Numerical calculations of ARROW structures by ... - OSA Publishing

Numerical calculations of ARROW structures by pseudospectral approach with Mur’s absorbing boundary conditions Chia-Chien Huang Department of Information Technology, Ling Tung University, No. 1, Ling-Tung Road, Taichung 408, Taiwan, China [email protected]

Abstract: The pseudospectral method, proposed in our previous work, has not yet been constructed for optical waveguides with leaky modes or anisotropic materials. Our present study focuses on antiresonant reflecting optical waveguides (ARROWS) made by anisotropic materials. In contrast to the fields in the outermost subdomain expanded by Laguerre-Gaussian functions for guided mode problems, the fields in the high-index outermost subdomain are expanded by the Chebyshev polynomials with Mur’s absorbing boundary condition (ABC). Accordingly, the traveling waves can leak freely out of the computational window, and the desirable properties of the pseudospectral scheme, i.e., provision of fast and accurate solutions, can be preserved. A number of numerical examples tested by the present approach are shown to be in good agreement with exact data and published results achieved by other numerical methods. ©2006 Optical Society of America OCIS codes: (130.2790) Guided waves; (230.7390) Waveguides, planar

References and links 1. 2. 3. 4. 5.

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M. A. Duguay, Y. Kokubun, T. L. Koch, and L. Pfeiffer, “Antiresonant reflecting optical waveguides in SiO2-Si multilayer structures,” Appl. Phys. Lett. 49, 13–15 (1986). T. Baba, Y. Kokubun, T. Sakaki, and K. Iga, “Loss reduction of an ARROW waveguide in shorter wavelength and its stack configuration,” J. Lightwave Technol. 6, 1440–1445 (1988). M. Mann, U. Trutschel, C. Wachter, L. Leine, and F. Lederer, “Directional coupler based on antiresonant reflecting optical waveguide,” Opt. Lett. 16, 805–807 (1991). Z. M. Mao and W. P. Huang, “An ARROW optical wavelength filter: design and analysis,” J. Lightwave Technol. 11, 1183–1188 (1992). F. Prieto, A. Llobera, D. Jimenez, C. Domenguez, A. Calle, and L. M. Lechuga, “Design and analysis of silicon antiresonant reflecting optical waveguides for evanescent field sensor,” J. Lightwave Technol. 18, 966–972 (2000). T. Baba and Y. Kokubun, “Dispersion and radiation loss characteristics of antiresonant reflecting optical waveguides-numerical results and analytical expressions,” IEEE J. Quantum Electron. 28, 1689–1700 (1992). W. P. Huang, R. M. Shubair, A. Nathan, and Y. L. Chow, “The modal characteristics of ARROW structures,” J. Lightwave Technol. 10, 1015–1022 (1992). T. Baba and Y. Kokubun, “New polarization-insensitive antiresonant reflecting optical waveguide (ARROW-B),” IEEE Photon. Technol. Lett. 1, 232–234 (1989). W. Jiang, J. Chrostowski, and M. Fontaine, “Analysis of ARROW waveguides,” Opt. Commun. 72, 180– 186 (1989). J. Chilwell and I. Hodgkinson, “Thin-film field-transfer matrix theory for planar multilayer waveguides and reflection from prism-loaded waveguides,” J .Opt. Soc. Am. A 1, 742–753 (1984). J. Kubica, D. Uttamchandani, and B. Culshaw, “Modal propagation within ARROW waveguides,” Opt. Commun. 78, 133–136 (1990). J. M. Kubica, “Numerical analysis of InP/InGaAsP ARROW waveguides using transfer matrix approach,” J. Lightwave Technol. 10, 767–771 (1992). E. Anemogiannis and E. N. Glytsis, “Multilayer waveguides: efficient numerical analysis of general structures,” J. Lightwave Technol. 10, 1344–1351 (1992). C. K. Chen, P. Berini, D. Feng, S. Tanev, and V. P. Tzolov, “Efficient and accurate numerical analysis of multilayer planar waveguides in lossy anisotropic media,” Opt. Express 7, 260–272 (2000).

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Received 18 September 2006; revised 23 October 2006; accepted 31 October 2006

