Efficient rational Chebyshev pseudo-spectral method with domain ...

Efficient rational Chebyshev pseudo-spectral method with domain decomposition for optical waveguides modal analysis ∗

Amgad Abdrabou,1 A. M. Heikal,1,2 and S. S. A. Obayya1, 1 Centre

for Photonics and Smart Materials, Zewail City of Science and Technology, Sheikh Zayed District, 6th of October City, Giza, Egypt 2 Electronics and Communication Department, Faculty of Engineering, Mansoura University, Egypt ∗ [email protected]

Abstract: We propose an accurate and computationally efficient rational Chebyshev multi-domain pseudo-spectral method (RC-MDPSM) for modal analysis of optical waveguides. For the first time, we introduce rational Chebyshev basis functions to efficiently handle semi-infinite computational subdomains. In addition, the efficiency of these basis functions is enhanced by employing an optimized algebraic map; thus, eliminating the use of PML-like absorbing boundary conditions. For leaky modes, we derived a leaky modes boundary condition at the guide-substrate interface providing an efficient technique to accurately model leaky modes with very small refractive index imaginary part. The efficiency and numerical precision of our technique are demonstrated through the analysis of high-index contrast dielectric and plasmonic waveguides, and the highly-leaky ARROW structure; where finding ARROW leaky modes using our technique clearly reflects its robustness. © 2016 Optical Society of America OCIS codes: (050.1755) Computational electromagnetic methods; (250.5403) Plasmonics; (230.7370) Waveguides.

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Received 1 Feb 2016; revised 14 Apr 2016; accepted 15 Apr 2016; published 4 May 2016 16 May 2016 | Vol. 24, No. 10 | DOI:10.1364/OE.24.010495 | OPTICS EXPRESS 10495

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1.

Introduction

Optical waveguides are the fundamental building blocks around which many photonic devices are developed. Prior to the fabrication of these devices, thorough understanding of the light propagation in optical waveguides is essential. Among the methods which provide analytical or semi-analytical solutions are the mode-matching method [1, 2] and the eigenmode expansion method [3]. However, the most commonly used modal solution techniques are based on the finite difference method [4, 5] and the finite element method [6, 7]. Although successful in handling many problems, they produce matrices of huge size and have a relatively slow convergence rate of order O(1/N p ), where N is the number grid points and p is a positive constant. On the other hand, multi-domain pseudo-spectral methods (MDPSMs) [8, 9, 10] emerged recently and proved to be very accurate and efficient with a significant reduction in memory usage and computation time in addition to the fast convergence rate. The MDPSM handles semi-infinite extensions of computational domains through different approaches. One technique was proposed in [11] using Laguerre-Gauss basis functions to mimic the semi-infinite extension of the

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computational domain. Although this approach was successful, it required prior estimation of a very important scaling factor which is related to the decay rate of the field in outer subdomains, and significantly affects the convergence of the method. Another alternative approach was proposed in [12] by using Chebyshev functions in all subdomains while truncating the computational domain. Moreover, perfectly-matched layers (PMLs) are also commonly used to truncate the computational domain. However, there are still serious simulation problems in which PMLs collapse, confront inevitable reflections or even exponential growth [13]. Other major drawbacks are the high sensitivity to the selection of the PML parameters such as conductivity profile, thickness of the layer, permittivity, and the distance to the structure. Moreover, non-physical Berenger PML modes appear in the numerical solution that cannot be wiped out [14]. In this paper, we introduce an efficient numerical technique that can alleviate the problems associated with the existing treatments of semi-infinite computation domains in the context of MDPSMs. Our approach is based on using rational Chebyshev functions in semi-infinite subdomains. The rational basis functions are constructed by composing cardinal Chebyshev functions with a carefully selected conformal map. The proposed approach renders itself to have advantages summarized as: • Unlike Laguerre functions [11], rational Chebyshev functions are bounded i.e. no exponential function with unknown scaling is required to ensure convergence. Hence, convergence is independent of any prior estimates. • Due to the properties of rational Chebyshev functions, we don’t need to employ PerfectlyMatched Layers (PMLs) or PML-like absorbing boundary conditions. • In addition to efficient handling of guided modes, the proposed scheme is very efficient in handling leaky modes. This is accomplished by deriving a nonlinear radiation (Robin-like) boundary condition at the guide-substrate interface, which is linearized using a Taylor series expansion. Through this treatment, exact analytical representation of leaky modes is newly proposed. 2.

