Abstract

It has been recently shown that guided modes in two-dimensional photonic crystal based structures can be fast and efficiently extracted by using the Galerkin’s method with Hermite–Gauss basis functions. Although quite efficient and reliable for photonic crystal line defect waveguides, difficulties are likely to arise for more complicated geometries, e.g., for coupled resonator optical waveguides. First, unwanted numerical instability may well occur if a large number of basis functions are retained in the calculation. Second, the method could have a slow convergence rate with respect to the truncation order of the electromagnetic field expansion. Third, spurious solutions are not unlikely to appear. All these three important issues are here resolved by applying the unconditionally stable S-matrix propagation method, by proposing an adaptive algorithm to expedite the convergence rate of the expansion through duly scaled Hermite–Gauss basis functions, and by providing an effective algorithm for the elimination of spurious modes.

© 2008 Optical Society of America

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References

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  1. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302-307 (1966).
    [CrossRef]
  2. Y. Naka and H. Ikuno, “Guided mode analysis of two-dimensional air-hole type photonic crystal optical waveguides,” IEICE Tech. Rep. of Japan EMT-00-78 (October, 2000), pp. 75-80.
  3. M. Qiu and S. He, “Guided modes in a two-dimensional metallic photonic crystal waveguide,” Phys. Lett. A 266, 425-429 (2000).
    [CrossRef]
  4. A. K. Abeeluck, N. M. Litchinitser, C. Headley, and B. J. Eggleton, “Analysis of spectral characteristics of photonic bandgap waveguides,” Opt. Express 10, 1320-1333 (2002).
    [PubMed]
  5. M. Koshiba, Y. Tsuji, and M. Hikari, “Time-domain beam propagation method and its application to photonic crystal circuits,” J. Lightwave Technol. 18, 102-110 (2000).
    [CrossRef]
  6. H. Jia and K. Yasumoto, “Rigorous mode analysis of coupled cavity waveguides in two-dimensional photonic crystals,” Int. J. Infrared Millim. Waves 26, 1291-1306 (2005).
    [CrossRef]
  7. H. Jia and K. Yasumoto, “Modal analysis of two-dimensional photonic crystal waveguides formed by rectangular cylinders using an improved Fourier series method,” IEEE Trans. Microwave Theory Tech. 54, 564-571 (2006).
    [CrossRef]
  8. B. Gralak, S. Enoch, and G. Tayeb, “Anomalous refractive properties of photonic crystals,” J. Opt. Soc. Am. A 17, 1012-1020 (2000).
    [CrossRef]
  9. S. Boscolo, C. Conti, M. Midrio, and C. G. Someda, “Numerical analysis of propagation and impedance matching in 2-D photonic crystal waveguides with finite length,” J. Lightwave Technol. 20, 403-310 (2002).
    [CrossRef]
  10. H. Benisty, “Modal analysis of optical guides with two-dimensional photonic bandgap boundaries,” J. Appl. Phys. 79, 7483-7492 (1996).
    [CrossRef]
  11. 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]
  12. Y. Xu, “Origin and elimination of spurious modes in the solution of field eigenvalue problems by the method of moments,” in Microwave Conference Proceedings, Asia-Pacific 1, 465-468 (1997).
    [CrossRef]
  13. L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024-1035 (1996).
    [CrossRef]
  14. M. Neviere and E. Popov, Light Propagation in Periodic Media: Differential Theory and Design (Marcel Dekker, 2003).
  15. R. L. Gallawa, C. Goyal, Y. Tu, and A. K. Ghatak, “Optical waveguide modes: an approximate solution using Galerkin's method with Hermite-Gauss basis functions,” IEEE J. Quantum Electron. 21, 518-522 (1991).
    [CrossRef]
  16. I. A. Erteza and J. W. Goodman, “A scalar variational analysis of rectangular dielectric waveguides using Hermite-Gaussian modal approximation,” J. Lightwave Technol. 13, 493-506 (1995).
    [CrossRef]
  17. A. Weisshaar, J. Li, R. L. Gallawa, and I. C. Goyal, “Vector and quasi-vector solutions for optical waveguide modes using efficient Galerkin's method with Hermite-Gauss basis functions,” J. Lightwave Technol. 13, 1795-1800 (1995).
    [CrossRef]
  18. A. Ortega-Monux, J. Gonzalo Wanguemert-Perez, and I. Molina-Fernandez, “Adaptive Hermite-Gauss decomposition method to analyze optical dielectric waveguides,” J. Opt. Soc. Am. A 20, 557-568 (2003).
    [CrossRef]
  19. V. García-Muñoz and M. A. Muriel, “Hermite-Gauss series expansions applied to arrayed waveguide gratings,” IEEE Photon. Technol. Lett. 17, 2331-2333 (2005).
    [CrossRef]
  20. J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112-147 (1998).
    [CrossRef]
  21. W. Schroeder and I. Wolff, “The origin of spurious modes in numerical solutions of electromagnetic Field eigenvalue problems,” IEEE Trans. Microwave Theory Tech. 42, 644-653 (1994).
    [CrossRef]

