Abstract

We design a photonic crystal T-junction waveguide on the basis of the nonradiative surface mode and further optimize the transmission ratio of our surface mode photonic crystal T-junction waveguide with the guide of coupled-mode theory. Compared with a traditional bandgap photonic crystal T-junction waveguide, our surface mode T-junction waveguide can obtain almost zero reflectance and good transmission. Better still, its size is just half of the size of bandgap photonic crystal waveguide branches. To the best of our knowledge, it is the first time that such a high-efficiency T-junction waveguide that uses the surface mode as the guided mode is proposed.

© 2011 Optical Society of America

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  1. A. Mekis, J. C. Chen, I. Kurland, S. H. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. B 77, 3787–3790(1996).
    [Crossref]
  2. S. G. Johnson, P. R. Villeneuve, S. H. Fan, and J. D. Joannopoulos, “Linear waveguides in photonic-crystal slabs,” Phys. Rev. B 62, 8212–8222 (2000).
    [Crossref]
  3. S. H. Fan, S. G. Johnson, J. D. Joannopoulos, C. Manolatou, and H. A. Haus, “Waveguide branches in photonic crystals,” J. Opt. Soc. Am. B 18, 162–165 (2001).
    [Crossref]
  4. S. H. Fan, “Sharp asymmetric line shapes in side-coupled waveguide-cavity systems,” Appl. Phys. Lett. 80, 908–910 (2002).
    [Crossref]
  5. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University Press, 2008).
  6. E. Moreno, F. J. García-Vidal, and L. Martín-Moreno, “Enhanced transmission and beaming of light via photonic crystal surface modes,” Phys. Rev. B 69, 121402 (2004).
    [Crossref]
  7. P. Kramper, M. Agio, C. M. Soukoulis, A. Birner, F. Müller, R. B. Wehrspohn, U. Gösele, and V. Sandoghdar, “Highly directional emission from photonic crystal waveguides of subwavelength width,” Phys. Rev. Lett. 92, 113903 (2004).
    [Crossref] [PubMed]
  8. S. K. Morrison and Y. S. Kivshar, “Engineering of directional emission from photonic-crystal waveguides,” Appl. Phys. Lett. 86, 081110 (2005).
    [Crossref]
  9. D. Y. Zhao and X. Y. Jiang, “Vertical surface emitting open coupled-cavities based on photonic crystal surface modes,” Appl. Opt. 47, 1729–1733 (2008).
    [Crossref] [PubMed]
  10. D. Zhao, C. Zhou, Y. Zhang, L. Shi, and X. Jiang, “Vertical cavity-surface emitting photonic crystal surface-mode lasers,” Appl. Phys. B 91, 475–478 (2008).
    [Crossref]
  11. R. Moussa, B. Wang, G. Tuttle, Th. Koschny, and C. M. Soukoulis, “Effect of beaming and enhanced transmission in photonic crystals,” Phys. Rev. B 76, 235417 (2007).
    [Crossref]
  12. W. Śmigaj, “Model of light collimation by photonic crystal surface modes,” Phys. Rev. B 75, 205430 (2007).
    [Crossref]
  13. H. B. Chen, Y. Zeng, X. S. Chen, J. Wang, and W. Lu, “Modulation of focus using photonic crystal waveguide,” Phys. Lett. A 372, 5096–5100 (2008).
    [Crossref]
  14. K. B. Chung, “Analysis of directional emission via surface modes on photonic crystals,” Opt. Commun. 281, 5349–5354 (2008).
    [Crossref]
  15. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984).
  16. C. Manolatou, S. G. Johnson, S. H. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “High-density integrated optics,” J. Lightwave Technol. 17, 1682–1692 (1999).
    [Crossref]
  17. S. H. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A 20, 569–572 (2003).
    [Crossref]

2008 (4)

D. Y. Zhao and X. Y. Jiang, “Vertical surface emitting open coupled-cavities based on photonic crystal surface modes,” Appl. Opt. 47, 1729–1733 (2008).
[Crossref] [PubMed]

D. Zhao, C. Zhou, Y. Zhang, L. Shi, and X. Jiang, “Vertical cavity-surface emitting photonic crystal surface-mode lasers,” Appl. Phys. B 91, 475–478 (2008).
[Crossref]

H. B. Chen, Y. Zeng, X. S. Chen, J. Wang, and W. Lu, “Modulation of focus using photonic crystal waveguide,” Phys. Lett. A 372, 5096–5100 (2008).
[Crossref]

