Abstract

The scalar wave equation, or Helmholtz equation, describes within a certain approximation the electromagnetic field distribution in a given system. In this paper we show how to solve the Helmholtz equation in complex geometries using conformal mapping and the homotopy perturbation method. The solution of the mapped Helmholtz equation is found by solving an infinite series of Poisson equations using two dimensional Fourier series. The solution is entirely based on analytical expressions and is not mesh dependent. The analytical results are compared to a numerical (finite element method) solution.

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References

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  1. R. P. Ratowsky, J. Fleck, and M. D. Feit, “Helmholtz beam propagation in rib waveguides and couplers by iterative Lanczos reduction,” J. Opt. Soc. Am. A 9, 265–273 (1992).
    [Crossref]
  2. M. Balagangadhar, T. Sarkar, I. Rejeb, and R. Boix, “Solution of the general Helmholtz equation in homogeneously filled waveguides using a static Green’s function,” IEEE Trans. Microwave Theory Tech. 46, 302–307 (1998).
    [Crossref]
  3. W. Ng and M. Stern, “Analysis of multiple-rib waveguide structures by the discrete-spectral-index method,” in Proceedings of IEEE Conference on Optoelectronics (IEEE, 1998), 365–371 (1998).
    [Crossref]
  4. C. T. Shih and S. Chao, “Simplified numerical method for analyzing TE-like modes in a three-dimensional circularly bent dielectric rib waveguide by solving two one-dimensional eigenvalue equations,” J. Opt. Soc. Am. B 25, 1031–1037 (2008).
    [Crossref]
  5. 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).
    [Crossref]
  6. 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).
    [Crossref]
  7. D. Yin, J. P. Barber, A. R. Hawkins, and H. Schmidt, “Low-loss integrated optical sensors based on hollow-core ARROW waveguide,” Proc. SPIE 5730, 218–225, (2005).
    [Crossref]
  8. D. Yin, D. W. Deamer, H. Schmidt, J. P. Barber, and A. R. Hawkins, “Integrated optical waveguides with liquid cores,” Appl. Phys. Lett. 85, 3477–3479 (2004).
    [Crossref]
  9. D. Yin, H. Schmidt, J. P. Barber, E. J. Lunt, and A. R. Hawkins, “Optical characterization of arch-shaped ARROW waveguides with liquid cores,” Opt. Express 13, 10564–10570 (2005).
    [Crossref] [PubMed]
  10. A. M. Young, C. L. Xu, W. Huang, and S. D. Senturia, “Design and analysis of an ARROW-waveguide-based silicon pressure transducer,” Proc. SPIE 1793, 42–53 (1993).
    [Crossref]
  11. K. J. Rowland, S. V. Afshar, and T. M. Monro, “Bandgaps and antiresonances in integrated-ARROWs and Bragg fibers; a simple model,” Opt. Express 16, 17935–17951 (2008).
    [Crossref] [PubMed]
  12. J.-L. Archambault, R. J. Black, S. Lacroix, and J. Bures, “Loss calculations for antiresonant waveguides,” J. Lightwave Technol. 11, 416–423 (1993).
    [Crossref]
  13. W. J. Gibbs, Conformal Transformations in Electrical Engineering (Chapman & Hall, 1958).
  14. R. Schinzinger and P. A. A. Laura, Conformal Mapping: Methods and Applications (Elsevier, 1991).
  15. C. Lee, M. Wu, and J. Hsu, “Beam propagation analysis for tapered waveguides: taking account of the curved phase-front effect in paraxial approximation,” J. Lightwave Technol. 15, 2183–2189 (1997).
    [Crossref]
  16. S. Liao, “An approximate solution technique not depending on small parameters: a special example,” Int. J. Non-Linear Mech. 30, 371–380 (1995).
    [Crossref]
  17. J. He, “Homotopy perturbation technique,” Comput. Methods Appl. Mech. Eng. 178, 257–262 (1999).
    [Crossref]
  18. I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products, Corrected and Enlarged Edition (Academic, 1980).
  19. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1972).
  20. M. Hazewinkel, ed., Encyclopaedia of Mathematics, Springer online Reference Works, http://eom.springer.de/default.htm .
  21. NIST Digital Library of Mathematical Functions, http://dlmf.nist.gov/ .