27 November 2006 / Vol. 14, No. 24 / OPTICS EXPRESS 11631

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W. P. Huang, C. L. Xu, W. Lui, and K. Yokoyama, “The perfectly matched layer boundary conditions for modal analysis of optical waveguides: leaky mode calculations,” IEEE Photon. Technol. Lett. 8, 652–654 (1996). J. C. Grant, J. C. Beal, and N. J. P. Frenette, “Finite element analysis of the ARROW leaky optical waveguide,” IEEE J. Quantum Electron. 30, 1250–1253 (1994). H. P. Uranus, H. J. W. M. Hoekstra, and E. V. Groesen, “Simple high-order Galerkin finite scheme for the investigation of both guided and leaky modes in anisotropic planar waveguides,” Opt. Quantum Electron. 36, 239–257 (2004). Y. Tsuji and M. Koshiba, “Guided-mode and leaky-mode analysis by imaginary distance beam propagation method based on finite element scheme,” J. Lightwave Technol. 18, 618–623 (2000). J. P. Boyd, “Chebyshev and Fourier Spectral methods,” in Lecture Notes in Engineering, 2nd ed. (Springer Verlag, 2001). C. C. Huang, C. C. Huang, and J. Y. Yang, “An efficient method for computing optical waveguides with discontinuous refractive index profiles using spectral collocation method with domain decomposition,” J. Lightwave Technol. 21, 2284–2296 (2003). C. C. Huang, C. C. Huang, and J. Y. Yang, “A full-vectorial pseudospectral modal analysis of dielectric optical waveguides with stepped refractive index profiles,” IEEE J. Sel. Top. Quantum Electron. 11, 457– 465 (2005). C. C. Huang and C. C. Huang, “An efficient and accurate semivectorial spectral collocation method for analyzing polarized modes of rib waveguides,” J. Lightwave Technol. 23, 2309–2317 (2005). G. Mur, “Absorbing boundary conditions for the finite difference approximation of the time-domain electromagnetic field equations,” IEEE Trans. Electromagn. Compat. 23, 377–382 (1981). A. Hardy and W. Streifer, “Coupled mode theory of parallel waveguides,” J. Lightwave Technol. 3, 1135– 1146 (1985). M. Mann, U. Trutschel, C. Wachter, L. Leine, and F. Lederer, “Directional coupler based on an antiresonant reflecting optical waveguide,” Opt. Lett. 16, 805–807 (1991). Y. H. Chen and Y. T. Huang, “Coupling-efficiency analysis and control of dual antiresonant reflecting optical waveguides,” J. Lightwave Technol. 14, 1507–1513 (1996).

1. Introduction Building optical integrated circuits for practical use in an optical communication system requires a variety of photonic components such as optical interconnectors, modulators, switches, filters, splitters, polarizers, and couplers. The diverse optical waveguides by using distinct materials and refractive index profiles, according to specifications, produce functional optical integrated circuits. To precisely compute propagation characteristics (i.e., dispersion, attenuation, mode distribution, and birefringence) of general optical waveguides with both guided and leaky modes, we may precisely tailor the expected optical devices. A low-loss waveguide antiresonant reflecting optical waveguide (ARROW), which utilizes two thin interference cladding films made on semiconductor substrates [1], has been proposed to construct a wide range of functional devices [2–5]. Unlike total internal reflection mechanisms for guided modes, ARROW structure supports leaky waves through high reflectivity corresponding to the antiresonant condition of a Fabry–Perot resonator. This kind of waveguide has several advantages [6,7] such as low loss, relatively large spot size for effectively coupling into optical fibers, high fabrication tolerance for thicknesses of interference claddings, high loss discrimination between the fundamental and higher-order modes, high polarization sensitivity, and polarization insensitivity [8]. In the viewpoint of computation, ARROW-type devices may be regarded as multilayer waveguide structures, and thus various analytical or numerical approaches including transverse resonance method (TRM) [7,9] and transfer matrix method (TMM) [10–12] have been developed to find the dispersion equations. To calculate propagation characteristics, except some traditional zero-search algorithms like Newton’s and scan methods, which cannot find all of the guided and leaky modes simultaneously, a more efficient and reliable technique is proposed for extracting the poles of dispersion equations in complex plane by the argument principle method (APM) [13,14]. However, the TRM and TMM are only accurate enough to be used with planar waveguides that have step-index profiles rather than graded-index profiles, because inhomogeneous materials are often approximated by a stack of dielectric layers with constant refractive indices. Moreover, the two schemes mentioned above are also unsuitable for the 2-D cross-section structures like rib ARROW providing also lateral #75140 - $15.00 USD

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confinement on mode profiles. Accordingly, it is necessary and desirable to pay close attention to the rigorous numerical solution methods. The schemes used most often such as the finite difference modal analysis [15], finite element modal analysis [16,17], and imaginary distance beam propagation method [18], have been built up to tackle various leaky waveguides. Recently, an efficient and accurate numerical method, based on a number of orthogonal basis functions termed as the pseudospectral method [19], has been proposed to solve dielectric optical waveguides supporting only the guided modes [20–22]. The purpose of this paper is to extend the ability of the pseudospectral scheme to the ARROW structures with leaky modes. At the present time, to modify the consideration representing outermost subdomains by the Laguerre Gaussian (LG) functions for guided mode problems with the exponential decay feature in our previous work [20–22], the outermost subdomains occupied by oscillatory leaky waves are represented by the Chebyshev polynomials with Mur’s absorbing boundary condition (ABC) [23]. Accordingly, the oscillatory behaviors of leaky waves extending to infinity can be well represented, and the superior performances of pseudospectral scheme for accuracy and efficiency may also be preserved. A number of isotropic and anisotropic ARROW structures are analyzed to verify the performances of the proposed solution method. Other issues of the paper are organized as follows. In Section 2, we formulate the wave equations for an anisotropic medium. The description of the pseudospectral scheme with Mur’s ABC is stated in Section 3. Section 4 simulates several numerical examples to validate the accuracy and efficiency of the present approach by comparing exact data. Finally, Section 5 concludes this work. 2. Formulations of planar waveguides Considering an anisotropic medium if the principal dielectric axes of the crystal are found to be parallel to the waveguide coordinate system, the dielectric tensor ε is expressed as follows:

ε

⎡n2 xx ⎢ = ε0 ⎢ 0 ⎢ ⎢⎣ 0

0 n 2yy 0

⎤ ⎥ 0⎥ ⎥ 2 nzz ⎥⎦

0

,

(1)

where ε 0 is the dielectric constant in free space, and nxx , n yy , and nzz are the refractive indices along the x , y , and z axes of the waveguide coordinate system, respectively. In this paper, we aim at analyzing leaky modes in planar ARROW of arbitrary index profiles with a diagonal dielectric tensor shown in expression (1). Assuming that ε is only a function of the independent x (i.e., planar waveguides with finite and infinite extension in x and y directions, respectively), for the source-free, time-harmonic of exp(iω t ) form, and nonmagnetic media, the modal equations in frequency domain derived from Maxwell’s two curl equations are given as follows: ∂ ∂E y ( x) 2 2 ( ) + k02 (n yy ( x) − neff ) E y ( x) = 0 , ∂x ∂x

(2)

for transverse-electric (TE) and that 2 ( x) nxx

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∂ 1 ∂H y ( x) 2 2 ( 2 ) + k02 (nxx ( x) − neff ) H y ( x) = 0 ∂x nzz ( x) ∂x

(3)



for transverse-magnetic (TM) polarizations. Here, k0 = 2π / λ0 , λ0 denotes the wavelength in free space, neff denotes the complex effective refractive index, and E y ( x ) and H y ( x ) represent, respectively, the components of the electric and magnetic fields in the y -direction. 3. Numerical scheme 3.1 Pseudospectral scheme and interfacial patching conditions The framework of the pseudospectral scheme is to divide computational domain into a number of subdomains determined by discontinuous material interfaces. In each subdomain, the optical field ϕ ( x) is expanded by a set of proper interpolation functions Ck ( x ) and the unknown grid point values ϕk as follows:

∑ C ( x)ϕ n

ϕ ( x) =

k =0

k

k

,

(4)

where Ck ( x) =

ψ n +1 ( x) ' ψ n +1 ( x)( x − xk )

, 0≤k ≤n.

(5)

Here, ψ n +1 ( x) denotes an arbitrary basis function of order n + 1 and is determined according to the feature of that subdomain, the prime denotes the first derivative of ψ n +1 ( x) with respect to x , and xk denotes the corresponding collocation point fulfilling the condition of Cl ( xk ) = δ lk , where δ lk denotes the Kronecker delta. The explicit forms of Ck ( x) for distinct basis functions are represented below. If we take the Chebyshev polynomials as basis functions, we have [19] Ck ( x) =

(−1)k +1 (1 − x 2 )Tn' ( x) ck n 2 ( x − xk )

, x ≠ xk

(6)

where Tn ( x) denotes the Chebyshev polynomial of order n , c0 = cn = 2 , and ck = 1 ( 1 ≤ k ≤ n − 1 ). For the LG functions, namely ψ n +1 (α x) = exp(−α x / 2)(α x) Ln (α x) where Ln (α x) denotes the Laguerre polynomial of order n , the explicit form of Ck ( x) is given as follows [19]: Ck (α x) =

e−α x / 2

(α x) Ln (α x)

e−α xk / 2 (α xLn )' (α xk )(α x − α xk )

, x ≠ xk .

(7)

Here, in LG functions, exists an important parameter α called as a scaling factor that can influence the accuracy for a given number of terms of basis functions. The definite procedure for determining α has been proposed in detail in our previous work [20, 21]. From Eqs. (2) and (3), we have an eigenvalue problem in the form

Φ = (k0 n eff )2ΙΦ ,

A

(8)

Φ

where I denotes a unit matrix, denotes the vector of unknown optical field, and A denotes the transverse differential operator. For obtaining the entries of the transverse operator to

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the e th subdomain, we substitute Eq. (4) with a suitable basis function into Eqs. (2) and (3) and obtain Alke = Ck(2) ( xl ) + k02 n 2yy ( xl )δ lk , (l , k = 0,1, 2,...n)

(9)

for TE and that Alke =

2 nxx ( xl ) 2 ( xl ) nzz

[Ck(2) ( xl ) −

2 2 ( xl )δ lk , (l , k = 0,1, 2,...n) C (1) ( xl )] + k02 nxx nzz ( xl ) k

(10)

for TM polarizations. In Eqs. (9) and (10), Ck( n ) ( xl ) means the n th derivative result of Ck ( x) with respect to x at collocation point xl . The global transverse operator A can be obtained by assembling the entries of the transverse operators to all subdomains. However, in matrix A , the rows located in the dielectric interfaces xr ( {r = 1, 2,...m} , where m denotes the number of discontinuous nodes) shared by two adjacent subdomains are imposed to be replaced by the interfacial boundary conditions

E y ( xr+ ) = E y ( xr- ) , ∂E y ( xr+ ) / ∂x = ∂E y ( xr− ) / ∂x

(11)

for TE and that 2 2 H y ( xl+ ) = H y ( xl- ) , nzz ( xr− )∂H y ( xr+ ) / ∂x = nzz ( xr+ )∂H y ( xr− ) / ∂x