K. B. Chung, “Analysis of directional emission via surface modes on photonic crystals,” Opt. Commun. 281, 5349–5354 (2008).
[Crossref]

2007 (2)

R. Moussa, B. Wang, G. Tuttle, Th. Koschny, and C. M. Soukoulis, “Effect of beaming and enhanced transmission in photonic crystals,” Phys. Rev. B 76, 235417 (2007).
[Crossref]

W. Śmigaj, “Model of light collimation by photonic crystal surface modes,” Phys. Rev. B 75, 205430 (2007).
[Crossref]

2005 (1)

S. K. Morrison and Y. S. Kivshar, “Engineering of directional emission from photonic-crystal waveguides,” Appl. Phys. Lett. 86, 081110 (2005).
[Crossref]

2004 (2)

E. Moreno, F. J. García-Vidal, and L. Martín-Moreno, “Enhanced transmission and beaming of light via photonic crystal surface modes,” Phys. Rev. B 69, 121402 (2004).
[Crossref]

P. Kramper, M. Agio, C. M. Soukoulis, A. Birner, F. Müller, R. B. Wehrspohn, U. Gösele, and V. Sandoghdar, “Highly directional emission from photonic crystal waveguides of subwavelength width,” Phys. Rev. Lett. 92, 113903 (2004).
[Crossref] [PubMed]

2003 (1)

2002 (1)

S. H. Fan, “Sharp asymmetric line shapes in side-coupled waveguide-cavity systems,” Appl. Phys. Lett. 80, 908–910 (2002).
[Crossref]

2001 (1)

2000 (1)

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

1999 (1)

1996 (1)

A. Mekis, J. C. Chen, I. Kurland, S. H. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. B 77, 3787–3790(1996).
[Crossref]

Agio, M.

P. Kramper, M. Agio, C. M. Soukoulis, A. Birner, F. Müller, R. B. Wehrspohn, U. Gösele, and V. Sandoghdar, “Highly directional emission from photonic crystal waveguides of subwavelength width,” Phys. Rev. Lett. 92, 113903 (2004).
[Crossref] [PubMed]

Birner, A.

P. Kramper, M. Agio, C. M. Soukoulis, A. Birner, F. Müller, R. B. Wehrspohn, U. Gösele, and V. Sandoghdar, “Highly directional emission from photonic crystal waveguides of subwavelength width,” Phys. Rev. Lett. 92, 113903 (2004).
[Crossref] [PubMed]

Chen, H. B.

H. B. Chen, Y. Zeng, X. S. Chen, J. Wang, and W. Lu, “Modulation of focus using photonic crystal waveguide,” Phys. Lett. A 372, 5096–5100 (2008).
[Crossref]

Chen, J. C.

A. Mekis, J. C. Chen, I. Kurland, S. H. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. B 77, 3787–3790(1996).
[Crossref]

Chen, X. S.

H. B. Chen, Y. Zeng, X. S. Chen, J. Wang, and W. Lu, “Modulation of focus using photonic crystal waveguide,” Phys. Lett. A 372, 5096–5100 (2008).
[Crossref]

Chung, K. B.

K. B. Chung, “Analysis of directional emission via surface modes on photonic crystals,” Opt. Commun. 281, 5349–5354 (2008).
[Crossref]

Fan, S. H.

S. H. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A 20, 569–572 (2003).
[Crossref]

S. H. Fan, “Sharp asymmetric line shapes in side-coupled waveguide-cavity systems,” Appl. Phys. Lett. 80, 908–910 (2002).
[Crossref]

S. H. Fan, S. G. Johnson, J. D. Joannopoulos, C. Manolatou, and H. A. Haus, “Waveguide branches in photonic crystals,” J. Opt. Soc. Am. B 18, 162–165 (2001).
[Crossref]

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

C. Manolatou, S. G. Johnson, S. H. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “High-density integrated optics,” J. Lightwave Technol. 17, 1682–1692 (1999).
[Crossref]

A. Mekis, J. C. Chen, I. Kurland, S. H. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. B 77, 3787–3790(1996).
[Crossref]

García-Vidal, F. J.

E. Moreno, F. J. García-Vidal, and L. Martín-Moreno, “Enhanced transmission and beaming of light via photonic crystal surface modes,” Phys. Rev. B 69, 121402 (2004).
[Crossref]

Gösele, U.

