Abstract

We present a semianalytical formulation for calculating the supermodes and corresponding Bloch factors of light in hexagonal lattice photonic crystal waveguide arrays. We then use this formulation to easily calculate dispersion curves and predict propagation in systems too large to calculate using standard numerical methods.

© 2012 Optical Society of America

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References

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  1. E. A. J. Marcatili, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell Syst. Tech. J. 48, 2071–2102 (1969).
  2. H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett. 85, 1863–1866 (2000).
    [CrossRef]
  3. P. Yeh, Optical Waves in Layered Media, Vol. 95 (Wiley, 1988).
  4. N. W. Ashcroft and N. D. Mermin, Solid State Physics (Saunders College, 1976).
  5. A. Locatelli, M. Conforti, D. Modotto, and C. De Angelis, “Discrete negative refraction in photonic crystal waveguide arrays,” Opt. Lett. 31, 1343–1345 (2006).
    [CrossRef]
  6. C. M. de Sterke, L. C. Botten, A. A. Asatryan, T. P. White, and R. C. McPhedran, “Modes of coupled photonic crystal waveguides,” Opt. Lett. 29, 1384–1386 (2004).
    [CrossRef]
  7. L. C. Botten, R. Hansen, and C. M. de Sterke, “Tight binding analysis of coupled photonic crystal waveguides,” in Australian Conference on Optical Fibre Technology/Australian Optical Society, 2006. ACOFT/AOS 2006 (IEEE, 2006), pp. 109–111.
  8. S. Ha, A. A. Sukhorukov, K. B. Dossou, L. C. Botten, A. V. Lavrinenko, D. N. Chigrin, and Y. S. Kivshar, “Dispersionless tunneling of slow light in antisymmetric photonic crystal couplers,” Opt. Express 16, 1104–1114 (2008).
    [CrossRef]
  9. A. A. Sukhorukov, S. Ha, A. S. Desyatnikov, A. V. Lavrinenko, and Y. S. Kivshar, “Slow-light vortices in periodic waveguides,” J. Opt. A 11, 094016 (2009).
    [CrossRef]
  10. J. S. Brownless, S. Mahmoodian, K. B. Dossou, F. J. Lawrence, L. C. Botten, and C. M. de Sterke, “Coupled waveguide modes in hexagonal photonic crystals,” Opt. Express 18, 25346–25360 (2010).
    [CrossRef]
  11. L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, “Bloch mode scattering matrix methods for modeling extended photonic crystal structures. I. Theory,” Phys. Rev. E 70, 056606 (2004).
    [CrossRef]
  12. L. C. Botten, N. A. Nicorovici, A. A. Asatryan, R. C. McPhedran, C. M. de Sterke, and P. A. Robinson, “Formulation for electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations. Part I. Method,” J. Opt. Soc. Am. A 17, 2165–2176(2000).
    [CrossRef]
  13. F. J. Lawrence, L. C. Botten, K. B. Dossou, C. M. de Sterke, and R. C. McPhedran, “Impedance of square and triangular lattice photonic crystals,” Phys. Rev. A 80, 023826 (2009).
    [CrossRef]
  14. L. C. Botten, K. B. Dossou, S. Wilcox, R. C. McPhedran, C. M. de Sterke, N. A. Nicorovici, and A. A. Asatryan, “Highly accurate modelling of generalized defect modes in photonic crystals using the fictitious source superposition method,” Int. J. Microw. Opt. Technol 1, 133–145 (2006).
  15. S. D. M. Adams, R. V. Craster, and S. Guenneau, “Guided and standing Bloch waves in periodic elastic strips,” Waves Random Complex Media 19, 321–346 (2009).
    [CrossRef]
  16. J.-H. Sun and T.-T. Wu, “Analyses of mode coupling in joined parallel phononic crystal waveguides,” Phys. Rev. B 71, 174303 (2005).
    [CrossRef]

2010 (1)

2009 (3)

A. A. Sukhorukov, S. Ha, A. S. Desyatnikov, A. V. Lavrinenko, and Y. S. Kivshar, “Slow-light vortices in periodic waveguides,” J. Opt. A 11, 094016 (2009).
[CrossRef]

F. J. Lawrence, L. C. Botten, K. B. Dossou, C. M. de Sterke, and R. C. McPhedran, “Impedance of square and triangular lattice photonic crystals,” Phys. Rev. A 80, 023826 (2009).
[CrossRef]

