## A high-accuracy pseudospectral full-vectorial leaky optical waveguide mode solver with carefully implemented UPML absorbing boundary conditions |

Optics Express, Vol. 19, Issue 2, pp. 1594-1608 (2011)

http://dx.doi.org/10.1364/OE.19.001594

Acrobat PDF (1885 KB)

### Abstract

The previously developed full-vectorial optical waveguide eigenmode solvers using pseudospectral frequency-domain (PSFD) formulations for optical waveguides with arbitrary step-index profile is further implemented with the uniaxial perfectly matched layer (UPML) absorption boundary conditions for treating leaky waveguides and calculating their complex modal effective indices. The role of the UPML reflection coefficient in achieving high-accuracy mode solution results is particularly investigated. A six-air-hole microstructured fiber is analyzed as an example to compare with published high-accuracy multipole method results for both the real and imaginary parts of the effective indices. It is shown that by setting the UPML reflection coefficient values as small as on the order of 10^{−40} ∼ 10^{−70}, relative errors in the calculated complex effective indices can be as small as on the order of 10^{−12}.

© 2011 Optical Society of America

## 1. Introduction

1. G. R. Hadley, “High-accuracy finite-difference equations for dielectric waveguide analysis I: Uniform regions and dielectric interfaces,” J. Lightwave Technol. **20**, 1210–1218 (2002). [CrossRef]

3. N. Thomas, P. Sewell, and T. M. Benson, “A new full-vectorial higher order finite-difference scheme for the modal analysis of rectangular dielectric waveguides,” J. Lightwave Technol. **25**, 2563–2570 (2002). [CrossRef]

4. Y. C. Chiang, Y. P. Chiou, and H. C. Chang, “Improved full-vectorial finite-difference mode solver for optical waveguides with step-index profiles,” J. Lightwave Technol. **20**, 1609–1618, (2002). [CrossRef]

1. G. R. Hadley, “High-accuracy finite-difference equations for dielectric waveguide analysis I: Uniform regions and dielectric interfaces,” J. Lightwave Technol. **20**, 1210–1218 (2002). [CrossRef]

2. G. R. Hadley, “High-accuracy finite-difference equations for dielectric waveguide analysis II: Dielectric corners,” J. Lightwave Technol. **20**, 1219–1231 (2002). [CrossRef]

4. Y. C. Chiang, Y. P. Chiou, and H. C. Chang, “Improved full-vectorial finite-difference mode solver for optical waveguides with step-index profiles,” J. Lightwave Technol. **20**, 1609–1618, (2002). [CrossRef]

5. M. Koshiba and Y. Tsuji, “Curvilinear hybrid edge/nodal elements with triangular shape for guided-wave problems,” J. Lightwave Technol. **18**, 737–743 (2000). [CrossRef]

7. P. J. Chiang, C. L. Wu, C. H. Teng, C. S. Yang, and H. C. Chang, “Full-vectorial optical waveguide mode solvers using multidomain pseudospectral frequency-domain (PSFD) formulations,” IEEE J. Quantum Electron. **44**, 56–66 (2008). [CrossRef]

^{−12}[7

7. P. J. Chiang, C. L. Wu, C. H. Teng, C. S. Yang, and H. C. Chang, “Full-vectorial optical waveguide mode solvers using multidomain pseudospectral frequency-domain (PSFD) formulations,” IEEE J. Quantum Electron. **44**, 56–66 (2008). [CrossRef]

*n*) is defined as the modal propagation constant divided by the free-space wavenumber.

_{eff}8. B. Yang, D. Gottlieb, and J. S. Hesthaven, “Spectral simulations of electromagnetic wave scattering,” J. Comput. Phys. **134**, 216–230 (1997). [CrossRef]

13. B. Y. Lin, H. C. Hsu, C. H. Teng, H. C. Chang, J. K. Wang, and Y. L. Wang, “Unraveling near-field origin of electromagnetic waves scattered from silver nanorod arrays using pseudo-spectral time-domain calculation,” Opt. Express17, 14211–14228 (2009). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-16-14211. [CrossRef] [PubMed]

14. Q. H. Liu, “A pseudospectral frequency-domain (PSFD) method for computational electromagnetics,” IEEE Antennas Wireless Propagat. Lett. **1**, 131–134 (2002). [CrossRef]

7. P. J. Chiang, C. L. Wu, C. H. Teng, C. S. Yang, and H. C. Chang, “Full-vectorial optical waveguide mode solvers using multidomain pseudospectral frequency-domain (PSFD) formulations,” IEEE J. Quantum Electron. **44**, 56–66 (2008). [CrossRef]

**44**, 56–66 (2008). [CrossRef]

15. P. J. Chiang, C. P. Yu, and H. C. Chang, “Analysis of two-dimensional photonic crystals using a multidomain pseudospectral method,” Phys. Rev. E **75**, 026703 (2007). [CrossRef]

