Abstract

An approximate circuit model is proposed for mode extraction in two-dimensional photonic crystal waveguides. The dispersion equation governing the complex propagation constant of the photonic crystal waveguide is then related to the resonance condition in the proposed circuit model. In this fashion, a scalar complex transcendental equation is given for mode extraction. To avoid searching for the complex roots of the derived scalar dispersion equation, however, the real and imaginary parts of the sought-after propagation constant are extracted by using the physical resonance condition and its corresponding quality factor in the here-proposed circuit model, respectively. All the necessary conditions for the accuracy of the proposed circuit model are discussed, and some numerical examples are given. It is fortunate that the proposed model can be applied for accurate mode extraction in most of the applications. Both major polarizations are considered.

© 2011 Optical Society of America

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References

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  1. S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Linear waveguides in photonic-crystal slabs,” Phys. Rev. B 62, 8212–8222 (2000).
    [CrossRef]
  2. C. Sauvan, P. Lalanne, J. C. Rodier, J. P. Hugonin, and A. Talneau, “Accurate modeling of line-defect photonic crystal waveguides,” IEEE Photon. Technol. Lett. 15, 1243–1245 (2003).
    [CrossRef]
  3. K. Yasumoto, H. Jia, and K. Sun, “Rigorous analysis of two-dimensional photonic crystal waveguide,” Radio Sci. 40, 1–7 (2005).
    [CrossRef]
  4. P. Sarrafi, A. Naqavi, K. Mehrany, S. Khorasani, and B. Rashidian, “An efficient approach toward guided mode extraction in two-dimensional photonic crystals,” Opt. Commun. 281, 2826–2833 (2008).
    [CrossRef]
  5. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).
  6. P. G. Ciarlet, The Finite Element Method for Elliptic Problems (Society for Industrial and Applied Mathematics, 2002).
  7. A. Khelif, B. Djafari-Rouhani, J. O. Vasseur, P. A. Deymier, Ph. Lambin, and L. Dobrzynski, “Transmittivity through straight and stublike waveguides in a two-dimensional phononic crystal,” Phys. Rev. B 65, 174308 (2002).
    [CrossRef]
  8. P. Sarrafi and K. Mehrany, “Fast convergent and unconditionally stable Galerkin’s method with adaptive Hermite–Gauss expansion for guided-mode extraction in two-dimensional photonic crystal based waveguides,” J. Opt. Soc. Am. B 26, 169–175 (2009).
    [CrossRef]
  9. E. Istrate and E. H. Sargent, “Photonic crystal waveguide analysis using interface boundary conditions,” IEEE J. Quantum Electron. 41, 461–467 (2005).
    [CrossRef]
  10. R. Sorrentino, “Transverse resonance technique,” in Numerical Techniques for Microwave and Millimeter—Wave Passive Structures, T. Itoh, ed. (Wiley, 1989), Chap. 1L.
  11. T. Rozzi and M. Mongiardo, Open Electromagnetic Waveguides (The Institution of Electrical Engineers, 1997), pp. 135–141.
  12. W. Kuang, W. J. Kim, A. Mock, and J. O’Brien, “Propagation loss of line-defect photonic crystal slab waveguides,” IEEE J. Sel. Top. Quantum Electron. 12, 1183–1195 (2006).
    [CrossRef]
  13. A. Khavasi, N. Habibi, A. H. Hosseinnia, and K. Mehrany, “A transmission line model for extraction of defect modes in two-dimensional photonic crystals,” in 2010 International Conference on Photonics (ICP) (IEEE, 2010), pp. 1–3.
  14. M. Miri, A. Khavasi, K. Mehrany, and B. Rashidian, “Transmission line model to design matching stage for light coupling into two-dimensional photonic crystals,” Opt. Lett. 35, 115–117 (2010).
    [CrossRef]
  15. S. Ramo, J. R. Whinnery, and T. Van Duzer, Fields and Waves in Communication Electronics (Wiley, 1965).
  16. E. Loewen and E. Popov, Diffraction Gratings and Applications (Dekker, 1997).
  17. A. Khavasi, A. K. Jahromi, and K. Mehrany, “Longitudinal Legendre polynomial expansion of electromagnetic fields for analysis of arbitrary-shaped gratings,” J. Opt. Soc. Am. A 25, 1564–1573 (2008).
    [CrossRef]
  18. G. Lifante, Integrated Photonics Fundamentals (Wiley, 2003).
  19. A. Agrawal and J. H. Lang, Foundations of Analog and Digital Electronic Circuits (Elsevier, 2005).
  20. Z. Szabó, G. Kádár, and J. Volk, “Band gaps in photonic crystals with dispersion,” Int. J. Comput. Math. Electr. Electron. Eng. (COMPEL) 24, 521–533 (2005).
    [CrossRef]
  21. A. H. Hosseinnia, “Migration of eigenmodes in photonic and quantum-well structures,” MSc thesis (Sharif University of Technology, 2008).
  22. E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, “Determination of guided and leaky modes in lossless and lossy planar multilayer optical waveguides: reflection pole method and wavevector density method,” J. Lightwave Technol. 17, 929–941(1999).
    [CrossRef]
  23. Y. Zhao and Y. Hao, “Finite-difference time-domain study of guided modes in nano-plasmonic waveguides,” IEEE Trans. Antennas Propag. 55, 3070–3077 (2007).
    [CrossRef]

