Abstract

To facilitate the analysis of radiation mode couplings, quasi leaky mode approximations were utilized in coupled-mode analysis. The key to effectively and accurately apply this approach is how to well approximate radiation modes by the quasi leaky modes in an equivalent closed waveguide model. In this paper, the principle, applicability and accuracy of the approximations are demonstrated, and the detailed implementation is also suggested by applying a unified coupled-mode analysis to fiber gratings. First of all, based on a thorough study on the characteristics of the complex modes, for the first time, quasi leaky modes are classified into guided-mode-like inner-cladding and radiation-mode-like outer-cladding leaky modes so as to explicitly establish equivalence relationships between the discrete leaky modes and the continuous radiation modes. With this new insight, the whole analysis process especially for some of the practically tricky issues such as the criteria for developing the proper equivalent waveguide model and the subsequent mode expansion basis are better understood and easier to be dealt with for different problems where radiation modes come into play. Moreover, as essential preconditions to extend the conventional coupled mode analysis to the present unified one, the couplings between the guided core mode and a leaky mode are studied in a systematic and consistent manner. An intuitive and then a deep understanding on the roles of complex modes on mode coupling and power exchanging are thus gained for further simulations. Lastly, the transmission spectra of fiber gratings with different surrounding indices are simulated. The simulated results agree well with those obtained theoretically and experimentally in the literatures, which strongly validate the principle of quasi leaky mode approximations and its implementation on the unified coupled-mode analysis expounded in this paper.

© 2010 OSA

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References

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  1. J. R. Qian, “Coupled-mode theory,” Textbook for Graduation in University of Science and Technology of China.
  2. S. L. Lee, Y. Chung, L. A. Coldren, and N. Dagli, “On Leaky Mode Approximations for modal expansion in multilayer open waveguides,” IEEE J. Quantum Electron. 31(10), 1790–1802 (1995).
    [CrossRef]
  3. J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114(2), 185–200 (1994).
    [CrossRef]
  4. F. Teixeira and W. Chew, “Systematic derivation of anisotropic PML absorbing media in cylindrical and spherical coordinates,” IEEE Microw. Guid. Wave Lett. 7(11), 371–373 (1997).
    [CrossRef]
  5. Y. C. Lu, L. Yang, W. P. Huang, and S. S. Jian, “Unified approach for coupling to cladding and radiation modes in fiber Bragg and long-period gratings,” J. Lightwave Technol. 27(11), 1461–1468 (2009).
    [CrossRef]
  6. H. Derudder, F. Olyslager, D. De Zutter, and S. Van den Berghe, “Efficient mode-matching analysis of discontinuities in finite planar substrates using perfectly matched layers,” IEEE Trans. Antenn. Propag. 49(2), 185–195 (2001).
    [CrossRef]
  7. H. Rogier and D. De Zutter, “Berenger and leaky modes in optical fibers terminated with a perfectly matched layer,” J. Lightwave Technol. 20(7), 1141–1148 (2002).
    [CrossRef]
  8. K. Jiang and W. P. Huang, “Finite-differene-based mode-matching method for 3-D waveguide structures under semivectorial approximation,” J. Lightwave Technol. 23(12), 4239–4248 (2005).
    [CrossRef]
  9. J. Mu and W. P. Huang, “Simulation of three-dimensional waveguide discontinuities by a full-vector mode-matching method based on finite-difference schemes,” Opt. Express 16(22), 18152–18163 (2008).
    [CrossRef] [PubMed]
  10. D. Vande Ginste, H. Rogier, and D. De Zutter, “Efficient computation of TM- and TE-polarized leaky modes in multilayered circular waveguides,” J. Lightwave Technol. 28(11), 1661–1669 (2010).
    [CrossRef]
  11. W. P. Huang and J. W. Mu, “Complex coupled-mode theory for optical waveguides,” Opt. Express 17(21), 19134–19152 (2009).
    [CrossRef]
  12. Y. C. Lu, W. P. Huang, and S. S. Jian, “Full vector complex coupled mode theory for tilted fiber gratings,” Opt. Express 18(2), 713–726 (2010).
    [CrossRef] [PubMed]
  13. N. Song, J. W. Mu, and W. P. Huang, “Application of the Complex Coupled-Mode Theory to Optical Fiber Grating Structures,” J. Lightwave Technol. 28(5), 761–767 (2010).
    [CrossRef]
  14. L. Yang and L. L. Xue, “Simplified Treatment for Radiation Mode in Coupled-mode Analysis”, ICMMT 2010, Chengdu, May, 2010.
  15. T. Erdogan, “Fiber gratings spectra,” J. Lightwave Technol. 15(8), 1277–1294 (1997).
    [CrossRef]
  16. T. Erdogan, “Cladding-mode resonances in short- and long-period fiber grating filters,” J. Opt. Soc. Am. A 14(8), 1760–1773 (1997) Errata: T. Erdogan, J. Opt. Soc. Am. A, 17(11), 2113–2113, (2000). Errata: Yinquan Yuan, J. Opt. Soc. Am. A, 26(10), 2199–2201, (2009).
    [CrossRef]
  17. H. Patrick, A. Kersey, and F. Bucholtz, “Analysis of the response of long period fiber gratings to external index of refraction,” J. Lightwave Technol. 16(9), 1606–1612 (1998).
    [CrossRef]
  18. D. B. Stegall and T. Erdogan, “Leaky cladding mode propagation in long-period fiber grating devices,” IEEE Photon. Technol. Lett. 11(3), 343–345 (1999).
    [CrossRef]
  19. Y. Koyamada, “Numerical analysis of core-mode to radiation-mode coupling in long-period fiber gratings,” IEEE Photon. Technol. Lett. 13(4), 308–310 (2001).
    [CrossRef]
  20. Y. Koyamada, “Analysis of core-mode to radiation-mode coupling in fiber Bragg gratings with finite cladding radius,” J. Lightwave Technol. 18(9), 1220–1225 (2000).
    [CrossRef]
  21. V. I. Kopp, V. M. Churikov, J. Singer, N. Chao, D. Neugroschl, and A. Z. Genack, “Chiral fiber gratings,” Science 305(5680), 74–75 (2004).
    [CrossRef] [PubMed]
  22. Y. Y. Shevchenko and J. Albert, “Plasmon resonances in gold-coated tilted fiber Bragg gratings,” Opt. Lett. 32(3), 211–213 (2007).
    [CrossRef] [PubMed]
  23. Y. C. Lu, L. Yang, W. P. Huang, and S. S. Jian, “Improved full-vector finite-difference complex mode solver for optical waveguides of circular symmetry,” J. Lightwave Technol. 26(13), 1868–1876 (2008).
    [CrossRef]
  24. R. E. Collin, Field Theory of Guided Waves, (A John Wiley & Sons. Inc., Publication, IEEE Press 1990), Chapter 1.
  25. W. Streifer, D. R. Scifres, and R. D. Burnham, “Coupled wave analysis of DFB and DBR lasers,” IEEE J. Quantum Electron. 13(4), 134–141 (1977).
    [CrossRef]