Rational Chebyshev multi-domain pseudo-spectral method

The wave equation for 2D optical waveguides is 1 ∂ n2 ∂ u ∂ 2u − θ + k02 (n2 (y) − n2e f f )u = 0, ∂ y2 n2 ∂ y ∂ y where

u(y) =

(1)

Ex (y) and θ = 0 for TE modes; Hx (y) and θ = 1 for TM modes.

(2)

To deal with the discontinuity of n2 (y), the whole domain is first divided into a finite number of subdomains at interfaces of discontinuity denoted by di , i = 1, 2, ..., M for a structure having M interfaces of different materials as shown in Fig. 1. This leads to a computational domain with (M + 1) subdomains, Ωi , i = 1, 2, ..., M + 1. The interface boundary conditions at the points of discontinuity, di are given by Ex (di− ) = Ex (di+ ),

∂ Ex − ∂ Ex + (d ) = (d ), ∂y i ∂y i

(3)

for TE modes, and Hx (di− ) = Hx (di+ ),

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1 ∂ Hx n2 ∂ y

(di− ) =

1 ∂ Hx n2 ∂ y

(di+ ),

(4)


x

z y

1

2

1

2

3

M

3

M+1

M

M+1

//

d1

d2

d3

dMͲ1

dM

Fig. 1. Schematic diagram of a multi-layer 2D planar waveguide.

for TM modes, where di− and di+ refer to locations infinitesimally to the left and to the right of di , respectively. 2.1. Rational Chebyshev functions with algebraic maps The proposed method is based on the cardinal Chebyshev function expansion [11, 15] defined for y˜ ∈ [−1, 1] as follows C j (y) ˜ =

(−1)N− j+1 (1 − y˜2 )TN0 (y) ˜ , σ j N 2 (y˜ − y˜ j )

y˜ 6= y˜ j ,

(5)

where TN (y) ˜ = cos(N cos−1 (y)) ˜ is the Chebyshev polynomial of degree N, σ0 = σN = 2, and σ j = 1 for j = 1, ..., N − 1. The basis functions in Eq. (5) are defined based on the GaussLobatto collocation points y˜k = − cos(kπ/N), k = 0, 1, .., N. For simplicity of reference, we define the subdomain Ω1 = (−∞, d1 ), Ωi = (di−1 , di ), i = 2, ..., M, and ΩM+1 = (dM , ∞). Since the cardinal Chebyshev basis functions are defined on the interval [−1, 1], conformal map are needed in order to define suitable basis functions in each subdomain, Ωi shown in Fig. 1. Referring to Fig. 1, we have two types of subdomains. First, bounded inner subdomains, Ω2 , ..., ΩM and two semi-infinite subdomains Ω1 and ΩM+1 . • For the bounded subdomains, the maps are simply linear and take the form y − di−1 φi (y) = 2 − 1, 2 ≤ i ≤ M, di − di−1 and their mapped Gauss-Lobatto collocation points are given by di − di−1 (i) y j = φi−1 (y˜ j ) = di−1 + (1 − cos( jπ/Ni )) . 2

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(6)

(7)


• For the semi-infinite subdomains Ω1 ≡ (−∞, d1 ) and ΩM+1 ≡ (dM , ∞), we define the algebraic maps φ1 (y; N1 ) and φM+1 (y; NM+1 ) in Ω1 and ΩM+1 , respectively. In Ω1 the map φ1 (y; N1 ) is given by [16] φ1 (y; N1 ) =

τ(N1 ) + (y − d1 ) , τ(N1 ) − (y − d1 )

(8)

where τ(N1 ) = κ1 N1 is an linear function with κ1 > 0 and N1 + 1 is the number of basis (1) functions used in Ω1 . The new collocation points, y j in the subdomain Ω1 are given by (1)

y j = d1 − τ(N1 )

1 + cos ( jπ/N1 ) , 1 − cos ( jπ/N1 )