2010 (1)

2009 (1)

2008 (2)

P. Sarrafi, A. Naqavi, K. Mehrany, S. Khorasani, and B. Rashidian, “An efficient approach toward guided mode extraction in two-dimensional photonic crystals,” Opt. Commun. 281, 2826–2833 (2008).
[CrossRef]

A. Khavasi, A. K. Jahromi, and K. Mehrany, “Longitudinal Legendre polynomial expansion of electromagnetic fields for analysis of arbitrary-shaped gratings,” J. Opt. Soc. Am. A 25, 1564–1573 (2008).
[CrossRef]

2007 (1)

Y. Zhao and Y. Hao, “Finite-difference time-domain study of guided modes in nano-plasmonic waveguides,” IEEE Trans. Antennas Propag. 55, 3070–3077 (2007).
[CrossRef]

2006 (1)

W. Kuang, W. J. Kim, A. Mock, and J. O’Brien, “Propagation loss of line-defect photonic crystal slab waveguides,” IEEE J. Sel. Top. Quantum Electron. 12, 1183–1195 (2006).
[CrossRef]

2005 (3)

K. Yasumoto, H. Jia, and K. Sun, “Rigorous analysis of two-dimensional photonic crystal waveguide,” Radio Sci. 40, 1–7 (2005).
[CrossRef]

E. Istrate and E. H. Sargent, “Photonic crystal waveguide analysis using interface boundary conditions,” IEEE J. Quantum Electron. 41, 461–467 (2005).
[CrossRef]

Z. Szabó, G. Kádár, and J. Volk, “Band gaps in photonic crystals with dispersion,” Int. J. Comput. Math. Electr. Electron. Eng. (COMPEL) 24, 521–533 (2005).
[CrossRef]

2003 (1)

C. Sauvan, P. Lalanne, J. C. Rodier, J. P. Hugonin, and A. Talneau, “Accurate modeling of line-defect photonic crystal waveguides,” IEEE Photon. Technol. Lett. 15, 1243–1245 (2003).
[CrossRef]

2002 (1)

A. Khelif, B. Djafari-Rouhani, J. O. Vasseur, P. A. Deymier, Ph. Lambin, and L. Dobrzynski, “Transmittivity through straight and stublike waveguides in a two-dimensional phononic crystal,” Phys. Rev. B 65, 174308 (2002).
[CrossRef]

2000 (1)

S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Linear waveguides in photonic-crystal slabs,” Phys. Rev. B 62, 8212–8222 (2000).
[CrossRef]

1999 (1)

Agrawal, A.

