Abstract

The reflection coefficient is one important parameter of the perfectly matched layer (PML). Here we investigate its effect on the modal analysis of leaky waveguide modes by examining three different leaky waveguide structures, i.e., the holey fiber, the air-core terahertz pipe waveguide, and the gain-guided and index-antiguided slab waveguide. Numerical results reveal that the typical values 10−8 ~10−12 are inadequate for obtaining the imaginary part of the complex propagation constant, and the suggested reflection coefficient would be much smaller, for example, 10−50 or 10−100. With such a small coefficient, both the computational window size and the PML thickness can be significantly reduced without loss of stability. Moreover, in some cases, the modal field profiles can only be accurately obtained with such a small coefficient.

© 2011 OSA

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  1. G. R. Hadley, “Transparent boundary condition for beam propagation,” Opt. Lett. 16(9), 624–626 (1991).
    [CrossRef] [PubMed]
  2. J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114(2), 185–200 (1994).
    [CrossRef]
  3. W. C. Chew and W. H. Weedon, “A 3-D perfectly matched medium from modified Maxwell’s equation with stretched coordinates,” Microw. Opt. Technol. Lett. 7(13), 599–604 (1994).
    [CrossRef]
  4. C. M. Rappaport, ““Perfectly matched absorbing boundary conditions based on anisotropic lossy mapping of space,”IEEE Microw. Guid. Wave Lett. 5(3), 90–92 (1995).
    [CrossRef]
  5. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).
  6. W. P. Huang, C. L. Xu, W. Lui, and K. Yokoyama, “The perfectly matched layer boundary condition for modal analysis of optical waveguides: leaky mode calculations,” IEEE Photon. Technol. Lett. 8(5), 652–654 (1996).
    [CrossRef]
  7. 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(4), 618–623 (2000).
    [CrossRef]
  8. 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(25), 6165–6177 (2004), http://www.opticsinfobase.org/oe/abstract.cfm?URI=OPEX-12-25-6165 .
    [CrossRef] [PubMed]
  9. W.-P. Huang and J. Mu, “Complex coupled-mode theory for optical waveguides,” Opt. Express 17(21), 19134–19152 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=OPEX-17-21-19134 .
    [CrossRef]
  10. P. L. Ho and Y. Y. Lu, “A mode-preserving perfectly matched layer for optical waveguides,” IEEE Photon. Technol. Lett. 15(9), 1234–1236 (2003).
    [CrossRef]
  11. B. Chen, D. G. Fang, and B. H. Zhou, “Modified Berenger PML absorbing boundary condition for FD-TD meshes,” IEEE Microw. Guid. Wave Lett. 5(11), 399–401 (1995).
    [CrossRef]
  12. 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(12), 908–910 (2010).
    [CrossRef]
  13. 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(10), 2322–2330 (2002).
    [CrossRef]
  14. C.-H. Lai, Y.-C. Hsueh, H.-W. Chen, Y.-J. Huang, H.-C. Chang, and C.-K. Sun, “Low-index terahertz pipe waveguides,” Opt. Lett. 34(21), 3457–3459 (2009).
    [CrossRef] [PubMed]
  15. T.-H. Her, X. Ao, and L. W. Casperson, “Gain saturation in gain-guided slab waveguides with large-index antiguiding,” Opt. Lett. 34(16), 2411–2413 (2009).
    [CrossRef] [PubMed]
  16. H. Cheng, W. Y. Crutchfield, M. Doery, and L. Greengard, “Fast, accurate integral equation methods for the analysis of photonic crystal fibers I: Theory,” Opt. Express 12(16), 3791–3805 (2004), http://www.opticsinfobase.org/oe/abstract.cfm?URI=OPEX-12-16-3791 .
    [CrossRef] [PubMed]
  17. P. Bienstman, S. Selleri, L. Rosa, H. P. Uranus, W. C. L. Hopman, R. Costa, A. Melloni, L. C. Andreani, J. P. Hugonin, P. Lalanne, D. Pinto, S. S. A. Obayya, M. Dems, and K. Panajotov, “Modelling leaky photonic wires: a mode solver comparison,” Opt. Quantum Electron. 38(9−11), 731–759 (2006).
    [CrossRef]
  18. M. A. Duguay, Y. Kokubun, T. L. Koch, and L. Pfeiffer, “Antiresonant reflecting optical waveguides in SiO2–Si multilayer structures,” Appl. Phys. Lett. 49(1), 13–15 (1986).
    [CrossRef]
  19. C.-H. Lai, B. You, J.-Y. Lu, T.-A. Liu, J.-L. Peng, C.-K. Sun, and H.-C. Chang, “Modal characteristics of antiresonant reflecting pipe waveguides for terahertz waveguiding,” Opt. Express 18(1), 309–322 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?URI=OPEX-18-1-309 .
    [CrossRef] [PubMed]
  20. A. E. Siegman, “Propagating modes in gain-guided optical fibers,” J. Opt. Soc. Am. A 20(8), 1617–1628 (2003).
    [CrossRef]