S. D. M. Adams, R. V. Craster, and S. Guenneau, “Guided and standing Bloch waves in periodic elastic strips,” Waves Random Complex Media 19, 321–346 (2009).
[CrossRef]

2008 (1)

2006 (2)

A. Locatelli, M. Conforti, D. Modotto, and C. De Angelis, “Discrete negative refraction in photonic crystal waveguide arrays,” Opt. Lett. 31, 1343–1345 (2006).
[CrossRef]

L. C. Botten, K. B. Dossou, S. Wilcox, R. C. McPhedran, C. M. de Sterke, N. A. Nicorovici, and A. A. Asatryan, “Highly accurate modelling of generalized defect modes in photonic crystals using the fictitious source superposition method,” Int. J. Microw. Opt. Technol 1, 133–145 (2006).

2005 (1)

J.-H. Sun and T.-T. Wu, “Analyses of mode coupling in joined parallel phononic crystal waveguides,” Phys. Rev. B 71, 174303 (2005).
[CrossRef]

2004 (2)

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, “Bloch mode scattering matrix methods for modeling extended photonic crystal structures. I. Theory,” Phys. Rev. E 70, 056606 (2004).
[CrossRef]

C. M. de Sterke, L. C. Botten, A. A. Asatryan, T. P. White, and R. C. McPhedran, “Modes of coupled photonic crystal waveguides,” Opt. Lett. 29, 1384–1386 (2004).
[CrossRef]

2000 (2)

1969 (1)

E. A. J. Marcatili, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell Syst. Tech. J. 48, 2071–2102 (1969).

Adams, S. D. M.

S. D. M. Adams, R. V. Craster, and S. Guenneau, “Guided and standing Bloch waves in periodic elastic strips,” Waves Random Complex Media 19, 321–346 (2009).
[CrossRef]

Aitchison, J. S.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett. 85, 1863–1866 (2000).
[CrossRef]

Asatryan, A. A.

L. C. Botten, K. B. Dossou, S. Wilcox, R. C. McPhedran, C. M. de Sterke, N. A. Nicorovici, and A. A. Asatryan, “Highly accurate modelling of generalized defect modes in photonic crystals using the fictitious source superposition method,” Int. J. Microw. Opt. Technol 1, 133–145 (2006).

C. M. de Sterke, L. C. Botten, A. A. Asatryan, T. P. White, and R. C. McPhedran, “Modes of coupled photonic crystal waveguides,” Opt. Lett. 29, 1384–1386 (2004).
[CrossRef]

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, “Bloch mode scattering matrix methods for modeling extended photonic crystal structures. I. Theory,” Phys. Rev. E 70, 056606 (2004).
[CrossRef]

L. C. Botten, N. A. Nicorovici, A. A. Asatryan, R. C. McPhedran, C. M. de Sterke, and P. A. Robinson, “Formulation for electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations. Part I. Method,” J. Opt. Soc. Am. A 17, 2165–2176(2000).
[CrossRef]

Ashcroft, N. W.

N. W. Ashcroft and N. D. Mermin, Solid State Physics (Saunders College, 1976).

Botten, L. C.

J. S. Brownless, S. Mahmoodian, K. B. Dossou, F. J. Lawrence, L. C. Botten, and C. M. de Sterke, “Coupled waveguide modes in hexagonal photonic crystals,” Opt. Express 18, 25346–25360 (2010).
[CrossRef]

F. J. Lawrence, L. C. Botten, K. B. Dossou, C. M. de Sterke, and R. C. McPhedran, “Impedance of square and triangular lattice photonic crystals,” Phys. Rev. A 80, 023826 (2009).
[CrossRef]

S. Ha, A. A. Sukhorukov, K. B. Dossou, L. C. Botten, A. V. Lavrinenko, D. N. Chigrin, and Y. S. Kivshar, “Dispersionless tunneling of slow light in antisymmetric photonic crystal couplers,” Opt. Express 16, 1104–1114 (2008).
[CrossRef]

L. C. Botten, K. B. Dossou, S. Wilcox, R. C. McPhedran, C. M. de Sterke, N. A. Nicorovici, and A. A. Asatryan, “Highly accurate modelling of generalized defect modes in photonic crystals using the fictitious source superposition method,” Int. J. Microw. Opt. Technol 1, 133–145 (2006).