**44**, 56–66 (2008). [CrossRef]

15. P. J. Chiang, C. P. Yu, and H. C. Chang, “Analysis of two-dimensional photonic crystals using a multidomain pseudospectral method,” Phys. Rev. E **75**, 026703 (2007). [CrossRef]

16. W. J. Gordon and C. A. Hall, “Transfinite element methods: blending-function interpolation over arbitrary curved element domains,” Numer. Math. **21**, 109–129 (1973). [CrossRef]

17. 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). [CrossRef]

18. T. P. White, B. T. Kuhlmey, R. C. Mcphedran, D. Maystre, G. Renversez, C. M. de Sterke, and L. C. Botten, “Multi-pole method for microstructured optical fibers. I. Formulation,” J. Opt. Soc. Am. B **19**, 2322–2330 (2002). [CrossRef]

19. B. T. Kuhlmey, T. P. White, G. Renversez, D. Maystre, L. C. Botten, C. M. de Sterke, and R. C. Mcphedran, “Multi-pole method for microstructured optical fibers. II. Implementation and results,” J. Opt. Soc. Am. B **19**, 2331–2340 (2002). [CrossRef]

20. P. Russell, “Photonic crystal fibers,” Science **299**, 358–362 (2003). [CrossRef] [PubMed]

*n*would become complex, by applying the perfectly matched layer (PML) [21

_{eff}21. J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comp. Phys. **114**, 185–200 (1994). [CrossRef]

22. Z. S. Sacks, D. M. Kingsland, R. Lee, and J. F. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antennas Propagat. **43**, 1460–1463 (1995). [CrossRef]

22. Z. S. Sacks, D. M. Kingsland, R. Lee, and J. F. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antennas Propagat. **43**, 1460–1463 (1995). [CrossRef]

24. C. P. Yu and H. C. Chang, “Yee-mesh-based finite difference eigenmode solver with PML absorbing boundary conditions for optical waveguides and photonic crystal fibers,” Opt. Express12, 6165–6177 (2004). http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-25-6165. [CrossRef] [PubMed]

25. 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). [CrossRef]

*n*would show oscillatory variation in the typically used PML reflection coefficient range of 10

_{eff}^{−7}∼ 10

^{−12}. In this paper, we particularly investigate the role of the UPML reflection coefficient in achieving high-accuracy effective-index results. We will consider a six-air-hole microstructured fiber as an example, which was studied in [18

18. T. P. White, B. T. Kuhlmey, R. C. Mcphedran, D. Maystre, G. Renversez, C. M. de Sterke, and L. C. Botten, “Multi-pole method for microstructured optical fibers. I. Formulation,” J. Opt. Soc. Am. B **19**, 2322–2330 (2002). [CrossRef]

19. B. T. Kuhlmey, T. P. White, G. Renversez, D. Maystre, L. C. Botten, C. M. de Sterke, and R. C. Mcphedran, “Multi-pole method for microstructured optical fibers. II. Implementation and results,” J. Opt. Soc. Am. B **19**, 2331–2340 (2002). [CrossRef]

18. T. P. White, B. T. Kuhlmey, R. C. Mcphedran, D. Maystre, G. Renversez, C. M. de Sterke, and L. C. Botten, “Multi-pole method for microstructured optical fibers. I. Formulation,” J. Opt. Soc. Am. B **19**, 2322–2330 (2002). [CrossRef]

19. B. T. Kuhlmey, T. P. White, G. Renversez, D. Maystre, L. C. Botten, C. M. de Sterke, and R. C. Mcphedran, “Multi-pole method for microstructured optical fibers. II. Implementation and results,” J. Opt. Soc. Am. B **19**, 2331–2340 (2002). [CrossRef]

*n*’s. We will show that by setting the UPML reflection coefficient values as small as on the order of 10

_{eff}^{−40}∼ 10

^{−70}, taking the advantage of the unprecedented accuracy of the PSMS, relative errors in the calculated complex

*n*’s can be as small as on the order of 10

_{eff}^{−12}. This issue regarding using small PML reflection coefficients was recently briefly discussed in [26

26. P. J. Chiang and Y. C. Chiang, “Pseudospectral frequency-domain formulae based on modified perfectly matched layers for calculating both guided and leaky modes,” IEEE Photon. Technol. Lett. **22**, 908–910 (2010). [CrossRef]

27. W. C. Chew and W. H. Weedon, “A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates,” Microwave Opt. Technol. Lett. **7**, 599–604 (1994). [CrossRef]

## 2. The H-formulation equations in an anisotropic Medium

*z*direction at the angular frequency

*ω*. The source-free time-harmonic Maxwell’s equations are written as where

*ɛ*

_{0}and

*μ*

_{0}are permittivity and permeability of free space, respectively, and the relative permittivity tensor [

