Abstract

We use the coupling matrix formalism to investigate continuous-wave and pulse propagation through microring coupled-resonator optical waveguides (CROWs). The dispersion relation agrees with that derived using the tight-binding model in the limit of weak inter-resonator coupling. We obtain an analytical expression for pulse propagation through a semi-infinite CROW in the case of weak coupling which fully accounts for the nonlinear dispersive characteristics. We also show that intensity of a pulse in a CROW is enhanced by a factor inversely proportional to the inter-resonator coupling. In finite CROWs, anomalous dispersions allows for a pulse to propagate with a negative group velocity such that the output pulse appears to emerge before the input as in “superluminal” propagation. The matrix formalism is a powerful approach for microring CROWs since it can be applied to structures and geometries for which analyses with the commonly used tight-binding approach are not applicable.

© 2004 Optical Society of America

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References

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    [CrossRef]
  2. Y. Xu, R. K. Lee, and A. Yariv, �??Propagation and second-harmonic generation of electromagnetic waves in a coupled-resonator optical waveguide,�?? J. Opt. Soc. Am. B 77, 387�??400 (2000).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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  8. G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, �??Optical delay lines based on optical filters,�?? IEEE J. Quantum Electron. 37, 525�??532 (2001).
    [CrossRef]
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    [CrossRef] [PubMed]
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  13. R. Orta, P. Savi, R. Tascone, and D. Trinchero, �??Synthesis of multiple-ring resonator filters for optical systems,�?? IEEE Photon. Technol. Lett. 7, 1447�??1449 (1995).
    [CrossRef]
  14. J. V. Hryniewicz, P. P. Absil, B. E. Little, R. A. Wilson, and P.-T. Ho, �??Higher order filter response in coupled microring resonators,�?? IEEE Photon. Technol. Lett. 12, 320�??322 (2000).
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    [CrossRef]
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    [CrossRef]
  18. A. Melloni and F. Morichetti, �??Linear and nonlinear pulse propagation in coupled resonator slow-wave optical structures,�?? Opt. Quantum Electron. 35, 365�??379 (2003).
    [CrossRef]
  19. J. E. Heeber, R. W. Boyd, and Q.-H. Park, �??SCISSOR solitons and other novel propagation effects in microresonator-modified waveguides,�?? J. Opt. Soc. Am. B 19, 722�??731 (2002).
    [CrossRef]
  20. S. Mookherjea and A. Yariv, �??Pulse propagation in a coupled-resonator optical waveguide to all orders of dispersion,�?? Phys. Rev. E 65, 056 601 (2002).
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    [CrossRef]
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  23. S. Longhi, M. Marano, M. Belmonte, and P. Laporta, �??Superluminal pulse propagation in linear and nonlinear photonic grating structures,�?? IEEE. J. Sel. Top. Quantum Electron. 9, 4�??16 (2003).
    [CrossRef]
  24. M. Bayindir, S. Tanriseven, and E. Ozbay, �??Propagation of light through localized coupled-cavity modes in one-dimensional photonic band-gap structures,�?? Appl. Phys. A 72, 117�??119 (2001).
    [CrossRef]
  25. W. Chen and D. L. Mills, �??Gap solitons and the nonlinear optical-response of superlattices,�?? Phys. Rev. Lett. 58, 160�??163 (1987).
    [CrossRef] [PubMed]
  26. C. M. de Sterke and J. E. Sipe, �??Envelope-function approach for the electrodynamics of nonlinear periodic structures,�?? Phys. Rev. A 38, 5149�??5165 (1988).
    [CrossRef] [PubMed]
  27. C. M. de Sterke, D. G. Salinas, and J. E. Sipe, �??Coupled-mode theory for light propagation through deep nonlinear gratings,�?? Phys. Rev. E 54, 1969�??1989 (1996).
    [CrossRef]
  28. B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, �??Bragg grating solitons,�?? Phys. Rev. Lett. 76, 1627�??1630 (1996).
    [CrossRef] [PubMed]
  29. D. N. Christodoulides and R. I. Joseph, �??Slow Bragg solitons in nonlinear periodic structures,�?? Phys. Rev. Lett. 62, 1746�??1749 (1989).
    [CrossRef] [PubMed]
  30. Little Optics press release, �??Higher order optical filters using microring resonators�?? (Little Optics, 2003), <a href="http://www.littleoptics.com/hofilter.pdf.">http://www.littleoptics.com/hofilter.pdf.</a>

Appl. Phys. A

M. Bayindir, S. Tanriseven, and E. Ozbay, �??Propagation of light through localized coupled-cavity modes in one-dimensional photonic band-gap structures,�?? Appl. Phys. A 72, 117�??119 (2001).
[CrossRef]

Electron. Lett.

