Abstract

A self-contained coupled-mode formulation for coupled two-dimensional photonic-crystal waveguides (PCWs) is discussed. Using a perturbation theory, the first-order coupled-mode equations are systematically derived, which govern the evolution of the modal amplitude of individual PCWs in isolation. The coupled-mode equations are used to analyze the coupled symmetric PCWs consisting of a square lattice of circular dielectric rods or air holes. It is shown that the results are in good agreement with those obtained by the rigorous direct analysis of the coupled waveguide system.

© 2012 Optical Society of America

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References

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  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]
  2. 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]
  3. M. Qiu and M. Swillo, “Contra-directional coupling between two-dimensional photonic crystal waveguides,” Photon. Nanostruct. Fundam. Applic. 1, 23–30 (2003).
    [CrossRef]
  4. 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]
  5. K. Yasumoto, H. Jia, and K. Sun, “Rigorous analysis of two-dimensional photonic crystal waveguides,” Radio Sci. 40, RS6S02 (2005).
    [CrossRef]
  6. L. C. Botten, R. A. Hansen, and C. Martijn de Sterke, “Supermodes in multiple coupled photonic crystal waveguides,” Opt. Express 14, 387–396 (2006).
    [CrossRef]
  7. M. Guasoni, A. Locatelli, and C. De Angelis, “Peculiar properties of photonic crystal binary waveguide arrays,” J. Opt. Soc. Am. B 25, 1515–1522 (2008).
    [CrossRef]
  8. W.-P. Huang, “Coupled-mode theory for optical waveguides: an overview,” J. Opt. Soc. Am. A 11, 963–983 (1994).
    [CrossRef]
  9. A. W. Snyder and J. D. Love, Optical Waveguide Theory(Chapman & Hall, 1983).
  10. 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]
  11. K. Watanabe and K. Yasumoto, “Coupled-mode analysis of coupled microstrip transmission lines using a singular perturbation technique,” Progr. Electromag. Res. PIER 25, 95–110 (2000).
    [CrossRef]

2008 (1)

2006 (1)

2005 (1)

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

2004 (2)

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]

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)

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]

M. Qiu and M. Swillo, “Contra-directional coupling between two-dimensional photonic crystal waveguides,” Photon. Nanostruct. Fundam. Applic. 1, 23–30 (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]

2000 (1)

K. Watanabe and K. Yasumoto, “Coupled-mode analysis of coupled microstrip transmission lines using a singular perturbation technique,” Progr. Electromag. Res. PIER 25, 95–110 (2000).
[CrossRef]

1994 (1)

Asatryan, A. A.

Benisty, H.

Botten, L. C.

De Angelis, C.

de Sterke, C. M.

de Sterke, C. Martijn

Guasoni, M.

Hansen, R. A.

Houdré, R.

Huang, W.-P.

Jia, H.

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

Locatelli, A.

Love, J. D.

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

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]

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. Nanostruct. Fundam. Applic. 1, 23–30 (2003).
[CrossRef]

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 & Hall, 1983).

Sun, K.

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

Swillo, M.

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

Watanabe, K.

K. Watanabe and K. Yasumoto, “Coupled-mode analysis of coupled microstrip transmission lines using a singular perturbation technique,” Progr. Electromag. Res. PIER 25, 95–110 (2000).
[CrossRef]

Weisbuch, C.

White, T. P.

Yasumoto, K.

K. Yasumoto, H. Jia, and K. Sun, “Rigorous analysis of two-dimensional photonic crystal waveguides,” Radio Sci. 40, RS6S02 (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]

K. Watanabe and K. Yasumoto, “Coupled-mode analysis of coupled microstrip transmission lines using a singular perturbation technique,” Progr. Electromag. Res. PIER 25, 95–110 (2000).
[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. Opt. Soc. Am. A (1)

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

Opt. Express (2)

Opt. Lett. (1)

Photon. Nanostruct. Fundam. Applic. (1)

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

Progr. Electromag. Res. PIER (1)

K. Watanabe and K. Yasumoto, “Coupled-mode analysis of coupled microstrip transmission lines using a singular perturbation technique,” Progr. Electromag. Res. PIER 25, 95–110 (2000).
[CrossRef]

Radio Sci. (1)

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

Other (1)

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

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Figures (3)

Fig. 1.
Fig. 1.

Schematics of the coupled two-parallel PCWs: (a) coupled PCW system, (b) isolated photonic crystal waveguide “a”, and (c) isolated photonic crystal waveguide “b”. The structures are two-dimensional.

Fig. 2.
Fig. 2.

Dispersion curves of even and odd modes in the coupled symmetric PCWs consisting of square lattice of dielectric rods in free space for (a) one-layered barrier and (b) two-layered barrier: εr=11.56, r/h=0.175, εsr=1.0, wa/h=wb/h=2.0. The solid line represents the dispersion curve for the fundamental mode in a single PCW in isolation.