2008 (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]

2006 (1)

H. Jia and K. Yasumoto, “Modal analysis of two-dimensional photonic crystal waveguides formed by rectangular cylinders using an improved Fourier series method,” IEEE Trans. Microwave Theory Tech. 54, 564-571 (2006).
[CrossRef]

2005 (2)

V. García-Muñoz and M. A. Muriel, “Hermite-Gauss series expansions applied to arrayed waveguide gratings,” IEEE Photon. Technol. Lett. 17, 2331-2333 (2005).
[CrossRef]

H. Jia and K. Yasumoto, “Rigorous mode analysis of coupled cavity waveguides in two-dimensional photonic crystals,” Int. J. Infrared Millim. Waves 26, 1291-1306 (2005).
[CrossRef]

2003 (1)

2002 (2)

A. K. Abeeluck, N. M. Litchinitser, C. Headley, and B. J. Eggleton, “Analysis of spectral characteristics of photonic bandgap waveguides,” Opt. Express 10, 1320-1333 (2002).
[PubMed]

S. Boscolo, C. Conti, M. Midrio, and C. G. Someda, “Numerical analysis of propagation and impedance matching in 2-D photonic crystal waveguides with finite length,” J. Lightwave Technol. 20, 403-310 (2002).
[CrossRef]

2000 (3)

1998 (1)

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112-147 (1998).
[CrossRef]

1996 (2)

L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024-1035 (1996).
[CrossRef]

H. Benisty, “Modal analysis of optical guides with two-dimensional photonic bandgap boundaries,” J. Appl. Phys. 79, 7483-7492 (1996).
[CrossRef]

1995 (2)

I. A. Erteza and J. W. Goodman, “A scalar variational analysis of rectangular dielectric waveguides using Hermite-Gaussian modal approximation,” J. Lightwave Technol. 13, 493-506 (1995).
[CrossRef]

A. Weisshaar, J. Li, R. L. Gallawa, and I. C. Goyal, “Vector and quasi-vector solutions for optical waveguide modes using efficient Galerkin's method with Hermite-Gauss basis functions,” J. Lightwave Technol. 13, 1795-1800 (1995).
[CrossRef]

1994 (1)

W. Schroeder and I. Wolff, “The origin of spurious modes in numerical solutions of electromagnetic Field eigenvalue problems,” IEEE Trans. Microwave Theory Tech. 42, 644-653 (1994).
[CrossRef]

1991 (1)

R. L. Gallawa, C. Goyal, Y. Tu, and A. K. Ghatak, “Optical waveguide modes: an approximate solution using Galerkin's method with Hermite-Gauss basis functions,” IEEE J. Quantum Electron. 21, 518-522 (1991).
[CrossRef]

1966 (1)

S. Yee, “Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302-307 (1966).
[CrossRef]

Abeeluck, A. K.

