Abstract

We consider the linear and nonlinear optical properties of an optical waveguide consisting of a side-coupled integrated spaced sequence of resonators (SCISSOR). This fully transmissive system possesses large and controllable dispersion because the phase shift imparted by each resonator is strongly frequency dependent. Additionally, near resonance, the circulating power in each resonator can greatly exceed the power carried by the waveguide, leading to greatly enhanced nonlinear effects. We show that the effects of nonlinearity and dispersion can be balanced to create temporal solitons and that many other novel and useful pulse propagation effects can occur over short propagation distances in such a structure.

© 2002 Optical Society of America

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  1. S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289–291 (1992).
    [CrossRef]
  2. Y. Yamamoto and R. E. Slusher, “Optical Processes in microcavities,” Phys. Today 46(6), 66–74 (1993).
    [CrossRef]
  3. J. C. Knight, H. S. T. Driver, R. J. Hutcheon, and G. N. Robertson, “Core-resonance capillary-fiber whispering-gallery-mode laser,” Opt. Lett. 17, 1280–1282 (1992).
    [CrossRef] [PubMed]
  4. J. Popp, M. H. Fields, and R. K. Chang, “Q-switching by saturable absorption in microdroplets: elastic scattering and laser emission,” Opt. Lett. 22, 1296–1298 (1997).
    [CrossRef]
  5. B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998–1005 (1997).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  8. C. K. Madsen and G. Lenz, “Optical all-pass filters for phase response design with applications for dispersion compensation,” IEEE Photon. Technol. Lett. 10, 994–996 (1998).
    [CrossRef]
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  10. F. C. Blom, D. R. van Dijk, H. J. Hoekstra, A. Driessen, and Th. J. A. Popma, “Experimental study of integrated-optics microcavity resonators: toward an all-optical switching device,” Appl. Phys. Lett. 71, 747–749 (1997).
    [CrossRef]
  11. D. W. Vernooy, V. S. Ilchenko, H. Mabuchi, E. W. Streed, and H. J. Kimble, “High-Q measurements of fused-silica microspheres in the near infrared,” Opt. Lett. 23, 247–249 (1998).
    [CrossRef]
  12. V. B. Braginsky and V. S. Ilchenko, “Properties of optical dielectric microresonators,” Sov. Phys. Dokl. 32, 306–307 (1987).
  13. We define the finesse as the FSR divided by the full width at half-depth (FWHD) of the resonance peak. Applying this definition to either the phase sensitivity or the intensity buildup, the finesse is calculated as F=FSRFWHD=2π2 arccos[2r/(1+r2)] →r≈1 π1−r.
  14. Implicit in this assumption is that each resonator is not strongly driven; i.e., the transmitted phase shift Φ per resonator is small with respect to unity.
  15. G. Lenz, B. J. Eggleton, C. R. Giles, C. K. Madsen, and R. E. Slusher, “Dispersive properties of optical filters for WDM systems,” IEEE J. Quantum Electron. 34, 1390–1402 (1998).
    [CrossRef]
  16. For the purpose of quoting the material nonlinearity of standard silica fiber, we have defaulted to a more intuitive definition of the nonlinear coefficient γ such that γPL is the nonlinear phase shift acquired for a power level of P over a distance L.
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  19. The values of keff and γeff are lowered by factors of 3/4 and 9/16, respectively, from their given maximum values when operating at dispersion extremum points.
  20. The simulations used to study pulse evolution in a sequence of waveguide-coupled resonators were carried out by an iterative method in which each iteration consisted of linear and nonlinear phase accumulation during one round trip within the resonator followed by interference at the coupler. Traditional beam or pulse propagation split-step Fourier methods are unnecessary, as structural dispersion possessing a discrete impulse response, is more readily treated in the time domain.
  21. Additionally, higher-order dispersive or nonlinear effects render the scattering of solitons inelastic. Under these conditions, the term “solitary wave” is more appropriate.
  22. To expand Eq. (18) correctly, the B term must also be expanded, which will generate more time derivative terms within the square brackets. Thus the SS contribution will consist of not only two m=2 terms but also one m=1 term. For terms such that m>1, the time derivatives implicitly appear to the far left of each term when the square brackets are expanded.
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    [CrossRef]
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  26. Including the effects of attenuation, the finesse is calculated as F=2π2 arccos{2ra/[1+(ra)2]} →ra≈1 π1−ra, and the transmission is given by T=a2−2ra cos ø+r21−2ra cos ø+(ra)2, where a is the transmission coefficient for a single pass around the resonator. If the attenuation is comparable with the cross coupling, light is resonantly attenuated strongly. Under the condition known as critical coupling (r=a), the finesse drops by a factor of 2 and, more important, the transmission drops to zero.
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    [CrossRef] [PubMed]
  28. G. Lenz, J. Zimmermann, T. Katsufuji, M. E. Lines, H. Y. Hwang, S. Spalter, R. E. Slusher, S.-W. Cheong, J. S. Sanghera, and I. D. Aggarwal, “Large Kerr effect in bulk Se-based chalcogenide glasses,” Opt. Lett. 25, 254–256 (2000).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  33. 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]
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    [CrossRef]
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    [CrossRef]
  36. S. Pereira, J. E. Sipe, J. E. Heebner, and R. W. Boyd, “Gap solitons in a two-channel side-coupled, integrated, space sequence of resonator structure,” Opt. Lett (to be published).
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    [CrossRef]
  38. A. B. Matsko, Y. V. Rostovtsev, H. Z. Cummins, and M. O. Scully, “Using slow light to enhance acousto-optical effects: application to squeezed light,” Phys. Rev. Lett. 84, 5752–5755 (2000).
    [CrossRef] [PubMed]
  39. A. B. Matsko, Y. V. Rostovtsev, M. Fleischhauer, and M. O. Scully, “Anomalous stimulated Brillouin scattering via ultraslow light,” Phys. Rev. Lett. 86, 2006–2009 (2001).
    [CrossRef] [PubMed]
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  41. N. Dubreuil, J. C. Knight, D. K. Leventhal, V. Sandoghdar, J. Hare, and V. Lefevre, “Eroded monomode optical fiber for whispering-gallery mode excitation in fused-silica microspheres,” Opt. Lett. 20, 813–815 (1995).
    [CrossRef] [PubMed]
  42. D. Rafizadeh, J. P. Zhang, S. C. Hagness, A. Taflove, K. A. Stair, S. T. Ho, and R. C. Tiberio, “Waveguide-coupled AlGaAs/GaAs microcavity ring and disk resonators with high finesse and 21.6 nm free-spectral range,” Opt. Lett. 22, 1244 (1997).
    [CrossRef] [PubMed]
  43. J.-P. Laine, B. E. Little, and H. A. Haus, “Etch-eroded fiber coupler for whispering-gallery-mode excitation in high-Q silica microspheres,” IEEE Photon. Technol. Lett. 11, 1429–1430 (1999).
    [CrossRef]
  44. M. Cai, O. Painter, and K. Vahala, “Observation of critical coupling in a fiber taper to a silica-microsphere whispering-gallery mode system,” Phys. Rev. Lett. 85, 74–77 (2000).
    [CrossRef] [PubMed]
  45. B. E. Little and S. T. Chu, “Toward very large-scale integrated photonics,” Opt. Photon. News 11, November2000, pp. 24–29.
    [CrossRef]
  46. 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]