A. Yariv, �??Universal relations for coupling of optical power between microresonators and dielectric waveguides,�?? Electron. Lett. 36, 321�??322 (2000).
[CrossRef]

IEEE J. Lightwave Technol.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, �??Microring resonator channel dropping filter,�?? IEEE J. Lightwave Technol. 15, 998�??1005 (1997).
[CrossRef]

C. K. Madsen, �??General IIR optical filter design for WDM applications using all-pass filters,�?? IEEE J. Lightwave Technol. 18, 860�??868 (2000).
[CrossRef]

IEEE J. of Lightwave Technol.

K. Oda, N. Takato, and H. Toba, �??A wide-FSR waveguide double-ring resonator for optical FDM transmission systems,�?? IEEE J. of Lightwave Technol. 9, 728�??736 (1991).
[CrossRef]

IEEE J. Quantum Electron.

G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, �??Optical delay lines based on optical filters,�?? IEEE J. Quantum Electron. 37, 525�??532 (2001).
[CrossRef]

IEEE Photon. Technol. Lett.

B. E. Little, S. T. Chu, W. Pan, D. Ripin, T. Kaneko, Y. Kokubun, and E. Ippen, �??Vertically coupled glass microring resonator channel dropping filters,�?? IEEE Photon. Technol. Lett. 11, 215�??217 (1999).
[CrossRef]

R. Orta, P. Savi, R. Tascone, and D. Trinchero, �??Synthesis of multiple-ring resonator filters for optical systems,�?? IEEE Photon. Technol. Lett. 7, 1447�??1449 (1995).
[CrossRef]

J. V. Hryniewicz, P. P. Absil, B. E. Little, R. A. Wilson, and P.-T. Ho, �??Higher order filter response in coupled microring resonators,�?? IEEE Photon. Technol. Lett. 12, 320�??322 (2000).
[CrossRef]

IEEE. J. Sel. Top. Quantum Electron.

S. Longhi, M. Marano, M. Belmonte, and P. Laporta, �??Superluminal pulse propagation in linear and nonlinear photonic grating structures,�?? IEEE. J. Sel. Top. Quantum Electron. 9, 4�??16 (2003).
[CrossRef]

J. Mod. Opt.

J. E. Heebner and R. W. Boyd, �??�??Slow�?? and �??fast�?? light in resonator-coupled waveguides,�?? J. Mod. Opt. 49, 2629�??2636 (2002).
[CrossRef]

J. Opt. Soc. Am. B

Y. Xu, R. K. Lee, and A. Yariv, �??Propagation and second-harmonic generation of electromagnetic waves in a coupled-resonator optical waveguide,�?? J. Opt. Soc. Am. B 77, 387�??400 (2000).
[CrossRef]

J. E. Heeber, R. W. Boyd, and Q.-H. Park, �??SCISSOR solitons and other novel propagation effects in microresonator-modified waveguides,�?? J. Opt. Soc. Am. B 19, 722�??731 (2002).
[CrossRef]

Opt. Express

Opt. Lett.

Opt. Quantum Electron.

A. Melloni and F. Morichetti, �??Linear and nonlinear pulse propagation in coupled resonator slow-wave optical structures,�?? Opt. Quantum Electron. 35, 365�??379 (2003).
[CrossRef]

Phys. Rev. A

C. M. de Sterke and J. E. Sipe, �??Envelope-function approach for the electrodynamics of nonlinear periodic structures,�?? Phys. Rev. A 38, 5149�??5165 (1988).
[CrossRef] [PubMed]

Phys. Rev. B

N. Stefanou and A. Modinos, �??Impurity bands in photonic insulators,�?? Phys. Rev. B 57, 12 127�??12 133 (1998).
[CrossRef]

Phys. Rev. E

S. Mookherjea and A. Yariv, �??Kerr-stabilized super-resonant modes in coupled-resonator optical waveguides,�?? Phys. Rev. E 66, 046 610 (2002).
[CrossRef]

C. M. de Sterke, D. G. Salinas, and J. E. Sipe, �??Coupled-mode theory for light propagation through deep nonlinear gratings,�?? Phys. Rev. E 54, 1969�??1989 (1996).
[CrossRef]

S. Mookherjea and A. Yariv, �??Pulse propagation in a coupled-resonator optical waveguide to all orders of dispersion,�?? Phys. Rev. E 65, 056 601 (2002).
[CrossRef]

Phys. Rev. Lett.

