Abstract

A self-contained coupled-mode theory for the coupled two asymmetric photonic crystal waveguides (PCWs) is presented. The first-order coupled-mode equations are derived under a weak coupling assumption. The coupling coefficients are obtained systematically by a matrix calculus using the modal solutions of each PCW in isolation. The coupled-mode equations are solved for contra-directional coupling between two asymmetric PCWs formed by a hexagonal lattice of circular air holes in a dielectric medium. The power transmission spectra at different output ports of the coupled PCWs are investigated. It is shown that the self-contained coupled-mode analysis is useful to characterize a peculiar feature of the contra-directionally coupled PCWs as a drop filter.

© 2014 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
    [CrossRef]
  2. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
    [CrossRef]
  3. A. Adibi, Y. Xu, R. Lee, A. Yariv, and A. Scherer, “Properties of the slab modes in photonic crystal optical waveguides,” J. Lightwave Technol. 18, 1554–1564 (2000).
    [CrossRef]
  4. M. Qiu, K. Azizi, A. Karlsson, M. Swillo, and B. Jaskorzynska, “Numerical studies of mode gaps and coupling efficiency for line-defect waveguides in two-dimensional photonic crystals,” Phys. Rev. B 64, 155113 (2001).
    [CrossRef]
  5. A. Sharkawy, S. Shi, J. Murakowski, and D. W. Prather, “Analysis and applications of photonic crystals coupled waveguide theory,” Proc. SPIE 4655, 356–367 (2002).
    [CrossRef]
  6. S. Olivier, H. Benisty, C. Weisbuch, C. J. M. Smith, T. F. Krauss, and R. Houdré, “Coupled-mode theory and propagation losses in photonic crystal waveguides,” Opt. Express 11, 1490–1496 (2003).
    [CrossRef]
  7. J. Zimmermann, M. Kamp, A. Forchel, and R. Marz, “Photonic crystal waveguide directional couplers as wavelength selective optical filters,” Opt. Commun. 230, 387–392 (2004).
    [CrossRef]
  8. Y. Tanaka, H. Nakamura, Y. Sugimoto, N. Ikeda, K. Asakawa, and K. Inoue, “Coupling properties in a 2-D photonic crystal slab directional coupler with a triangular lattice of air holes,” IEEE J. Quantum Electron. 41, 76–84 (2005).
    [CrossRef]
  9. M. Qiu and M. Swillo, “Contra-directional coupling between two-dimensional photonic crystal waveguides,” Photon. Nanostr. Fundam. Appl. 1, 23–30 (2003).
    [CrossRef]
  10. Z. Xu, J. Wang, Q. He, L. Cao, P. Su, and G. Jin, “Optical filter based on contra-directional waveguide coupling in a 2D photonic crystal with square lattice of dielectric rods,” Opt. Express 13, 5608–5613 (2005).
    [CrossRef]
  11. E. Engin, J. L. O’Brien, and M. J. Cryan, “Design and analysis of a gallium nitride-on-sapphire tunable photonic crystal directional coupler,” J. Opt. Soc. Am. B 29, 1157–1164 (2012).
    [CrossRef]
  12. 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]
  13. K. Yasumoto, H. Jia, and K. Sun, “Rigorous modal analysis of two-dimensional photonic crystal waveguides,” Radio Sci. 40(6), RS6S02, 1–7 (2005).
    [CrossRef]
  14. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).
  15. W.-P. Huang, “Coupled-mode theory for optical waveguides: an overview,” J. Opt. Soc. Am. A 11, 963–983 (1994).
    [CrossRef]
  16. K. Yasumoto, V. Jandieri, and Y. Liu, “Coupled-mode formulation of two-parallel photonic-crystal waveguides,” J. Opt. Soc. Am. A 30, 96–101 (2013).
    [CrossRef]
  17. K. Yasumoto, H. Toyama, and T. Kushta, “Accurate analysis of two-dimensional electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique,” IEEE Trans. Antennas Propag. 52, 2603–2611 (2004).
    [CrossRef]

2013 (1)

2012 (1)

2005 (3)

Z. Xu, J. Wang, Q. He, L. Cao, P. Su, and G. Jin, “Optical filter based on contra-directional waveguide coupling in a 2D photonic crystal with square lattice of dielectric rods,” Opt. Express 13, 5608–5613 (2005).
[CrossRef]