2010

2009

2008

2007

2005

2004

V. I. Kopp, V. M. Churikov, J. Singer, N. Chao, D. Neugroschl, and A. Z. Genack, “Chiral fiber gratings,” Science 305(5680), 74–75 (2004).
[CrossRef] [PubMed]

2002

2001

Y. Koyamada, “Numerical analysis of core-mode to radiation-mode coupling in long-period fiber gratings,” IEEE Photon. Technol. Lett. 13(4), 308–310 (2001).
[CrossRef]

H. Derudder, F. Olyslager, D. De Zutter, and S. Van den Berghe, “Efficient mode-matching analysis of discontinuities in finite planar substrates using perfectly matched layers,” IEEE Trans. Antenn. Propag. 49(2), 185–195 (2001).
[CrossRef]

2000

1999

D. B. Stegall and T. Erdogan, “Leaky cladding mode propagation in long-period fiber grating devices,” IEEE Photon. Technol. Lett. 11(3), 343–345 (1999).
[CrossRef]

1998

1997

T. Erdogan, “Cladding-mode resonances in short- and long-period fiber grating filters,” J. Opt. Soc. Am. A 14(8), 1760–1773 (1997) Errata: T. Erdogan, J. Opt. Soc. Am. A, 17(11), 2113–2113, (2000). Errata: Yinquan Yuan, J. Opt. Soc. Am. A, 26(10), 2199–2201, (2009).
[CrossRef]

F. Teixeira and W. Chew, “Systematic derivation of anisotropic PML absorbing media in cylindrical and spherical coordinates,” IEEE Microw. Guid. Wave Lett. 7(11), 371–373 (1997).
[CrossRef]

T. Erdogan, “Fiber gratings spectra,” J. Lightwave Technol. 15(8), 1277–1294 (1997).
[CrossRef]

1995

S. L. Lee, Y. Chung, L. A. Coldren, and N. Dagli, “On Leaky Mode Approximations for modal expansion in multilayer open waveguides,” IEEE J. Quantum Electron. 31(10), 1790–1802 (1995).
[CrossRef]

1994

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

1977

W. Streifer, D. R. Scifres, and R. D. Burnham, “Coupled wave analysis of DFB and DBR lasers,” IEEE J. Quantum Electron. 13(4), 134–141 (1977).
[CrossRef]

Albert, J.