2010 (2)

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(12), 908–910 (2010).
[CrossRef]

C.-H. Lai, B. You, J.-Y. Lu, T.-A. Liu, J.-L. Peng, C.-K. Sun, and H.-C. Chang, “Modal characteristics of antiresonant reflecting pipe waveguides for terahertz waveguiding,” Opt. Express 18(1), 309–322 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?URI=OPEX-18-1-309 .
[CrossRef] [PubMed]

2009 (3)

C.-H. Lai, Y.-C. Hsueh, H.-W. Chen, Y.-J. Huang, H.-C. Chang, and C.-K. Sun, “Low-index terahertz pipe waveguides,” Opt. Lett. 34(21), 3457–3459 (2009).
[CrossRef] [PubMed]

T.-H. Her, X. Ao, and L. W. Casperson, “Gain saturation in gain-guided slab waveguides with large-index antiguiding,” Opt. Lett. 34(16), 2411–2413 (2009).
[CrossRef] [PubMed]

W.-P. Huang and J. Mu, “Complex coupled-mode theory for optical waveguides,” Opt. Express 17(21), 19134–19152 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=OPEX-17-21-19134 .
[CrossRef]

2006 (1)

P. Bienstman, S. Selleri, L. Rosa, H. P. Uranus, W. C. L. Hopman, R. Costa, A. Melloni, L. C. Andreani, J. P. Hugonin, P. Lalanne, D. Pinto, S. S. A. Obayya, M. Dems, and K. Panajotov, “Modelling leaky photonic wires: a mode solver comparison,” Opt. Quantum Electron. 38(9−11), 731–759 (2006).
[CrossRef]

2004 (2)

H. Cheng, W. Y. Crutchfield, M. Doery, and L. Greengard, “Fast, accurate integral equation methods for the analysis of photonic crystal fibers I: Theory,” Opt. Express 12(16), 3791–3805 (2004), http://www.opticsinfobase.org/oe/abstract.cfm?URI=OPEX-12-16-3791 .
[CrossRef] [PubMed]

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(25), 6165–6177 (2004), http://www.opticsinfobase.org/oe/abstract.cfm?URI=OPEX-12-25-6165 .
[CrossRef] [PubMed]

2003 (2)

A. E. Siegman, “Propagating modes in gain-guided optical fibers,” J. Opt. Soc. Am. A 20(8), 1617–1628 (2003).
[CrossRef]

P. L. Ho and Y. Y. Lu, “A mode-preserving perfectly matched layer for optical waveguides,” IEEE Photon. Technol. Lett. 15(9), 1234–1236 (2003).
[CrossRef]

2002 (1)

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(10), 2322–2330 (2002).
[CrossRef]

2000 (1)

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(4), 618–623 (2000).
[CrossRef]

1996 (1)

W. P. Huang, C. L. Xu, W. Lui, and K. Yokoyama, “The perfectly matched layer boundary condition for modal analysis of optical waveguides: leaky mode calculations,” IEEE Photon. Technol. Lett. 8(5), 652–654 (1996).
[CrossRef]

1995 (2)

B. Chen, D. G. Fang, and B. H. Zhou, “Modified Berenger PML absorbing boundary condition for FD-TD meshes,” IEEE Microw. Guid. Wave Lett. 5(11), 399–401 (1995).
[CrossRef]

C. M. Rappaport, ““Perfectly matched absorbing boundary conditions based on anisotropic lossy mapping of space,”IEEE Microw. Guid. Wave Lett. 5(3), 90–92 (1995).
[CrossRef]

1994 (2)

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114(2), 185–200 (1994).
[CrossRef]

W. C. Chew and W. H. Weedon, “A 3-D perfectly matched medium from modified Maxwell’s equation with stretched coordinates,” Microw. Opt. Technol. Lett. 7(13), 599–604 (1994).
[CrossRef]

1991 (1)

G. R. Hadley, “Transparent boundary condition for beam propagation,” Opt. Lett. 16(9), 624–626 (1991).
[CrossRef] [PubMed]

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(1), 13–15 (1986).
[CrossRef]

Andreani, L. C.