*ɛ*] and the relative permeability tensor [

*μ*] are defined respectively as By taking the curl of Eq. (4) and using Eq. (3), we obtain the vectorial wave equation where

*k*

_{0}is the wavenumber in free space and

*z*direction,

*H⃗*can be expressed as where

*β*is the modal propagation constant. From Eq. (2), we have By substituting Eq. (6) into Eq. (5) and using Eq. (7), we obtain where the differential operators in the matrix are defined as

*x*-

*y*coordinate system defined in the figure, the available boundary conditions (DBCs and NBCs) are, respectively, expressed as and

*n*and

_{x}*n*are the

_{y}*x*- and

*y*-components of the normal unit vector at the interface in Fig. 1 and the superscripts

*a*and

*b*refer to regions a and b, respectively. Of course, all quantities in Eqs. (13)–(15) are evaluated at the interface.

## 3. The PSMS-UPML Formulation

*μ*,

_{x}*μ*,

_{y}*μ*,

_{z}*ɛ*,

_{x}*ɛ*, and

_{y}*ɛ*are constants, and that in region b (considered as the UPML region) has the corresponding tensors as [

_{z}*μ*]

_{UPML}and [

*ɛ*]

_{UPML}.

*μ*]

_{UPML}and [

*ɛ*]

_{UPML}such that [28

28. K. Saitoh and M. Koshiba, “Full-vectorial finite element beam propagation method with perfectly matched layers for anisotropic optical waveguides,” J. Lightwave Technol. **19**, 405–413 (2001). [CrossRef]

*μ*] and [

*ɛ*] in the whole computational domain including UPML regions can be expressed as with where

*x*and

*y*directions, respectively, in UPML regions. Assuming the appropriate values of the

*m*-power profiles as where

*d*and

_{x}*d*are the thicknesses of the UPML as shown in Fig. 2 which shows the cross-section of an arbitrary leaky waveguide with the computational domain surrounded by UPML regions, and

_{y}*ρ*and

_{x}*ρ*are the distances from the beginning of the UPML along the

_{y}*x*and

*y*directions, respectively. Using the theoretical reflection coefficient

*R*[21

21. J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comp. Phys. **114**, 185–200 (1994). [CrossRef]

*R*approaches to zero) at the inner interface of the UPML, the maximum values of

*α*and

_{x}*α*can be determined as where

_{y}*λ*is the wavelength corresponding to

*ω*and

*ɛ*) being the smallest of

_{i}*ɛ*,

_{x}*ɛ*, and

_{y}*ɛ*of the anisotropic material region interfaced to the UPML. Note that

_{z}*α*

_{max(x)}and

*α*

_{max(y)}are large enough corresponding to

*R*, as for the isotropic material region case [21

21. J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comp. Phys. **114**, 185–200 (1994). [CrossRef]

*m*= 2,

*s*and

_{x}*s*in Eqs. (18) and (19) become By substituting Eqs. (16) and (17) into Eqs. (9)–(12), we obtain

_{y}*s*′

*=*

_{x}*∂*(1/

*s*)/

_{x}*∂x*,

*s*′

*=*

_{y}*∂*(1/

*s*)/

_{y}*∂y*,

*s*″

*=*

_{x}*∂*

^{2}(1/

*s*)/

_{x}*∂x*

^{2}, and

*s*″

*=*

_{y}*∂*

^{2}(1/

*s*)/

_{y}*∂y*

^{2}.

*p*subdomains, the matrix eigenvalue equation before imposing boundary conditions across the subdomain interfaces would be similar to that of Eq. (27) of [7

**44**, 56–66 (2008). [CrossRef]

*pk*× 2

*pk*one. The adjacent subdomains are connected by imposing DBCs and NBCs on the matrix elements, corresponding to the interface grids of the subdomains, in this 2

*pk*× 2

*pk*matrix eigenvalue equation. How such modification of the matrix equation is conducted can be referred to the Appendix in [15

15. P. J. Chiang, C. P. Yu, and H. C. Chang, “Analysis of two-dimensional photonic crystals using a multidomain pseudospectral method,” Phys. Rev. E **75**, 026703 (2007). [CrossRef]

*x*-axis and

*y*-axis, respectively, and regions III are the four corner regions. The

*s*and

_{x}*s*values for different regions are listed in Table 1.