(12)

for TM polarizations, where the xr+ and xr− are, respectively, referred to the locations at the infinitesimal right and left of the interface xr . 3.2 Determining basis functions and boundary conditions The best choice is that the interior and outermost subdomains are expanded, respectively, by the Chebyshev polynomials and LG functions, if only guided modes exist [20–22]. However, the outermost subdomains represented by the LG functions are inappropriate while tackling the waveguide structures with leaky modes. In this work, we mainly propose an efficient solution method for handling leaky waveguides. In general, the optical fields in the outermost subdomains exhibit two conditions: one is with evanescent wave, and the other is with traveling wave. Certainly, the outermost subdomains with the evanescent decay characteristic can be efficiently represented by the LG functions as well as those considered in Refs. [20– 22]. As for that with traveling field, since the oscillatory behavior resulting from leaky modes penetrating to substrate, which have a larger refractive index than that in the core layer, this kind of subdomain is expanded by the Chebyshev polynomials at collocation points but has to be fulfilled by the Mur’s ABC [23] at the boundary point (i. e., outmost collocation point), which allows the fields to radiate freely out of the computational window. The perfectly matched layer (PML) condition may be a more efficient scheme than Mur’s ABC, but the implementation of PML to the present method is very complicated. In this paper, we mainly aim to validate the accuracy and efficiency of the pseudospectral approach to leaky mode problems. As a result, we prefer choosing a simpler boundary condition like Mur’s ABC to accomplish the work. In this work, we adopt the second-order formula of Mur’s schemes in frequency domain, and it can be given as follows: (

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∂ + ikr )2 ϕ = 0 , ∂r

(13)



where i = −1 , ϕ denotes the unknown field and is expanded by the Chebyshev polynomials, r denotes the outward direction normal to the boundary, and kr denotes the wavenumber in r -direction. In the paper, r and kr can be replaced by x and k x in the present coordinate system, respectively. Further, Eq. (13) can be expanded as follows: (

∂2 ∂x 2

+ 2ik x

∂ − k x2 )ϕ | x = xbp = 0 , ∂x

(14)

where xbp denotes the boundary point in the outermost subdomain. Note that the chosen computational interval of the outermost subdomain expanded by the Chebyshev polynomials slightly affects the convergence. Observing Eq. (14), we find that the plane wave solution of ϕ ( x) = ϕ (0) exp(−ik x x) is fulfilled [17]. After that, we substitute the plane wave solution into Eqs. (2) and (3) for the outermost subdomain with homogeneous refractive index; we have 2 k x = k0 n 2yy − neff

(15)

for TE, and k x = k0

nzz nxx

2 2 nxx − neff ,

(16)

for TM polarizations. Through Eqs. (2) and (3), the first term in Eq. (14) can be replaced by the terms of −k x2ϕ for both TE and TM polarizations. As a result, Eq. (14) can be further modified as follows: (2ik x

∂ − 2k x2 )ϕ | x = xbp = 0 ∂x

(17)

for TE and TM polarizations. For ensuring only the outgoing plane wave, we must restrict that the real part of k x is positive. In numerical implementation, in order to keep the unit matrix I in 2 )ϕ is added simultaneously in two sides of Eq. (17). Therefore a Eq. (8), the term (k02 neff

matrix eigenvalue problem is obtained by evading a generalized matrix eigenvalue problem. Since the introduction of the Mur’s ABC, the eigenvalue neff resides within the global matrix and is required to be solved by iteration scheme. The computational procedure is that we first 0 assign the initial guess of neff to A , and then a new value of n1eff is found by solving a general linear matrix eigenvalue. Subsequently, n1eff is used as a new value to calculate the next 2 . We do not accept the convergent results until the differences of real and imaginary one neff

part of neff between the adjacent iterations are simultaneously less than the order of 10-6 . Besides the consideration of choosing basis functions in the present work, the Chebyshev polynomials with Mur’s ABC can also be used to represent the evanescent field having decaying characteristics at the boundary as well as that used in the finite element scheme with Sommerfeld’s ABC [17]. However, in our numerical experiments concerning computational efficiency (the required number of basis functions), the preferable choice is still the LG functions. Moreover, no supplementary boundary conditions are demanded for evanescent fields [20–22], and thus the numerical implementation may be significantly simplified.