Benisty, H.

H. Benisty, “Modal analysis of optical guides with two-dimensional photonic bandgap boundaries,” J. Appl. Phys. 79, 7483-7492 (1996).
[CrossRef]

Boscolo, S.

S. Boscolo, C. Conti, M. Midrio, and C. G. Someda, “Numerical analysis of propagation and impedance matching in 2-D photonic crystal waveguides with finite length,” J. Lightwave Technol. 20, 403-310 (2002).
[CrossRef]

Conti, C.

S. Boscolo, C. Conti, M. Midrio, and C. G. Someda, “Numerical analysis of propagation and impedance matching in 2-D photonic crystal waveguides with finite length,” J. Lightwave Technol. 20, 403-310 (2002).
[CrossRef]

Eggleton, B. J.

Enoch, S.

Erteza, I. A.

I. A. Erteza and J. W. Goodman, “A scalar variational analysis of rectangular dielectric waveguides using Hermite-Gaussian modal approximation,” J. Lightwave Technol. 13, 493-506 (1995).
[CrossRef]

Gallawa, R. L.

A. Weisshaar, J. Li, R. L. Gallawa, and I. C. Goyal, “Vector and quasi-vector solutions for optical waveguide modes using efficient Galerkin's method with Hermite-Gauss basis functions,” J. Lightwave Technol. 13, 1795-1800 (1995).
[CrossRef]

R. L. Gallawa, C. Goyal, Y. Tu, and A. K. Ghatak, “Optical waveguide modes: an approximate solution using Galerkin's method with Hermite-Gauss basis functions,” IEEE J. Quantum Electron. 21, 518-522 (1991).
[CrossRef]

García-Muñoz, V.

V. García-Muñoz and M. A. Muriel, “Hermite-Gauss series expansions applied to arrayed waveguide gratings,” IEEE Photon. Technol. Lett. 17, 2331-2333 (2005).
[CrossRef]

Ghatak, A. K.

R. L. Gallawa, C. Goyal, Y. Tu, and A. K. Ghatak, “Optical waveguide modes: an approximate solution using Galerkin's method with Hermite-Gauss basis functions,” IEEE J. Quantum Electron. 21, 518-522 (1991).
[CrossRef]

Goodman, J. W.

I. A. Erteza and J. W. Goodman, “A scalar variational analysis of rectangular dielectric waveguides using Hermite-Gaussian modal approximation,” J. Lightwave Technol. 13, 493-506 (1995).
[CrossRef]

Goyal, C.

R. L. Gallawa, C. Goyal, Y. Tu, and A. K. Ghatak, “Optical waveguide modes: an approximate solution using Galerkin's method with Hermite-Gauss basis functions,” IEEE J. Quantum Electron. 21, 518-522 (1991).
[CrossRef]

Goyal, I. C.

A. Weisshaar, J. Li, R. L. Gallawa, and I. C. Goyal, “Vector and quasi-vector solutions for optical waveguide modes using efficient Galerkin's method with Hermite-Gauss basis functions,” J. Lightwave Technol. 13, 1795-1800 (1995).
[CrossRef]

Gralak, B.

He, S.

M. Qiu and S. He, “Guided modes in a two-dimensional metallic photonic crystal waveguide,” Phys. Lett. A 266, 425-429 (2000).
[CrossRef]

Headley, C.

Hikari, M.

Ikuno, H.

Y. Naka and H. Ikuno, “Guided mode analysis of two-dimensional air-hole type photonic crystal optical waveguides,” IEICE Tech. Rep. of Japan EMT-00-78 (October, 2000), pp. 75-80.

Jia, H.