A. Agrawal and J. H. Lang, Foundations of Analog and Digital Electronic Circuits (Elsevier, 2005).

Anemogiannis, E.

Ciarlet, P. G.

P. G. Ciarlet, The Finite Element Method for Elliptic Problems (Society for Industrial and Applied Mathematics, 2002).

Deymier, P. A.

A. Khelif, B. Djafari-Rouhani, J. O. Vasseur, P. A. Deymier, Ph. Lambin, and L. Dobrzynski, “Transmittivity through straight and stublike waveguides in a two-dimensional phononic crystal,” Phys. Rev. B 65, 174308 (2002).
[CrossRef]

Djafari-Rouhani, B.

A. Khelif, B. Djafari-Rouhani, J. O. Vasseur, P. A. Deymier, Ph. Lambin, and L. Dobrzynski, “Transmittivity through straight and stublike waveguides in a two-dimensional phononic crystal,” Phys. Rev. B 65, 174308 (2002).
[CrossRef]

Dobrzynski, L.

A. Khelif, B. Djafari-Rouhani, J. O. Vasseur, P. A. Deymier, Ph. Lambin, and L. Dobrzynski, “Transmittivity through straight and stublike waveguides in a two-dimensional phononic crystal,” Phys. Rev. B 65, 174308 (2002).
[CrossRef]

Fan, S.

S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Linear waveguides in photonic-crystal slabs,” Phys. Rev. B 62, 8212–8222 (2000).
[CrossRef]

Gaylord, T. K.

Glytsis, E. N.

Habibi, N.

A. Khavasi, N. Habibi, A. H. Hosseinnia, and K. Mehrany, “A transmission line model for extraction of defect modes in two-dimensional photonic crystals,” in 2010 International Conference on Photonics (ICP) (IEEE, 2010), pp. 1–3.

Hagness, S. C.

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

Hao, Y.

Y. Zhao and Y. Hao, “Finite-difference time-domain study of guided modes in nano-plasmonic waveguides,” IEEE Trans. Antennas Propag. 55, 3070–3077 (2007).
[CrossRef]

Hosseinnia, A. H.

A. Khavasi, N. Habibi, A. H. Hosseinnia, and K. Mehrany, “A transmission line model for extraction of defect modes in two-dimensional photonic crystals,” in 2010 International Conference on Photonics (ICP) (IEEE, 2010), pp. 1–3.

A. H. Hosseinnia, “Migration of eigenmodes in photonic and quantum-well structures,” MSc thesis (Sharif University of Technology, 2008).

Hugonin, J. P.

C. Sauvan, P. Lalanne, J. C. Rodier, J. P. Hugonin, and A. Talneau, “Accurate modeling of line-defect photonic crystal waveguides,” IEEE Photon. Technol. Lett. 15, 1243–1245 (2003).
[CrossRef]

Istrate, E.

E. Istrate and E. H. Sargent, “Photonic crystal waveguide analysis using interface boundary conditions,” IEEE J. Quantum Electron. 41, 461–467 (2005).
[CrossRef]

Jahromi, A. K.

Jia, H.

K. Yasumoto, H. Jia, and K. Sun, “Rigorous analysis of two-dimensional photonic crystal waveguide,” Radio Sci. 40, 1–7 (2005).
[CrossRef]

Joannopoulos, J. D.

S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Linear waveguides in photonic-crystal slabs,” Phys. Rev. B 62, 8212–8222 (2000).
[CrossRef]

Johnson, S. G.

S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Linear waveguides in photonic-crystal slabs,” Phys. Rev. B 62, 8212–8222 (2000).
[CrossRef]

Kádár, G.

Z. Szabó, G. Kádár, and J. Volk, “Band gaps in photonic crystals with dispersion,” Int. J. Comput. Math. Electr. Electron. Eng. (COMPEL) 24, 521–533 (2005).
[CrossRef]

Khavasi, A.

Khelif, A.