P. Kramper, M. Agio, C. M. Soukoulis, A. Birner, F. Müller, R. B. Wehrspohn, U. Gösele, and V. Sandoghdar, “Highly directional emission from photonic crystal waveguides of subwavelength width,” Phys. Rev. Lett. 92, 113903 (2004).
[Crossref] [PubMed]

Haus, H. A.

Jiang, X.

D. Zhao, C. Zhou, Y. Zhang, L. Shi, and X. Jiang, “Vertical cavity-surface emitting photonic crystal surface-mode lasers,” Appl. Phys. B 91, 475–478 (2008).
[Crossref]

Jiang, X. Y.

Joannopoulos, J. D.

S. H. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A 20, 569–572 (2003).
[Crossref]

S. H. Fan, S. G. Johnson, J. D. Joannopoulos, C. Manolatou, and H. A. Haus, “Waveguide branches in photonic crystals,” J. Opt. Soc. Am. B 18, 162–165 (2001).
[Crossref]

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

C. Manolatou, S. G. Johnson, S. H. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “High-density integrated optics,” J. Lightwave Technol. 17, 1682–1692 (1999).
[Crossref]

A. Mekis, J. C. Chen, I. Kurland, S. H. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. B 77, 3787–3790(1996).
[Crossref]

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University Press, 2008).

Johnson, S. G.

S. H. Fan, S. G. Johnson, J. D. Joannopoulos, C. Manolatou, and H. A. Haus, “Waveguide branches in photonic crystals,” J. Opt. Soc. Am. B 18, 162–165 (2001).
[Crossref]

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

C. Manolatou, S. G. Johnson, S. H. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “High-density integrated optics,” J. Lightwave Technol. 17, 1682–1692 (1999).
[Crossref]

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University Press, 2008).

Kivshar, Y. S.

S. K. Morrison and Y. S. Kivshar, “Engineering of directional emission from photonic-crystal waveguides,” Appl. Phys. Lett. 86, 081110 (2005).
[Crossref]

Koschny, Th.

R. Moussa, B. Wang, G. Tuttle, Th. Koschny, and C. M. Soukoulis, “Effect of beaming and enhanced transmission in photonic crystals,” Phys. Rev. B 76, 235417 (2007).
[Crossref]

Kramper, P.

P. Kramper, M. Agio, C. M. Soukoulis, A. Birner, F. Müller, R. B. Wehrspohn, U. Gösele, and V. Sandoghdar, “Highly directional emission from photonic crystal waveguides of subwavelength width,” Phys. Rev. Lett. 92, 113903 (2004).
[Crossref] [PubMed]

Kurland, I.

A. Mekis, J. C. Chen, I. Kurland, S. H. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. B 77, 3787–3790(1996).
[Crossref]

Lu, W.

H. B. Chen, Y. Zeng, X. S. Chen, J. Wang, and W. Lu, “Modulation of focus using photonic crystal waveguide,” Phys. Lett. A 372, 5096–5100 (2008).
[Crossref]

Manolatou, C.

Martín-Moreno, L.

E. Moreno, F. J. García-Vidal, and L. Martín-Moreno, “Enhanced transmission and beaming of light via photonic crystal surface modes,” Phys. Rev. B 69, 121402 (2004).
[Crossref]

Meade, R. D.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University Press, 2008).

Mekis, A.

A. Mekis, J. C. Chen, I. Kurland, S. H. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. B 77, 3787–3790(1996).
[Crossref]

Moreno, E.

E. Moreno, F. J. García-Vidal, and L. Martín-Moreno, “Enhanced transmission and beaming of light via photonic crystal surface modes,” Phys. Rev. B 69, 121402 (2004).
[Crossref]

Morrison, S. K.

S. K. Morrison and Y. S. Kivshar, “Engineering of directional emission from photonic-crystal waveguides,” Appl. Phys. Lett. 86, 081110 (2005).
[Crossref]

Moussa, R.

R. Moussa, B. Wang, G. Tuttle, Th. Koschny, and C. M. Soukoulis, “Effect of beaming and enhanced transmission in photonic crystals,” Phys. Rev. B 76, 235417 (2007).
[Crossref]

Müller, F.

P. Kramper, M. Agio, C. M. Soukoulis, A. Birner, F. Müller, R. B. Wehrspohn, U. Gösele, and V. Sandoghdar, “Highly directional emission from photonic crystal waveguides of subwavelength width,” Phys. Rev. Lett. 92, 113903 (2004).
[Crossref] [PubMed]

Sandoghdar, V.