H. Jia and K. Yasumoto, “Modal analysis of two-dimensional photonic crystal waveguides formed by rectangular cylinders using an improved Fourier series method,” IEEE Trans. Microwave Theory Tech. 54, 564-571 (2006).
[CrossRef]

H. Jia and K. Yasumoto, “Rigorous mode analysis of coupled cavity waveguides in two-dimensional photonic crystals,” Int. J. Infrared Millim. Waves 26, 1291-1306 (2005).
[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]

Koshiba, M.

Lagarias, J. C.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112-147 (1998).
[CrossRef]

Li, J.

A. Weisshaar, J. Li, R. L. Gallawa, and I. C. Goyal, “Vector and quasi-vector solutions for optical waveguide modes using efficient Galerkin's method with Hermite-Gauss basis functions,” J. Lightwave Technol. 13, 1795-1800 (1995).
[CrossRef]

Li, L.

Litchinitser, N. M.

Mehrany, K.

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]

Midrio, M.

S. Boscolo, C. Conti, M. Midrio, and C. G. Someda, “Numerical analysis of propagation and impedance matching in 2-D photonic crystal waveguides with finite length,” J. Lightwave Technol. 20, 403-310 (2002).
[CrossRef]

Molina-Fernandez, I.

Muriel, M. A.

V. García-Muñoz and M. A. Muriel, “Hermite-Gauss series expansions applied to arrayed waveguide gratings,” IEEE Photon. Technol. Lett. 17, 2331-2333 (2005).
[CrossRef]

Naka, Y.

Y. Naka and H. Ikuno, “Guided mode analysis of two-dimensional air-hole type photonic crystal optical waveguides,” IEICE Tech. Rep. of Japan EMT-00-78 (October, 2000), pp. 75-80.

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]

Neviere, M.

M. Neviere and E. Popov, Light Propagation in Periodic Media: Differential Theory and Design (Marcel Dekker, 2003).

Ortega-Monux, A.

Popov, E.

M. Neviere and E. Popov, Light Propagation in Periodic Media: Differential Theory and Design (Marcel Dekker, 2003).

Qiu, M.

M. Qiu and S. He, “Guided modes in a two-dimensional metallic photonic crystal waveguide,” Phys. Lett. A 266, 425-429 (2000).
[CrossRef]

Rashidian, B.

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]

Reeds, J. A.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112-147 (1998).
[CrossRef]

Sarrafi, P.

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]

Schroeder, W.

W. Schroeder and I. Wolff, “The origin of spurious modes in numerical solutions of electromagnetic Field eigenvalue problems,” IEEE Trans. Microwave Theory Tech. 42, 644-653 (1994).
[CrossRef]

Someda, C. G.

S. Boscolo, C. Conti, M. Midrio, and C. G. Someda, “Numerical analysis of propagation and impedance matching in 2-D photonic crystal waveguides with finite length,” J. Lightwave Technol. 20, 403-310 (2002).
[CrossRef]

Tayeb, G.

Tsuji, Y.

Tu, Y.

R. L. Gallawa, C. Goyal, Y. Tu, and A. K. Ghatak, “Optical waveguide modes: an approximate solution using Galerkin's method with Hermite-Gauss basis functions,” IEEE J. Quantum Electron. 21, 518-522 (1991).
[CrossRef]

Wanguemert-Perez, J. Gonzalo

Weisshaar, A.

A. Weisshaar, J. Li, R. L. Gallawa, and I. C. Goyal, “Vector and quasi-vector solutions for optical waveguide modes using efficient Galerkin's method with Hermite-Gauss basis functions,” J. Lightwave Technol. 13, 1795-1800 (1995).
[CrossRef]

Wolff, I.

W. Schroeder and I. Wolff, “The origin of spurious modes in numerical solutions of electromagnetic Field eigenvalue problems,” IEEE Trans. Microwave Theory Tech. 42, 644-653 (1994).
[CrossRef]

Wright, M. H.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112-147 (1998).
[CrossRef]

Wright, P. E.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112-147 (1998).
[CrossRef]

Xu, Y.