K. Yasumoto, H. Jia, and K. Sun, “Rigorous modal analysis of two-dimensional photonic crystal waveguides,” Radio Sci. 40(6), RS6S02, 1–7 (2005).
[CrossRef]

Y. Tanaka, H. Nakamura, Y. Sugimoto, N. Ikeda, K. Asakawa, and K. Inoue, “Coupling properties in a 2-D photonic crystal slab directional coupler with a triangular lattice of air holes,” IEEE J. Quantum Electron. 41, 76–84 (2005).
[CrossRef]

2004 (3)

K. Yasumoto, H. Toyama, and T. Kushta, “Accurate analysis of two-dimensional electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique,” IEEE Trans. Antennas Propag. 52, 2603–2611 (2004).
[CrossRef]

J. Zimmermann, M. Kamp, A. Forchel, and R. Marz, “Photonic crystal waveguide directional couplers as wavelength selective optical filters,” Opt. Commun. 230, 387–392 (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]

2003 (2)

M. Qiu and M. Swillo, “Contra-directional coupling between two-dimensional photonic crystal waveguides,” Photon. Nanostr. Fundam. Appl. 1, 23–30 (2003).
[CrossRef]

S. Olivier, H. Benisty, C. Weisbuch, C. J. M. Smith, T. F. Krauss, and R. Houdré, “Coupled-mode theory and propagation losses in photonic crystal waveguides,” Opt. Express 11, 1490–1496 (2003).
[CrossRef]

2002 (1)

A. Sharkawy, S. Shi, J. Murakowski, and D. W. Prather, “Analysis and applications of photonic crystals coupled waveguide theory,” Proc. SPIE 4655, 356–367 (2002).
[CrossRef]

2001 (1)

M. Qiu, K. Azizi, A. Karlsson, M. Swillo, and B. Jaskorzynska, “Numerical studies of mode gaps and coupling efficiency for line-defect waveguides in two-dimensional photonic crystals,” Phys. Rev. B 64, 155113 (2001).
[CrossRef]

2000 (1)

1994 (1)

1987 (2)

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef]

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef]

Adibi, A.

Asakawa, K.

Y. Tanaka, H. Nakamura, Y. Sugimoto, N. Ikeda, K. Asakawa, and K. Inoue, “Coupling properties in a 2-D photonic crystal slab directional coupler with a triangular lattice of air holes,” IEEE J. Quantum Electron. 41, 76–84 (2005).
[CrossRef]

Asatryan, A. A.

Azizi, K.

M. Qiu, K. Azizi, A. Karlsson, M. Swillo, and B. Jaskorzynska, “Numerical studies of mode gaps and coupling efficiency for line-defect waveguides in two-dimensional photonic crystals,” Phys. Rev. B 64, 155113 (2001).
[CrossRef]

Benisty, H.

Botten, L. C.

Cao, L.

Cryan, M. J.

de Sterke, C. M.

Engin, E.

Forchel, A.

J. Zimmermann, M. Kamp, A. Forchel, and R. Marz, “Photonic crystal waveguide directional couplers as wavelength selective optical filters,” Opt. Commun. 230, 387–392 (2004).
[CrossRef]

He, Q.

Houdré, R.

Huang, W.-P.

Ikeda, N.

Y. Tanaka, H. Nakamura, Y. Sugimoto, N. Ikeda, K. Asakawa, and K. Inoue, “Coupling properties in a 2-D photonic crystal slab directional coupler with a triangular lattice of air holes,” IEEE J. Quantum Electron. 41, 76–84 (2005).
[CrossRef]

Inoue, K.

Y. Tanaka, H. Nakamura, Y. Sugimoto, N. Ikeda, K. Asakawa, and K. Inoue, “Coupling properties in a 2-D photonic crystal slab directional coupler with a triangular lattice of air holes,” IEEE J. Quantum Electron. 41, 76–84 (2005).
[CrossRef]

Jandieri, V.

Jaskorzynska, B.

M. Qiu, K. Azizi, A. Karlsson, M. Swillo, and B. Jaskorzynska, “Numerical studies of mode gaps and coupling efficiency for line-defect waveguides in two-dimensional photonic crystals,” Phys. Rev. B 64, 155113 (2001).
[CrossRef]

Jia, H.