2008 (2)

2005 (2)

D. Yin, H. Schmidt, J. P. Barber, E. J. Lunt, and A. R. Hawkins, “Optical characterization of arch-shaped ARROW waveguides with liquid cores,” Opt. Express 13, 10564–10570 (2005).
[Crossref] [PubMed]

D. Yin, J. P. Barber, A. R. Hawkins, and H. Schmidt, “Low-loss integrated optical sensors based on hollow-core ARROW waveguide,” Proc. SPIE 5730, 218–225, (2005).
[Crossref]

2004 (1)

D. Yin, D. W. Deamer, H. Schmidt, J. P. Barber, and A. R. Hawkins, “Integrated optical waveguides with liquid cores,” Appl. Phys. Lett. 85, 3477–3479 (2004).
[Crossref]

1999 (1)

J. He, “Homotopy perturbation technique,” Comput. Methods Appl. Mech. Eng. 178, 257–262 (1999).
[Crossref]

1998 (1)

M. Balagangadhar, T. Sarkar, I. Rejeb, and R. Boix, “Solution of the general Helmholtz equation in homogeneously filled waveguides using a static Green’s function,” IEEE Trans. Microwave Theory Tech. 46, 302–307 (1998).
[Crossref]

1997 (1)

C. Lee, M. Wu, and J. Hsu, “Beam propagation analysis for tapered waveguides: taking account of the curved phase-front effect in paraxial approximation,” J. Lightwave Technol. 15, 2183–2189 (1997).
[Crossref]

1995 (1)

S. Liao, “An approximate solution technique not depending on small parameters: a special example,” Int. J. Non-Linear Mech. 30, 371–380 (1995).
[Crossref]

1993 (2)

A. M. Young, C. L. Xu, W. Huang, and S. D. Senturia, “Design and analysis of an ARROW-waveguide-based silicon pressure transducer,” Proc. SPIE 1793, 42–53 (1993).
[Crossref]

J.-L. Archambault, R. J. Black, S. Lacroix, and J. Bures, “Loss calculations for antiresonant waveguides,” J. Lightwave Technol. 11, 416–423 (1993).
[Crossref]

1992 (2)

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).
[Crossref]

R. P. Ratowsky, J. Fleck, and M. D. Feit, “Helmholtz beam propagation in rib waveguides and couplers by iterative Lanczos reduction,” J. Opt. Soc. Am. A 9, 265–273 (1992).
[Crossref]

1986 (1)

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).
[Crossref]

1980 (1)

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products, Corrected and Enlarged Edition (Academic, 1980).

1972 (1)

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1972).

Afshar, S. V.

Archambault, J.-L.

J.-L. Archambault, R. J. Black, S. Lacroix, and J. Bures, “Loss calculations for antiresonant waveguides,” J. Lightwave Technol. 11, 416–423 (1993).
[Crossref]

Baba, T.

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).
[Crossref]

Balagangadhar, M.

M. Balagangadhar, T. Sarkar, I. Rejeb, and R. Boix, “Solution of the general Helmholtz equation in homogeneously filled waveguides using a static Green’s function,” IEEE Trans. Microwave Theory Tech. 46, 302–307 (1998).
[Crossref]

Barber, J. P.

D. Yin, H. Schmidt, J. P. Barber, E. J. Lunt, and A. R. Hawkins, “Optical characterization of arch-shaped ARROW waveguides with liquid cores,” Opt. Express 13, 10564–10570 (2005).
[Crossref] [PubMed]

D. Yin, J. P. Barber, A. R. Hawkins, and H. Schmidt, “Low-loss integrated optical sensors based on hollow-core ARROW waveguide,” Proc. SPIE 5730, 218–225, (2005).
[Crossref]

D. Yin, D. W. Deamer, H. Schmidt, J. P. Barber, and A. R. Hawkins, “Integrated optical waveguides with liquid cores,” Appl. Phys. Lett. 85, 3477–3479 (2004).
[Crossref]

Black, R. J.

J.-L. Archambault, R. J. Black, S. Lacroix, and J. Bures, “Loss calculations for antiresonant waveguides,” J. Lightwave Technol. 11, 416–423 (1993).
[Crossref]

Boix, R.

M. Balagangadhar, T. Sarkar, I. Rejeb, and R. Boix, “Solution of the general Helmholtz equation in homogeneously filled waveguides using a static Green’s function,” IEEE Trans. Microwave Theory Tech. 46, 302–307 (1998).
[Crossref]

Bures, J.