_{y}## 4. Numerical example: analysis of a six-air-hole microstructured fiber

**19**, 2322–2330 (2002). [CrossRef]

**19**, 2331–2340 (2002). [CrossRef]

*n*’s of the modes would be obtained by searching zeros of a complex function resulting from the determinant of a large complex matrix corresponding to a derived homogeneous system of equations. Achievement of the solutions of high accuracy requires special techniques and careful procedures [18

_{eff}**19**, 2322–2330 (2002). [CrossRef]

**19**, 2331–2340 (2002). [CrossRef]

**19**, 2322–2330 (2002). [CrossRef]

**19**, 2331–2340 (2002). [CrossRef]

*p*= 1) mode were provided, with the imaginary part of

*n*pushed down to 10

_{eff}^{−11}. We thus compare first our analysis of this sixth mode with that of [19

**19**, 2331–2340 (2002). [CrossRef]

**19**, 2322–2330 (2002). [CrossRef]

**19**, 2331–2340 (2002). [CrossRef]

*d*= 5

*μ*m, the hole pitch Λ = 6.75

*μ*m, and the silica background with index

*n*= 1.45. Figure 3(b) shows the domain and mesh division profile including the UPML regions employing the Chebyshev-Gauss-Lobatto grid points, where 32 subdomains are adopted and the mesh pattern of each subdomain depicted is for polynomials of degree

*M*=

*N*= 8, corresponding to (8 + 1)

^{2}× 2 × 32 = 5184 total unknowns in the eigenvector for the transverse magnetic field components. According to [18

**19**, 2322–2330 (2002). [CrossRef]

*n*for the sixth mode (with class

_{eff}*p*= 1 as designated in [18

**19**, 2322–2330 (2002). [CrossRef]

*μ*m would be 1.438364934 –

*j*1.41647 ×10

^{−6}and the real and imaginary parts converge to 10

^{−9}and 10

^{−11}, respectively. In our calculations, we obtain the corresponding

*n*values as 1.43836493417887 –

_{eff}*j*1.41647611 × 10

^{−6}when

*M*=

*N*= 20 is used and 1.43836493417887 –

*j*1.41647598 × 10

^{−6}when

*M*=

*N*= 22 is used, which agree with that of [19

**19**, 2331–2340 (2002). [CrossRef]

*n*

_{eff, ref}= 1.438364934178 –

*j*1.416476 ×10

^{−6}(with 10

^{−12}precision in both the real and the imaginary parts) as a reference to discuss our calculation results using the PSMS-UPML under different UPML parameters. In Fig. 3(a), the computational window sizes are

*W*=

_{x}*W*= 15.75

_{y}*μ*m, the UMPL thicknesses are

*d*=

_{x}*d*= 2

_{y}*μ*m, and the operating wavelength is

*λ*= 1.45

*μ*m. Figures 4(a) and (b), respectively, show the relative errors (RErs) in the real and the imagniary parts of

*n*of the sixth mode relative to the above-mentioned

_{eff}*n*

_{eff, ref}with respect to the number of unknowns for different reflection coefficient values (

*R*= 10

^{−6}, 10

^{−7}, 10

^{−9}, 10

^{−10}, 10

^{−15}, 10

^{−17}, 10

^{−20}, 10

^{−40}, 10

^{−50}, 10

^{−60}, and 10

^{−70}). Note that the reference value is obtained with

*R*= 10

^{−70}. It is seen that in both figures the RErs with

*R*≤ 10

^{−40}can drop to as small as on the order of 10

^{−10}but not for

*R*> 10

^{−40}. Note that in Fig. 4(b), the REr of Im[

*n*] for

_{eff}*R*= 10

^{−7}appears to be smaller than those using 10

^{−9}≥

*R*≥ 10

^{−20}. However, Fig. 4(a) shows that the REr of Re[

*n*] for

_{eff}*R*= 10

^{−7}is only better than that for

*R*= 10

^{−6}. We thus examine the total relative error of the complex

*n*, expressed as TREr(

_{eff}*n*) and defined as TREr(

_{eff}*n*) = |

_{eff}*n*–

_{eff}*n*

_{eff, ref}|. The TREr(

*n*)’s versus the number of unknowns are shown in Fig. 5(a), where it is seen that the TREr(

_{eff}*n*) for large number of unknowns is better with the smaller

_{eff}*R*, i.e., TREr(

*n*)

_{eff}_{R=10−6}> TREr(

*n*)

_{eff}_{R=10−7}> TREr(

*n*)

_{eff}_{R=10−9}> TREr(

*n*)

_{eff}_{R=10−10}> ⋯ > TREr(

*n*)

_{eff}_{R=10−70}. However, in Fig. 5(a), the dependence on

*R*of the convergence behavior does not follow this trend when the number of unknowns is smaller than 6400 for TREr(

*n*) < 10

_{eff}^{−7}. This fact could lead to the wrong conclusion that the optimal

*R*values are within 10

^{−6}∼ 10

^{−12}when analysis methods of limited accuracy are employed since they could not achieve sufficiently low TREr values to reach better numerical convergency. The maximum number of unknowns used in Figs. 4 and 5(a) is 28224, corresponding to

*M*=

*N*= 20. For this enough high polynomial degree, the TREr(

*n*) versus

_{eff}*R*is plotted in Fig. 5(b), which shows the decrease of TREr(

*n*) with decreasing

_{eff}*R*until

*R*= 10

^{−40}, in consistent with the PML principle that

*R*→ 0 for the ideal PML.