2001

A. B. Matsko, Y. V. Rostovtsev, M. Fleischhauer, and M. O. Scully, “Anomalous stimulated Brillouin scattering via ultraslow light,” Phys. Rev. Lett. 86, 2006–2009 (2001).
[CrossRef] [PubMed]

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]

2000

1999

1998

G. Lenz, B. J. Eggleton, C. R. Giles, C. K. Madsen, and R. E. Slusher, “Dispersive properties of optical filters for WDM systems,” IEEE J. Quantum Electron. 34, 1390–1402 (1998).
[CrossRef]

D. W. Vernooy, V. S. Ilchenko, H. Mabuchi, E. W. Streed, and H. J. Kimble, “High-Q measurements of fused-silica microspheres in the near infrared,” Opt. Lett. 23, 247–249 (1998).
[CrossRef]

V. V. Vassiliev, V. L. Velichansky, V. S. Ilchenko, M. L. Gorodetsky, L. Hollberg, and A. V. Yarovitsky, “Narrow-line-width diode laser with a high-Q microsphere resonator,” Opt. Commun. 158, 305–312 (1998).
[CrossRef]

C. K. Madsen and G. Lenz, “Optical all-pass filters for phase response design with applications for dispersion compensation,” IEEE Photon. Technol. Lett. 10, 994–996 (1998).
[CrossRef]

1997

F. C. Blom, D. R. van Dijk, H. J. Hoekstra, A. Driessen, and Th. J. A. Popma, “Experimental study of integrated-optics microcavity resonators: toward an all-optical switching device,” Appl. Phys. Lett. 71, 747–749 (1997).
[CrossRef]

J. Popp, M. H. Fields, and R. K. Chang, “Q-switching by saturable absorption in microdroplets: elastic scattering and laser emission,” Opt. Lett. 22, 1296–1298 (1997).
[CrossRef]