J.-L. Archambault, R. J. Black, S. Lacroix, and J. Bures, “Loss calculations for antiresonant waveguides,” J. Lightwave Technol. 11, 416–423 (1993).
[Crossref]

Chao, S.

Deamer, D. W.

D. Yin, D. W. Deamer, H. Schmidt, J. P. Barber, and A. R. Hawkins, “Integrated optical waveguides with liquid cores,” Appl. Phys. Lett. 85, 3477–3479 (2004).
[Crossref]

Duguay, M. A.

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).
[Crossref]

Feit, M. D.

Fleck, J.

Gibbs, W. J.

W. J. Gibbs, Conformal Transformations in Electrical Engineering (Chapman & Hall, 1958).

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products, Corrected and Enlarged Edition (Academic, 1980).

Hawkins, A. R.

D. Yin, J. P. Barber, A. R. Hawkins, and H. Schmidt, “Low-loss integrated optical sensors based on hollow-core ARROW waveguide,” Proc. SPIE 5730, 218–225, (2005).
[Crossref]

D. Yin, H. Schmidt, J. P. Barber, E. J. Lunt, and A. R. Hawkins, “Optical characterization of arch-shaped ARROW waveguides with liquid cores,” Opt. Express 13, 10564–10570 (2005).
[Crossref] [PubMed]

D. Yin, D. W. Deamer, H. Schmidt, J. P. Barber, and A. R. Hawkins, “Integrated optical waveguides with liquid cores,” Appl. Phys. Lett. 85, 3477–3479 (2004).
[Crossref]

He, J.

J. He, “Homotopy perturbation technique,” Comput. Methods Appl. Mech. Eng. 178, 257–262 (1999).
[Crossref]

Hsu, J.

C. Lee, M. Wu, and J. Hsu, “Beam propagation analysis for tapered waveguides: taking account of the curved phase-front effect in paraxial approximation,” J. Lightwave Technol. 15, 2183–2189 (1997).
[Crossref]

Huang, W.

A. M. Young, C. L. Xu, W. Huang, and S. D. Senturia, “Design and analysis of an ARROW-waveguide-based silicon pressure transducer,” Proc. SPIE 1793, 42–53 (1993).
[Crossref]

Koch, T. L.

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).
[Crossref]

Kokubun, Y.

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).
[Crossref]

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).
[Crossref]

Lacroix, S.

J.-L. Archambault, R. J. Black, S. Lacroix, and J. Bures, “Loss calculations for antiresonant waveguides,” J. Lightwave Technol. 11, 416–423 (1993).
[Crossref]

Laura, P. A. A.

R. Schinzinger and P. A. A. Laura, Conformal Mapping: Methods and Applications (Elsevier, 1991).

Lee, C.

C. Lee, M. Wu, and J. Hsu, “Beam propagation analysis for tapered waveguides: taking account of the curved phase-front effect in paraxial approximation,” J. Lightwave Technol. 15, 2183–2189 (1997).
[Crossref]

Liao, S.

S. Liao, “An approximate solution technique not depending on small parameters: a special example,” Int. J. Non-Linear Mech. 30, 371–380 (1995).
[Crossref]

Lunt, E. J.

Monro, T. M.

Ng, W.

W. Ng and M. Stern, “Analysis of multiple-rib waveguide structures by the discrete-spectral-index method,” in Proceedings of IEEE Conference on Optoelectronics (IEEE, 1998), 365–371 (1998).
[Crossref]

Pfeiffer, L.

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).
[Crossref]

Ratowsky, R. P.

Rejeb, I.

M. Balagangadhar, T. Sarkar, I. Rejeb, and R. Boix, “Solution of the general Helmholtz equation in homogeneously filled waveguides using a static Green’s function,” IEEE Trans. Microwave Theory Tech. 46, 302–307 (1998).
[Crossref]

Rowland, K. J.

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products, Corrected and Enlarged Edition (Academic, 1980).

Sarkar, T.

M. Balagangadhar, T. Sarkar, I. Rejeb, and R. Boix, “Solution of the general Helmholtz equation in homogeneously filled waveguides using a static Green’s function,” IEEE Trans. Microwave Theory Tech. 46, 302–307 (1998).
[Crossref]

Schinzinger, R.

R. Schinzinger and P. A. A. Laura, Conformal Mapping: Methods and Applications (Elsevier, 1991).

Schmidt, H.