*n*]’s and Im[

_{eff}*n*]’s corresponding to the

_{eff}*R*= 10

^{−70}results in Fig. 5(a) are listed in Table 2 for different polynomial degrees and numbers of unknowns. In Table 2 and the following Tables, the digits of each effective index are shown down to the place of 10

^{−14}. It can be seen that our results with

*M*=

*N*= 18 have already agreed with the obtained value of [19

**19**, 2331–2340 (2002). [CrossRef]

^{−9}for Re[

*n*] and to 10

_{eff}^{−11}for Im[

*n*]. The calculated Re[

_{eff}*n*]’s and Im[

_{eff}*n*]’s with

_{eff}*M*=

*N*= 20 using different

*R*s from 10

^{−6}to 10

^{−70}, corresponding to Fig. 5(b), are listed in Table 3. We also examine the effect of the distance between the six air-holes and the UPML layers, or the size of the computational window. With fixed

*d*=

_{x}*d*= 2

_{y}*μ*m,

*W*=

_{x}*W*in Fig. 3 is varied from 13 to 15.75

_{y}*μ*m. The number of unknowns is taken to be 28224 and

*M*=

*N*= 20. The obtained (Re[

*n*] – 1.438364934) and Im[

_{eff}*n*] are shown in Figs. 6(a) and (b), respectively, versus

_{eff}*W*=

_{x}*W*for

_{y}*R*= 10

^{−7}and 10

^{−70}. It is seen that both Re[

*n*] and Im[

_{eff}*n*] for

_{eff}*R*= 10

^{−70}almost do not vary with

*W*=

_{x}*W*under the scales shown while those for

_{y}*R*= 10

^{−7}show significant variations. The ranges of variation for both Re[

*n*] and Im[

_{eff}*n*] values for

_{eff}*R*= 10

^{−7}are almost on the same order and are about 2 × 10

^{−7}. And it is interesting to observe that for

*R*= 10

^{−7}, when Re[

*n*] reaches the most accurate value (near zero in Fig. 6(a)), Im[

_{eff}*n*] in Fig. 6(b) would be with the largest (positive or negative) error, and vice versa. Finally, using

_{eff}*R*= 10

^{−70}, TREr(

*n*)’s versus the number of unknowns for

_{eff}*W*=

_{x}*W*= 13.75

_{y}*μ*m and 15.75

*μ*m are plotted in Fig. 7. It is seen that for both window sizes, the TREr(

*n*) reaches the lowest value when the number of unknowns increases to 14400. No better accuracy can be revealed for more unknowns due to reaching the accuracy limit of the reference value. The results with

_{eff}*W*=

_{x}*W*= 13.75

_{y}*μ*m are seen to be a little better than those with

*W*=

_{x}*W*= 15.75

_{y}*μ*m because smaller window size gives better mesh resolution for the same number of unknowns. According to the above discussions, we can conclude that the reflection coefficient

*R*is the key UPML parameter for achieving high-accuracy

*n*and its value of as small as 10

_{eff}^{−70}is suggested.

*W*=

_{x}*W*= 13.75

_{y}*μ*m. Only the first five modes are considered. The seventh to the tenth modes studied in [18

**19**, 2322–2330 (2002). [CrossRef]

*p*= 3) mode, the calculated Re[

*n*]’s and Im[

_{eff}*n*]’s with

_{eff}*M*=

*N*= 20 using

*R*= 10

^{−50}, 10

^{−60}, and 10

^{−70}, respectively, are listed in Table 4. Then, using

*R*= 10

^{−70}, the calculated Re[

*n*]’s and Im[

_{eff}*n*]’s are listed in Table 5 for different polynomial degrees and numbers of unknowns. We can safely conclude that the converged

_{eff}*n*is 1.445395256948 –

_{eff}*j*3.1947 × 10

^{−8}. We then list in Table 6 the calculated Re[

*n*]’s and Im[

_{eff}*n*]’s, with the corresponding loss values in dB/m, of the first six modes which are the

_{eff}*p*= 3, 4, 2, 5, 6, and 1 modes in sequence designated in [18

**19**, 2322–2330 (2002). [CrossRef]

*M*=

*N*= 20 and

*R*= 10

^{−70}. Note that

*p*= 3 and

*p*= 4 modes are twofold degenerate, and so are

*p*= 5 and

*p*= 6 modes. The loss in dB/m is obtained by multiplying Im[

*n*] by (20/[ln(10)])(2

_{eff}*π*/

*λ*) × 10

^{6}[18

**19**, 2322–2330 (2002). [CrossRef]

*λ*is in micrometers. Although the numerical accuracy of the results in [18

**19**, 2322–2330 (2002). [CrossRef]

**19**, 2331–2340 (2002). [CrossRef]

**19**, 2322–2330 (2002). [CrossRef]

*n*] between [18

_{eff}**19**, 2322–2330 (2002). [CrossRef]