K. Yasumoto, H. Jia, and K. Sun, “Rigorous modal analysis of two-dimensional photonic crystal waveguides,” Radio Sci. 40(6), RS6S02, 1–7 (2005).
[CrossRef]

Jin, G.

John, S.

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef]

Kamp, M.

J. Zimmermann, M. Kamp, A. Forchel, and R. Marz, “Photonic crystal waveguide directional couplers as wavelength selective optical filters,” Opt. Commun. 230, 387–392 (2004).
[CrossRef]

Karlsson, A.

M. Qiu, K. Azizi, A. Karlsson, M. Swillo, and B. Jaskorzynska, “Numerical studies of mode gaps and coupling efficiency for line-defect waveguides in two-dimensional photonic crystals,” Phys. Rev. B 64, 155113 (2001).
[CrossRef]

Krauss, T. F.

Kushta, T.

K. Yasumoto, H. Toyama, and T. Kushta, “Accurate analysis of two-dimensional electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique,” IEEE Trans. Antennas Propag. 52, 2603–2611 (2004).
[CrossRef]

Lee, R.

Liu, Y.

Love, J. D.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).

Marz, R.

J. Zimmermann, M. Kamp, A. Forchel, and R. Marz, “Photonic crystal waveguide directional couplers as wavelength selective optical filters,” Opt. Commun. 230, 387–392 (2004).
[CrossRef]

McPhedran, R. C.

Murakowski, J.

A. Sharkawy, S. Shi, J. Murakowski, and D. W. Prather, “Analysis and applications of photonic crystals coupled waveguide theory,” Proc. SPIE 4655, 356–367 (2002).
[CrossRef]

Nakamura, H.

Y. Tanaka, H. Nakamura, Y. Sugimoto, N. Ikeda, K. Asakawa, and K. Inoue, “Coupling properties in a 2-D photonic crystal slab directional coupler with a triangular lattice of air holes,” IEEE J. Quantum Electron. 41, 76–84 (2005).
[CrossRef]

O’Brien, J. L.

Olivier, S.

Prather, D. W.

A. Sharkawy, S. Shi, J. Murakowski, and D. W. Prather, “Analysis and applications of photonic crystals coupled waveguide theory,” Proc. SPIE 4655, 356–367 (2002).
[CrossRef]

Qiu, M.

M. Qiu and M. Swillo, “Contra-directional coupling between two-dimensional photonic crystal waveguides,” Photon. Nanostr. Fundam. Appl. 1, 23–30 (2003).
[CrossRef]

M. Qiu, K. Azizi, A. Karlsson, M. Swillo, and B. Jaskorzynska, “Numerical studies of mode gaps and coupling efficiency for line-defect waveguides in two-dimensional photonic crystals,” Phys. Rev. B 64, 155113 (2001).
[CrossRef]

Scherer, A.

Sharkawy, A.

A. Sharkawy, S. Shi, J. Murakowski, and D. W. Prather, “Analysis and applications of photonic crystals coupled waveguide theory,” Proc. SPIE 4655, 356–367 (2002).
[CrossRef]

Shi, S.

A. Sharkawy, S. Shi, J. Murakowski, and D. W. Prather, “Analysis and applications of photonic crystals coupled waveguide theory,” Proc. SPIE 4655, 356–367 (2002).
[CrossRef]

Smith, C. J. M.

Snyder, A. W.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).

Su, P.

Sugimoto, Y.

Y. Tanaka, H. Nakamura, Y. Sugimoto, N. Ikeda, K. Asakawa, and K. Inoue, “Coupling properties in a 2-D photonic crystal slab directional coupler with a triangular lattice of air holes,” IEEE J. Quantum Electron. 41, 76–84 (2005).
[CrossRef]

Sun, K.

K. Yasumoto, H. Jia, and K. Sun, “Rigorous modal analysis of two-dimensional photonic crystal waveguides,” Radio Sci. 40(6), RS6S02, 1–7 (2005).
[CrossRef]

Swillo, M.

M. Qiu and M. Swillo, “Contra-directional coupling between two-dimensional photonic crystal waveguides,” Photon. Nanostr. Fundam. Appl. 1, 23–30 (2003).
[CrossRef]

M. Qiu, K. Azizi, A. Karlsson, M. Swillo, and B. Jaskorzynska, “Numerical studies of mode gaps and coupling efficiency for line-defect waveguides in two-dimensional photonic crystals,” Phys. Rev. B 64, 155113 (2001).
[CrossRef]

Tanaka, Y.