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]

Bucholtz, F.

Burnham, R. D.

W. Streifer, D. R. Scifres, and R. D. Burnham, “Coupled wave analysis of DFB and DBR lasers,” IEEE J. Quantum Electron. 13(4), 134–141 (1977).
[CrossRef]

Chao, N.

V. I. Kopp, V. M. Churikov, J. Singer, N. Chao, D. Neugroschl, and A. Z. Genack, “Chiral fiber gratings,” Science 305(5680), 74–75 (2004).
[CrossRef] [PubMed]

Chew, W.

F. Teixeira and W. Chew, “Systematic derivation of anisotropic PML absorbing media in cylindrical and spherical coordinates,” IEEE Microw. Guid. Wave Lett. 7(11), 371–373 (1997).
[CrossRef]

Chung, Y.

S. L. Lee, Y. Chung, L. A. Coldren, and N. Dagli, “On Leaky Mode Approximations for modal expansion in multilayer open waveguides,” IEEE J. Quantum Electron. 31(10), 1790–1802 (1995).
[CrossRef]

Churikov, V. M.

V. I. Kopp, V. M. Churikov, J. Singer, N. Chao, D. Neugroschl, and A. Z. Genack, “Chiral fiber gratings,” Science 305(5680), 74–75 (2004).
[CrossRef] [PubMed]

Coldren, L. A.

S. L. Lee, Y. Chung, L. A. Coldren, and N. Dagli, “On Leaky Mode Approximations for modal expansion in multilayer open waveguides,” IEEE J. Quantum Electron. 31(10), 1790–1802 (1995).
[CrossRef]

Dagli, N.

S. L. Lee, Y. Chung, L. A. Coldren, and N. Dagli, “On Leaky Mode Approximations for modal expansion in multilayer open waveguides,” IEEE J. Quantum Electron. 31(10), 1790–1802 (1995).
[CrossRef]

De Zutter, D.

Derudder, H.

H. Derudder, F. Olyslager, D. De Zutter, and S. Van den Berghe, “Efficient mode-matching analysis of discontinuities in finite planar substrates using perfectly matched layers,” IEEE Trans. Antenn. Propag. 49(2), 185–195 (2001).
[CrossRef]

Erdogan, T.

D. B. Stegall and T. Erdogan, “Leaky cladding mode propagation in long-period fiber grating devices,” IEEE Photon. Technol. Lett. 11(3), 343–345 (1999).
[CrossRef]

T. Erdogan, “Fiber gratings spectra,” J. Lightwave Technol. 15(8), 1277–1294 (1997).
[CrossRef]

T. Erdogan, “Cladding-mode resonances in short- and long-period fiber grating filters,” J. Opt. Soc. Am. A 14(8), 1760–1773 (1997) Errata: T. Erdogan, J. Opt. Soc. Am. A, 17(11), 2113–2113, (2000). Errata: Yinquan Yuan, J. Opt. Soc. Am. A, 26(10), 2199–2201, (2009).
[CrossRef]

Genack, A. Z.

V. I. Kopp, V. M. Churikov, J. Singer, N. Chao, D. Neugroschl, and A. Z. Genack, “Chiral fiber gratings,” Science 305(5680), 74–75 (2004).
[CrossRef] [PubMed]

Huang, W. P.

Jian, S. S.

Jiang, K.

Kersey, A.

Kopp, V. I.

V. I. Kopp, V. M. Churikov, J. Singer, N. Chao, D. Neugroschl, and A. Z. Genack, “Chiral fiber gratings,” Science 305(5680), 74–75 (2004).
[CrossRef] [PubMed]

Koyamada, Y.

Y. Koyamada, “Numerical analysis of core-mode to radiation-mode coupling in long-period fiber gratings,” IEEE Photon. Technol. Lett. 13(4), 308–310 (2001).
[CrossRef]

Y. Koyamada, “Analysis of core-mode to radiation-mode coupling in fiber Bragg gratings with finite cladding radius,” J. Lightwave Technol. 18(9), 1220–1225 (2000).
[CrossRef]

Lee, S. L.

S. L. Lee, Y. Chung, L. A. Coldren, and N. Dagli, “On Leaky Mode Approximations for modal expansion in multilayer open waveguides,” IEEE J. Quantum Electron. 31(10), 1790–1802 (1995).
[CrossRef]

Lu, Y. C.

Mu, J.

Mu, J. W.

Neugroschl, D.