P. Bienstman, S. Selleri, L. Rosa, H. P. Uranus, W. C. L. Hopman, R. Costa, A. Melloni, L. C. Andreani, J. P. Hugonin, P. Lalanne, D. Pinto, S. S. A. Obayya, M. Dems, and K. Panajotov, “Modelling leaky photonic wires: a mode solver comparison,” Opt. Quantum Electron. 38(9−11), 731–759 (2006).
[CrossRef]

Ao, X.

T.-H. Her, X. Ao, and L. W. Casperson, “Gain saturation in gain-guided slab waveguides with large-index antiguiding,” Opt. Lett. 34(16), 2411–2413 (2009).
[CrossRef] [PubMed]

Berenger, J. P.

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114(2), 185–200 (1994).
[CrossRef]

Bienstman, P.

P. Bienstman, S. Selleri, L. Rosa, H. P. Uranus, W. C. L. Hopman, R. Costa, A. Melloni, L. C. Andreani, J. P. Hugonin, P. Lalanne, D. Pinto, S. S. A. Obayya, M. Dems, and K. Panajotov, “Modelling leaky photonic wires: a mode solver comparison,” Opt. Quantum Electron. 38(9−11), 731–759 (2006).
[CrossRef]

Botten, L. C.

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(10), 2322–2330 (2002).
[CrossRef]

Casperson, L. W.

T.-H. Her, X. Ao, and L. W. Casperson, “Gain saturation in gain-guided slab waveguides with large-index antiguiding,” Opt. Lett. 34(16), 2411–2413 (2009).
[CrossRef] [PubMed]

Chang, H.-C.

C.-H. Lai, B. You, J.-Y. Lu, T.-A. Liu, J.-L. Peng, C.-K. Sun, and H.-C. Chang, “Modal characteristics of antiresonant reflecting pipe waveguides for terahertz waveguiding,” Opt. Express 18(1), 309–322 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?URI=OPEX-18-1-309 .
[CrossRef] [PubMed]

C.-H. Lai, Y.-C. Hsueh, H.-W. Chen, Y.-J. Huang, H.-C. Chang, and C.-K. Sun, “Low-index terahertz pipe waveguides,” Opt. Lett. 34(21), 3457–3459 (2009).
[CrossRef] [PubMed]

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(25), 6165–6177 (2004), http://www.opticsinfobase.org/oe/abstract.cfm?URI=OPEX-12-25-6165 .
[CrossRef] [PubMed]

Chen, B.

B. Chen, D. G. Fang, and B. H. Zhou, “Modified Berenger PML absorbing boundary condition for FD-TD meshes,” IEEE Microw. Guid. Wave Lett. 5(11), 399–401 (1995).
[CrossRef]

Chen, H.-W.

C.-H. Lai, Y.-C. Hsueh, H.-W. Chen, Y.-J. Huang, H.-C. Chang, and C.-K. Sun, “Low-index terahertz pipe waveguides,” Opt. Lett. 34(21), 3457–3459 (2009).
[CrossRef] [PubMed]

Cheng, H.

H. Cheng, W. Y. Crutchfield, M. Doery, and L. Greengard, “Fast, accurate integral equation methods for the analysis of photonic crystal fibers I: Theory,” Opt. Express 12(16), 3791–3805 (2004), http://www.opticsinfobase.org/oe/abstract.cfm?URI=OPEX-12-16-3791 .
[CrossRef] [PubMed]

Chew, W. C.

W. C. Chew and W. H. Weedon, “A 3-D perfectly matched medium from modified Maxwell’s equation with stretched coordinates,” Microw. Opt. Technol. Lett. 7(13), 599–604 (1994).
[CrossRef]

Chiang, P.-J.