Y. Xu, “Origin and elimination of spurious modes in the solution of field eigenvalue problems by the method of moments,” in Microwave Conference Proceedings, Asia-Pacific 1, 465-468 (1997).
[CrossRef]

Yasumoto, K.

H. Jia and K. Yasumoto, “Modal analysis of two-dimensional photonic crystal waveguides formed by rectangular cylinders using an improved Fourier series method,” IEEE Trans. Microwave Theory Tech. 54, 564-571 (2006).
[CrossRef]

H. Jia and K. Yasumoto, “Rigorous mode analysis of coupled cavity waveguides in two-dimensional photonic crystals,” Int. J. Infrared Millim. Waves 26, 1291-1306 (2005).
[CrossRef]

Yee, S.

S. Yee, “Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302-307 (1966).
[CrossRef]

IEEE J. Quantum Electron. (1)

R. L. Gallawa, C. Goyal, Y. Tu, and A. K. Ghatak, “Optical waveguide modes: an approximate solution using Galerkin's method with Hermite-Gauss basis functions,” IEEE J. Quantum Electron. 21, 518-522 (1991).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

V. García-Muñoz and M. A. Muriel, “Hermite-Gauss series expansions applied to arrayed waveguide gratings,” IEEE Photon. Technol. Lett. 17, 2331-2333 (2005).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

S. Yee, “Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302-307 (1966).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (2)

W. Schroeder and I. Wolff, “The origin of spurious modes in numerical solutions of electromagnetic Field eigenvalue problems,” IEEE Trans. Microwave Theory Tech. 42, 644-653 (1994).
[CrossRef]

H. Jia and K. Yasumoto, “Modal analysis of two-dimensional photonic crystal waveguides formed by rectangular cylinders using an improved Fourier series method,” IEEE Trans. Microwave Theory Tech. 54, 564-571 (2006).
[CrossRef]

Int. J. Infrared Millim. Waves (1)

H. Jia and K. Yasumoto, “Rigorous mode analysis of coupled cavity waveguides in two-dimensional photonic crystals,” Int. J. Infrared Millim. Waves 26, 1291-1306 (2005).
[CrossRef]

J. Appl. Phys. (1)

H. Benisty, “Modal analysis of optical guides with two-dimensional photonic bandgap boundaries,” J. Appl. Phys. 79, 7483-7492 (1996).
[CrossRef]

J. Lightwave Technol. (4)

S. Boscolo, C. Conti, M. Midrio, and C. G. Someda, “Numerical analysis of propagation and impedance matching in 2-D photonic crystal waveguides with finite length,” J. Lightwave Technol. 20, 403-310 (2002).
[CrossRef]

I. A. Erteza and J. W. Goodman, “A scalar variational analysis of rectangular dielectric waveguides using Hermite-Gaussian modal approximation,” J. Lightwave Technol. 13, 493-506 (1995).
[CrossRef]

A. Weisshaar, J. Li, R. L. Gallawa, and I. C. Goyal, “Vector and quasi-vector solutions for optical waveguide modes using efficient Galerkin's method with Hermite-Gauss basis functions,” J. Lightwave Technol. 13, 1795-1800 (1995).
[CrossRef]

M. Koshiba, Y. Tsuji, and M. Hikari, “Time-domain beam propagation method and its application to photonic crystal circuits,” J. Lightwave Technol. 18, 102-110 (2000).
[CrossRef]

J. Opt. Soc. Am. A (3)

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. Express (1)

Phys. Lett. A (1)

M. Qiu and S. He, “Guided modes in a two-dimensional metallic photonic crystal waveguide,” Phys. Lett. A 266, 425-429 (2000).
[CrossRef]

SIAM J. Optim. (1)

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112-147 (1998).
[CrossRef]

Other (3)

Y. Naka and H. Ikuno, “Guided mode analysis of two-dimensional air-hole type photonic crystal optical waveguides,” IEICE Tech. Rep. of Japan EMT-00-78 (October, 2000), pp. 75-80.