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

D. Rafizadeh, J. P. Zhang, S. C. Hagness, A. Taflove, K. A. Stair, S. T. Ho, and R. C. Tiberio, “Waveguide-coupled AlGaAs/GaAs microcavity ring and disk resonators with high finesse and 21.6 nm free-spectral range,” Opt. Lett. 22, 1244 (1997).
[CrossRef] [PubMed]

M. D. Lukin, M. Fleischhauer, A. S. Zibrov, H. G. Robinson, V. L. Velichansky, L. Hollberg, and M. O. Scully, “Spectroscopy in dense coherent media: line narrowing and interference effects,” Phys. Rev. Lett. 79, 2959–2962 (1997).
[CrossRef]

1996

1995

1993

Y. Yamamoto and R. E. Slusher, “Optical Processes in microcavities,” Phys. Today 46(6), 66–74 (1993).
[CrossRef]

1992

J. C. Knight, H. S. T. Driver, R. J. Hutcheon, and G. N. Robertson, “Core-resonance capillary-fiber whispering-gallery-mode laser,” Opt. Lett. 17, 1280–1282 (1992).
[CrossRef] [PubMed]

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289–291 (1992).
[CrossRef]

1991

1989

V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, “Quality-factor and nonlinear properties of optical whispering-gallery modes,” Phys. Rev. A 137, 393–397 (1989).

1987

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

V. B. Braginsky and V. S. Ilchenko, “Properties of optical dielectric microresonators,” Sov. Phys. Dokl. 32, 306–307 (1987).

Absil, P. P.

Aggarwal, I. D.

Arnold, S.

Blom, F. C.

F. C. Blom, D. R. van Dijk, H. J. Hoekstra, A. Driessen, and Th. J. A. Popma, “Experimental study of integrated-optics microcavity resonators: toward an all-optical switching device,” Appl. Phys. Lett. 71, 747–749 (1997).
[CrossRef]

Boyd, R. W.

Braginsky, V. B.

V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, “Quality-factor and nonlinear properties of optical whispering-gallery modes,” Phys. Rev. A 137, 393–397 (1989).

V. B. Braginsky and V. S. Ilchenko, “Properties of optical dielectric microresonators,” Sov. Phys. Dokl. 32, 306–307 (1987).

Byer, R. L.

Cai, M.

M. Cai, O. Painter, and K. Vahala, “Observation of critical coupling in a fiber taper to a silica-microsphere whispering-gallery mode system,” Phys. Rev. Lett. 85, 74–77 (2000).
[CrossRef] [PubMed]

Chang, R. K.

Chen, W.

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

Cheong, S.-W.

Cho, P. S.

Chu, S. T.

B. E. Little, S. T. Chu, J. V. Hryniewicz, and P. P. Absil, “Filter synthesis for periodically coupled microring resonators,” Opt. Lett. 25, 344–346 (2000).
[CrossRef]

B. E. Little and S. T. Chu, “Toward very large-scale integrated photonics,” Opt. Photon. News 11, November2000, pp. 24–29.
[CrossRef]

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

B. E. Little and S. T. Chu, “Estimating surface roughness loss and output coupling in microdisk resonators,” Opt. Lett. 21, 1390–1392 (1996).
[CrossRef] [PubMed]

Cummins, H. Z.

A. B. Matsko, Y. V. Rostovtsev, H. Z. Cummins, and M. O. Scully, “Using slow light to enhance acousto-optical effects: application to squeezed light,” Phys. Rev. Lett. 84, 5752–5755 (2000).
[CrossRef] [PubMed]

de Sterke, C. M.

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]

Driessen, A.

F. C. Blom, D. R. van Dijk, H. J. Hoekstra, A. Driessen, and Th. J. A. Popma, “Experimental study of integrated-optics microcavity resonators: toward an all-optical switching device,” Appl. Phys. Lett. 71, 747–749 (1997).
[CrossRef]

Driver, H. S. T.

Dubreuil, N.

Eggleton, B. J.

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]

G. Lenz, B. J. Eggleton, C. R. Giles, C. K. Madsen, and R. E. Slusher, “Dispersive properties of optical filters for WDM systems,” IEEE J. Quantum Electron. 34, 1390–1402 (1998).
[CrossRef]

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]

Fields, M. H.

Fleischhauer, M.

A. B. Matsko, Y. V. Rostovtsev, M. Fleischhauer, and M. O. Scully, “Anomalous stimulated Brillouin scattering via ultraslow light,” Phys. Rev. Lett. 86, 2006–2009 (2001).
[CrossRef] [PubMed]

M. D. Lukin, M. Fleischhauer, A. S. Zibrov, H. G. Robinson, V. L. Velichansky, L. Hollberg, and M. O. Scully, “Spectroscopy in dense coherent media: line narrowing and interference effects,” Phys. Rev. Lett. 79, 2959–2962 (1997).
[CrossRef]

Foresi, J.