D. Yin, J. P. Barber, A. R. Hawkins, and H. Schmidt, “Low-loss integrated optical sensors based on hollow-core ARROW waveguide,” Proc. SPIE 5730, 218–225, (2005).
[Crossref]

D. Yin, H. Schmidt, J. P. Barber, E. J. Lunt, and A. R. Hawkins, “Optical characterization of arch-shaped ARROW waveguides with liquid cores,” Opt. Express 13, 10564–10570 (2005).
[Crossref] [PubMed]

D. Yin, D. W. Deamer, H. Schmidt, J. P. Barber, and A. R. Hawkins, “Integrated optical waveguides with liquid cores,” Appl. Phys. Lett. 85, 3477–3479 (2004).
[Crossref]

Senturia, S. D.

A. M. Young, C. L. Xu, W. Huang, and S. D. Senturia, “Design and analysis of an ARROW-waveguide-based silicon pressure transducer,” Proc. SPIE 1793, 42–53 (1993).
[Crossref]

Shih, C. T.

Stern, M.

W. Ng and M. Stern, “Analysis of multiple-rib waveguide structures by the discrete-spectral-index method,” in Proceedings of IEEE Conference on Optoelectronics (IEEE, 1998), 365–371 (1998).
[Crossref]

Wu, M.

C. Lee, M. Wu, and J. Hsu, “Beam propagation analysis for tapered waveguides: taking account of the curved phase-front effect in paraxial approximation,” J. Lightwave Technol. 15, 2183–2189 (1997).
[Crossref]

Xu, C. L.

A. M. Young, C. L. Xu, W. Huang, and S. D. Senturia, “Design and analysis of an ARROW-waveguide-based silicon pressure transducer,” Proc. SPIE 1793, 42–53 (1993).
[Crossref]

Yin, D.

D. Yin, J. P. Barber, A. R. Hawkins, and H. Schmidt, “Low-loss integrated optical sensors based on hollow-core ARROW waveguide,” Proc. SPIE 5730, 218–225, (2005).
[Crossref]

D. Yin, H. Schmidt, J. P. Barber, E. J. Lunt, and A. R. Hawkins, “Optical characterization of arch-shaped ARROW waveguides with liquid cores,” Opt. Express 13, 10564–10570 (2005).
[Crossref] [PubMed]

D. Yin, D. W. Deamer, H. Schmidt, J. P. Barber, and A. R. Hawkins, “Integrated optical waveguides with liquid cores,” Appl. Phys. Lett. 85, 3477–3479 (2004).
[Crossref]

Young, A. M.

A. M. Young, C. L. Xu, W. Huang, and S. D. Senturia, “Design and analysis of an ARROW-waveguide-based silicon pressure transducer,” Proc. SPIE 1793, 42–53 (1993).
[Crossref]

Appl. Phys. Lett. (2)

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).
[Crossref]

D. Yin, D. W. Deamer, H. Schmidt, J. P. Barber, and A. R. Hawkins, “Integrated optical waveguides with liquid cores,” Appl. Phys. Lett. 85, 3477–3479 (2004).
[Crossref]

Comput. Methods Appl. Mech. Eng. (1)

J. He, “Homotopy perturbation technique,” Comput. Methods Appl. Mech. Eng. 178, 257–262 (1999).
[Crossref]

Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (1)

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1972).

IEEE J. Quantum Electron. (1)

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).
[Crossref]

IEEE Trans. Microwave Theory Tech. (1)

M. Balagangadhar, T. Sarkar, I. Rejeb, and R. Boix, “Solution of the general Helmholtz equation in homogeneously filled waveguides using a static Green’s function,” IEEE Trans. Microwave Theory Tech. 46, 302–307 (1998).
[Crossref]

Int. J. Non-Linear Mech. (1)

S. Liao, “An approximate solution technique not depending on small parameters: a special example,” Int. J. Non-Linear Mech. 30, 371–380 (1995).
[Crossref]

J. Lightwave Technol. (2)

J.-L. Archambault, R. J. Black, S. Lacroix, and J. Bures, “Loss calculations for antiresonant waveguides,” J. Lightwave Technol. 11, 416–423 (1993).
[Crossref]

C. Lee, M. Wu, and J. Hsu, “Beam propagation analysis for tapered waveguides: taking account of the curved phase-front effect in paraxial approximation,” J. Lightwave Technol. 15, 2183–2189 (1997).
[Crossref]