Y. Xu, “Origin and elimination of spurious modes in the solution of field eigenvalue problems by the method of moments,” in Microwave Conference Proceedings, Asia-Pacific 1, 465-468 (1997).
[CrossRef]

M. Neviere and E. Popov, Light Propagation in Periodic Media: Differential Theory and Design (Marcel Dekker, 2003).

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Figures (7)

Fig. 1
Fig. 1

Typical photonic crystal based waveguide.

Fig. 2
Fig. 2

Propagation constant at the normalized frequency ω a 2 π c = 0.26 versus the number of basis function kept in the calculation; circles: the generalized transfer matrix based method proposed in [11]; dots: the here-proposed generalized scattering matrix approach.

Fig. 3
Fig. 3

(a) Schematic of the elementary cell in the analyzed CROW. (b) The propagation constant versus the truncation order N, and the scaling factor L. The bold line designates the optimum scaling factor.

Fig. 4
Fig. 4

Adaptive updating of the scaling factor versus number of iteration steps.

Fig. 5
Fig. 5

Convergence of the solution in terms of the truncation order and for different values of scaling factors: L = 0.4 (crosses), L = 0.54 (dots), and L = 1.5 (squares). The encircled crosses, dots, and squares are unwanted spurious modes.

Fig. 6
Fig. 6

Band structure extracted by retaining N = 80 (crosses), and N = 81 (circles) basis functions. The unwanted spurious modes are encircled to be discriminated from the correct solutions.

Fig. 7
Fig. 7

Band structure calculated by using the proposed strategy of removing the unwanted spurious modes (crosses), and by applying the PWE method (circles).

Equations (41)