P. Kramper, M. Agio, C. M. Soukoulis, A. Birner, F. Müller, R. B. Wehrspohn, U. Gösele, and V. Sandoghdar, “Highly directional emission from photonic crystal waveguides of subwavelength width,” Phys. Rev. Lett. 92, 113903 (2004).
[Crossref] [PubMed]

Shi, L.

D. Zhao, C. Zhou, Y. Zhang, L. Shi, and X. Jiang, “Vertical cavity-surface emitting photonic crystal surface-mode lasers,” Appl. Phys. B 91, 475–478 (2008).
[Crossref]

Smigaj, W.

W. Śmigaj, “Model of light collimation by photonic crystal surface modes,” Phys. Rev. B 75, 205430 (2007).
[Crossref]

Soukoulis, C. M.

R. Moussa, B. Wang, G. Tuttle, Th. Koschny, and C. M. Soukoulis, “Effect of beaming and enhanced transmission in photonic crystals,” Phys. Rev. B 76, 235417 (2007).
[Crossref]

P. Kramper, M. Agio, C. M. Soukoulis, A. Birner, F. Müller, R. B. Wehrspohn, U. Gösele, and V. Sandoghdar, “Highly directional emission from photonic crystal waveguides of subwavelength width,” Phys. Rev. Lett. 92, 113903 (2004).
[Crossref] [PubMed]

Suh, W.

Tuttle, G.

R. Moussa, B. Wang, G. Tuttle, Th. Koschny, and C. M. Soukoulis, “Effect of beaming and enhanced transmission in photonic crystals,” Phys. Rev. B 76, 235417 (2007).
[Crossref]

Villeneuve, P. R.

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

C. Manolatou, S. G. Johnson, S. H. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “High-density integrated optics,” J. Lightwave Technol. 17, 1682–1692 (1999).
[Crossref]

A. Mekis, J. C. Chen, I. Kurland, S. H. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. B 77, 3787–3790(1996).
[Crossref]

Wang, B.

R. Moussa, B. Wang, G. Tuttle, Th. Koschny, and C. M. Soukoulis, “Effect of beaming and enhanced transmission in photonic crystals,” Phys. Rev. B 76, 235417 (2007).
[Crossref]

Wang, J.

H. B. Chen, Y. Zeng, X. S. Chen, J. Wang, and W. Lu, “Modulation of focus using photonic crystal waveguide,” Phys. Lett. A 372, 5096–5100 (2008).
[Crossref]

Wehrspohn, R. B.

P. Kramper, M. Agio, C. M. Soukoulis, A. Birner, F. Müller, R. B. Wehrspohn, U. Gösele, and V. Sandoghdar, “Highly directional emission from photonic crystal waveguides of subwavelength width,” Phys. Rev. Lett. 92, 113903 (2004).
[Crossref] [PubMed]

Winn, J. N.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University Press, 2008).

Zeng, Y.

H. B. Chen, Y. Zeng, X. S. Chen, J. Wang, and W. Lu, “Modulation of focus using photonic crystal waveguide,” Phys. Lett. A 372, 5096–5100 (2008).
[Crossref]

Zhang, Y.

D. Zhao, C. Zhou, Y. Zhang, L. Shi, and X. Jiang, “Vertical cavity-surface emitting photonic crystal surface-mode lasers,” Appl. Phys. B 91, 475–478 (2008).
[Crossref]

Zhao, D.

D. Zhao, C. Zhou, Y. Zhang, L. Shi, and X. Jiang, “Vertical cavity-surface emitting photonic crystal surface-mode lasers,” Appl. Phys. B 91, 475–478 (2008).
[Crossref]

Zhao, D. Y.

Zhou, C.

D. Zhao, C. Zhou, Y. Zhang, L. Shi, and X. Jiang, “Vertical cavity-surface emitting photonic crystal surface-mode lasers,” Appl. Phys. B 91, 475–478 (2008).
[Crossref]

Appl. Opt. (1)

Appl. Phys. B (1)

D. Zhao, C. Zhou, Y. Zhang, L. Shi, and X. Jiang, “Vertical cavity-surface emitting photonic crystal surface-mode lasers,” Appl. Phys. B 91, 475–478 (2008).
[Crossref]

Appl. Phys. Lett. (2)

S. H. Fan, “Sharp asymmetric line shapes in side-coupled waveguide-cavity systems,” Appl. Phys. Lett. 80, 908–910 (2002).
[Crossref]