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (1)

Opt. Express (2)

Proc. SPIE (2)

A. M. Young, C. L. Xu, W. Huang, and S. D. Senturia, “Design and analysis of an ARROW-waveguide-based silicon pressure transducer,” Proc. SPIE 1793, 42–53 (1993).
[Crossref]

D. Yin, J. P. Barber, A. R. Hawkins, and H. Schmidt, “Low-loss integrated optical sensors based on hollow-core ARROW waveguide,” Proc. SPIE 5730, 218–225, (2005).
[Crossref]

Tables of Integrals, Series, and Products, Corrected and Enlarged Edition (1)

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products, Corrected and Enlarged Edition (Academic, 1980).

Other (5)

M. Hazewinkel, ed., Encyclopaedia of Mathematics, Springer online Reference Works, http://eom.springer.de/default.htm .

NIST Digital Library of Mathematical Functions, http://dlmf.nist.gov/ .

W. Ng and M. Stern, “Analysis of multiple-rib waveguide structures by the discrete-spectral-index method,” in Proceedings of IEEE Conference on Optoelectronics (IEEE, 1998), 365–371 (1998).
[Crossref]

W. J. Gibbs, Conformal Transformations in Electrical Engineering (Chapman & Hall, 1958).

R. Schinzinger and P. A. A. Laura, Conformal Mapping: Methods and Applications (Elsevier, 1991).

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Figures (8)

Fig. 1
Fig. 1

Sketch of the rib ARROW waveguide cross-section in the z-plane and the upper half-plane w to which the rib ARROW structure is mapped conformally. Below, the square in the χ-plane to which the upper half-plane w is mapped in a second conformal mapping step.

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Fig. 2
Fig. 2

The Jacobian of the square to rib waveguide conformal mapping. The vertical axis of the plot has been truncated at 15.

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Fig. 3
Fig. 3

Eigenvalue λ2 for the second mode in a rib waveguide with aspect ratios t/s = 2 and g/s = 1 as a function of the HPM order with the number of Fourier terms as parameter. Calculations are shown for 2, 4, 6 and 12 Fourier terms and compared to the result from FEM.

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Fig. 4
Fig. 4

The initial guess and first three solutions to the homotopy equations (Equation 21). The effect of the two peaks of the Jacobian close to the real axis is clearly seen in the HPM solutions.

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Fig. 5
Fig. 5

Sixth order homotopy solution in model domain (a), physical domain using HPM (b) and using FEM (c) for t/s = 1 and g/s = 1.

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Fig. 6
Fig. 6

Sixth order homotopy solution in model domain (a), physical domain using HPM (b) and using FEM (c) for t/s = 2 and g/s = 1.

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Fig. 7
Fig. 7

Second eigenfunction for sixth order homotopy solution in model domain (a), physical domain using HPM (b) and using FEM (c) for t/s = 1 and g/s = 1.

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Fig. 8
Fig. 8

The eigenvalue λ2 and the HPM field amplitude ψ(χref) for increasing number of homotopy terms, where χref is a specific reference point in the square domain (here χref = 1/2 + i3/5). The changes in eigenvalue and field amplitude are plotted on the right axis.

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Tables (2)

Tables Icon

Table 1 Eigenvalues for 6th order HPM λ HPM 2 compared to eigenvalues from FEM λ FEM 2 for two different rib waveguides and the two first modes. The 2nd mode eigen-values are based on a higher order initial guess. The relative L2 norm deviations ɛ of 6th order HPM solutions from the FEM solutions are also listed. In the calculations 12 Fourier terms were used.

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Tables Icon

Table 2 Comparison of the performance (memory used and CPU time) of FEM and conformal mapping/homotopy (HPM) solutions.

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Equations (50)

Equations on this page are rendered with MathJax. Learn more.