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4. Numerical results and discussion In order to verify the performances of the present scheme as applied to various ARROW structures, we first analyze in detail the propagation characteristics of an isotropic example, which has been studied by Baba and Kokubun [6] using the system interference matrix and by Chen et al. [14] using TMM with APM. Except for the anisotropy, the second example is the same as the first example. Finally, we study the coupling length and field distributions of even and odd modes for an ARROW-based directional coupler device. The accuracy of these examples calculated by the proposed approach is discussed and compared with other published results. 4.1 Isotropic ARROW structure The schematic diagrams of the geometry and refractive index profile of the ARROW structure are depicted in Fig. 1. The relevant parameters used are those with an operating wavelength of λ=0.6328μm , the refractive indices are n a =1 for air and n s =3.85 for substrate, and the thicknesses (refractive indices) for the first cladding, second cladding, and core layers are, respectively, d1 =0.142λ (n1 =2.3) , d 2 =3.15λ (n 2 =1.46) , and d c =6.3λ (n c =1.46) under the antiresonant condition. The computational domain is divided into five subdomains. The calculated results of the complex effective indices for the first i time iterations n ieff are shown in Table 1, which also shows that n 0eff =1.459 denotes the chosen initial guess of effective index. In the example, the number of terms of LG basis functions for the outermost subdomain occupied by air is N a = 10 while the computational interval estimated according to our criterion [21] for spreading of guided mode is Sa = 1μ m , the number of Chebyshev polynomials for the first cladding, second cladding, and core layers are N1 =N 2 =N c =20 , and the Chebyshev polynomials for substrate layer is Ns = 40 while the computational interval of the outermost subdomain chosen is S sub = 1μ m . More terms of basis functions in the substrate layer are necessary due to the sharply oscillatory behavior of leaky waves. In addition, considering the effect caused by the computational interval of the outermost subdomain with higher refractive index, the differences of the convergent results between S sub = 1μ m and S sub = 2 μ m for all of the modes are negligible while using Ns = 50 for S sub = 2 μ m . We conclude that the major factor in obtaining accurate results is use of the pseudospectral scheme. Table 1 also shows that with only three iterations, our method achieves the convergent values. In addition, the exact four-digit values for n eff can be obtained even when only one iteration is executed. In our numerical experiments, the present approach obtains the same convergent rate for using different n 0eff , as long as the chosen value of n 0eff is slightly smaller than n c . The solutions of this work, obtained by the third iteration, and the results calculated by Chen et al. [14] are shown in Table 2. We can see that the two schemes are in excellent agreement. na

Sa

n c ,d c n 1 ,d1 n 2 ,d 2 Se m ico nd uct or su b.

ns

S sub

n Fig. 1. Schematic diagrams of geometry and refractive index profile of the ARROW structure.

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Table 1. Complex Effective Indices of the TE and TM Modes of an Isotropic ARROW Structure by the Present Method for the First Three Iterations 0 (Ns =40, N1 =N 2 =N c =20, N a =10, neff = 1.459)

Mode

TE0

1

Re( neff )

-Im( neff )( ×10−3 ) Re( neff ) 1

1.802689419

-Im( neff )( ×10−3 ) Re( neff )

2

0

2

1.802689419

3

0

-Im( neff )( ×10−3 )

1.802689419

3

0

TE1 1.457941265 0.000054189

1.457941265 0.000054189

1.457941265 0.000054189

TE2 1.451919137 0.052846592

1.451919174 0.052870681

1.451919174 0.052870681

TE3 1.451173877 0.192047522

1.451174055 0.192035341

1.451174055 0.192035341

TE4 1.441371354 0.004372767

1.441371363 0.004374469

1.441371363 0.004374469

TE5 1.427413547 0.213184218

1.427414119 0.213733418

1.427414119 0.213733398

TE6 1.424444291 0.766252682

1.424447391 0.766727773

1.424447391 0.766727684

TM0 1.585174751 0.000000004

1.585174679 0.000000004

1.585174679 0.000000004

TM1 1.457890856 0.002450742

1.457890856 0.002450742

1.457890856 0.002450742

TM2 1.451755121 0.552864213

1.451754691 0.553891766

1.451754691 0.553891897

TM3 1.451302333 1.152697420

1.451304282 1.151032482

1.451304282 1.151033285

TM4 1.440916234 0.190565224

1.440916633 0.190617140

1.440916633 0.190617138

TM5 1.426481018 2.169331527

1.426462968 2.181427554

1.426462961 2.181440421

TM6 1.425511013 5.078997130

1.425568355 5.056058275

1.425568264 5.056121818

The initial guess of effective index is

n 0eff =1.459 ,

and the number of terms of basis function in each subdomain

is Na = 10 , N1 =N 2 =N c =20 , and Ns = 40 . Table 2. Calculated Results of the Present Method and TMM Using APM [14]

This work

TMM using APM [14] −3

Mode Re( neff ) -Im( neff )( ×10 ) Re( neff ) -Im( neff )( ×10−3 ) TE0 1.802689419

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-

0

TE1 1.457941265 0.000054189

1.457941265 0.000054189

TE2 1.451919174 0.052870681

1.451919174 0.052870681

TE3 1.451174055 0.192035341

1.451174055 0.192035341

TE4 1.441371363 0.004374469

1.441371363 0.004374469

TE5 1.427414119 0.213733398

-

TE6 1.424447391 0.766727684

-

TM0 1.585174679 0.000000004

-

TM1 1.457890856 0.002450742

1.457890856 0.002450742

TM2 1.451754691 0.553891897

1.451754691 0.553891897

TM3 1.451304282 1.151033285

1.451304282 1.151033285

TM4 1.440916633 0.190617138

1.440916633 0.19061714?