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, �??Bragg grating solitons,�?? Phys. Rev. Lett. 76, 1627�??1630 (1996).
[CrossRef] [PubMed]

D. N. Christodoulides and R. I. Joseph, �??Slow Bragg solitons in nonlinear periodic structures,�?? Phys. Rev. Lett. 62, 1746�??1749 (1989).
[CrossRef] [PubMed]

W. Chen and D. L. Mills, �??Gap solitons and the nonlinear optical-response of superlattices,�?? Phys. Rev. Lett. 58, 160�??163 (1987).
[CrossRef] [PubMed]

M. Bayindir, B. Temelkuran, and E. Ozbay, �??Tight-binding description of the coupled defect modes in three-dimensional photonic crystals,�?? Phys. Rev. Lett. 84, 2140�??2143 (2000).
[CrossRef] [PubMed]

Other

A. Yariv and P. Yeh, Optical waves in crystals: Propagation and control of laser radiation (Wiley, New York, 1984).

Little Optics press release, �??Higher order optical filters using microring resonators�?? (Little Optics, 2003), <a href="http://www.littleoptics.com/hofilter.pdf.">http://www.littleoptics.com/hofilter.pdf.</a>

A. D. Poularikas, The handbook af formulas and tables for signal processing (IEEE Press, New York, 1998).
[CrossRef]

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Figures (10)

Fig. 1.
Fig. 1.

An infinitely long chain of coupled ring resonators, with the forward and backward propagating field components labelled.

Fig. 2.
Fig. 2.

The exact and cosine-approximate (i.e. tight-binding-approximate) dispersion relations for m=100 and κ=-0.8i.

Fig. 3.
Fig. 3.

A CROW consisting of N ring resonators with input and output waveguides.

Fig. 4.
Fig. 4.

The exact dispersion relation for an infinite CROW and the dispersion relation as extracted from 20 coupled resonators. The rings have a radius of 16.4µm and the inter-resonator coupling is -0.5i.

Fig. 5.
Fig. 5.

A semi-infinite CROW.

Fig. 6.
Fig. 6.

Evolution of a 2.4ps (FWHM) Gaussian pulse centered about 1.5µm in a semi-infinite CROW with κ2=0.0016. The fields are normalized to the maximum field amplitude in the first resonator. (a) Theoretical results computed using Eq. (30). (b) Results computed numerically with the transfer matrices using a chain of 100 ring resonators (n eff=1.5, R=16µm).

Fig. 7.
Fig. 7.

The transmission characteristics of a 10 ring long CROW. The ring radius is 164.5µm and n eff=1.5. Inter-resonator coupling is -0.3i and the waveguide-CROW coupling is -0.5i. A 30.5 ps (FWHM) long pulse centered at 1.55µm is input into the CROW. (a) Transmittance of at the drop port. The dashed line shows the spectrum of the input pulse. (b) Phase response at the drop port. (c) Transmittance at the through port. (d) Phase response at the through port.

Fig. 8.
Fig. 8.

The pulse transmission through the CROW described in Fig. (7). The 0th resonator is the input pulse and the 11th resonator is the output pulse at the drop port.

Fig. 9.
Fig. 9.

The input pulse and the output pulses at the drop and through ports of the the CROW described in Fig. (7). The solid vertical line marks the maximum of the input pulse, and the dashed vertical line marks the maximum of the output pulse at the through port. The peak of the through port pulse occurs about 5ps sooner than the peak of the input.

Fig. 10.
Fig. 10.

FDTD simulation of 2 coupled ring resonators with input and output waveguides. The radius of the rings is 5µm, and the effective index is n eff=3.617-0.5539λ. The inter-resonator coupling is -0.32i and the waveguide-resonator coupling is -0.4i. The input pulse is a 2.4ps (FWHM) Gaussian centered at 1.55µm. (a) Comparison between the FDTD simulation and the transfer matrix method. Output refers to the drop port. (b) Intensity built-up and anomalous dispersion as confirmed by the FDTD simulation.