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(12), 908–910 (2010).
[CrossRef]

Chiang, Y.-C.

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(12), 908–910 (2010).
[CrossRef]

Costa, R.

P. Bienstman, S. Selleri, L. Rosa, H. P. Uranus, W. C. L. Hopman, R. Costa, A. Melloni, L. C. Andreani, J. P. Hugonin, P. Lalanne, D. Pinto, S. S. A. Obayya, M. Dems, and K. Panajotov, “Modelling leaky photonic wires: a mode solver comparison,” Opt. Quantum Electron. 38(9−11), 731–759 (2006).
[CrossRef]

Crutchfield, W. Y.

H. Cheng, W. Y. Crutchfield, M. Doery, and L. Greengard, “Fast, accurate integral equation methods for the analysis of photonic crystal fibers I: Theory,” Opt. Express 12(16), 3791–3805 (2004), http://www.opticsinfobase.org/oe/abstract.cfm?URI=OPEX-12-16-3791 .
[CrossRef] [PubMed]

de Sterke, C. M.

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(10), 2322–2330 (2002).
[CrossRef]

Dems, M.

P. Bienstman, S. Selleri, L. Rosa, H. P. Uranus, W. C. L. Hopman, R. Costa, A. Melloni, L. C. Andreani, J. P. Hugonin, P. Lalanne, D. Pinto, S. S. A. Obayya, M. Dems, and K. Panajotov, “Modelling leaky photonic wires: a mode solver comparison,” Opt. Quantum Electron. 38(9−11), 731–759 (2006).
[CrossRef]

Doery, M.

H. Cheng, W. Y. Crutchfield, M. Doery, and L. Greengard, “Fast, accurate integral equation methods for the analysis of photonic crystal fibers I: Theory,” Opt. Express 12(16), 3791–3805 (2004), http://www.opticsinfobase.org/oe/abstract.cfm?URI=OPEX-12-16-3791 .
[CrossRef] [PubMed]

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(1), 13–15 (1986).
[CrossRef]

Fang, D. G.

B. Chen, D. G. Fang, and B. H. Zhou, “Modified Berenger PML absorbing boundary condition for FD-TD meshes,” IEEE Microw. Guid. Wave Lett. 5(11), 399–401 (1995).
[CrossRef]

Greengard, L.

H. Cheng, W. Y. Crutchfield, M. Doery, and L. Greengard, “Fast, accurate integral equation methods for the analysis of photonic crystal fibers I: Theory,” Opt. Express 12(16), 3791–3805 (2004), http://www.opticsinfobase.org/oe/abstract.cfm?URI=OPEX-12-16-3791 .
[CrossRef] [PubMed]

Hadley, G. R.

G. R. Hadley, “Transparent boundary condition for beam propagation,” Opt. Lett. 16(9), 624–626 (1991).
[CrossRef] [PubMed]

Her, T.-H.

T.-H. Her, X. Ao, and L. W. Casperson, “Gain saturation in gain-guided slab waveguides with large-index antiguiding,” Opt. Lett. 34(16), 2411–2413 (2009).
[CrossRef] [PubMed]

Ho, P. L.

P. L. Ho and Y. Y. Lu, “A mode-preserving perfectly matched layer for optical waveguides,” IEEE Photon. Technol. Lett. 15(9), 1234–1236 (2003).
[CrossRef]

Hopman, W. C. L.

P. Bienstman, S. Selleri, L. Rosa, H. P. Uranus, W. C. L. Hopman, R. Costa, A. Melloni, L. C. Andreani, J. P. Hugonin, P. Lalanne, D. Pinto, S. S. A. Obayya, M. Dems, and K. Panajotov, “Modelling leaky photonic wires: a mode solver comparison,” Opt. Quantum Electron. 38(9−11), 731–759 (2006).
[CrossRef]

Hsueh, Y.-C.

C.-H. Lai, Y.-C. Hsueh, H.-W. Chen, Y.-J. Huang, H.-C. Chang, and C.-K. Sun, “Low-index terahertz pipe waveguides,” Opt. Lett. 34(21), 3457–3459 (2009).
[CrossRef] [PubMed]

Huang, W. P.