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

[ ϴ Ω ] = [ a 0 ( z ) a 0 ( z ) a N ( z ) b 0 ( z ) b 1 ( z ) b N ( z ) ] [ Ψ 0 ( 2 x L ) Ψ 1 ( 2 x L ) Ψ N ( 2 x L ) ] ,
ϴ = j ε μ E y ,
Ω = H x ,
ϴ = H y ,
Ω = j ε μ E x ,
d d z [ a ¯ b ¯ ] = [ 0 Q ( z ; ω ) H ( z ; ω ) 0 ] [ a ¯ b ¯ ] ,
[ a ¯ ( z 0 + L z ) b ¯ ( z 0 + L z ) ] = m = 1 M exp ( [ 0 Q ̃ m ( ω ) H ̃ m ( ω ) 0 ] ) [ a ¯ ( z 0 ) b ¯ ( z 0 ) ] ,
Q ̃ m ( ω ) = z 0 + [ ( m 1 ) M ] L z z 0 + ( m M ) L z Q ( z ; ω ) d z ,
H ̃ m ( ω ) = z 0 + [ ( m 1 ) M ] L z z 0 + ( m M ) L z H ( z ; ω ) d z .
k = j L z ln ( eig { m = 1 M exp ( [ 0 Q ̃ m ( ω ) H ̃ m ( ω ) 0 ] ) } ) .
d 2 d z 2 [ a ¯ ] = Q ( z ) H ( z ) [ a ¯ ] .
[ a ¯ ] = n = 1 N ( c n + [ v ¯ n ] e λ n z + c n [ v ¯ n ] e + λ n z ) ,
( Q ̃ m × M L z ) [ b ¯ ] = n = 1 N ( c n + λ n [ v ¯ n ] e λ n z + c n λ n [ v ¯ n ] e + λ n z ) ,
( M L z ) 2 Q ̃ m H ̃ m ,
[ a ¯ b ¯ ] = [ V exp ( L + ) V exp ( L ) ( Q ̃ m × M L z ) 1 V L + exp ( L + ) ( Q ̃ m × M L z ) 1 V L exp ( L ) ] [ c ¯ n + c ¯ n ] ,
( M L z ) 2 Q ̃ m H ̃ m ,
[ [ c ¯ n + ] ( m + 1 ) [ c ¯ n ] ( m + 1 ) ] = t ̃ ( m ) [ [ c ¯ n + ] ( m ) [ c ¯ n ] ( m ) ] ,
t ̃ ( m ) = W ( m + 1 ) 1 W ( m ) Φ ( m ) ,
W ( m ) = [ W 11 ( m ) W 12 ( m ) W 21 ( m ) W 22 ( m ) ] = [ V ( m ) V ( m ) Q ( m ) 1 V ( m ) L ( m ) + Q ( m ) 1 V ( m ) L ( m ) ] ,
Φ ( m ) = [ exp ( L ( m ) + ) z = Δ z 0 0 exp ( L ( m ) ) z = Δ z ] .
[ a ¯ ( z + L z ) b ¯ ( z + L z ) ] = W ( 1 ) T W ( 1 ) 1 [ a ¯ ( z ) b ¯ ( z ) ] ,
T = m = 1 M t ̃ ( m ) ,
[ [ c ¯ n + ] ( m + 1 ) [ c ¯ n ] ( m ) ] = s ̃ ( m ) [ [ c ¯ n + ] ( m ) [ c ¯ n ] ( m + 1 ) ] ,
s ̃ ( m ) = [ s ̃ 11 s ̃ 12 s ̃ 21 s ̃ 22 ] ,
[ [ c ¯ n + ] ( m + 1 ) [ c ¯ n ] ( m ) exp ( L ( m ) ) z = Δ z ] = s ( m ) [ [ c ¯ n + ] ( m ) exp ( L ( m ) + ) z = Δ z [ c ¯ n ] ( m + 1 ) ] .
s ̃ ( m ) = [ I 0 0 exp ( L ( m ) + ) z = Δ z ] s ( m ) [ exp ( L ( m ) + ) z = Δ z 0 0 I ] .
s ( m ) = [ W 11 ( m + 1 ) W 12 ( m ) W 21 ( m + 1 ) W 22 ( m ) ] 1 [ W 11 ( m ) W 12 ( m + 1 ) W 21 ( m ) W 22 ( m + 1 ) ] .
[ [ c ¯ n + ] ( M + 1 ) [ c ¯ n ] ( 1 ) ] = S [ [ c ¯ n + ] ( 1 ) [ c ¯ n ] ( M + 1 ) ] ,
S ( m ) = [ S 11 S 12 S 21 S 22 ] .
[ [ c ¯ n + ] ( M + 1 ) [ c ¯ n ] ( 1 ) ] = [ D 0 0 D + ] [ [ c ¯ n + ] ( 1 ) [ c ¯ n ] ( M + 1 ) ] ,
e j k L z [ S 11 0 S 21 I ] [ [ c ¯ n + ] ( 1 ) [ c ¯ n ] ( 1 ) ] = [ I S 12 0 S 22 ] [ [ c ¯ n + ] ( 1 ) [ c ¯ n ] ( 1 ) ] ,
k = j L z ln ( eig { [ I S 12 0 S 22 ] , [ S 11 0 S 21 I ] } ) .
σ ( L ) = M N n 2 α m n ( L ) 2 M N α m n ( L ) 2 ,
α m n ( L ) = + ϴ ( x , ( m 1 2 ) L z M ) Ψ n ( 2 x L z ) d x .
Q = k 0 I ,
H = 1 k 0 G 2 k 0 D ,
Q = k 0 F 1 ,
H = 1 k 0 G D 1 G k 0 I ,
G = [ g m n ] ; g m n = 2 L Ψ n ( x ) Ψ m ( x ) d x ,
D = [ d m n ( z ) ] ; d m n ( z ) = Ψ m ( x ) Ψ n ( x ) ε r ( L x 2 , z ) d x ,
F = [ f m n ( z ) ] ; f m n ( z ) = Ψ m ( x ) Ψ n ( x ) ε r ( L x 2 , z ) d x ,

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