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

Giles, C. R.

G. Lenz, B. J. Eggleton, C. R. Giles, C. K. Madsen, and R. E. Slusher, “Dispersive properties of optical filters for WDM systems,” IEEE J. Quantum Electron. 34, 1390–1402 (1998).
[CrossRef]

Gorodetsky, M. L.

V. V. Vassiliev, V. L. Velichansky, V. S. Ilchenko, M. L. Gorodetsky, L. Hollberg, and A. V. Yarovitsky, “Narrow-line-width diode laser with a high-Q microsphere resonator,” Opt. Commun. 158, 305–312 (1998).
[CrossRef]

M. L. Gorodetsky, A. A. Savchenkov, and V. S. Ilchenko, “Ultimate Q of optical microsphere resonators,” Opt. Lett. 21, 453–455 (1996).
[CrossRef] [PubMed]

V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, “Quality-factor and nonlinear properties of optical whispering-gallery modes,” Phys. Rev. A 137, 393–397 (1989).

Hagness, S. C.

Hare, J.

Haus, H. A.

J.-P. Laine, B. E. Little, and H. A. Haus, “Etch-eroded fiber coupler for whispering-gallery-mode excitation in high-Q silica microspheres,” IEEE Photon. Technol. Lett. 11, 1429–1430 (1999).
[CrossRef]

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

Heebner, J. E.

Ho, P.-T.

Ho, S. T.

Hoekstra, H. J.

F. C. Blom, D. R. van Dijk, H. J. Hoekstra, A. Driessen, and Th. J. A. Popma, “Experimental study of integrated-optics microcavity resonators: toward an all-optical switching device,” Appl. Phys. Lett. 71, 747–749 (1997).
[CrossRef]

Hollberg, L.

V. V. Vassiliev, V. L. Velichansky, V. S. Ilchenko, M. L. Gorodetsky, L. Hollberg, and A. V. Yarovitsky, “Narrow-line-width diode laser with a high-Q microsphere resonator,” Opt. Commun. 158, 305–312 (1998).
[CrossRef]

M. D. Lukin, M. Fleischhauer, A. S. Zibrov, H. G. Robinson, V. L. Velichansky, L. Hollberg, and M. O. Scully, “Spectroscopy in dense coherent media: line narrowing and interference effects,” Phys. Rev. Lett. 79, 2959–2962 (1997).
[CrossRef]

Hryniewicz, J. V.

Hutcheon, R. J.

Hwang, H. Y.

Ilchenko, V. S.

D. W. Vernooy, V. S. Ilchenko, H. Mabuchi, E. W. Streed, and H. J. Kimble, “High-Q measurements of fused-silica microspheres in the near infrared,” Opt. Lett. 23, 247–249 (1998).
[CrossRef]

V. V. Vassiliev, V. L. Velichansky, V. S. Ilchenko, M. L. Gorodetsky, L. Hollberg, and A. V. Yarovitsky, “Narrow-line-width diode laser with a high-Q microsphere resonator,” Opt. Commun. 158, 305–312 (1998).
[CrossRef]

M. L. Gorodetsky, A. A. Savchenkov, and V. S. Ilchenko, “Ultimate Q of optical microsphere resonators,” Opt. Lett. 21, 453–455 (1996).
[CrossRef] [PubMed]

V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, “Quality-factor and nonlinear properties of optical whispering-gallery modes,” Phys. Rev. A 137, 393–397 (1989).

V. B. Braginsky and V. S. Ilchenko, “Properties of optical dielectric microresonators,” Sov. Phys. Dokl. 32, 306–307 (1987).

Jonekis, L. G.

Katsufuji, T.

Kimble, H. J.

Knight, J. C.

Krug, P. A.

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]

Laine, J.-P.

J.-P. Laine, B. E. Little, and H. A. Haus, “Etch-eroded fiber coupler for whispering-gallery-mode excitation in high-Q silica microspheres,” IEEE Photon. Technol. Lett. 11, 1429–1430 (1999).
[CrossRef]

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

Lee, R. K.

Lefevre, V.

Lenz, G.