Fig. 3.
Fig. 3.

Dispersion lines of even and odd modes in the coupled symmetric PCWs consisting of square lattice of air holes in a background dielectric medium for (a) one-layered barrier and (b) two-layered barrier: εr=1.0, r/h=0.45, εsr=12.96, wa/h=wb/h=1.275. The solid line represents the dispersion line for the fundamental mode in a single PCW.

Tables (2)

Tables Icon

Table 1. Normalized Propagation Constants βh/2π of Even and Odd Modes Calculated for the Coupled Symmetric PCWs Consisting of a Square Lattice of Dielectric Rods in Free Spacea

Tables Icon

Table 2. Normalized Propagation Constants βh/2π of Even and Edd Modes Calculated for the Coupled Symmetric PCWs Consisting of a Square Lattice of Air Holes in a Background Dielectric Mediuma

Equations (36)

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ψ(x,z)=m=[cm+exp(iγmx)+cmexp(iγmx)]exp(iβmz),
a+=Wa(ω,β)[R¯¯BNB(ω,β)a+T¯¯BNB(ω,β)b+],a=Wa(ω,β)R¯¯UNU(ω,β)a+,
b=Wb(ω,β)[T¯¯BNB(ω,β)a+R¯¯BNB(ω,β)b+],b+=Wb(ω,β)R¯¯LNL(ω,β)b
Wa(ω,β)=[exp(iγmwa)δmn],Wb(ω,β)=[exp(iγmwb)δmn],
[IWa(ω,β)R¯¯UNU(ω,β)Wa(ω,β)R¯¯BNB(ω,β)]a=Wa(ω,β)R¯¯UNU(ω,β)Wa(ω,β)T¯¯BNB(ω,β)b+,
[IWb(ω,β)R¯¯LNL(ω,β)Wb(ω,β)R¯¯BNB(ω,β)]b+=Wb(ω,β)R¯¯LNL(ω,β)Wb(ω,β)T¯¯BNB(ω,β)a.
Da(ω,β)a=Dab(ω,β)b++Daa(ω,β)a,
Db(ω,β)b+=Dba(ω,β)a+Dbb(ω,β)b+
Da(ω,β)=IWa(ω,β)R¯¯UNU(ω,β)Wa(ω,β)R¯¯B(ω,β),
Db(ω,β)=IWb(ω,β)R¯¯LNL(ω,β)Wb(ω,β)R¯¯B(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),
Daa(ω,β)=Wa(ω,β)R¯¯UNU(ω,β)Wa(ω,β)[R¯¯BNB(ω,β)R¯¯B(ω,β)],
Dbb(ω,β)=Wb(ω,β)R¯¯LNL(ω,β)Wb(ω,β)[R¯¯BNB(ω,β)R¯¯B(ω,β)],
R¯¯BNB(ω,β)=Z2NBY2Z1NBΛNBY1,
T¯¯BNB(ω,β)=Z1NBY2+Z2NBΛNBY1
Z1,2NB=12(Y1+ΛNBY2)112(Y1ΛNBY2)1,
Y1,2=12(X1+X2)1±12(X1X2)1,
R¯¯BNB(ω,β)=Y11Y2+O(δ2Λ2NB),
T¯¯BNB(ω,β)=Y11δΛNB(Y1Y2Y11Y2)+O(δ2Λ2NB).
R¯¯B(ω,β)=Y11Y2,T¯¯B(ω,β)=0.
Da(ω,β)a=Dab(ω,β)b+,
Db(ω,β)b+=Dba(ω,β)a.
β=β0+δβ,a=a0+δa,b+=b0+δb,
Da(ω,β0)a0=0,Db(ω,β0)b0=0.
Da(ω,βa)δa=δβDa(ω,βa)βaa0+Dab(ω,βb)b0,
Db(ω,βb)δb=δβDb(ω,βb)βbb0+Dba(ω,βa)a0.
a0=A(z)eiβazfa(z),b0=B(z)eiβbzfb(z),
Da(ω,βa)δa=iexp(iβaz)Da(ω,βa)βafaddzA(z)+exp(iβbz)Dab(ω,βb)fbB(z),
Db(ω,βb)δb=iexp(iβbz)Db(ω,βb)βbfbddzB(z)+exp(iβaz)Dba(ω,βa)faA(z),
ddzA(z)=iexp(iΔβz)κabB(z),
ddzB(z)=iexp(iΔβz)κbaA(z)
κab=gaTDab(ω,βb)fbgaTDa(ω,βa)βafa,
κba=gbTDba(ω,βa)fagbTDb(ω,βb)βbfb,
βeven=β0+κfor even mode,
βodd=β0κfor odd mode.

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