C. M. de Sterke, L. C. Botten, A. A. Asatryan, T. P. White, and R. C. McPhedran, “Modes of coupled photonic crystal waveguides,” Opt. Lett. 29, 1384–1386 (2004).
[CrossRef]

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, “Bloch mode scattering matrix methods for modeling extended photonic crystal structures. I. Theory,” Phys. Rev. E 70, 056606 (2004).
[CrossRef]

L. C. Botten, N. A. Nicorovici, A. A. Asatryan, R. C. McPhedran, C. M. de Sterke, and P. A. Robinson, “Formulation for electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations. Part I. Method,” J. Opt. Soc. Am. A 17, 2165–2176(2000).
[CrossRef]

L. C. Botten, R. Hansen, and C. M. de Sterke, “Tight binding analysis of coupled photonic crystal waveguides,” in Australian Conference on Optical Fibre Technology/Australian Optical Society, 2006. ACOFT/AOS 2006 (IEEE, 2006), pp. 109–111.

Brownless, J. S.

Chigrin, D. N.

Conforti, M.

Craster, R. V.

S. D. M. Adams, R. V. Craster, and S. Guenneau, “Guided and standing Bloch waves in periodic elastic strips,” Waves Random Complex Media 19, 321–346 (2009).
[CrossRef]

De Angelis, C.

de Sterke, C. M.

J. S. Brownless, S. Mahmoodian, K. B. Dossou, F. J. Lawrence, L. C. Botten, and C. M. de Sterke, “Coupled waveguide modes in hexagonal photonic crystals,” Opt. Express 18, 25346–25360 (2010).
[CrossRef]

F. J. Lawrence, L. C. Botten, K. B. Dossou, C. M. de Sterke, and R. C. McPhedran, “Impedance of square and triangular lattice photonic crystals,” Phys. Rev. A 80, 023826 (2009).
[CrossRef]

L. C. Botten, K. B. Dossou, S. Wilcox, R. C. McPhedran, C. M. de Sterke, N. A. Nicorovici, and A. A. Asatryan, “Highly accurate modelling of generalized defect modes in photonic crystals using the fictitious source superposition method,” Int. J. Microw. Opt. Technol 1, 133–145 (2006).

C. M. de Sterke, L. C. Botten, A. A. Asatryan, T. P. White, and R. C. McPhedran, “Modes of coupled photonic crystal waveguides,” Opt. Lett. 29, 1384–1386 (2004).
[CrossRef]

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, “Bloch mode scattering matrix methods for modeling extended photonic crystal structures. I. Theory,” Phys. Rev. E 70, 056606 (2004).
[CrossRef]

L. C. Botten, N. A. Nicorovici, A. A. Asatryan, R. C. McPhedran, C. M. de Sterke, and P. A. Robinson, “Formulation for electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations. Part I. Method,” J. Opt. Soc. Am. A 17, 2165–2176(2000).
[CrossRef]

L. C. Botten, R. Hansen, and C. M. de Sterke, “Tight binding analysis of coupled photonic crystal waveguides,” in Australian Conference on Optical Fibre Technology/Australian Optical Society, 2006. ACOFT/AOS 2006 (IEEE, 2006), pp. 109–111.

Desyatnikov, A. S.

A. A. Sukhorukov, S. Ha, A. S. Desyatnikov, A. V. Lavrinenko, and Y. S. Kivshar, “Slow-light vortices in periodic waveguides,” J. Opt. A 11, 094016 (2009).
[CrossRef]

Dossou, K. B.

J. S. Brownless, S. Mahmoodian, K. B. Dossou, F. J. Lawrence, L. C. Botten, and C. M. de Sterke, “Coupled waveguide modes in hexagonal photonic crystals,” Opt. Express 18, 25346–25360 (2010).
[CrossRef]

F. J. Lawrence, L. C. Botten, K. B. Dossou, C. M. de Sterke, and R. C. McPhedran, “Impedance of square and triangular lattice photonic crystals,” Phys. Rev. A 80, 023826 (2009).
[CrossRef]

S. Ha, A. A. Sukhorukov, K. B. Dossou, L. C. Botten, A. V. Lavrinenko, D. N. Chigrin, and Y. S. Kivshar, “Dispersionless tunneling of slow light in antisymmetric photonic crystal couplers,” Opt. Express 16, 1104–1114 (2008).
[CrossRef]

L. C. Botten, K. B. Dossou, S. Wilcox, R. C. McPhedran, C. M. de Sterke, N. A. Nicorovici, and A. A. Asatryan, “Highly accurate modelling of generalized defect modes in photonic crystals using the fictitious source superposition method,” Int. J. Microw. Opt. Technol 1, 133–145 (2006).