A. Khelif, B. Djafari-Rouhani, J. O. Vasseur, P. A. Deymier, Ph. Lambin, and L. Dobrzynski, “Transmittivity through straight and stublike waveguides in a two-dimensional phononic crystal,” Phys. Rev. B 65, 174308 (2002).
[CrossRef]

Khorasani, S.

P. Sarrafi, A. Naqavi, K. Mehrany, S. Khorasani, and B. Rashidian, “An efficient approach toward guided mode extraction in two-dimensional photonic crystals,” Opt. Commun. 281, 2826–2833 (2008).
[CrossRef]

Kim, W. J.

W. Kuang, W. J. Kim, A. Mock, and J. O’Brien, “Propagation loss of line-defect photonic crystal slab waveguides,” IEEE J. Sel. Top. Quantum Electron. 12, 1183–1195 (2006).
[CrossRef]

Kuang, W.

W. Kuang, W. J. Kim, A. Mock, and J. O’Brien, “Propagation loss of line-defect photonic crystal slab waveguides,” IEEE J. Sel. Top. Quantum Electron. 12, 1183–1195 (2006).
[CrossRef]

Lalanne, P.

C. Sauvan, P. Lalanne, J. C. Rodier, J. P. Hugonin, and A. Talneau, “Accurate modeling of line-defect photonic crystal waveguides,” IEEE Photon. Technol. Lett. 15, 1243–1245 (2003).
[CrossRef]

Lambin, Ph.

A. Khelif, B. Djafari-Rouhani, J. O. Vasseur, P. A. Deymier, Ph. Lambin, and L. Dobrzynski, “Transmittivity through straight and stublike waveguides in a two-dimensional phononic crystal,” Phys. Rev. B 65, 174308 (2002).
[CrossRef]

Lang, J. H.

A. Agrawal and J. H. Lang, Foundations of Analog and Digital Electronic Circuits (Elsevier, 2005).

Lifante, G.

G. Lifante, Integrated Photonics Fundamentals (Wiley, 2003).

Loewen, E.

E. Loewen and E. Popov, Diffraction Gratings and Applications (Dekker, 1997).

Mehrany, K.

Miri, M.

Mock, A.

W. Kuang, W. J. Kim, A. Mock, and J. O’Brien, “Propagation loss of line-defect photonic crystal slab waveguides,” IEEE J. Sel. Top. Quantum Electron. 12, 1183–1195 (2006).
[CrossRef]

Mongiardo, M.

T. Rozzi and M. Mongiardo, Open Electromagnetic Waveguides (The Institution of Electrical Engineers, 1997), pp. 135–141.

Naqavi, A.

P. Sarrafi, A. Naqavi, K. Mehrany, S. Khorasani, and B. Rashidian, “An efficient approach toward guided mode extraction in two-dimensional photonic crystals,” Opt. Commun. 281, 2826–2833 (2008).
[CrossRef]

O’Brien, J.

W. Kuang, W. J. Kim, A. Mock, and J. O’Brien, “Propagation loss of line-defect photonic crystal slab waveguides,” IEEE J. Sel. Top. Quantum Electron. 12, 1183–1195 (2006).
[CrossRef]

Popov, E.

E. Loewen and E. Popov, Diffraction Gratings and Applications (Dekker, 1997).

Ramo, S.

S. Ramo, J. R. Whinnery, and T. Van Duzer, Fields and Waves in Communication Electronics (Wiley, 1965).

Rashidian, B.

M. Miri, A. Khavasi, K. Mehrany, and B. Rashidian, “Transmission line model to design matching stage for light coupling into two-dimensional photonic crystals,” Opt. Lett. 35, 115–117 (2010).
[CrossRef]

P. Sarrafi, A. Naqavi, K. Mehrany, S. Khorasani, and B. Rashidian, “An efficient approach toward guided mode extraction in two-dimensional photonic crystals,” Opt. Commun. 281, 2826–2833 (2008).
[CrossRef]

Rodier, J. C.

