	$1.535 - j 1.3 \times 10^{- 3}$	$1.279 - j 5.3 \times 10^{- 3}$
12	$1.53336435738 - j 1.30418841 \times 10^{- 3}$	$1.535 - j 2.5 \times 10^{- 3}$	$1.279 - j 5.3 \times 10^{- 3}$
16	$1.53335860157 - j 1.29818233 \times 10^{- 3}$	$1.535 - j 2.5 \times 10^{- 3}$	$1.279 - j 5.3 \times 10^{- 3}$
18	$1.53335884637 - j 1.29781743 \times 10^{- 3}$	—	—
20	$1.53335886015 - j 1.29782630 \times 10^{- 3}$	—	—
22	$1.53335885986 - j 1.29782673 \times 10^{- 3}$	—	—
24	$1.53335885992 - j 1.29782676 \times 10^{- 3}$	—	—
Exact	$1.53335885990 - j 1.29782676 \times 10^{- 3}$	—	—

Degree N	n _eff
8	$1.54483429399659 - j 1.265152742 \times 10^{- 5}$
10	$1.54483258251423 - j 1.265442819 \times 10^{- 5}$
12	$1.54483259502135 - j 1.265453801 \times 10^{- 5}$
14	$1.54483259437432 - j 1.265453810 \times 10^{- 5}$
16	$1.54483259443511 - j 1.265453818 \times 10^{- 5}$
18	$1.54483259441965 - j 1.265453822 \times 10^{- 5}$
20	$1.54483259441919 - j 1.265453820 \times 10^{- 5}$

n _eff
Modes	TM	TE
0	—	$1.5567815098 - j 8.1056 \times 10^{- 5}$
1	$1.5554262224 - j 9.0718 \times 10^{- 5}$	$1.5505122072 - j 5.3734 \times 10^{- 4}$
2	$1.5471557818 - j 2.7100 \times 10^{- 5}$	$1.5473735385 - j 8.5139 \times 10^{- 8}$
3	$1.5400049837 - j 2.3581 \times 10^{- 4}$	$1.5350911715 - j 4.6175 \times 10^{- 4}$

Degree N	n _eff
8	$1.45796234298 - j 4.3468 \times 10^{- 5}$
10	$1.45794641710 - j 4.2449 \times 10^{- 5}$
12	$1.45794316464 - j 4.2391 \times 10^{- 5}$
14	$1.45794241019 - j 4.2606 \times 10^{- 5}$
16	$1.45794226876 - j 4.2594 \times 10^{- 5}$
18	$1.45794230985 - j 4.2354 \times 10^{- 5}$
20	$1.45794230125 - j 4.2353 \times 10^{- 5}$

n _eff
Degree N	MPML	PML	Dirichlet
4	$1.53374449563 - j 1.01414068 \times 10^{- 3}$	$1.531 - j 1.9 \times 10^{- 3}$	$1.278 - j 5.5 \times 10^{- 3}$
8	$1.53345862709 - j 1.28117983 \times 10^{- 3}$	$1.535 - j 1.3 \times 10^{- 3}$	$1.279 - j 5.3 \times 10^{- 3}$
12	$1.53336435738 - j 1.30418841 \times 10^{- 3}$	$1.535 - j 2.5 \times 10^{- 3}$	$1.279 - j 5.3 \times 10^{- 3}$
16	$1.53335860157 - j 1.29818233 \times 10^{- 3}$	$1.535 - j 2.5 \times 10^{- 3}$	$1.279 - j 5.3 \times 10^{- 3}$
18	$1.53335884637 - j 1.29781743 \times 10^{- 3}$	—	—
20	$1.53335886015 - j 1.29782630 \times 10^{- 3}$	—	—
22	$1.53335885986 - j 1.29782673 \times 10^{- 3}$	—	—
24	$1.53335885992 - j 1.29782676 \times 10^{- 3}$	—	—
Exact	$1.53335885990 - j 1.29782676 \times 10^{- 3}$	—	—

Analysis of optical waveguides with ultra-thin metal film based on the multidomain pseudospectral frequency-domain method

Abstract

1. Introduction

2. Numerical examples and discussion

2.1 Dielectric-metal waveguide

2.2 Mode selective waveguide

2.3 Mode selective coupler and polarizing beam splitter

3. Conclusion

Acknowledgments

References and links

Cited By

Figures (12)

Tables (4)

Equations (5)

Optics Express