Equations (39)

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E K ( r , t ) = E 0 exp ( i ω K t ) n exp ( i n K Λ ) E Ω ( r n Λ z ̂ ) ,
ω K = Ω [ 1 Δ α 2 + κ 1 cos ( K Λ ) ] ,
Δ α = d 3 r [ ε ( r ) ε 0 ( r ) ] E Ω ( r ) · E Ω ( r )
κ 1 = d 3 r [ ε 0 ( r Λ z ̂ ) ε ( r Λ z ̂ ) ] × E Ω ( r ) · E Ω ( r Λ z ̂ ) .
[ b n b n + 1 ] = [ t κ κ * t * ] [ a n a n + 1 ] , [ d n d n + 1 ] = [ t κ κ * t * ] [ c n c n + 1 ]
x n = [ a b c d ] n ,
x n + 1 = [ P 0 0 P ] x n x n
P = 1 κ [ t 1 1 t * ]
x n = [ 0 Q Q 0 ] x n x n
Q = [ 0 e i β R π e i β R π 0 ]
x n + 1 = x n
𝓔 ( ρ , ϕ ) = E ( ρ ) × { a n exp [ i β R ( π ϕ ) ] + d n exp [ i β R ( π ϕ ) ] 0 < ϕ < π b n exp [ i β R ( π + ϕ ) ] + c n exp [ i β R ( π + ϕ ) ] π < ϕ < 0
x n + 1 = exp ( i K Λ ) x n ,
Det exp ( i K Λ ) U = Det ( P Q ) 2 exp ( i 2 K Λ ) U = 0 ,
sin ( ω K Ω m π ) = ± Im ( κ ) cos ( K Λ ) ,
ω K Ω = 1 ± κ 2 cos ( K Λ ) ,
q ̂ ξ 1 = [ ζ + γ 1 ζ + γ 1 ] , q ̂ ξ 1 = [ ( ζ + γ ) 1 ζ + γ 1 ] , q ̂ ξ 2 = [ ζ γ 1 ζ γ 1 ] , q ̂ ξ 2 = [ ( ζ γ ) 1 ζ γ 1 ] ,
γ = 1 2 t 1 + exp ( 4 i m π ω Ω ) + 2 exp ( 2 i m π ω Ω ) ( 1 2 t 2 ) ,
ζ = 1 2 t [ 1 + exp ( 2 i m π ω Ω ) ] .
q ̂ ξ 1 = q ̂ ξ 2 = [ 1 1 1 1 ] , q ̂ ξ 1 = q ̂ ξ 2 = [ 1 1 1 1 ] .
q ̂ ξ 1 = [ 2 t 1 2 t 1 ] , q ̂ ξ 1 = [ 2 t 1 2 t 1 ] , q ̂ ξ 2 = [ 0 1 0 1 ] , q ̂ ξ 2 = [ 0 1 0 1 ] .
[ a b ] n + 1 = P Q [ a b ] n ,
[ a N + 1 b N + 1 ] = P out Q ( P Q ) N 1 P in [ a 0 b 0 ] [ A B C D ] [ a 0 b 0 ] ,
b 0 a 0 = A B T thr ( ω ) ,
b N + 1 a 0 = C AD B T out ( ω ) .
v g , max = κ Λ Ω m π .
S = c n eff v g , max ,
S = π 2 κ .
𝓔 ( t , z = 0 ) = band d ω b 1 ( ω ) exp ( i ω t ) ,
b 1 ( ω ) = d t 2 π 𝓔 ( t , z = 0 ) exp ( i ω t ) .
𝓔 ( t , z = N Λ ) = band d ω exp ( i ω t ) d t 2 π 𝓔 ( t , z = 0 ) exp [ i ( ω t + K ( ω ) N Λ ) ] .
𝓔 ( t , z = N Λ ) = ΛΩ κ 2 e i Ω t 0 π Λ d K sin ( K Λ ) e i K N Λ e i Ω κ 2 cos ( K Λ ) ( t t )
d t 2 π 𝓔 ( t , z = 0 ) e i Ω t .
e i Ω κ cos ( x ) ( t t ) = m c m J m [ Ω κ 2 ( t t ) ] cos ( mx )
c m = { 1 if m = 0 2 i m if m > 0 ,
𝓔 ( t , z = N Λ ) = Ω κ 2 e i Ω t m c m 0 π d x sin ( x ) cos ( mx ) e ixN
d t 2 π J m [ Ω κ 2 ( t t ) ] 𝓔 ( t , z = 0 ) e i Ω t .
α m , N = { i π 4 for N = m 1 and N = m 1 i π 4 for N = m + 1 and N = m + 1 2 ( m 2 + N 2 1 ) ( m 2 + N 2 1 ) 2 4 m 2 N 2 for N + m = even
E ( t , z = N Λ ) = κ 2 Ω 2 π m c m α m , N d t J m [ Ω κ 2 ( t t ) ] E ( t , z = 0 ) .

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