W. P. Huang, C. L. Xu, W. Lui, and K. Yokoyama, “The perfectly matched layer boundary condition for modal analysis of optical waveguides: leaky mode calculations,” IEEE Photon. Technol. Lett. 8(5), 652–654 (1996).
[CrossRef]

Huang, W.-P.

W.-P. Huang and J. Mu, “Complex coupled-mode theory for optical waveguides,” Opt. Express 17(21), 19134–19152 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=OPEX-17-21-19134 .
[CrossRef]

Huang, Y.-J.

C.-H. Lai, Y.-C. Hsueh, H.-W. Chen, Y.-J. Huang, H.-C. Chang, and C.-K. Sun, “Low-index terahertz pipe waveguides,” Opt. Lett. 34(21), 3457–3459 (2009).
[CrossRef] [PubMed]

Hugonin, J. P.

P. Bienstman, S. Selleri, L. Rosa, H. P. Uranus, W. C. L. Hopman, R. Costa, A. Melloni, L. C. Andreani, J. P. Hugonin, P. Lalanne, D. Pinto, S. S. A. Obayya, M. Dems, and K. Panajotov, “Modelling leaky photonic wires: a mode solver comparison,” Opt. Quantum Electron. 38(9−11), 731–759 (2006).
[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(1), 13–15 (1986).
[CrossRef]

Kokubun, Y.

M. A. Duguay, Y. Kokubun, T. L. Koch, and L. Pfeiffer, “Antiresonant reflecting optical waveguides in SiO2–Si multilayer structures,” Appl. Phys. Lett. 49(1), 13–15 (1986).
[CrossRef]

Koshiba, M.

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(4), 618–623 (2000).
[CrossRef]

Kuhlmey, B. T.

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(10), 2322–2330 (2002).
[CrossRef]

Lai, C.-H.

C.-H. Lai, B. You, J.-Y. Lu, T.-A. Liu, J.-L. Peng, C.-K. Sun, and H.-C. Chang, “Modal characteristics of antiresonant reflecting pipe waveguides for terahertz waveguiding,” Opt. Express 18(1), 309–322 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?URI=OPEX-18-1-309 .
[CrossRef] [PubMed]

C.-H. Lai, Y.-C. Hsueh, H.-W. Chen, Y.-J. Huang, H.-C. Chang, and C.-K. Sun, “Low-index terahertz pipe waveguides,” Opt. Lett. 34(21), 3457–3459 (2009).
[CrossRef] [PubMed]

Lalanne, P.

P. Bienstman, S. Selleri, L. Rosa, H. P. Uranus, W. C. L. Hopman, R. Costa, A. Melloni, L. C. Andreani, J. P. Hugonin, P. Lalanne, D. Pinto, S. S. A. Obayya, M. Dems, and K. Panajotov, “Modelling leaky photonic wires: a mode solver comparison,” Opt. Quantum Electron. 38(9−11), 731–759 (2006).
[CrossRef]

Liu, T.-A.

C.-H. Lai, B. You, J.-Y. Lu, T.-A. Liu, J.-L. Peng, C.-K. Sun, and H.-C. Chang, “Modal characteristics of antiresonant reflecting pipe waveguides for terahertz waveguiding,” Opt. Express 18(1), 309–322 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?URI=OPEX-18-1-309 .
[CrossRef] [PubMed]

Lu, J.-Y.

C.-H. Lai, B. You, J.-Y. Lu, T.-A. Liu, J.-L. Peng, C.-K. Sun, and H.-C. Chang, “Modal characteristics of antiresonant reflecting pipe waveguides for terahertz waveguiding,” Opt. Express 18(1), 309–322 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?URI=OPEX-18-1-309 .
[CrossRef] [PubMed]

Lu,, Y. Y.

P. L. Ho and Y. Y. Lu, “A mode-preserving perfectly matched layer for optical waveguides,” IEEE Photon. Technol. Lett. 15(9), 1234–1236 (2003).
[CrossRef]

Lui, W.

W. P. Huang, C. L. Xu, W. Lui, and K. Yokoyama, “The perfectly matched layer boundary condition for modal analysis of optical waveguides: leaky mode calculations,” IEEE Photon. Technol. Lett. 8(5), 652–654 (1996).
[CrossRef]

Maystre, D.

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(10), 2322–2330 (2002).
[CrossRef]

McPhedran, R. C.