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]

G. Lenz, J. Zimmermann, T. Katsufuji, M. E. Lines, H. Y. Hwang, S. Spalter, R. E. Slusher, S.-W. Cheong, J. S. Sanghera, and I. D. Aggarwal, “Large Kerr effect in bulk Se-based chalcogenide glasses,” Opt. Lett. 25, 254–256 (2000).
[CrossRef]

G. Lenz, B. J. Eggleton, C. R. Giles, C. K. Madsen, and R. E. Slusher, “Dispersive properties of optical filters for WDM systems,” IEEE J. Quantum Electron. 34, 1390–1402 (1998).
[CrossRef]

C. K. Madsen and G. Lenz, “Optical all-pass filters for phase response design with applications for dispersion compensation,” IEEE Photon. Technol. Lett. 10, 994–996 (1998).
[CrossRef]

Leventhal, D. K.

Levi, A. F. J.

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289–291 (1992).
[CrossRef]

Lines, M. E.

Little, B. E.

P. P. Absil, J. V. Hryniewicz, B. E. Little, P. S. Cho, R. A. Wilson, L. G. Jonekis, and P.-T. Ho, “Wavelength conversion in GaAs micro-ring resonators,” Opt. Lett. 25, 554–556 (2000).
[CrossRef]

B. E. Little and S. T. Chu, “Toward very large-scale integrated photonics,” Opt. Photon. News 11, November2000, pp. 24–29.
[CrossRef]

B. E. Little, S. T. Chu, J. V. Hryniewicz, and P. P. Absil, “Filter synthesis for periodically coupled microring resonators,” Opt. Lett. 25, 344–346 (2000).
[CrossRef]

J.-P. Laine, B. E. Little, and H. A. Haus, “Etch-eroded fiber coupler for whispering-gallery-mode excitation in high-Q silica microspheres,” IEEE Photon. Technol. Lett. 11, 1429–1430 (1999).
[CrossRef]

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

B. E. Little and S. T. Chu, “Estimating surface roughness loss and output coupling in microdisk resonators,” Opt. Lett. 21, 1390–1392 (1996).
[CrossRef] [PubMed]

Liu, C. T.

Logan, R. A.

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289–291 (1992).
[CrossRef]

Lukin, M. D.

M. D. Lukin, M. Fleischhauer, A. S. Zibrov, H. G. Robinson, V. L. Velichansky, L. Hollberg, and M. O. Scully, “Spectroscopy in dense coherent media: line narrowing and interference effects,” Phys. Rev. Lett. 79, 2959–2962 (1997).
[CrossRef]

Mabuchi, H.

Madsen, C. K.

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]

G. Lenz, B. J. Eggleton, C. R. Giles, C. K. Madsen, and R. E. Slusher, “Dispersive properties of optical filters for WDM systems,” IEEE J. Quantum Electron. 34, 1390–1402 (1998).
[CrossRef]

C. K. Madsen and G. Lenz, “Optical all-pass filters for phase response design with applications for dispersion compensation,” IEEE Photon. Technol. Lett. 10, 994–996 (1998).
[CrossRef]

Matsko, A. B.

A. B. Matsko, Y. V. Rostovtsev, M. Fleischhauer, and M. O. Scully, “Anomalous stimulated Brillouin scattering via ultraslow light,” Phys. Rev. Lett. 86, 2006–2009 (2001).
[CrossRef] [PubMed]

A. B. Matsko, Y. V. Rostovtsev, H. Z. Cummins, and M. O. Scully, “Using slow light to enhance acousto-optical effects: application to squeezed light,” Phys. Rev. Lett. 84, 5752–5755 (2000).
[CrossRef] [PubMed]

McCall, S. L.

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289–291 (1992).
[CrossRef]

Mills, D. L.

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

Painter, O.

M. Cai, O. Painter, and K. Vahala, “Observation of critical coupling in a fiber taper to a silica-microsphere whispering-gallery mode system,” Phys. Rev. Lett. 85, 74–77 (2000).
[CrossRef] [PubMed]

Pearton, S. J.

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289–291 (1992).
[CrossRef]

Popma, Th. J. A.

F. C. Blom, D. R. van Dijk, H. J. Hoekstra, A. Driessen, and Th. J. A. Popma, “Experimental study of integrated-optics microcavity resonators: toward an all-optical switching device,” Appl. Phys. Lett. 71, 747–749 (1997).
[CrossRef]

Popp, J.

Rafizadeh, D.

Ramsey, J. M.

Robertson, G. N.

Robinson, H. G.

M. D. Lukin, M. Fleischhauer, A. S. Zibrov, H. G. Robinson, V. L. Velichansky, L. Hollberg, and M. O. Scully, “Spectroscopy in dense coherent media: line narrowing and interference effects,” Phys. Rev. Lett. 79, 2959–2962 (1997).
[CrossRef]

Rostovtsev, Y. V.