C. Sauvan, P. Lalanne, J. C. Rodier, J. P. Hugonin, and A. Talneau, “Accurate modeling of line-defect photonic crystal waveguides,” IEEE Photon. Technol. Lett. 15, 1243–1245 (2003).
[CrossRef]

Rozzi, T.

T. Rozzi and M. Mongiardo, Open Electromagnetic Waveguides (The Institution of Electrical Engineers, 1997), pp. 135–141.

Sargent, E. H.

E. Istrate and E. H. Sargent, “Photonic crystal waveguide analysis using interface boundary conditions,” IEEE J. Quantum Electron. 41, 461–467 (2005).
[CrossRef]

Sarrafi, P.

P. Sarrafi and K. Mehrany, “Fast convergent and unconditionally stable Galerkin’s method with adaptive Hermite–Gauss expansion for guided-mode extraction in two-dimensional photonic crystal based waveguides,” J. Opt. Soc. Am. B 26, 169–175 (2009).
[CrossRef]

P. Sarrafi, A. Naqavi, K. Mehrany, S. Khorasani, and B. Rashidian, “An efficient approach toward guided mode extraction in two-dimensional photonic crystals,” Opt. Commun. 281, 2826–2833 (2008).
[CrossRef]

Sauvan, C.

C. Sauvan, P. Lalanne, J. C. Rodier, J. P. Hugonin, and A. Talneau, “Accurate modeling of line-defect photonic crystal waveguides,” IEEE Photon. Technol. Lett. 15, 1243–1245 (2003).
[CrossRef]

Sorrentino, R.

R. Sorrentino, “Transverse resonance technique,” in Numerical Techniques for Microwave and Millimeter—Wave Passive Structures, T. Itoh, ed. (Wiley, 1989), Chap. 1L.

Sun, K.

K. Yasumoto, H. Jia, and K. Sun, “Rigorous analysis of two-dimensional photonic crystal waveguide,” Radio Sci. 40, 1–7 (2005).
[CrossRef]

Szabó, Z.

Z. Szabó, G. Kádár, and J. Volk, “Band gaps in photonic crystals with dispersion,” Int. J. Comput. Math. Electr. Electron. Eng. (COMPEL) 24, 521–533 (2005).
[CrossRef]

Taflove, A.

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

Talneau, A.

C. Sauvan, P. Lalanne, J. C. Rodier, J. P. Hugonin, and A. Talneau, “Accurate modeling of line-defect photonic crystal waveguides,” IEEE Photon. Technol. Lett. 15, 1243–1245 (2003).
[CrossRef]

Van Duzer, T.

S. Ramo, J. R. Whinnery, and T. Van Duzer, Fields and Waves in Communication Electronics (Wiley, 1965).

Vasseur, J. O.

A. Khelif, B. Djafari-Rouhani, J. O. Vasseur, P. A. Deymier, Ph. Lambin, and L. Dobrzynski, “Transmittivity through straight and stublike waveguides in a two-dimensional phononic crystal,” Phys. Rev. B 65, 174308 (2002).
[CrossRef]

Villeneuve, P. R.

S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Linear waveguides in photonic-crystal slabs,” Phys. Rev. B 62, 8212–8222 (2000).
[CrossRef]

Volk, J.

Z. Szabó, G. Kádár, and J. Volk, “Band gaps in photonic crystals with dispersion,” Int. J. Comput. Math. Electr. Electron. Eng. (COMPEL) 24, 521–533 (2005).
[CrossRef]

Whinnery, J. R.

S. Ramo, J. R. Whinnery, and T. Van Duzer, Fields and Waves in Communication Electronics (Wiley, 1965).

Yasumoto, K.

K. Yasumoto, H. Jia, and K. Sun, “Rigorous analysis of two-dimensional photonic crystal waveguide,” Radio Sci. 40, 1–7 (2005).
[CrossRef]

Zhao, Y.