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(10), 2322–2330 (2002).
[CrossRef]

Melloni, A.

P. Bienstman, S. Selleri, L. Rosa, H. P. Uranus, W. C. L. Hopman, R. Costa, A. Melloni, L. C. Andreani, J. P. Hugonin, P. Lalanne, D. Pinto, S. S. A. Obayya, M. Dems, and K. Panajotov, “Modelling leaky photonic wires: a mode solver comparison,” Opt. Quantum Electron. 38(9−11), 731–759 (2006).
[CrossRef]

Mu, J.

W.-P. Huang and J. Mu, “Complex coupled-mode theory for optical waveguides,” Opt. Express 17(21), 19134–19152 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=OPEX-17-21-19134 .
[CrossRef]

Obayya, S. S. A.

P. Bienstman, S. Selleri, L. Rosa, H. P. Uranus, W. C. L. Hopman, R. Costa, A. Melloni, L. C. Andreani, J. P. Hugonin, P. Lalanne, D. Pinto, S. S. A. Obayya, M. Dems, and K. Panajotov, “Modelling leaky photonic wires: a mode solver comparison,” Opt. Quantum Electron. 38(9−11), 731–759 (2006).
[CrossRef]

Panajotov, K.

P. Bienstman, S. Selleri, L. Rosa, H. P. Uranus, W. C. L. Hopman, R. Costa, A. Melloni, L. C. Andreani, J. P. Hugonin, P. Lalanne, D. Pinto, S. S. A. Obayya, M. Dems, and K. Panajotov, “Modelling leaky photonic wires: a mode solver comparison,” Opt. Quantum Electron. 38(9−11), 731–759 (2006).
[CrossRef]

Peng, J.-L.

C.-H. Lai, B. You, J.-Y. Lu, T.-A. Liu, J.-L. Peng, C.-K. Sun, and H.-C. Chang, “Modal characteristics of antiresonant reflecting pipe waveguides for terahertz waveguiding,” Opt. Express 18(1), 309–322 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?URI=OPEX-18-1-309 .
[CrossRef] [PubMed]

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(1), 13–15 (1986).
[CrossRef]

Pinto, D.

P. Bienstman, S. Selleri, L. Rosa, H. P. Uranus, W. C. L. Hopman, R. Costa, A. Melloni, L. C. Andreani, J. P. Hugonin, P. Lalanne, D. Pinto, S. S. A. Obayya, M. Dems, and K. Panajotov, “Modelling leaky photonic wires: a mode solver comparison,” Opt. Quantum Electron. 38(9−11), 731–759 (2006).
[CrossRef]

Rappaport, C. M.

C. M. Rappaport, ““Perfectly matched absorbing boundary conditions based on anisotropic lossy mapping of space,”IEEE Microw. Guid. Wave Lett. 5(3), 90–92 (1995).
[CrossRef]

Renversez, G.

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(10), 2322–2330 (2002).
[CrossRef]

Rosa, L.

P. Bienstman, S. Selleri, L. Rosa, H. P. Uranus, W. C. L. Hopman, R. Costa, A. Melloni, L. C. Andreani, J. P. Hugonin, P. Lalanne, D. Pinto, S. S. A. Obayya, M. Dems, and K. Panajotov, “Modelling leaky photonic wires: a mode solver comparison,” Opt. Quantum Electron. 38(9−11), 731–759 (2006).
[CrossRef]

Selleri, S.

P. Bienstman, S. Selleri, L. Rosa, H. P. Uranus, W. C. L. Hopman, R. Costa, A. Melloni, L. C. Andreani, J. P. Hugonin, P. Lalanne, D. Pinto, S. S. A. Obayya, M. Dems, and K. Panajotov, “Modelling leaky photonic wires: a mode solver comparison,” Opt. Quantum Electron. 38(9−11), 731–759 (2006).
[CrossRef]

Siegman, A. E.

A. E. Siegman, “Propagating modes in gain-guided optical fibers,” J. Opt. Soc. Am. A 20(8), 1617–1628 (2003).
[CrossRef]

Sun, C.-K.