A. B. Matsko, Y. V. Rostovtsev, M. Fleischhauer, and M. O. Scully, “Anomalous stimulated Brillouin scattering via ultraslow light,” Phys. Rev. Lett. 86, 2006–2009 (2001).
[CrossRef] [PubMed]

A. B. Matsko, Y. V. Rostovtsev, H. Z. Cummins, and M. O. Scully, “Using slow light to enhance acousto-optical effects: application to squeezed light,” Phys. Rev. Lett. 84, 5752–5755 (2000).
[CrossRef] [PubMed]

Sandoghdar, V.

Sanghera, J. S.

Savchenkov, A. A.

Scherer, A.

Schiller, S.

Scully, M. O.

A. B. Matsko, Y. V. Rostovtsev, M. Fleischhauer, and M. O. Scully, “Anomalous stimulated Brillouin scattering via ultraslow light,” Phys. Rev. Lett. 86, 2006–2009 (2001).
[CrossRef] [PubMed]

A. B. Matsko, Y. V. Rostovtsev, H. Z. Cummins, and M. O. Scully, “Using slow light to enhance acousto-optical effects: application to squeezed light,” Phys. Rev. Lett. 84, 5752–5755 (2000).
[CrossRef] [PubMed]

M. D. Lukin, M. Fleischhauer, A. S. Zibrov, H. G. Robinson, V. L. Velichansky, L. Hollberg, and M. O. Scully, “Spectroscopy in dense coherent media: line narrowing and interference effects,” Phys. Rev. Lett. 79, 2959–2962 (1997).
[CrossRef]

Sipe, J. E.

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]

Slusher, R. E.

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]

G. Lenz, J. Zimmermann, T. Katsufuji, M. E. Lines, H. Y. Hwang, S. Spalter, R. E. Slusher, S.-W. Cheong, J. S. Sanghera, and I. D. Aggarwal, “Large Kerr effect in bulk Se-based chalcogenide glasses,” Opt. Lett. 25, 254–256 (2000).
[CrossRef]

G. Lenz, B. J. Eggleton, C. R. Giles, C. K. Madsen, and R. E. Slusher, “Dispersive properties of optical filters for WDM systems,” IEEE J. Quantum Electron. 34, 1390–1402 (1998).
[CrossRef]

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]

Y. Yamamoto and R. E. Slusher, “Optical Processes in microcavities,” Phys. Today 46(6), 66–74 (1993).
[CrossRef]

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289–291 (1992).
[CrossRef]

Spalter, S.

Stair, K. A.

Streed, E. W.

Taflove, A.

Tiberio, R. C.

Vahala, K.

M. Cai, O. Painter, and K. Vahala, “Observation of critical coupling in a fiber taper to a silica-microsphere whispering-gallery mode system,” Phys. Rev. Lett. 85, 74–77 (2000).
[CrossRef] [PubMed]

van Dijk, D. R.

F. C. Blom, D. R. van Dijk, H. J. Hoekstra, A. Driessen, and Th. J. A. Popma, “Experimental study of integrated-optics microcavity resonators: toward an all-optical switching device,” Appl. Phys. Lett. 71, 747–749 (1997).
[CrossRef]

Vassiliev, V. V.

V. V. Vassiliev, V. L. Velichansky, V. S. Ilchenko, M. L. Gorodetsky, L. Hollberg, and A. V. Yarovitsky, “Narrow-line-width diode laser with a high-Q microsphere resonator,” Opt. Commun. 158, 305–312 (1998).
[CrossRef]

Velichansky, V. L.

V. V. Vassiliev, V. L. Velichansky, V. S. Ilchenko, M. L. Gorodetsky, L. Hollberg, and A. V. Yarovitsky, “Narrow-line-width diode laser with a high-Q microsphere resonator,” Opt. Commun. 158, 305–312 (1998).
[CrossRef]

M. D. Lukin, M. Fleischhauer, A. S. Zibrov, H. G. Robinson, V. L. Velichansky, L. Hollberg, and M. O. Scully, “Spectroscopy in dense coherent media: line narrowing and interference effects,” Phys. Rev. Lett. 79, 2959–2962 (1997).
[CrossRef]

Vernooy, D. W.

Whitten, W. B.

Wilson, R. A.

Xu, Y.

Yamamoto, Y.

Y. Yamamoto and R. E. Slusher, “Optical Processes in microcavities,” Phys. Today 46(6), 66–74 (1993).
[CrossRef]

Yariv, A.

Yarovitsky, A. V.

V. V. Vassiliev, V. L. Velichansky, V. S. Ilchenko, M. L. Gorodetsky, L. Hollberg, and A. V. Yarovitsky, “Narrow-line-width diode laser with a high-Q microsphere resonator,” Opt. Commun. 158, 305–312 (1998).
[CrossRef]

Zhang, J. P.