TM5 1.426462961 2.181440421

-

TM6 1.425568264 5.056121818

-



To further demonstrate the accuracy of our method, we calculate the dispersion and radiation loss characteristics of the ARROW structure versus the thickness of each layer while the thicknesses of other layers are at corresponding antiresonant conditions. First, Figs. 2(a) and 2(b) show the dispersion and loss characteristics of the first six ARROW modes of TE polarization, labeled TE1–TE6, versus the thickness of the first cladding (d1/λ) , respectively, while other parameters are fixed. Here, referring to Ref. [6], we call TE 0 as the first cladding mode. In Fig. 2(a), the ranges of d1/λ with flat portions of effective index for each mode correspond to the antiresonances of Fabry–Perot cavities formed in the two interference claddings. The same portions represent the low-loss portions while referring to Fig. 2(b). It is clear that the low-loss portion of TE1 is broad so as to allow large fabrication tolerance. In contrast to the flat portions, the transitional portions in Fig. 2(a) illustrate the resonances subject to high-loss portions in Fig. 2(b). For comparing the order of losses between different polarized waves, Fig. 2(b) also illustrates the first two ARROW modes of TM polarization. It is clear that the losses of TM modes are larger than TE ones at a corresponding mode number. As a result, the higher-order TE and TM modes can be easily filtered out to obtain singlemode ( TE1 ) propagation. Besides, in Fig. 2(b), the loss characteristics display periodic variety as a function of (d1/λ) . Likewise, the propagation characteristics versus the thickness of the second cladding (d 2 /λ) and core (d c /λ) layers are shown in Figs. 3 and 4, respectively. In Figs. 3(b) and 4(b), at corresponding first antiresonant conditions, high loss discrimination as that observed in Fig. 2(b) is also shown.

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TE1

1.46 TE2

TE3

Real Part of neff

1.45 TE4

TE5

1.44

1.43

TE6

1.42

1.41 0

0.1

0.2

0.3

0.4

0.5

0.6

Thickness of the First Cladding d1/λ

(a) 3

10

TM2

TE6

2

Attenuation(dB⋅λ /cm)

10

TE2

TE3

TE4

TE5

1

10

0

10

TM1

-1

10

TE1 -2

10

-3

10

0

0.1

0.2

0.3

0.4

0.5

0.6

Thickness of the First Cladding d1/λ

(b) Fig. 2. (a). Dispersion characteristics; (b).Radiation loss characteristics of various modes of an isotropic ARROW versus the thickness of the first cladding.

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TE1

1.46 TE2

TE3

Real Part of neff

1.45 TE4

1.44

1.43

TE5

1.42

TE6

1.41

1.4 0

2

4

6

8

10

Thickness of the Second Claddingd2/λ

(a) 4

10

TE6 3

10


TM2 2

TE5

10

1

10

TM1 0

10

TE3

TE2

TE4

-1

10

TE1 -2

10

0

2

4

6

8

10

Thickness of the Second Cladding d2/λ

(b) Fig. 3. (a). Dispersion characteristics; (b). Radiation loss characteristics of various modes of an isotropic ARROW versus the thickness of the second cladding.

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1.46 TE6

1.45

Real Part of neff

TE5

1.44

TE4 TE3 TE2

1.43

TE1

1.42

1.41

1.40 0

2

4

6

8

10

Thickness of the Core dc/λ

(a) 4

10

TE5

3

10

TM2

TE6 TE3


2

TE4

10

TE2

1

10

0

10

TM1 -1

10

TE1

-2

10

-3

10

0

2

4

6

8

10

Thickness of the Core dc/λ

(b) Fig. 4. (a). Dispersion characteristics; (b). Radiation loss characteristics of various modes of an isotropic ARROW versus the thickness of the core.

Furthermore, Figs. 5(a)–5(f) show the real part of relative field profiles of various TE (solid curve) and TM (dotted curve) polarized waves simultaneously at the first antiresonant condition (i. e., d1 =0.142λ , d 2 =3.15λ , and d c =6.3λ ). Here, “relative” means that the field profiles are normalized by their maximum values [6]. In this paper, Figs. 5(a)–5(f) are #75140 - $15.00 USD

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sequentially labeled as TE 0 (TM 0 ) to TE 6 (TM 6 ) . We can observe that the fields in Fig. 5(a) are confined to the vicinity of the first cladding layer, and the confinement of the TE mode is tighter than that of the TM mode. Figure 5(b) illustrates that the lowest ARROW mode (TE1) is most localized in the core layer, which is of practical interest, and that it is similar to the fundamental mode of conventional waveguides. The oscillatory characteristics in the substrate layer are also clearly shown, and the larger loss of TM mode is observed, as well as the values listed in Table 2. From Figs. 5(b)–5(f), excluding Figs. 5(b) and 5(e), the fields penetrating the interference claddings are fairly large.

TE Mode TM Mode

Relative Field Amplitude (a.u.)

1

0.5

0

-0.5

-4

-3

-2

-1

0

1

2

3

1

2

3

Depth(μ m)

(a)


1

-0.5

0

-0.5

-4

-3

-2

-1

0

Depth(μ m)

(b)

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1

0.5

0

-0.5

-1 -4

-3

-2

-1

0

1

2

3

1

2

3

Depth(μ m)

(c)


1

0.5

0

-0.5

-1 -4

-3

-2

-1

0

Depth(μ m)

(d)

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1

0.5

0

-0.5

-1 -4

-3

-2

-1

0

1

2

3

1

2

3

Depth(μ m)

(e)


1

0.5

0

-0.5

-1 -4

-3

-2

-1

0

Depth(μ m)

(f) Fig. 5. Relative field profiles of TE and TM modes of isotropic ARROW on the first antiresonant condition ( d1 =0.142λ , d 2 =3.15λ , and d c =6.3λ ): (a) TE0 and TM0 (the first cladding mode); (b) TE1 and TM1; (c)TE2 and TM2; (d) TE3 and TM3; (e) TE4 and TM4; (f) TE5 and TM5.