∇ 2 ϕ ( r ) + ( n r 2 k 0 2 − β 2 ) ϕ ( r ) = 0 ,
∂ 2 ϕ ( x , y ) ∂ x 2 + ∂ 2 ϕ ( x , y ) ∂ y 2 + λ 2 ϕ ( x , y ) = 0 ,
ϕ ( Γ ) = 0 ,
z ( w ) = A ∫ ∏ j = 1 n ( w − a j ) ( α j / π ) − 1 d w ,
χ ( w ) − χ ( 0 ) = C sq ∫ d w 1 − w 2 1 − k r 2 w 2 = 1 2 K ( k r ) ∫ 0 w d ϑ 1 − ϑ 2 1 − k r 2 ϑ 2 = arcsn ( w , k r ) 2 K ( k r ) ,
w ( χ ) = sn ( [ 2 χ − 1 ] K ( k r ) , k r ) .
z ( w ) = C ∫ 1 − k 2 w 2 ( 1 − k 1 2 w 2 ) 1 − w 2 d w = s k 1 π 1 − k 1 2 k 2 − k 1 2 ∫ 0 w 1 − k 2 ϑ 2 ( 1 − k 1 2 ϑ 2 ) 1 − ϑ 2 d ϑ ,
z ( ζ ) = s π sn ( α ) dn ( α ) cn α ∫ 0 ζ 1 − k 2 sn 2 τ 1 − k 1 2 sn 2 τ d τ = s π ( sn ( α ) dn ( α ) cn α ζ − Π J ( ζ , α ) ) ,
g s = K ( k ) π ( sn ( α ) dn ( α ) cn α − Z ( α , k ) )
t s = 2 K ′ ( k ) π ( sn ( α ) dn ( α ) cn α − Z ( α , k ) ) − α K ( k ) ,
∇ x , y 2 ϕ ( x , y ) = ∇ υ , ω 2 ψ ( υ , ω ) | d χ d z | 2 ,
∂ 2 ψ ( υ , ω ) ∂ υ 2 + ∂ 2 ψ ( υ , ω ) ∂ ω 2 + | d χ d z | − 2 λ 2 ψ ( υ , ω ) = 0 .
d χ d z = d χ d w d w d z = 1 2 K ( k r ) π s cn α sn ( α ) dn ( α ) 1 − k 1 2 w 2 1 − w 2 k r 2 1 − k 2 w 2 ,
𝒜 ( u ) − g ( r ) = 0 , r ∈ Ω ,
𝒝 ( u , ∂ u ∂ n ) = 0 , r ∈ Γ ,
ℒ ( u ) + 𝒩 ( u ) − g ( r ) = 0 .
ℋ ( v , p ) = ( 1 − p ) { ℒ ( v ) − ℒ ( u 0 ) } + p { A ( v ) − g ( r ) } = 0 ,
v = ∑ n = 0 ∞ v n p n .
u = lim p → 1 v = ∑ n = 0 ∞ v n .
ℋ = ( 1 − p ) ( ∂ 2 ψ ∂ υ 2 + ∂ 2 ψ ∂ ω 2 − ∂ 2 ψ 0 ∂ υ 2 − ∂ 2 ψ 0 ∂ ω 2 ) + p ( ∂ 2 ψ ∂ υ 2 + ∂ 2 ψ ∂ ω 2 + | d χ d z | − 2 λ 2 ψ ) = 0 ,
p 0 : ψ 0 p 1 : ∂ 2 ψ 0 ∂ υ 2 + ∂ 2 ψ 0 ∂ ω 2 + | d χ d z | − 2 λ 2 ψ 0 + ∂ 2 ψ 1 ∂ υ 2 + ∂ 2 ψ 1 ∂ ω 2 = 0 p 2 : | d χ d z | − 2 λ 2 ψ 1 + ∂ 2 ψ 2 ∂ υ 2 + ∂ 2 ψ 2 ∂ ω 2 = 0 p 3 : | d χ d z | − 2 λ 2 ψ 2 + ∂ 2 ψ 3 ∂ υ 2 + ∂ 2 ψ 3 ∂ ω 2 = 0 ⋮ p n : | d χ d z | − 2 λ 2 ψ n − 1 + ∂ 2 ψ n ∂ υ 2 + ∂ 2 ψ n ∂ ω 2 = 0
ψ = ∑ n = 0 ∞ ψ n p n
∇ υ , ω 2 ψ n ( υ , ω ) = h n ( υ , ω ) ,
ψ n ( υ , ω ) = ∑ j = 1 ∞ ∑ m = 1 ∞ E m j sin ( m π a υ ) sin ( j π b ω ) ,
∫ 0 a sin ( m π υ a ) sin ( q π υ a ) d υ = a 2 δ m q
∫ 0 b sin ( j π ω b ) sin ( r π ω b ) d ω = b 2 δ j r .