Zibrov, A. S.

M. D. Lukin, M. Fleischhauer, A. S. Zibrov, H. G. Robinson, V. L. Velichansky, L. Hollberg, and M. O. Scully, “Spectroscopy in dense coherent media: line narrowing and interference effects,” Phys. Rev. Lett. 79, 2959–2962 (1997).
[CrossRef]

Zimmermann, J.

Appl. Phys. Lett.

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289–291 (1992).
[CrossRef]

F. C. Blom, D. R. van Dijk, H. J. Hoekstra, A. Driessen, and Th. J. A. Popma, “Experimental study of integrated-optics microcavity resonators: toward an all-optical switching device,” Appl. Phys. Lett. 71, 747–749 (1997).
[CrossRef]

IEEE J. Quantum Electron.

G. Lenz, B. J. Eggleton, C. R. Giles, C. K. Madsen, and R. E. Slusher, “Dispersive properties of optical filters for WDM systems,” IEEE J. Quantum Electron. 34, 1390–1402 (1998).
[CrossRef]

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.

J.-P. Laine, B. E. Little, and H. A. Haus, “Etch-eroded fiber coupler for whispering-gallery-mode excitation in high-Q silica microspheres,” IEEE Photon. Technol. Lett. 11, 1429–1430 (1999).
[CrossRef]

C. K. Madsen and G. Lenz, “Optical all-pass filters for phase response design with applications for dispersion compensation,” IEEE Photon. Technol. Lett. 10, 994–996 (1998).
[CrossRef]

J. Lightwave Technol.

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

J. Opt. Soc. Am. B

Opt. Commun.

V. V. Vassiliev, V. L. Velichansky, V. S. Ilchenko, M. L. Gorodetsky, L. Hollberg, and A. V. Yarovitsky, “Narrow-line-width diode laser with a high-Q microsphere resonator,” Opt. Commun. 158, 305–312 (1998).
[CrossRef]

Opt. Lett.

P. P. Absil, J. V. Hryniewicz, B. E. Little, P. S. Cho, R. A. Wilson, L. G. Jonekis, and P.-T. Ho, “Wavelength conversion in GaAs micro-ring resonators,” Opt. Lett. 25, 554–556 (2000).
[CrossRef]

M. L. Gorodetsky, A. A. Savchenkov, and V. S. Ilchenko, “Ultimate Q of optical microsphere resonators,” Opt. Lett. 21, 453–455 (1996).
[CrossRef] [PubMed]

G. Lenz, J. Zimmermann, T. Katsufuji, M. E. Lines, H. Y. Hwang, S. Spalter, R. E. Slusher, S.-W. Cheong, J. S. Sanghera, and I. D. Aggarwal, “Large Kerr effect in bulk Se-based chalcogenide glasses,” Opt. Lett. 25, 254–256 (2000).
[CrossRef]

B. E. Little and S. T. Chu, “Estimating surface roughness loss and output coupling in microdisk resonators,” Opt. Lett. 21, 1390–1392 (1996).
[CrossRef] [PubMed]

A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled resonator optical waveguide: a proposal and analysis,” Opt. Lett. 24, 711–713 (1999).
[CrossRef]

B. E. Little, S. T. Chu, J. V. Hryniewicz, and P. P. Absil, “Filter synthesis for periodically coupled microring resonators,” Opt. Lett. 25, 344–346 (2000).
[CrossRef]

S. Schiller and R. L. Byer, “High-resolution spectroscopy of whispering gallery modes in large dielectric spheres,” Opt. Lett. 16, 1138–1140 (1991).
[CrossRef] [PubMed]

J. C. Knight, H. S. T. Driver, R. J. Hutcheon, and G. N. Robertson, “Core-resonance capillary-fiber whispering-gallery-mode laser,” Opt. Lett. 17, 1280–1282 (1992).
[CrossRef] [PubMed]

J. Popp, M. H. Fields, and R. K. Chang, “Q-switching by saturable absorption in microdroplets: elastic scattering and laser emission,” Opt. Lett. 22, 1296–1298 (1997).
[CrossRef]

D. W. Vernooy, V. S. Ilchenko, H. Mabuchi, E. W. Streed, and H. J. Kimble, “High-Q measurements of fused-silica microspheres in the near infrared,” Opt. Lett. 23, 247–249 (1998).
[CrossRef]

J. E. Heebner and R. W. Boyd, “Enhanced all-optical switching by use of a nonlinear fiber ring resonator,” Opt. Lett. 24, 847–849 (1999).
[CrossRef]