Y. Zhao and Y. Hao, “Finite-difference time-domain study of guided modes in nano-plasmonic waveguides,” IEEE Trans. Antennas Propag. 55, 3070–3077 (2007).
[CrossRef]

IEEE J. Quantum Electron. (1)

E. Istrate and E. H. Sargent, “Photonic crystal waveguide analysis using interface boundary conditions,” IEEE J. Quantum Electron. 41, 461–467 (2005).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

W. Kuang, W. J. Kim, A. Mock, and J. O’Brien, “Propagation loss of line-defect photonic crystal slab waveguides,” IEEE J. Sel. Top. Quantum Electron. 12, 1183–1195 (2006).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

C. Sauvan, P. Lalanne, J. C. Rodier, J. P. Hugonin, and A. Talneau, “Accurate modeling of line-defect photonic crystal waveguides,” IEEE Photon. Technol. Lett. 15, 1243–1245 (2003).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

Y. Zhao and Y. Hao, “Finite-difference time-domain study of guided modes in nano-plasmonic waveguides,” IEEE Trans. Antennas Propag. 55, 3070–3077 (2007).
[CrossRef]

Int. J. Comput. Math. Electr. Electron. Eng. (COMPEL) (1)

Z. Szabó, G. Kádár, and J. Volk, “Band gaps in photonic crystals with dispersion,” Int. J. Comput. Math. Electr. Electron. Eng. (COMPEL) 24, 521–533 (2005).
[CrossRef]

J. Lightwave Technol. (1)

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

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

Opt. Commun. (1)

P. Sarrafi, A. Naqavi, K. Mehrany, S. Khorasani, and B. Rashidian, “An efficient approach toward guided mode extraction in two-dimensional photonic crystals,” Opt. Commun. 281, 2826–2833 (2008).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. B (2)

S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Linear waveguides in photonic-crystal slabs,” Phys. Rev. B 62, 8212–8222 (2000).
[CrossRef]

A. Khelif, B. Djafari-Rouhani, J. O. Vasseur, P. A. Deymier, Ph. Lambin, and L. Dobrzynski, “Transmittivity through straight and stublike waveguides in a two-dimensional phononic crystal,” Phys. Rev. B 65, 174308 (2002).
[CrossRef]

Radio Sci. (1)

K. Yasumoto, H. Jia, and K. Sun, “Rigorous analysis of two-dimensional photonic crystal waveguide,” Radio Sci. 40, 1–7 (2005).
[CrossRef]

Other (10)

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

P. G. Ciarlet, The Finite Element Method for Elliptic Problems (Society for Industrial and Applied Mathematics, 2002).

R. Sorrentino, “Transverse resonance technique,” in Numerical Techniques for Microwave and Millimeter—Wave Passive Structures, T. Itoh, ed. (Wiley, 1989), Chap. 1L.

T. Rozzi and M. Mongiardo, Open Electromagnetic Waveguides (The Institution of Electrical Engineers, 1997), pp. 135–141.

S. Ramo, J. R. Whinnery, and T. Van Duzer, Fields and Waves in Communication Electronics (Wiley, 1965).

E. Loewen and E. Popov, Diffraction Gratings and Applications (Dekker, 1997).

A. Khavasi, N. Habibi, A. H. Hosseinnia, and K. Mehrany, “A transmission line model for extraction of defect modes in two-dimensional photonic crystals,” in 2010 International Conference on Photonics (ICP) (IEEE, 2010), pp. 1–3.

G. Lifante, Integrated Photonics Fundamentals (Wiley, 2003).

A. Agrawal and J. H. Lang, Foundations of Analog and Digital Electronic Circuits (Elsevier, 2005).

A. H. Hosseinnia, “Migration of eigenmodes in photonic and quantum-well structures,” MSc thesis (Sharif University of Technology, 2008).

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Figures (7)

Fig. 1.
Fig. 1.

(a) Typical photonic crystal waveguide in a square lattice photonic crystal. The lattice constant and radii of rods in the photonic crystal and defect regions are denoted by a , r , and r 1 , respectively. (b) Proposed transmission line model. (c) Defect photonic crystal that is formed by periodically repeating the defect region along the x direction. (d) Problem to be solved for extraction of the photonic crystal impedance.