V. I. Kopp, V. M. Churikov, J. Singer, N. Chao, D. Neugroschl, and A. Z. Genack, “Chiral fiber gratings,” Science 305(5680), 74–75 (2004).
[CrossRef] [PubMed]

Olyslager, F.

H. Derudder, F. Olyslager, D. De Zutter, and S. Van den Berghe, “Efficient mode-matching analysis of discontinuities in finite planar substrates using perfectly matched layers,” IEEE Trans. Antenn. Propag. 49(2), 185–195 (2001).
[CrossRef]

Patrick, H.

Rogier, H.

Scifres, D. R.

W. Streifer, D. R. Scifres, and R. D. Burnham, “Coupled wave analysis of DFB and DBR lasers,” IEEE J. Quantum Electron. 13(4), 134–141 (1977).
[CrossRef]

Shevchenko, Y. Y.

Singer, J.

V. I. Kopp, V. M. Churikov, J. Singer, N. Chao, D. Neugroschl, and A. Z. Genack, “Chiral fiber gratings,” Science 305(5680), 74–75 (2004).
[CrossRef] [PubMed]

Song, N.

Stegall, D. B.

D. B. Stegall and T. Erdogan, “Leaky cladding mode propagation in long-period fiber grating devices,” IEEE Photon. Technol. Lett. 11(3), 343–345 (1999).
[CrossRef]

Streifer, W.

W. Streifer, D. R. Scifres, and R. D. Burnham, “Coupled wave analysis of DFB and DBR lasers,” IEEE J. Quantum Electron. 13(4), 134–141 (1977).
[CrossRef]

Teixeira, F.

F. Teixeira and W. Chew, “Systematic derivation of anisotropic PML absorbing media in cylindrical and spherical coordinates,” IEEE Microw. Guid. Wave Lett. 7(11), 371–373 (1997).
[CrossRef]

Van den Berghe, S.

H. Derudder, F. Olyslager, D. De Zutter, and S. Van den Berghe, “Efficient mode-matching analysis of discontinuities in finite planar substrates using perfectly matched layers,” IEEE Trans. Antenn. Propag. 49(2), 185–195 (2001).
[CrossRef]

Vande Ginste, D.

Yang, L.

IEEE J. Quantum Electron.

S. L. Lee, Y. Chung, L. A. Coldren, and N. Dagli, “On Leaky Mode Approximations for modal expansion in multilayer open waveguides,” IEEE J. Quantum Electron. 31(10), 1790–1802 (1995).
[CrossRef]

W. Streifer, D. R. Scifres, and R. D. Burnham, “Coupled wave analysis of DFB and DBR lasers,” IEEE J. Quantum Electron. 13(4), 134–141 (1977).
[CrossRef]

IEEE Microw. Guid. Wave Lett.

F. Teixeira and W. Chew, “Systematic derivation of anisotropic PML absorbing media in cylindrical and spherical coordinates,” IEEE Microw. Guid. Wave Lett. 7(11), 371–373 (1997).
[CrossRef]

IEEE Photon. Technol. Lett.

D. B. Stegall and T. Erdogan, “Leaky cladding mode propagation in long-period fiber grating devices,” IEEE Photon. Technol. Lett. 11(3), 343–345 (1999).
[CrossRef]

Y. Koyamada, “Numerical analysis of core-mode to radiation-mode coupling in long-period fiber gratings,” IEEE Photon. Technol. Lett. 13(4), 308–310 (2001).
[CrossRef]

IEEE Trans. Antenn. Propag.

H. Derudder, F. Olyslager, D. De Zutter, and S. Van den Berghe, “Efficient mode-matching analysis of discontinuities in finite planar substrates using perfectly matched layers,” IEEE Trans. Antenn. Propag. 49(2), 185–195 (2001).
[CrossRef]

J. Comput. Phys.

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.

T. Erdogan, “Fiber gratings spectra,” J. Lightwave Technol. 15(8), 1277–1294 (1997).
[CrossRef]

Y. Koyamada, “Analysis of core-mode to radiation-mode coupling in fiber Bragg gratings with finite cladding radius,” J. Lightwave Technol. 18(9), 1220–1225 (2000).
[CrossRef]

H. Patrick, A. Kersey, and F. Bucholtz, “Analysis of the response of long period fiber gratings to external index of refraction,” J. Lightwave Technol. 16(9), 1606–1612 (1998).
[CrossRef]

H. Rogier and D. De Zutter, “Berenger and leaky modes in optical fibers terminated with a perfectly matched layer,” J. Lightwave Technol. 20(7), 1141–1148 (2002).
[CrossRef]