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4.2 Anisotropic ARROW structure Following the same structure analyzed in Subsection 4.1, but with anisotropy in inner layers sandwiched between air and substrate layers, the anisotropic material with a diagonal form of dielectric tensor is expressed in Eq. (1). The refractive indices are n zzc =1.46 for the core, n zz1 =2.3 for the first cladding, and n zz2 =1.46 for the second cladding. Other values follow the relations of n xxj =n yyj =1.03n zzj , where (j=c,1,2) . Here, the initial guess of effective index of n 0eff =1.459 is chosen, and three iterations are executed to achieve convergent solutions. The results obtained by our method, using 140 unknowns ( N a = 10 , N1 =N 2 =N c =30 , N s = 40 ), are shown in Table 3 with the six-order accuracy of the finite element method [17] and TMM using APM [14]. The number of unknowns in Ref. [17] is nearly 1300 because of the effective mesh size of 0.005μm . This shows that the computational efficiency of the present scheme is superior to that of the finite element method with six-order accuracy [17]. In addition, the results of complex effective indices obtained by our work show excellent agreement with exact solutions [14]. However, the present scheme obtains a full matrix. The computational time is 0.18 sec (each iteration) executed on Pentium IV PC with a CPU clock rate of 3.0 GHz in a double precision. We obtain that the effective indices and losses as a function of thickness of each inner layer are similar to its isotropic counterpart. As for that the principal dielectric axes of the crystal are not parallel to the waveguide coordinate system, the more complicated hybrid modes and the propagation characteristics depend strongly on the included angle between the optics axis and the waveguide coordinate system. Table 3. Calculated Results of an Anisotropic ARROW Structure by the Present Method, Six-Order Accuracy FEM [17], and TMM Using APM [14] 0 (Ns =40, N1 =N 2 =N c =30, N a =10, neff = 1.459)

This work

Mode

Re( neff )

Six-order FEM [17] −3

-Im( neff )( ×10 ) Re( neff )

0

TMM using APM [14] −3

-Im( neff )( ×10 )

0

1.867833876

Re( neff ) -Im( neff )( ×10−3 )

TE0

1.867833877

TE1

1.501798936 0.000050179

1.501798936 0.000050179

1.501798936 0.000050179

-

TE2

1.495945499 0.053815143

1.495945499 0.053815130

1.495945499 0.053815143

TE3

1.495255344 0.184243873

1.495255344 0.184243812

1.495255344 0.184243873

TE4

1.485698165 0.004051178

1.485698165 0.004051177

1.485698165 0.004051178

TE5

1.472140044 0.217484450

-

-

TE6

1.469389780 0.735702355

-

-

TM0 1.632729920 0.000000004

1.632729919 0.000000004

-

TM1 1.501625054 0.002544521

1.501625054 0.002544521

1.501625054 0.002544521

TM2 1.495287895 0.576101022

1.495287895 0.576101022

1.495287895 0.576101022

TM3 1.494855078 1.189339701

1.494855078 1.189339701

1.494855078 1.189339701

TM4 1.484121307 0.197863211

1.484121307 0.197863211

1.484121307 0.197863211

TM5 1.469174801 0.224772521

-

-

TM6 1.468390303 5.256321451

-

-

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4.3 ARROW-based directional coupler The directional coupler, which consists of two parallel waveguides, is an interesting photonic structure to build various optical communication devices such as a modulator, filter, power divider, and polarizer. The coupling mechanism of a conventional directional coupler is well studied by the coupled mode theory [24], which considers merely the weakly coupled approximation. In addition, the coupling strength decreases as an exponential function of waveguide separation. Because of the widespread applications of ARROW-based structures, a novel dual ARROW-based directional coupler utilizing antiresonant reflection as the guiding mechanism has been proposed to investigate the propagation characteristics [25]. In comparison with a conventional directional coupler that utilizes total internal reflection, the ARROW structures have strong coupling, which is due to their having leaky waves rather than the evanescent waves found in conventional two parallel waveguides; furthermore, the variation of the coupling length versus waveguide separation displays periodicity [25]. In Fig. 6, the configuration of the dual ARROW structure grown on a Si substrate with a refractive index n s =3.5 consists of two ARROW structures separated by a separation cladding layer with a refractive index n sep =1.46 and thickness dsep =2μm [26]. At the operating wavelength λ=0.6328μm , the two waveguide cores are with n c1 =n c2 =1.46 and d c1 =d c2 =4μm . The core 1 is sandwiched by the two thin cladding layers with n h1 =n h2 =2.3 and d h1 =d h2 =0.089μm , and the same circumstance is also encountered by core 2, which is sandwiched by two cladding layers with n h3 =n h4 =2.3 and d h3 =d h4 =0.089μm . For the upper cladding layer, which is grown over the thin cladding layer of d h1 and in contact with air having n a =1 , the parameters are n l1 =1.46 and d l1 =2μm . As for the lower cladding layer grown over the Si substrate, it is with n l2 =1.46 and d l2 =2μm . The purpose of the different interference claddings is to accomplish the antiresonant conditions. According to the conclusion found by Chen and Huang [26], the coupling efficiency can be flexibly controlled through the adjustment of outermost cladding thickness d l1 . If we consider the conditions of d l1 =d l2 =2μm and d l1 =0,d h1 =0.03μm , the maximum coupling efficiency of Co =99.87% (while the coupling length is L c =λ/(N s -N a )=59mm , where Ns and N a denote lowest order symmetric (even) and asymmetric (odd) modes, respectively) and decoupled phenomenon can be achieved [26], respectively.

