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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 2 — Jan. 17, 2011
  • pp: 562–569
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Effect of perfectly matched layer reflection coefficient on modal analysis of leaky waveguide modes

Chih-Hsien Lai and Hung-chun Chang  »View Author Affiliations


Optics Express, Vol. 19, Issue 2, pp. 562-569 (2011)
http://dx.doi.org/10.1364/OE.19.000562


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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

1. Introduction

2. The case of holey fiber

We first investigate the holey fiber with a ring of six air holes symmetrically arranged around the core. The structure is shown in Fig. 1
Fig. 1 Structure of the six-hole holey fiber.
, with the hole diameter D=5 μm, the hole pitch Λ=6.75 μm, and the refractive indices ns=1.45 and na=1. The computational window size is denoted as W and the PML thickness is denoted as d. For the FDFD simulation, the grid sizes Δxy=0.05 μm and λ=1.45 μm are assumed.

We calculate the complex effective index neff, defined as neff=β/k 0, where k 0 is the free-space wavenumber. Calculated results of Re(neff) and Im(neff) of the fundamental mode are shown in Figs. 2(a)
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.
and 2(b), respectively, as a function of W with a fixed d=1 μm. Obviously, results using R=10−8 are the least stable because they vary and oscillate with W most seriously, and the oscillation amplitude becomes smaller when W increases. However, results of Re(neff) are consistent over the whole simulation range of W with up to the first eight digits (1.4453975), while those of Im(neff) differ even in the first digit. Hence the typical value R=10−8 might be acceptable to calculate Re(neff), but is inadequate for Im(neff). The latter is so highly affected by the computational window size that it is difficult to decide what size of W should be adopted to obtain the reasonable converged magnitude. (Of course a very large computational window size far above those shown in Fig. 2 might lead to the converged result, but it is unpractical if the limited computing resource is taken into account.) Results using R=10−12 show similar characteristics but vary with a less significant oscillation. In general, as the value of R decreases, the variation and oscillation become gradually diminished. In the case of R=10−50, it is clear both Re(neff) and Im(neff) are extremely consistent over the whole W range. Note that in Fig. 2, W starts from 20 μm, which is very close to the required minimum computation window size 18.5μm in this case (2Λ + D, the distance between the outmost boundaries of the two air holes located in the central horizontal line). In other words, if a sufficiently small value of R like 10−50 is used for simulation, only a small W near the minimum size requirement would be enough to obtain the converged magnitude of neff, which significantly saves the computation cost in terms of memory and time.

Now we fix W=24 μm and then change d to calculate neff. Results are shown in Figs. 3(a)
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.
and 3(b) for Re(neff) and Im(neff), respectively. Again, the results using R=10−8 exhibit the worst performance, and the variation and oscillation get diminished as R decreases. In the case of R=10−50, both Re(neff) and Im(neff) are consistent over the simulated range of d. Note that for R=10−50, a very thin PML thickness d=0.2 μm, which is equivalent to only 4 grids in PML, is sufficient to obtain the converged magnitude. Therefore, the computation cost can be saved when such a small R value is used for simulation.

Figure 4
Fig. 4 Im(neff) of the holey fiber as a function of R with W=24 μm and d=1 μm.
shows Im(neff) as a function of R, where W and d are fixed to 24 μm and 1 μm, respectively. It is observed that Im(neff) will converge to a stable magnitude when R is smaller than some threshold value, and in this case of holey fiber it is about 10−50. The calculated result of neff with R=10−50 is 1.445397590 − j3.23×10−8 under the adopted grid size of 0.05 μm (it is 1.445395151 − j3.1965×10−8 when the grid size decreases to 0.0125 μm), which is close to 1.445395345 − j3.15×10−8 obtained by the multipole method [13

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]

] and 1.445395232 − j3.1945×10−8 by the integral equation method [16

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]

]. Note that the degree of accuracy depends on the inherent property of the numerical scheme used for simulation, and is out of the scope of this paper. (Interested readers might refer to [17

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]

], which compares several mode solvers by using a photonic wire structure as a common example. As a reference, our result to that example is 2.412290 − j2.9202×10−8 with the parameters R=10−50 and the grid size of 0.005 μm.) What we aim to emphasize here is the effect of R, as discussed above and in the following, which is illustrated in this work using the finite difference method. This effect would be general to other numerical techniques as long as the PML is employed for leaky-mode analysis.

3. The case of air-core pipe waveguide

The second leaky structure is the air-core pipe waveguide recently proposed for THz waveguiding [14

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]

]. It consists of a large air core and a thin low-index dielectric cladding, as shown in Fig. 5
Fig. 5 Structure of the air-core pipe waveguide.
, and its guiding mechanism is similar to that of the antiresonant reflecting optical waveguide (ARROW) [18

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

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]

].

By assuming the core diameter D=9 mm, the cladding thickness t=1 mm, the refractive indices n 1=1 and n 2=1.4, Δxy=0.025 mm, and the operating frequency being 840 GHz, we calculate neff of the fundamental mode by changing W and d, respectively. Since Re(neff) is usually much easier to converge, only the results of Im(neff) are shown in Figs. 6(a)
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.
and 6(b), respectively. Calculated results exhibit the same characteristics as that of the previously discussed holey fiber. For larger R such as 10−10, significant variation and oscillation occur, and the oscillation becomes smaller as W or d increases; therefore, a larger W or a thicker d is required to obtain the converged Im(neff). While for R=10−100, calculated Im(neff)’s are stable over the simulation range, so that both W or d can be significantly reduced to obtain the converged magnitude without loss of stability. From 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.
, which shows Im(neff) as a function of R, it is clear Im(neff) converges if R is smaller than the threshold value of 10−100.

4. The case of gain-guided and index-antiguided slab waveguide

5. Conclusion

Acknowledgement

This work was supported in part by the National Science Council of the Republic of China under grant NSC97-2221-E-002-043-MY2, in part by the Excellent Research Projects of National Taiwan University under grant 98R0062-07, and in part by the Ministry of Education of the Republic of China under “The Aim of Top University Plan” grant.

References and links

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]

OCIS Codes
(000.4430) General : Numerical approximation and analysis
(230.7370) Optical devices : Waveguides
(050.1755) Diffraction and gratings : Computational electromagnetic methods

ToC Category:
Optical Devices

History
Original Manuscript: October 21, 2010
Revised Manuscript: December 22, 2010
Manuscript Accepted: December 22, 2010
Published: January 4, 2011

Citation
Chih-Hsien Lai and Hung-chun Chang, "Effect of perfectly matched layer reflection coefficient on modal analysis of leaky waveguide modes," Opt. Express 19, 562-569 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-2-562


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References

  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]

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