Fig. 2.
Fig. 2.

Ray path followed by a guided mode in the homogeneous defect region of a photonic crystal waveguide made by removing a row of rods in the same square lattice photonic crystal shown in Fig. 1(a).

Fig. 3.
Fig. 3.

(a) Reflection coefficient of the zeroth-order diffracted wave used for finding the scalar impedance of the defect region and the photonic crystal region in the same typical photonic crystal waveguide shown in Fig. 1(a). (b) Equivalent circuit model when the corresponding impedance of the defect region and the photonic crystal behind it is directly calculated.

Fig. 4.
Fig. 4.

Dispersion diagram for an E-polarized guided mode supported by a porous silicon square lattice photonic crystal waveguide. (a) Real part of the propagation constant extracted by the rigorous approach (solid line), the here-proposed technique (dashed line), and the FDTD (dots). (b) Imaginary part of the propagation constant extracted by the rigorous approach (solid line), using Eq. (13) (dashed line), Eq. (8) (dotted line), Eq. (10) of this paper (dotted–dashed line) and the FDTD (dots).

Fig. 5.
Fig. 5.

Dispersion diagram for an E-polarized guided mode supported by a porous silicon square lattice photonic crystal waveguide. (a) Real part of the propagation constant extracted by the rigorous approach (solid line), the here-proposed technique (dashed line), and the FDTD (dots). (b) Imaginary part of the propagation constant extracted by the rigorous approach (solid line), using Eq. (13) (dashed line), Eq. (8) (dotted line), Eq. (10) of this paper (dotted–dashed line), and the FDTD (dots).

Fig. 6.
Fig. 6.

Resonant nature of the proposed circuit model. (a) Temporal transfer function, Y ( j ω ) , versus angular frequency. (b) Spatial transfer function, Y ( j β ) , versus the real part of the propagation constant.

Fig. 7.
Fig. 7.

Dispersion diagram for an H-polarized guided mode supported by a row of silver rods: the rigorous approach (solid line), the here-proposed technique (dashed line), and the FDTD (dots).

Equations (18)

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

Z PC / η = ( 1 + R PC ( ω , β ) ) ( 1 R PC ( ω , β ) ) .
Z in = Z 0 Z PC + j Z 0 tan ( κ x l ) Z 0 + j Z PC tan ( κ x l ) .
( r in ( ω , β ) + r PC ( ω , β ) ) + j ( x in ( ω , β ) + x PC ( ω , β ) ) = 0.
Z in / η = ( 1 + R in ( ω , β ) ) ( 1 R in ( ω , β ) ) .
R in ( ω , β ) R pc ( ω , β ) = 1.
x in ( ω 0 , β ) = x PC ( ω 0 , β ) .
Y ( j ω ) = 1 r in ( ω , β r 0 ) + r PC ( ω , β r 0 ) + j [ x in ( ω , β r 0 ) + x PC ( ω , β r 0 ) ] ,
β i 0 = Δ ω 3 dB 2 v g ,
Y ( j β ) = 1 r in ( ω 0 , β ) + r PC ( ω 0 , β ) + j [ x in ( ω 0 , β ) + x PC ( ω 0 , β ) ] ,
β i 0 = Δ β 3 dB 2 .
2 l n 2 k 0 2 β r 0 2 + arg ( R 1 ) + arg ( R 2 ) = 2 ν π ,
exp ( 2 β i 0 L ) = | R 1 | | R 2 | ,
β i 0 = ln ( | R | 2 ) 4 l β r 0 / Re [ κ x ] .
tan ( κ x l ) + 2 x PC Z 0 ( x PC 2 Z 0 2 ) = 0.
ϕ = 2 tan 1 ( x PC Z 0 ) .
κ x l + ϕ = γ π .
ε r ( ω ) = ε + k = 1 N p a k 1 + j δ k ω ω 0 k ( ω ω 0 k ) 2 .
ε r ( ω ) = 1 ω p 2 ω 2 j ω γ ,

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