**19**, 2322–2330 (2002). [CrossRef]

*E*| and |

_{z}*ZH*| were plotted for

_{z}*p*= 3,

*p*= 4, and

*p*= 2 modes, where

*Z*denotes the impedance of free space. In this PSMS-UPML, which is based on the transverse-magnetic-field formulation,

*H*and

_{x}*H*are first obtained. Then,

_{y}*E*and

_{z}*H*can be calculated with high accuracy, although involving the derivatives of

_{z}*H*and

_{x}*H*, since the derivative is simply performed by the differential matrix operation under the pseudospectral theory. In Fig. 8, for each mode, the |

_{y}*E*|, |

_{z}*H*|, |

_{z}*E*|, and |

_{x}*H*| mode-field profiles are shown, with the maximum of each profile set to be unity. The |

_{y}*E*| and |

_{z}*H*| profiles for the first three modes (

_{z}*p*= 3,

*p*= 4, and

*p*= 2) are found to be consistent with those shown in Figs. 4 and 5 of [18

**19**, 2322–2330 (2002). [CrossRef]

*H*| and |

_{x}*H*| profiles of the

_{y}*p*= 2 and

*p*= 6 modes appear to have very little differences but their |

*E*| and |

_{z}*H*| profiles have obvious different features. The same can be observed between the

_{z}*p*= 5 and

*p*= 1 modes. Moreover, it can be seen that the |

*H*| (|

_{x}*H*|) profile of the

_{y}*p*= 2 mode resembes the |

*H*| (|

_{y}*H*|) profile of the

_{x}*p*= 5 mode, and the same feature exists between the

*p*= 6 and

*p*= 1 modes. Such fact should not be surprising since these four modes actually possess very close Re[

*n*]’s with the same leading digits, 1.438, as seen in Table 6. And more noticeable differences appear in the |

_{eff}*E*| and |

_{z}*H*| profiles.

_{z}## 5. Conclusion

*H*–

_{x}*H*formulation by considering the Helmholtz equations. The spatial derivatives in the equations are approximated at collocation points by utilizing Chebyshev-Lagrange interpolating polynomials, leading to a matrix eigenvalue equation with the squared complex propagation constant as the eigenvalue. We have shown other field components can be determined with high accuracy from the obtained

_{y}*H*and

_{x}*H*(eigen vector) with the required spatial derivatives treated by the same approximation at collocation points. The high-accuracy performance of the PSMS-UPML is demonstrated by using a six-air-hole microstructured fiber as an analysis example, with the comparison made with published high-accuracy multipole method results [19

_{y}**19**, 2331–2340 (2002). [CrossRef]

*R*, in achieving high-accuracy mode solution results is investigated in detail. It is shown that by setting the values of

*R*as small as on the order of 10

^{−40}∼ 10

^{−70}, relative errors in the calculated complex effective indices can be as small as on the order of 10

^{−12}.

## Acknowledgments

## References and links

1. | G. R. Hadley, “High-accuracy finite-difference equations for dielectric waveguide analysis I: Uniform regions and dielectric interfaces,” J. Lightwave Technol. |

2. | G. R. Hadley, “High-accuracy finite-difference equations for dielectric waveguide analysis II: Dielectric corners,” J. Lightwave Technol. |

3. | N. Thomas, P. Sewell, and T. M. Benson, “A new full-vectorial higher order finite-difference scheme for the modal analysis of rectangular dielectric waveguides,” J. Lightwave Technol. |

4. | Y. C. Chiang, Y. P. Chiou, and H. C. Chang, “Improved full-vectorial finite-difference mode solver for optical waveguides with step-index profiles,” J. Lightwave Technol. |

5. | M. Koshiba and Y. Tsuji, “Curvilinear hybrid edge/nodal elements with triangular shape for guided-wave problems,” J. Lightwave Technol. |

6. | P. J. Chiang, C. S. Yang, C. L. Wu, C. H. Teng, and H. C. Chang, “Application of pseudospectral methods to optical waveguide mode solvers,” |

7. | P. J. Chiang, C. L. Wu, C. H. Teng, C. S. Yang, and H. C. Chang, “Full-vectorial optical waveguide mode solvers using multidomain pseudospectral frequency-domain (PSFD) formulations,” IEEE J. Quantum Electron. |

8. | B. Yang, D. Gottlieb, and J. S. Hesthaven, “Spectral simulations of electromagnetic wave scattering,” J. Comput. Phys. |

9. | B. Yang and J. S. Hesthaven, “A pseudospectral method for time-domain computation of electromagnetic scattering by bodies of revolution,” IEEE Trans. Antennas Propagat. |

10. | J. S. Hesthaven, P. G. Dinesen, and J. P. Lynov, “Spectral collocation time-domain modeling of diffractive optical elements,” J. Comput. Phys. |