E m j = − 4 a b κ m j ∫ 0 b ∫ 0 a h n ( υ , ω ) sin ( m π a υ ) sin ( j π b ω ) d υ d ω
κ m j = ( m π a ) 2 + ( j π b ) 2 ,   m , j ∈ [ 1 , 2 , … ] .
ψ 0 ( υ , ω ) = sin ( π a υ ) sin ( π b ω ) .
ψ 1 ( υ , ω ) = ∑ j = 1 ∞ ∑ m = 1 ∞ 4 sin ( m π υ ) sin ( j π ω ) κ m j × ∫ 0 1 ∫ 0 1 ( | d χ d z | − 2 λ 2 − 2 π 2 ) sin ( π υ ) sin ( π ω ) sin ( m π υ ) sin ( j π ω ) d υ d ω .
ψ n ( υ , ω ) = ∑ j = 1 ∞ ∑ m = 1 ∞ 4 sin ( m π υ ) sin ( j π ω ) κ m j × ∫ 0 1 ∫ 0 1 ( | d χ d z | − 2 λ 2 ψ n − 1 ( υ , ω ) ) sin ( m π υ ) sin ( j π ω ) d υ d ω ,
λ 2 = − ∫ ψ ∇ ν , ω 2 ψ d Ω ∫ | d χ d z | − 2 ψ 2 d Ω ,
ɛ = ‖ ψ HPM − ψ FEM ‖ L 2 ‖ ψ FEM ‖ L 2 = [ ∫ ( ψ HPM − ψ FEM ) 2 d Ω ∫ ψ FEM 2 d Ω ] 1 2 .
u = arcsn ( z , k ) = ∫ 0 z d t ( 1 − t 2 ) ( 1 − k 2 t 2 ) = F ( ϕ , k ) ,   with   z = sin ϕ
am ( u , k ) = am ( u ) = ϕ ,
sn ( u , k ) = sn ( u ) = z = sin ϕ = sin ( am ( u ) ) ,
cn ( u , k ) = cn ( u ) = cos ( am ( u ) ) ,
dn ( u , k ) = dn ( u ) = ∂ am ( u ) ∂ u .
sn 2 ( u ) + cn 2 ( u ) = 1 ,   and k 2 sn 2 ( u ) + dn 2 ( u ) = 1 .
F ( ϕ , k ) = ∫ 0 ϕ d θ 1 − k 2 sin 2 θ = ∫ 0 sin ϕ d t ( 1 − t 2 ) ( 1 − k 2 t 2 ) ,
F 1 ( z , k ) = ∫ 0 z d t ( 1 − t 2 ) ( 1 − k 2 t 2 ) ,   with   z = sin ϕ ,
K = K ( k ) = ∫ 0 π / 2 d θ 1 − k 2 sin 2 θ = ∫ 0 1 d t ( 1 − t 2 ) ( 1 − k 2 t 2 ) ,
K ′ = K ′ ( k ) = K ( k ′ ) = K ( 1 − k 2 ) .
E ( ϕ , k ) = ∫ 0 ϕ 1 − k 2 sin 2 θ d θ = ∫ 0 sin ϕ 1 − k 2 t 2 1 − t 2 d t
E 1 ( z , k ) = ∫ 0 z 1 − k 2 t 2 1 − t 2 d t ,   with   z = sin ϕ ,
E = E ( k ) = E ( π 2 , k ) = ∫ 0 π / 2 1 − k 2 sin 2 θ d θ = ∫ 0 1 1 − k 2 t 2 1 − t 2 d t .
Π J ( z , α , k ) = k 2 sn ( α , k ) cn ( α , k ) dn ( α , k ) ∫ 0 z sn 2 ( u , k ) 1 − k 2 sn 2 ( α , k ) sn 2 ( u , k ) d u
Π ( z , α , k ) = ∫ 0 z d u 1 − k 2 sn 2 ( α , k ) sn 2 ( u , k ) = ∫ 0 z d t ( 1 − k 2 sn 2 ( α , k ) t 2 ) 1 − t 2 1 − k 2 t 2 .
Π ( z , α , k ) = z + sn ( α , k ) cn ( α , k ) dn ( α , k ) Π J ( z , α , k ) .
Z ( u , k ) = Z ( u ) = E 1 ( u , k ) − F 1 ( u , k ) E ( k ) K ( k ) .

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