Eisenberg, H. S.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett. 85, 1863–1866 (2000).
[CrossRef]

Guenneau, S.

S. D. M. Adams, R. V. Craster, and S. Guenneau, “Guided and standing Bloch waves in periodic elastic strips,” Waves Random Complex Media 19, 321–346 (2009).
[CrossRef]

Ha, S.

Hansen, R.

L. C. Botten, R. Hansen, and C. M. de Sterke, “Tight binding analysis of coupled photonic crystal waveguides,” in Australian Conference on Optical Fibre Technology/Australian Optical Society, 2006. ACOFT/AOS 2006 (IEEE, 2006), pp. 109–111.

Kivshar, Y. S.

Langtry, T. N.

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, “Bloch mode scattering matrix methods for modeling extended photonic crystal structures. I. Theory,” Phys. Rev. E 70, 056606 (2004).
[CrossRef]

Lavrinenko, A. V.

Lawrence, F. J.

J. S. Brownless, S. Mahmoodian, K. B. Dossou, F. J. Lawrence, L. C. Botten, and C. M. de Sterke, “Coupled waveguide modes in hexagonal photonic crystals,” Opt. Express 18, 25346–25360 (2010).
[CrossRef]

F. J. Lawrence, L. C. Botten, K. B. Dossou, C. M. de Sterke, and R. C. McPhedran, “Impedance of square and triangular lattice photonic crystals,” Phys. Rev. A 80, 023826 (2009).
[CrossRef]

Locatelli, A.

Mahmoodian, S.

Marcatili, E. A. J.

E. A. J. Marcatili, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell Syst. Tech. J. 48, 2071–2102 (1969).

McPhedran, R. C.

F. J. Lawrence, L. C. Botten, K. B. Dossou, C. M. de Sterke, and R. C. McPhedran, “Impedance of square and triangular lattice photonic crystals,” Phys. Rev. A 80, 023826 (2009).
[CrossRef]

L. C. Botten, K. B. Dossou, S. Wilcox, R. C. McPhedran, C. M. de Sterke, N. A. Nicorovici, and A. A. Asatryan, “Highly accurate modelling of generalized defect modes in photonic crystals using the fictitious source superposition method,” Int. J. Microw. Opt. Technol 1, 133–145 (2006).

C. M. de Sterke, L. C. Botten, A. A. Asatryan, T. P. White, and R. C. McPhedran, “Modes of coupled photonic crystal waveguides,” Opt. Lett. 29, 1384–1386 (2004).
[CrossRef]

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, “Bloch mode scattering matrix methods for modeling extended photonic crystal structures. I. Theory,” Phys. Rev. E 70, 056606 (2004).
[CrossRef]

L. C. Botten, N. A. Nicorovici, A. A. Asatryan, R. C. McPhedran, C. M. de Sterke, and P. A. Robinson, “Formulation for electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations. Part I. Method,” J. Opt. Soc. Am. A 17, 2165–2176(2000).
[CrossRef]

Mermin, N. D.

N. W. Ashcroft and N. D. Mermin, Solid State Physics (Saunders College, 1976).

Modotto, D.

Morandotti, R.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett. 85, 1863–1866 (2000).
[CrossRef]

Nicorovici, N. A.

L. C. Botten, K. B. Dossou, S. Wilcox, R. C. McPhedran, C. M. de Sterke, N. A. Nicorovici, and A. A. Asatryan, “Highly accurate modelling of generalized defect modes in photonic crystals using the fictitious source superposition method,” Int. J. Microw. Opt. Technol 1, 133–145 (2006).

L. C. Botten, N. A. Nicorovici, A. A. Asatryan, R. C. McPhedran, C. M. de Sterke, and P. A. Robinson, “Formulation for electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations. Part I. Method,” J. Opt. Soc. Am. A 17, 2165–2176(2000).
[CrossRef]

Robinson, P. A.

Silberberg, Y.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett. 85, 1863–1866 (2000).
[CrossRef]

Sukhorukov, A. A.

Sun, J.-H.

J.-H. Sun and T.-T. Wu, “Analyses of mode coupling in joined parallel phononic crystal waveguides,” Phys. Rev. B 71, 174303 (2005).
[CrossRef]

White, T. P.

C. M. de Sterke, L. C. Botten, A. A. Asatryan, T. P. White, and R. C. McPhedran, “Modes of coupled photonic crystal waveguides,” Opt. Lett. 29, 1384–1386 (2004).
[CrossRef]

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, “Bloch mode scattering matrix methods for modeling extended photonic crystal structures. I. Theory,” Phys. Rev. E 70, 056606 (2004).
[CrossRef]

Wilcox, S.