S. K. Morrison and Y. S. Kivshar, “Engineering of directional emission from photonic-crystal waveguides,” Appl. Phys. Lett. 86, 081110 (2005).
[Crossref]

J. Lightwave Technol. (1)

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

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

Opt. Commun. (1)

K. B. Chung, “Analysis of directional emission via surface modes on photonic crystals,” Opt. Commun. 281, 5349–5354 (2008).
[Crossref]

Phys. Lett. A (1)

H. B. Chen, Y. Zeng, X. S. Chen, J. Wang, and W. Lu, “Modulation of focus using photonic crystal waveguide,” Phys. Lett. A 372, 5096–5100 (2008).
[Crossref]

Phys. Rev. B (5)

R. Moussa, B. Wang, G. Tuttle, Th. Koschny, and C. M. Soukoulis, “Effect of beaming and enhanced transmission in photonic crystals,” Phys. Rev. B 76, 235417 (2007).
[Crossref]

W. Śmigaj, “Model of light collimation by photonic crystal surface modes,” Phys. Rev. B 75, 205430 (2007).
[Crossref]

E. Moreno, F. J. García-Vidal, and L. Martín-Moreno, “Enhanced transmission and beaming of light via photonic crystal surface modes,” Phys. Rev. B 69, 121402 (2004).
[Crossref]

A. Mekis, J. C. Chen, I. Kurland, S. H. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. B 77, 3787–3790(1996).
[Crossref]

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

Phys. Rev. Lett. (1)

P. Kramper, M. Agio, C. M. Soukoulis, A. Birner, F. Müller, R. B. Wehrspohn, U. Gösele, and V. Sandoghdar, “Highly directional emission from photonic crystal waveguides of subwavelength width,” Phys. Rev. Lett. 92, 113903 (2004).
[Crossref] [PubMed]

Other (2)

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University Press, 2008).

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984).

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Figures (5)

Fig. 1
Fig. 1

(a) Structure of PhC SMW1 T-junction waveguide with smaller surface rods; (b)  1 a × 30 a supercell; (c) surface mode ( 7 × supercell , seven repetitious periods in the x direction).

Fig. 2
Fig. 2

(a) Dispersion relation curves of the surface modes for the different radii of surface rods; (b) frequencies of surface modes obtained by the PWE method and the FDTD method.

Fig. 3
Fig. 3

Schematic of SMW1 T-junction waveguide using coupled-mode theory.

Fig. 4
Fig. 4

(a) Transmission ratio of the unmodified SMW1 T-junction waveguide; (b) steady-state field distribution of the unmodified SMW1 T-junction waveguide in Fig. 1a with frequency 0.391 ( 2 π c / a ).

Fig. 5
Fig. 5

(a) Improved SMW1 T-junction waveguide; (b) transmission ratio (black line) and reflection ratio (black dots) of traditional PhC T-waveguide and the transmission ratio (red line) and reflection ratio (red dots) of our improved SMW1 T-junction waveguide; (c) steady-state field distribution of the improved SMW1 T-junction waveguide with frequency 0.397 ( 2 π c / a ).

Equations (2)

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

d F d t = i ω 0 F F / τ 4 n 3 F / τ n + n 3 2 τ n s n + s n = s n + + 2 τ i F .
R ( ω ) == | s 1 | 2 | s 1 + | 2 = ( ω ω 0 ) 2 + ( 1 τ 1 1 τ 2 1 τ 3 1 τ 4 ) 2 ( ω ω 0 ) 2 + ( 1 τ 1 + 1 τ 2 + 1 τ 3 + 1 τ 4 ) 2 , T 1 2 == | s 2 | 2 | s 1 + | 2 = 4 τ 1 τ 2 ( ω ω 0 ) 2 + ( 1 τ 1 + 1 τ 2 + 1 τ 3 + 1 τ 4 ) 2 , T 1 3 == | s 3 | 2 | s 1 + | 2 = 4 τ 1 τ 3 ( ω ω 0 ) 2 + ( 1 τ 1 + 1 τ 2 + 1 τ 3 + 1 τ 4 ) 2 , T 1 4 == | s 4 | 2 | s 1 + | 2 = 4 τ 1 τ 4 ( ω ω 0 ) 2 + ( 1 τ 1 + 1 τ 2 + 1 τ 3 + 1 τ 4 ) 2 .

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