(9)

at which we obtain the physical values of the electric or magnetic field components. Similarly, for the semi-infinite subdomain ΩM+1 , we employ the map φM+1 (y; NM+1 ) given by (y − dm ) − α(NM+1 ) , (10) φM+1 (y; NM+1 ) = (y − dm ) + α(NM+1 ) (M+1)

where α(NM+1 ) = κ2 NM+1 , κ2 > 0. The corresponding collocation points, y j subdomain ΩM+1 are then given by (M+1)

yj

= dm + α(NM+1 )

1 − cos ( jπNM+1 ) . 1 + cos ( jπNM+1 )

in the

(11)

The solution of the wave Eq. (1) is obtained by expanding the field in terms of mapped basis functions obtained by composing Chebyshev functions by the suitable conformal map for each subdomain Ωi , as we have described and imposing Dirichlet zero-boundary conditions at y = ±∞ as shown in Fig. (2). x

z ZeroͲboundaryconditions

y

1

2

M

3

M+1

//

y=

d1

d2

d3

dMͲ1

dM

y=

Fig. 2. Application of zero-boundary conditions for guided modes at y = ±∞.

The field expansion in a subdomain Ωi takes the form u(i) (y) =

Ni

(i)

∑ uj

(C j ◦ φi )(y),

y ∈ Ωi ,

(12)

j=0

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where Ni + 1 denotes the number of basis functions in a subdomain Ωi and the circle “◦” in Eq. (12) denotes a compositions of the functions C j (y) ˜ with the map φi (y), namely (C j ◦ φi )(y) = C j (φi (y)) . The maps φi (y) all satisfy φi (Ωi ) = [−1, 1] i.e. the function φi maps the (i) subdomain Ωi onto the interval [−1, 1]. The unknown coefficients u j in Eq. (12) represent the field values in the subdomain Ωi at the mapped Gauss-Lobatto collocation points for that (i) subdomain, given by y j = φi−1 (y˜ j ). Cardinality of the basis functions in Eq. (12) is preserved (i)

i.e. we have (C j ◦ φi )(yk ) = δk j , where δk j is the Kronecker delta. It is worth noting that composing the algebraic maps φ1 (y) and φM+1 (y) defined by Eqs. (8) and (10), respectively, leads to rational Chebyshev basis functions defined on the semi-infinite subdomains Ω1 and ΩM+1 . The proposed treatment accounts for semi-infinite extensions of the outer subdomains, hence non-physical modes have no chance to appear in the solution spectrum. It also leads to an efficient replacement of the non-physical perfectly-matched layers (PMLs). The multi-domain pseudo-spectral method reported [12] applies a domain truncation approach. In that approach, the outer semi-infinite subdomains Ω1 and ΩM+1 shown in Fig. 1 are truncated by replacing the original computational domain (−∞, ∞) by a computational window [−W,W ] for W > max {|d1 |, |dM |} and imposing Dirichlet zero-boundary conditions at y = ±W . Truncation of the domain assumes rapid decay rate of the field in outer subdomains. Although this approach is successful, the appropriate value of W should vary according to the refractive index profile (RIP) to accurately describe the field in outer regions. Further, the value of W should increase as the number of basis functions in the outermost subdomains increases to maintain high convergence rate (see [16] and references therein). Another approach to handle the subdomains Ω1 and ΩM+1 was proposed in [11]. It was based on using Laguerre-Gauss basis functions. However, that approach requires a priori estimate for a scaling parameter whose value significantly affects both the accuracy and rate of convergence. 2.2.

Leaky modes treatment

For multi-layer planar waveguides where the substrate has a high refractive index, leaky modes arise and accurate calculation of the effective index imaginary part is challenging. Instead of expanding the field in the outer subdomain representing the substrate with high refractive index, we use the analytic expression of the field u(y) in the substrate region. From this expression we derive the suitable boundary conditions at the guide-substrate interface. This approach can be described follows For a substrate with refractive index ns in the subdomain ΩM+1 , the field u(y) denoted by u(M+1) can be expressed as u(M+1) (y) = u0 e− jγs (y−yM ) , (13) q − , where γs = k02 n2s − β 2 . Differentiating Eq. (13) with respect to y and evaluating at y = dM yield q du(M+1) = − j k02 n2s − β 2 u(M+1) − (14) dy − y=dM y=dM

Since the condition β 2 /k02 n2s