L. C. Botten, K. B. Dossou, S. Wilcox, R. C. McPhedran, C. M. de Sterke, N. A. Nicorovici, and A. A. Asatryan, “Highly accurate modelling of generalized defect modes in photonic crystals using the fictitious source superposition method,” Int. J. Microw. Opt. Technol 1, 133–145 (2006).

Wu, T.-T.

J.-H. Sun and T.-T. Wu, “Analyses of mode coupling in joined parallel phononic crystal waveguides,” Phys. Rev. B 71, 174303 (2005).
[CrossRef]

Yeh, P.

P. Yeh, Optical Waves in Layered Media, Vol. 95 (Wiley, 1988).

Bell Syst. Tech. J. (1)

E. A. J. Marcatili, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell Syst. Tech. J. 48, 2071–2102 (1969).

Int. J. Microw. Opt. Technol (1)

L. C. Botten, K. B. Dossou, S. Wilcox, R. C. McPhedran, C. M. de Sterke, N. A. Nicorovici, and A. A. Asatryan, “Highly accurate modelling of generalized defect modes in photonic crystals using the fictitious source superposition method,” Int. J. Microw. Opt. Technol 1, 133–145 (2006).

J. Opt. A (1)

A. A. Sukhorukov, S. Ha, A. S. Desyatnikov, A. V. Lavrinenko, and Y. S. Kivshar, “Slow-light vortices in periodic waveguides,” J. Opt. A 11, 094016 (2009).
[CrossRef]

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

Opt. Express (2)

Opt. Lett. (2)

Phys. Rev. A (1)

F. J. Lawrence, L. C. Botten, K. B. Dossou, C. M. de Sterke, and R. C. McPhedran, “Impedance of square and triangular lattice photonic crystals,” Phys. Rev. A 80, 023826 (2009).
[CrossRef]

Phys. Rev. B (1)

J.-H. Sun and T.-T. Wu, “Analyses of mode coupling in joined parallel phononic crystal waveguides,” Phys. Rev. B 71, 174303 (2005).
[CrossRef]

Phys. Rev. E (1)

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, “Bloch mode scattering matrix methods for modeling extended photonic crystal structures. I. Theory,” Phys. Rev. E 70, 056606 (2004).
[CrossRef]

Phys. Rev. Lett. (1)

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett. 85, 1863–1866 (2000).
[CrossRef]

Waves Random Complex Media (1)

S. D. M. Adams, R. V. Craster, and S. Guenneau, “Guided and standing Bloch waves in periodic elastic strips,” Waves Random Complex Media 19, 321–346 (2009).
[CrossRef]

Other (3)

P. Yeh, Optical Waves in Layered Media, Vol. 95 (Wiley, 1988).

N. W. Ashcroft and N. D. Mermin, Solid State Physics (Saunders College, 1976).

L. C. Botten, R. Hansen, and C. M. de Sterke, “Tight binding analysis of coupled photonic crystal waveguides,” in Australian Conference on Optical Fibre Technology/Australian Optical Society, 2006. ACOFT/AOS 2006 (IEEE, 2006), pp. 109–111.

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Figures (7)

Fig. 1.
Fig. 1.

(a) Unit cell for a hexagonal lattice PC and corresponding lattice vectors. (b) The two different coupled waveguide geometries in a hexagonal lattice. Staggered (left), separated by an even number of defect rows, and in-line (right), separated by an odd number of defect rows.

Fig. 2.
Fig. 2.

(a) Bloch mode amplitudes defined across an interface between PC 1 and PC 2, described in Eq. (1). (b) Bloch mode amplitudes defined in a PCW comprised of PC 2 sandwiched between layers of PC 1. (c) Shifted Bloch mode amplitudes defined in the same PCW as (b).

Fig. 3.
Fig. 3.

(a) Bloch mode amplitudes defined across a barrier consisting of rows of PC 1 between two waveguides. The broken lines indicate a perfectly matched layer, rather than an interface. (b) Bloch mode amplitudes defined in a section of the PCW array.

Fig. 4.
Fig. 4.

Comparison of the dispersion relation calculated using the FSS (green dots), and our semianalytic modal method (red lines) for a four-waveguide array with waveguides separated by four rows of inclusions (shown in inset). Background refractive index, n=3, cylindrical inclusions of index n=1 with the waveguide created by changing the refractive index of the waveguide inclusions to n=1.5.