C.-H. Lai, B. You, J.-Y. Lu, T.-A. Liu, J.-L. Peng, C.-K. Sun, and H.-C. Chang, “Modal characteristics of antiresonant reflecting pipe waveguides for terahertz waveguiding,” Opt. Express 18(1), 309–322 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?URI=OPEX-18-1-309 .
[CrossRef] [PubMed]

C.-H. Lai, Y.-C. Hsueh, H.-W. Chen, Y.-J. Huang, H.-C. Chang, and C.-K. Sun, “Low-index terahertz pipe waveguides,” Opt. Lett. 34(21), 3457–3459 (2009).
[CrossRef] [PubMed]

Tsuji, Y.

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(4), 618–623 (2000).
[CrossRef]

Uranus, H. P.

P. Bienstman, S. Selleri, L. Rosa, H. P. Uranus, W. C. L. Hopman, R. Costa, A. Melloni, L. C. Andreani, J. P. Hugonin, P. Lalanne, D. Pinto, S. S. A. Obayya, M. Dems, and K. Panajotov, “Modelling leaky photonic wires: a mode solver comparison,” Opt. Quantum Electron. 38(9−11), 731–759 (2006).
[CrossRef]

Weedon, W. H.

W. C. Chew and W. H. Weedon, “A 3-D perfectly matched medium from modified Maxwell’s equation with stretched coordinates,” Microw. Opt. Technol. Lett. 7(13), 599–604 (1994).
[CrossRef]

White, T. P.

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(10), 2322–2330 (2002).
[CrossRef]

Xu, C. L.

W. P. Huang, C. L. Xu, W. Lui, and K. Yokoyama, “The perfectly matched layer boundary condition for modal analysis of optical waveguides: leaky mode calculations,” IEEE Photon. Technol. Lett. 8(5), 652–654 (1996).
[CrossRef]

Yokoyama, K.

W. P. Huang, C. L. Xu, W. Lui, and K. Yokoyama, “The perfectly matched layer boundary condition for modal analysis of optical waveguides: leaky mode calculations,” IEEE Photon. Technol. Lett. 8(5), 652–654 (1996).
[CrossRef]

You, B.

C.-H. Lai, B. You, J.-Y. Lu, T.-A. Liu, J.-L. Peng, C.-K. Sun, and H.-C. Chang, “Modal characteristics of antiresonant reflecting pipe waveguides for terahertz waveguiding,” Opt. Express 18(1), 309–322 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?URI=OPEX-18-1-309 .
[CrossRef] [PubMed]

Yu, C.-P.

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(25), 6165–6177 (2004), http://www.opticsinfobase.org/oe/abstract.cfm?URI=OPEX-12-25-6165 .
[CrossRef] [PubMed]

Zhou, B. H.

B. Chen, D. G. Fang, and B. H. Zhou, “Modified Berenger PML absorbing boundary condition for FD-TD meshes,” IEEE Microw. Guid. Wave Lett. 5(11), 399–401 (1995).
[CrossRef]

Appl. Phys. Lett. (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(1), 13–15 (1986).
[CrossRef]

IEEE Microw. Guid. Wave Lett. (2)

C. M. Rappaport, ““Perfectly matched absorbing boundary conditions based on anisotropic lossy mapping of space,”IEEE Microw. Guid. Wave Lett. 5(3), 90–92 (1995).
[CrossRef]

B. Chen, D. G. Fang, and B. H. Zhou, “Modified Berenger PML absorbing boundary condition for FD-TD meshes,” IEEE Microw. Guid. Wave Lett. 5(11), 399–401 (1995).
[CrossRef]

IEEE Photon. Technol. Lett. (3)

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(12), 908–910 (2010).
[CrossRef]

P. L. Ho and Y. Y. Lu, “A mode-preserving perfectly matched layer for optical waveguides,” IEEE Photon. Technol. Lett. 15(9), 1234–1236 (2003).
[CrossRef]

W. P. Huang, C. L. Xu, W. Lui, and K. Yokoyama, “The perfectly matched layer boundary condition for modal analysis of optical waveguides: leaky mode calculations,” IEEE Photon. Technol. Lett. 8(5), 652–654 (1996).
[CrossRef]

J. Comput. Phys. (1)

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114(2), 185–200 (1994).
[CrossRef]

J. Lightwave Technol. (1)

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(4), 618–623 (2000).
[CrossRef]