Y. Tanaka, H. Nakamura, Y. Sugimoto, N. Ikeda, K. Asakawa, and K. Inoue, “Coupling properties in a 2-D photonic crystal slab directional coupler with a triangular lattice of air holes,” IEEE J. Quantum Electron. 41, 76–84 (2005).
[CrossRef]

Toyama, H.

K. Yasumoto, H. Toyama, and T. Kushta, “Accurate analysis of two-dimensional electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique,” IEEE Trans. Antennas Propag. 52, 2603–2611 (2004).
[CrossRef]

Wang, J.

Weisbuch, C.

White, T. P.

Xu, Y.

Xu, Z.

Yablonovitch, E.

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef]

Yariv, A.

Yasumoto, K.

K. Yasumoto, V. Jandieri, and Y. Liu, “Coupled-mode formulation of two-parallel photonic-crystal waveguides,” J. Opt. Soc. Am. A 30, 96–101 (2013).
[CrossRef]

K. Yasumoto, H. Jia, and K. Sun, “Rigorous modal analysis of two-dimensional photonic crystal waveguides,” Radio Sci. 40(6), RS6S02, 1–7 (2005).
[CrossRef]

K. Yasumoto, H. Toyama, and T. Kushta, “Accurate analysis of two-dimensional electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique,” IEEE Trans. Antennas Propag. 52, 2603–2611 (2004).
[CrossRef]

Zimmermann, J.

J. Zimmermann, M. Kamp, A. Forchel, and R. Marz, “Photonic crystal waveguide directional couplers as wavelength selective optical filters,” Opt. Commun. 230, 387–392 (2004).
[CrossRef]

IEEE J. Quantum Electron. (1)

Y. Tanaka, H. Nakamura, Y. Sugimoto, N. Ikeda, K. Asakawa, and K. Inoue, “Coupling properties in a 2-D photonic crystal slab directional coupler with a triangular lattice of air holes,” IEEE J. Quantum Electron. 41, 76–84 (2005).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

K. Yasumoto, H. Toyama, and T. Kushta, “Accurate analysis of two-dimensional electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique,” IEEE Trans. Antennas Propag. 52, 2603–2611 (2004).
[CrossRef]

J. Lightwave Technol. (1)

J. Opt. Soc. Am. A (2)

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

Opt. Commun. (1)

J. Zimmermann, M. Kamp, A. Forchel, and R. Marz, “Photonic crystal waveguide directional couplers as wavelength selective optical filters,” Opt. Commun. 230, 387–392 (2004).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Photon. Nanostr. Fundam. Appl. (1)

M. Qiu and M. Swillo, “Contra-directional coupling between two-dimensional photonic crystal waveguides,” Photon. Nanostr. Fundam. Appl. 1, 23–30 (2003).
[CrossRef]

Phys. Rev. B (1)

M. Qiu, K. Azizi, A. Karlsson, M. Swillo, and B. Jaskorzynska, “Numerical studies of mode gaps and coupling efficiency for line-defect waveguides in two-dimensional photonic crystals,” Phys. Rev. B 64, 155113 (2001).
[CrossRef]

Phys. Rev. Lett. (2)

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef]

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef]

Proc. SPIE (1)

A. Sharkawy, S. Shi, J. Murakowski, and D. W. Prather, “Analysis and applications of photonic crystals coupled waveguide theory,” Proc. SPIE 4655, 356–367 (2002).
[CrossRef]

Radio Sci. (1)

K. Yasumoto, H. Jia, and K. Sun, “Rigorous modal analysis of two-dimensional photonic crystal waveguides,” Radio Sci. 40(6), RS6S02, 1–7 (2005).
[CrossRef]

Other (1)

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1.
Fig. 1.

Schematics of the coupled parallel PCWs: (a) coupled PCW system, (b) isolated PCW a, and (c) isolated PCW b. The circular rods forming the PCW are parallel and infinitely long in the y direction. The waveguide problem is 2D.

Fig. 2.
Fig. 2.