Fig. 5.
Fig. 5.

Dispersion curve for a M=31 PCW array with four rows of inclusions between each waveguide. The background index is n=3, the bulk photonic crystal inclusions have index n=1, and the waveguide is formed with inclusions of index n=1.5.

Fig. 6.
Fig. 6.

Comparison of methods for calculating the field throughout the waveguide array at frequency d/λ=0.293. One point is taken per unit cell of the array shown in Fig. 4. The colors represent the different waveguides; the open markers are from Bloch mode supercell calculations (SCM), and the solid markers are results from our modal method (MM).

Fig. 7.
Fig. 7.

Propagation of light through the M=31 PCW array described in Fig. 5. The initial condition is that all the energy begins in the PCW mode centered at waveguide j=16 with a frequency of d/λ=0.293.

Equations (55)

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

c1+=R12c1+T21c2+,
c2=R21c2++T12c1.
c+=R21Λ2c,
c=eikxdxR21Λ2c+,
c˜±=c±eikxxROW,
Λ˜=Λeikxdx/2,
c˜+=R21Λ˜2c˜,
c˜=R21Λ˜2c˜+,
c˜=R21Λ˜2R21Λ˜2c˜,
0=(I±R21Λ˜2)c˜.
det(I±R21Λ˜2)=0.
c˜+=+c˜,or
c˜+=c˜,
e˜=12(c˜+c˜+),
Hswg(r)=(Λ˜2c˜+)TΨ+(r)+c˜TΨ(r),
=e˜T(Λ˜2Ψ+(r)+Ψ(r)),
c˜j+=R˜Λ˜2c˜j+T˜Λ˜2c˜j+1+,
c˜j+1=T˜Λ˜2c˜j+R˜Λ˜2c˜j+1+,
c˜j+=R21Λ˜2c˜j+T12Λ˜1c˜b+,
c˜j+1=T12Λ˜1c˜b,
c˜b+=R12Λ˜1c˜b,
c˜b=R12Λ˜1c˜b++T21Λ˜2c˜j.
c˜j+1=T12Λ˜1(IR12Λ˜1R12Λ˜1)1T21Λ˜2c˜j=T˜Λ˜2c˜j,
T˜=T12Λ˜1(IR12Λ˜1R12Λ˜1)1T21.
R˜=R21+T12Λ˜1R12Λ˜1(IR12Λ˜1R12Λ˜1)1T21.
T˜=T12Λ˜1T21,
R˜=R21.
c˜1=R21Λ˜2c˜1+,
c˜1+=R21Λ˜2c˜1+T˜Λ˜2c˜2+.
c˜M=R21Λ˜2c˜M++T˜Λ˜2c˜M1,
c˜M+=R21Λ˜2c˜M.
c˜j=R21Λ˜2c˜j++T˜Λ˜2c˜j1,
c˜j+=R21Λ˜2c˜j+T˜Λ˜2c˜j+1+.
e˜j=12(c˜j+c˜j+).
e˜1=R˜Λ˜2e˜1+12T˜Λ˜2c˜2+,
e˜j=R˜Λ˜2e˜j+12T˜Λ˜2c˜j+1++12T˜Λ˜2c˜j1,
e˜M=R˜Λ˜2e˜M+12T˜Λ˜2c˜M1.
(AB00BAB00BA0000A)(e˜1e˜2e˜3e˜M)=(e˜1e˜2e˜3e˜M),
Φjs=ϕjsI,
ϕjs=2M+1sin(jsπM+1).
(ABBA)(e˜1e˜2)=(e˜1e˜2).
Φ1(ABBA)ΦΦ1(e˜1e˜2)=Φ1(e˜1e˜2),
(A+B00AB)(e˜1+e˜2e˜1e˜2)=(e˜1+e˜2e˜1e˜2)(x˜1x˜2).
(AσsB)x˜s=x˜s,
(e˜1e˜2)=Φsxs,
(Ieikxd/2R21Λ2f+σs2eikxd(+1)/2T12Λ1fT21Λ2f)x˜s=0.
e˜j=Φjse˜=ϕjse˜.
a˜j=sϕjsws,
i.e.a˜=ϕw,
a˜j(p)=sϕjsμ1spws(0),
i.e.,a˜(p)=ϕLpw(0),
a˜(p)=ϕLϕ1a˜(0),
aj=κja˜j,
H(r)=jκja˜jHswg(rRj),
Hj(r)=κja˜jHswg(rRj).

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