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

A. E. Siegman, “Propagating modes in gain-guided optical fibers,” J. Opt. Soc. Am. A 20(8), 1617–1628 (2003).
[CrossRef]

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

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(10), 2322–2330 (2002).
[CrossRef]

Microw. Opt. Technol. Lett. (1)

W. C. Chew and W. H. Weedon, “A 3-D perfectly matched medium from modified Maxwell’s equation with stretched coordinates,” Microw. Opt. Technol. Lett. 7(13), 599–604 (1994).
[CrossRef]

Opt. Express (4)

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(25), 6165–6177 (2004), http://www.opticsinfobase.org/oe/abstract.cfm?URI=OPEX-12-25-6165 .
[CrossRef] [PubMed]

W.-P. Huang and J. Mu, “Complex coupled-mode theory for optical waveguides,” Opt. Express 17(21), 19134–19152 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=OPEX-17-21-19134 .
[CrossRef]

H. Cheng, W. Y. Crutchfield, M. Doery, and L. Greengard, “Fast, accurate integral equation methods for the analysis of photonic crystal fibers I: Theory,” Opt. Express 12(16), 3791–3805 (2004), http://www.opticsinfobase.org/oe/abstract.cfm?URI=OPEX-12-16-3791 .
[CrossRef] [PubMed]

C.-H. Lai, B. You, J.-Y. Lu, T.-A. Liu, J.-L. Peng, C.-K. Sun, and H.-C. Chang, “Modal characteristics of antiresonant reflecting pipe waveguides for terahertz waveguiding,” Opt. Express 18(1), 309–322 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?URI=OPEX-18-1-309 .
[CrossRef] [PubMed]

Opt. Lett. (3)

G. R. Hadley, “Transparent boundary condition for beam propagation,” Opt. Lett. 16(9), 624–626 (1991).
[CrossRef] [PubMed]

C.-H. Lai, Y.-C. Hsueh, H.-W. Chen, Y.-J. Huang, H.-C. Chang, and C.-K. Sun, “Low-index terahertz pipe waveguides,” Opt. Lett. 34(21), 3457–3459 (2009).
[CrossRef] [PubMed]

T.-H. Her, X. Ao, and L. W. Casperson, “Gain saturation in gain-guided slab waveguides with large-index antiguiding,” Opt. Lett. 34(16), 2411–2413 (2009).
[CrossRef] [PubMed]

Opt. Quantum Electron. (1)

P. Bienstman, S. Selleri, L. Rosa, H. P. Uranus, W. C. L. Hopman, R. Costa, A. Melloni, L. C. Andreani, J. P. Hugonin, P. Lalanne, D. Pinto, S. S. A. Obayya, M. Dems, and K. Panajotov, “Modelling leaky photonic wires: a mode solver comparison,” Opt. Quantum Electron. 38(9−11), 731–759 (2006).
[CrossRef]

Other (1)

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).

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Figures (9)

Fig. 1
Fig. 1

Structure of the six-hole holey fiber.

Fig. 2
Fig. 2

Calculated effective index neff of the holey fiber as a function of W with d=1 μm. (a) Real part. (b) Imaginary part.

Fig. 3
Fig. 3

Calculated effective index neff of the holey fiber as a function of d with W=24 μm. (a) Real part. (b) Imaginary part.

Fig. 4
Fig. 4

Im(neff ) of the holey fiber as a function of R with W=24 μm and d=1 μm.

Fig. 5
Fig. 5

Structure of the air-core pipe waveguide.

Fig. 6
Fig. 6

(a) Im(neff ) of the pipe waveguide as a function of W with d=0.4 mm. (b) Im(neff ) of the pipe waveguide as a function of d with W=14 mm.

Fig. 7
Fig. 7

Im(neff ) of the pipe waveguide as a function of R with W=15.6 mm and d=0.4 mm.

Fig. 8
Fig. 8

Im(neff ) of the gain-guided and index-antiguided slab waveguide as a function of R. (a) g=0 (IAG case). (b) g=1 cm−1 (GG + IAG case).

Fig. 9
Fig. 9

Fundamental TE modal field profiles of the gain-guided and index-antiguided slab waveguide. (a) g=0 (IAG case). (b) g=1 cm−1 (GG + IAG case).

Equations (2)

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α 1 s α ,
s = 1 j ( m + 1 ) λ 4 π n d ( ρ d ) m ln 1 R ,

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