K. Jiang and W. P. Huang, “Finite-differene-based mode-matching method for 3-D waveguide structures under semivectorial approximation,” J. Lightwave Technol. 23(12), 4239–4248 (2005).
[CrossRef]

Y. C. Lu, L. Yang, W. P. Huang, and S. S. Jian, “Improved full-vector finite-difference complex mode solver for optical waveguides of circular symmetry,” J. Lightwave Technol. 26(13), 1868–1876 (2008).
[CrossRef]

Y. C. Lu, L. Yang, W. P. Huang, and S. S. Jian, “Unified approach for coupling to cladding and radiation modes in fiber Bragg and long-period gratings,” J. Lightwave Technol. 27(11), 1461–1468 (2009).
[CrossRef]

N. Song, J. W. Mu, and W. P. Huang, “Application of the Complex Coupled-Mode Theory to Optical Fiber Grating Structures,” J. Lightwave Technol. 28(5), 761–767 (2010).
[CrossRef]

D. Vande Ginste, H. Rogier, and D. De Zutter, “Efficient computation of TM- and TE-polarized leaky modes in multilayered circular waveguides,” J. Lightwave Technol. 28(11), 1661–1669 (2010).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Express

Opt. Lett.

Science

V. I. Kopp, V. M. Churikov, J. Singer, N. Chao, D. Neugroschl, and A. Z. Genack, “Chiral fiber gratings,” Science 305(5680), 74–75 (2004).
[CrossRef] [PubMed]

Other

R. E. Collin, Field Theory of Guided Waves, (A John Wiley & Sons. Inc., Publication, IEEE Press 1990), Chapter 1.

L. Yang and L. L. Xue, “Simplified Treatment for Radiation Mode in Coupled-mode Analysis”, ICMMT 2010, Chengdu, May, 2010.

J. R. Qian, “Coupled-mode theory,” Textbook for Graduation in University of Science and Technology of China.

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Figures (11)

Fig. 1
Fig. 1

Theoretical model of fiber grating and its reference waveguide models. (a) sketch of fiber grating; (b) index profile of open reference waveguide; (c) index profile of equivalent closed reference waveguide.

Fig. 2
Fig. 2

Mode spectrum of the closed waveguide with an outer-cladding of noc = 1.46.

Fig. 3
Fig. 3

Amplitudes and phases of the radial components of the electric fields Er, for the complex modes with an outer-cladding of noc = 1.46. (a) the guided core mode HE11; (b) the inner-cladding leaky mode HE16; (c) the lowest order outer-cladding leaky mode; (d) a higher order outer-cladding leaky mode.

Fig. 4
Fig. 4

Variations of the effective refractive indices of the inner-cladding leaky modes in the closed waveguide with an outer-cladding of noc = 1.46 with the change of the absorption coefficient of PML, RPML.

Fig. 5
Fig. 5

Variations of effective refractive indices of inner- and outer-cladding leaky modes in the closed waveguide with an outer-cladding of noc = 1.46 with the change of the starting position of PML, roc.

Fig. 6
Fig. 6

Mode spectrum of the closed waveguide with an outer-cladding of noc = 1.45.

Fig. 7
Fig. 7

Variations of effective refractive indices of the closed waveguide with an outer-cladding of noc = 1.45 with the change of the starting position of PML roc.

Fig. 8
Fig. 8

Comparison of coupling strengths between the core mode and leaky modes with angular order m = 1 at λ = 1.55μm (noc = 1.48).

Fig. 9
Fig. 9

Power exchanging between the core mode and a leaky mode at different wavelength (noc = 1.75).

Fig. 10
Fig. 10

Simulated transmission spectra of LPGs with different surrounding indices.

Fig. 11
Fig. 11

Simulated transmission spectra of FBGs with different surrounding indices.

Equations (6)

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

n ( z ) = n co + Δn [ 1 + ν cos ( 2 π Λ z ) ] ,
1 2 ( e t m × h t n * ) z ^ d A = δ m n ,
1 2 ( e t m × h t n ) z ^ d A = 0           m n ,
1 2 ( e t m × h t m ) z ^ d A = N m ,
d a m d z + j β m a m = j n = 1 N + M κ m n a n j n = 1 N + M χ m n b n , d b m d z j β m b m = + j n = 1 N + M χ m n a n + j n = 1 N + M κ m n b n .
κ m n = ω ε 0 4 N m ( n 2 n 0 2 ) ( e t m e t n n 0 2 n 2 e z m e z n ) d s , χ m n = ω ε 0 4 N m ( n 2 n 0 2 ) ( e t m e t n + n 0 2 n 2 e z m e z n ) d s ,

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