11. | G. Zhao and Q. H. Liu, “The 3-D multidomain pseudospectral time-domain algorithm for inhomogeneous conductive media,” IEEE Trans. Antennas Propagat. |

12. | C. H. Teng, B. Y. Lin, H. C. Chang, H. C. Hsu, C. N. Lin, and K. A. Feng, “A Legendre pseudospectral penalty scheme for solving time-domain Maxwell’s equations,” J. Sci. Comput. |

13. | B. Y. Lin, H. C. Hsu, C. H. Teng, H. C. Chang, J. K. Wang, and Y. L. Wang, “Unraveling near-field origin of electromagnetic waves scattered from silver nanorod arrays using pseudo-spectral time-domain calculation,” Opt. Express17, 14211–14228 (2009). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-16-14211. [CrossRef] [PubMed] |

14. | Q. H. Liu, “A pseudospectral frequency-domain (PSFD) method for computational electromagnetics,” IEEE Antennas Wireless Propagat. Lett. |

15. | P. J. Chiang, C. P. Yu, and H. C. Chang, “Analysis of two-dimensional photonic crystals using a multidomain pseudospectral method,” Phys. Rev. E |

16. | W. J. Gordon and C. A. Hall, “Transfinite element methods: blending-function interpolation over arbitrary curved element domains,” Numer. Math. |

17. | 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. |

18. | T. P. White, B. T. Kuhlmey, R. C. Mcphedran, D. Maystre, G. Renversez, C. M. de Sterke, and L. C. Botten, “Multi-pole method for microstructured optical fibers. I. Formulation,” J. Opt. Soc. Am. B |

19. | B. T. Kuhlmey, T. P. White, G. Renversez, D. Maystre, L. C. Botten, C. M. de Sterke, and R. C. Mcphedran, “Multi-pole method for microstructured optical fibers. II. Implementation and results,” J. Opt. Soc. Am. B |

20. | P. Russell, “Photonic crystal fibers,” Science |

21. | J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comp. Phys. |

22. | Z. S. Sacks, D. M. Kingsland, R. Lee, and J. F. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antennas Propagat. |

23. | P. J. Chiang and H. C. Chang, “Analysis of leaky optical waveguides using pseudospectral methods,” |

24. | C. P. Yu and H. C. Chang, “Yee-mesh-based finite difference eigenmode solver with PML absorbing boundary conditions for optical waveguides and photonic crystal fibers,” Opt. Express12, 6165–6177 (2004). http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-25-6165. [CrossRef] [PubMed] |

25. | 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. |

26. | P. J. Chiang and Y. C. Chiang, “Pseudospectral frequency-domain formulae based on modified perfectly matched layers for calculating both guided and leaky modes,” IEEE Photon. Technol. Lett. |

27. | W. C. Chew and W. H. Weedon, “A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates,” Microwave Opt. Technol. Lett. |

28. | K. Saitoh and M. Koshiba, “Full-vectorial finite element beam propagation method with perfectly matched layers for anisotropic optical waveguides,” J. Lightwave Technol. |

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(230.7370) Optical devices : Waveguides

(050.1755) Diffraction and gratings : Computational electromagnetic methods

(060.4005) Fiber optics and optical communications : Microstructured fibers

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: November 16, 2010

Revised Manuscript: January 7, 2011

Manuscript Accepted: January 9, 2011

Published: January 13, 2011

**Citation**

Po-jui Chiang and Hung-chun Chang, "A high-accuracy pseudospectral full-vectorial leaky optical waveguide mode solver with carefully implemented UPML absorbing boundary conditions," Opt. Express **19**, 1594-1608 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-2-1594