Schematic view of a contra-directionally coupled two asymmetric PCWs consisting of the hexagonal lattice of circular air holes with radius r=0.36h being located in a background medium with εs=10.5ε0. The widths of guiding layers in the upper and lower PCWs are wa=0.83h and wb=3h, respectively. The number of barrier layers between two PCWs is NB=1. The initial excitation is launched in the upper PCW at z=l.

Fig. 3.
Fig. 3.

Dispersion diagrams for the upper PCW “a” in isolation, which consists of the hexagonal lattice of circular air-holes with radius r=0.36h located in a background dielectric with εs=10.5ε0, where wa=0.83h and NU=N=10.

Fig. 4.
Fig. 4.

Dispersion diagrams for the lower PCW b in isolation, which consists of the hexagonal lattice of circular air holes with radius r=0.36h located in a background dielectric with εs=10.5ε0, where wb=3h and NL=N=10.

Fig. 5.
Fig. 5.

(a) Dispersion diagrams of the guided modes for the coupled asymmetric PCWs as shown in Fig. 2, where the number of barrier layers between two PCWs is NB=1. (b) Enlarged dispersion diagrams for the coupling region surrounded by a gray rectangle in (a). The solid lines are obtained by the rigorous analysis [13], and the dashed lines present the results of the coupled-mode analysis given by Eqs. (17) and (18).

Fig. 6.
Fig. 6.

The same as in Fig. 2, but for three-layered barrier structure with NB=3.

Fig. 7.
Fig. 7.

(a) Dispersion diagrams of the guided mode for the coupled PCWs with NB=3 as shown in Fig. 6. (b) Enlarged dispersion diagrams for the coupling region surrounded by a gray rectangle in (a). The solid lines are obtained by rigorous analysis [13], and the dashed lines present the results of the coupled-mode analysis given by Eqs. (17) and (18).

Fig. 8.
Fig. 8.

Transmission power spectrum at different output ports calculated by the coupled-mode analysis in Eqs. (20) and (21) for the coupled PCWs with NB=3. The length of PCWs is l=2000h, and the output port numbers correspond to those given in Fig. 6.

Equations (21)

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

ψ(x,z)=A(z)m=αaca,mpa,m(x)exp(iβa,mz)+B(z)m=αbcb,mpb,m(x)exp(iβb,mz),
12ωμ0m=αj2βj,m|cj,m|2|pj,m(x)|2dx=±1(j=a,b),
ddzA(z)=iexp(iΔβz)κabB(z),
ddzB(z)=iexp(iΔβz)κbaA(z),
κab=αbgaT·Dab(ω,βb)·cbαagaT·Da(ω,βa)βa·ca,
κba=αagbT·Dba(ω,βa)·caαbgbT·Db(ω,βb)βb·cb,
Da(ω,β)=IWa(ω,β)R¯¯UNU(ω,β)Wa(ω,β)R¯¯BN(ω,β),
Db(ω,β)=IWb(ω,β)R¯¯LNL(ω,β)Wb(ω,β)R¯¯BN(ω,β),
Dab(ω,β)=Wa(ω,β)R¯¯UNU(ω,β)Wa(ω,β)T¯¯BNB(ω,β),
Dba(ω,β)=Wb(ω,β)R¯¯LNL(ω,β)Wb(ω,β)T¯¯BNB(ω,β),
Wa(ω,β)=[exp(iγa,mwa)δmn],Wb(ω,β)=[exp(iγb,mwb)δmn],
A(z)=A1exp(iηa,1z)+A2exp(iηa,2z),
B(z)=A1κbaηb,1exp(iηb,1z)+A2κbaηb,2exp(iηb,2z),
ηa,1=Δβ2+q2,ηa,2=Δβ2q2,
ηb,1=Δβ2+q2,ηb,2=Δβ2q2,
q=(Δβ)2+4κabκba,
β1,2(ω)=βa(ω)Δβ(ω)2±q(ω)2,
β3,4(ω)=βb(ω)+Δβ(ω)2±q(ω)2,
A(z=l)=1,B(z=0)=0.
A(z)=qcos(q2z)+iΔβsin(q2z)qcos(q2l)+iΔβsin(q2l)exp[iΔβ2(lz)],
B(z)=i2κbasin(q2z)qcos(q2l)+iΔβsin(q2l)exp[iΔβ2(z+l)].

Metrics