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a ir

u p per clad d in g w ith lo w ind ex

w avegu ide core 1

w avegu id e separation

w avegu id e core 2

na n l1 , d l1 n h1 , d h1 n g1 , d g1 n h2 , d h2 n sep , d sep n h3 , d h3 n g2 , d g2 n h4 , d h4

lower cladd ing with low index

x Si-sub .

n l2 , d l2 ns

z Fig. 6. Schematic diagrams of the coupling structure and refractive index profile of the ARROW-based directional coupler with Si substrate, where dg1, dg1, and dsep are the thicknesses of core 1, core 2, and waveguide separation, respectively. Others layers, excluding the half space of air and substrate layers, dl1, dh1, dh2 dh3, dh4, and dl2 are the interference cladding layers.

To validate the characteristics studied in Ref. [26] and demonstrate the ability of our scheme, by the present scheme, the dual ARROW structure is considered as an 11-layer waveguide structure, and the computational window is divided into 11 subdomains. The numbers of terms of LG functions used for air is N a = 10 , of Chebyshev polynomials for substrate is N s =40 , and of others are N i =20 (where i represents other layers). Compared with the coupling length of L c =59mm , obtained in Ref. [26], the calculated coupling length is L c =58.60mm (where N s =1.45785857 and N a =1.45785317 ). The coupling length of TM mode obtained by our scheme is L c =9.51mm (where N s =1.45787059 and N a =1.45783732 ), which is far smaller than that of the TE mode. This is because the coupling resonances of leaky waves of TM modes are larger than that of TE modes. In our calculation, the imaginary parts (radiation loss) of complex effective indices of TM modes are two orders larger than that of TE modes (namely, TE: ≈ 10-6 dB/cm, TM: ≈ 10-4 dB/cm). From the dispersion characteristics of TE modes illustrated in Fig. 7 it can be clearly seen that the maximum coupling is located at d l1 =2μm , making the dual ARROW acts as a symmetric coupler, and the decoupling portions are close to d l1 =0μm and d l1 =4μm . Additionally, the field profiles of the even and odd TE modes for three cases of d l1 =2μm (maximum coupling), d l1 =0.5μm (half coupling), and d l1 =0 (decoupling, while d h1 =0.03μ m is used [26]) are clearly shown by the decreased order of coupling in Figs. 8(a)–8(c), respectively. Finally, in Fig. 9, the coupling characteristics of the TE and TM modes as a function of the waveguide separation are also illustrated. Obviously, the coupling lengths of the TE and TM modes appear to be a periodic function, as predicted in Ref. [25]. Consequently, this shows that we may apply a dual ARROW-based coupler to easily accomplish a variety of applications in a remote coupler [25].

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1.45789 Even Mode Odd Mode

1.45788

Effective index neff

1.45787

1.45786

1.45785

1.45784

1.45783

1.45782 0

1

2

3

4

Thickness of the Upper Cladding Layer dl1(μ m)

Fig. 7. Dispersion characteristics versus the upper cladding layer dl1 for the lowest symmetric (even) and asymmetric (odd) modes.

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1 Even Mode Odd Mode

0.8

Relative Field Amplitude(a.u.)

0.6 ns

nl2

ng2

nsep

ng1

nl1

na

0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 0

5

10

15

20

x-Direction(μ m)

(a) 1 Even Mode Odd Mode

0.8


0.6 ns

ng2

nl2

nsep

nl1

ng1

na

0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 0

5

10

15

20

x-Direction(μ m)

(b)

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1 Even Mode Odd Mode


0.8 0.6 ng2

nl2

ns

nsep

nl1

ng1

na

0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 0

5

10

15

20

x-Direction(μ m) (c)

Fig. 8. Relative field profiles of the lowest symmetric (even) and asymmetric (odd) modes at different order of coupling: (a) maximum coupling, (b) half coupling, (c) decoupling. 60 TE Mode

Coupling Length Lc (μ m)

50

40

30

20

TM Mode

10

0 0

2

4 6 8 10 Waveguide Separation of dsep (μ m)

12

Fig. 9. The coupling lengths of TE and TM modes versus waveguide separation dsep.

5. Conclusion We successfully demonstrate the ability of the present method to deal with the leaky modes supported by the isotropic and anisotropic ARROW structures. The main consideration is to represent the outermost subdomains with strongly oscillatory characteristics by the efficient Chebyshev polynomials with Mur’s ABC allowing outgoing waves to freely penetrate into

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substrate, evading most reflected waves. As a result, the oscillatory characteristics are accurately represented. The results calculated by the present scheme show excellent agreement with exact solutions and the computational efficiency is far higher than the finite element method. The application of our scheme to a dual ARROW structure is considered as an ARROW-based directional coupler; the accurate values of the coupling length are obtained, and periodicity of the coupling length versus waveguide separation is also validated. The extension of the present method to three-dimensional ARROW structures of practical interests will be reported elsewhere. Acknowledgments This work was supported by the National Science Council, Taiwan, Republic of China under contract No. NSC 95-2221-E-275-005.

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