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### References

- G. R. Hadley, “High-accuracy finite-difference equations for dielectric waveguide analysis I: Uniform regions and dielectric interfaces,” J. Lightwave Technol. 20, 1210–1218 (2002). [CrossRef]
- G. R. Hadley, “High-accuracy finite-difference equations for dielectric waveguide analysis II: Dielectric corners,” J. Lightwave Technol. 20, 1219–1231 (2002). [CrossRef]
- N. Thomas, P. Sewell, and T. M. Benson, “A new full-vectorial higher order finite-difference scheme for the modal analysis of rectangular dielectric waveguides,” J. Lightwave Technol. 25, 2563–2570 (2002). [CrossRef]
- Y. C. Chiang, Y. P. Chiou, and H. C. Chang, “Improved full-vectorial finite-difference mode solver for optical waveguides with step-index profiles,” J. Lightwave Technol. 20, 1609–1618 (2002). [CrossRef]
- M. Koshiba, and Y. Tsuji, “Curvilinear hybrid edge/nodal elements with triangular shape for guided-wave problems,” J. Lightwave Technol. 18, 737–743 (2000). [CrossRef]
- P. J. Chiang, C. S. Yang, C. L. Wu, C. H. Teng, and H. C. Chang, “Application of pseudospectral methods to optical waveguide mode solvers,” OSA 2005 Integrated Photonics Research and Applications (IPRA ’05) Technical Digest (Optical Society of America, 2005), paper IMG4.
- P. J. Chiang, C. L. Wu, C. H. Teng, C. S. Yang, and H. C. Chang, “Full-vectorial optical waveguide mode solvers using multidomain pseudospectral frequency-domain (PSFD) formulations,” IEEE J. Quantum Electron. 44, 56–66 (2008). [CrossRef]
- B. Yang, D. Gottlieb, and J. S. Hesthaven, “Spectral simulations of electromagnetic wave scattering,” J. Comput. Phys. 134, 216–230 (1997). [CrossRef]
- B. Yang, and J. S. Hesthaven, “A pseudospectral method for time-domain computation of electromagnetic scattering by bodies of revolution,” IEEE Trans. Antenn. Propag. 47, 132–141 (1999). [CrossRef]
- J. S. Hesthaven, P. G. Dinesen, and J. P. Lynov, “Spectral collocation time-domain modeling of diffractive optical elements,” J. Comput. Phys. 155, 287–306 (1999). [CrossRef]
- G. Zhao, and Q. H. Liu, “The 3-D multidomain pseudospectral time-domain algorithm for inhomogeneous conductive media,” IEEE Trans. Antenn. Propag. 52, 742–749 (2004). [CrossRef]
- C. H. Teng, B. Y. Lin, H. C. Chang, H. C. Hsu, C. N. Lin, and K. A. Feng, “A Legendre pseudospectral penalty scheme for solving time-domain Maxwell’s equations,” J. Sci. Comput. 36, 351–390 (2008). [CrossRef]
- B. Y. Lin, H. C. Hsu, C. H. Teng, H. C. Chang, J. K. Wang, and Y. L. Wang, “Unraveling near-field origin of electromagnetic waves scattered from silver nanorod arrays using pseudo-spectral time-domain calculation,” Opt. Express 17, 14211–14228 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-16-14211. [CrossRef] [PubMed]
- Q. H. Liu, “A pseudospectral frequency-domain (PSFD) method for computational electromagnetics,” IEEE Antennas Wirel. Propag. Lett. 1, 131–134 (2002). [CrossRef]
- P. J. Chiang, C. P. Yu, and H. C. Chang, “Analysis of two-dimensional photonic crystals using a multidomain pseudospectral method,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75, 026703 (2007). [CrossRef]
- W. J. Gordon, and C. A. Hall, “Transfinite element methods: blending-function interpolation over arbitrary curved element domains,” Numer. Math. 21, 109–129 (1973). [CrossRef]
- 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). [CrossRef]
- T. P. White, B. T. Kuhlmey, R. C. Mcphedran, D. Maystre, G. Renversez, C. M. de Sterke, and L. C. Botten, “Multipole method for microstructured optical fibers. I. Formulation,” J. Opt. Soc. Am. B 19, 2322–2330 (2002). [CrossRef]
- B. T. Kuhlmey, T. P. White, G. Renversez, D. Maystre, L. C. Botten, C. M. de Sterke, and R. C. Mcphedran, “Multipole method for microstructured optical fibers. II. Implementation and results,” J. Opt. Soc. Am. B 19, 2331–2340 (2002). [CrossRef]
- P. Russell, “Photonic crystal fibers,” Science 299, 358–362 (2003). [CrossRef] [PubMed]
- J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994). [CrossRef]
- Z. S. Sacks, D. M. Kingsland, R. Lee, and J. F. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antenn. Propag. 43, 1460–1463 (1995). [CrossRef]
- P. J. Chiang, and H. C. Chang, “Analysis of leaky optical waveguides using pseudospectral methods,” OSA 2006 Integrated Photonics Research and Applications (IPRA ’06) Technical Digest (Optical Society of America, 2005), paper ITuA3.
- C. P. Yu, and H. C. Chang, “Yee-mesh-based finite difference eigenmode solver with PML absorbing boundary conditions for optical waveguides and photonic crystal fibers,” Opt. Express 12, 6165–6177 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-25-6165. [CrossRef] [PubMed]
- 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). [CrossRef]
- P. J. Chiang, and Y. C. Chiang, “Pseudospectral frequency-domain formulae based on modified perfectly matched layers for calculating both guided and leaky modes,” IEEE Photon. Technol. Lett. 22, 908–910 (2010). [CrossRef]
- W. C. Chew, and W. H. Weedon, “A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates,” Microw. Opt. Technol. Lett. 7, 599–604 (1994). [CrossRef]
- K. Saitoh, and M. Koshiba, “Full-vectorial finite element beam propagation method with perfectly matched layers for anisotropic optical waveguides,” J. Lightwave Technol. 19, 405–413 (2001). [CrossRef]

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