S. Arnold, C. T. Liu, W. B. Whitten, and J. M. Ramsey, “Room-temperature microparticle-based persistent spectral hole burning memory,” Opt. Lett. 16, 420–422 (1991).
[CrossRef] [PubMed]

N. Dubreuil, J. C. Knight, D. K. Leventhal, V. Sandoghdar, J. Hare, and V. Lefevre, “Eroded monomode optical fiber for whispering-gallery mode excitation in fused-silica microspheres,” Opt. Lett. 20, 813–815 (1995).
[CrossRef] [PubMed]

D. Rafizadeh, J. P. Zhang, S. C. Hagness, A. Taflove, K. A. Stair, S. T. Ho, and R. C. Tiberio, “Waveguide-coupled AlGaAs/GaAs microcavity ring and disk resonators with high finesse and 21.6 nm free-spectral range,” Opt. Lett. 22, 1244 (1997).
[CrossRef] [PubMed]

Opt. Photon. News

B. E. Little and S. T. Chu, “Toward very large-scale integrated photonics,” Opt. Photon. News 11, November2000, pp. 24–29.
[CrossRef]

Phys. Rev. A

V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, “Quality-factor and nonlinear properties of optical whispering-gallery modes,” Phys. Rev. A 137, 393–397 (1989).

Phys. Rev. Lett.

M. D. Lukin, M. Fleischhauer, A. S. Zibrov, H. G. Robinson, V. L. Velichansky, L. Hollberg, and M. O. Scully, “Spectroscopy in dense coherent media: line narrowing and interference effects,” Phys. Rev. Lett. 79, 2959–2962 (1997).
[CrossRef]

A. B. Matsko, Y. V. Rostovtsev, H. Z. Cummins, and M. O. Scully, “Using slow light to enhance acousto-optical effects: application to squeezed light,” Phys. Rev. Lett. 84, 5752–5755 (2000).
[CrossRef] [PubMed]

A. B. Matsko, Y. V. Rostovtsev, M. Fleischhauer, and M. O. Scully, “Anomalous stimulated Brillouin scattering via ultraslow light,” Phys. Rev. Lett. 86, 2006–2009 (2001).
[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]

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]

M. Cai, O. Painter, and K. Vahala, “Observation of critical coupling in a fiber taper to a silica-microsphere whispering-gallery mode system,” Phys. Rev. Lett. 85, 74–77 (2000).
[CrossRef] [PubMed]

Phys. Today

Y. Yamamoto and R. E. Slusher, “Optical Processes in microcavities,” Phys. Today 46(6), 66–74 (1993).
[CrossRef]

Sov. Phys. Dokl.

V. B. Braginsky and V. S. Ilchenko, “Properties of optical dielectric microresonators,” Sov. Phys. Dokl. 32, 306–307 (1987).

Other

We define the finesse as the FSR divided by the full width at half-depth (FWHD) of the resonance peak. Applying this definition to either the phase sensitivity or the intensity buildup, the finesse is calculated as F=FSRFWHD=2π2 arccos[2r/(1+r2)] →r≈1 π1−r.

Implicit in this assumption is that each resonator is not strongly driven; i.e., the transmitted phase shift Φ per resonator is small with respect to unity.

G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, San Diego, Calif., 2001).

The values of keff and γeff are lowered by factors of 3/4 and 9/16, respectively, from their given maximum values when operating at dispersion extremum points.

The simulations used to study pulse evolution in a sequence of waveguide-coupled resonators were carried out by an iterative method in which each iteration consisted of linear and nonlinear phase accumulation during one round trip within the resonator followed by interference at the coupler. Traditional beam or pulse propagation split-step Fourier methods are unnecessary, as structural dispersion possessing a discrete impulse response, is more readily treated in the time domain.

Additionally, higher-order dispersive or nonlinear effects render the scattering of solitons inelastic. Under these conditions, the term “solitary wave” is more appropriate.

To expand Eq. (18) correctly, the B term must also be expanded, which will generate more time derivative terms within the square brackets. Thus the SS contribution will consist of not only two m=2 terms but also one m=1 term. For terms such that m>1, the time derivatives implicitly appear to the far left of each term when the square brackets are expanded.

H. M. Gibbs, Optical Bistability: Controlling Light with Light (Academic, New York, 1985).

Including the effects of attenuation, the finesse is calculated as F=2π2 arccos{2ra/[1+(ra)2]} →ra≈1 π1−ra, and the transmission is given by T=a2−2ra cos ø+r21−2ra cos ø+(ra)2, where a is the transmission coefficient for a single pass around the resonator. If the attenuation is comparable with the cross coupling, light is resonantly attenuated strongly. Under the condition known as critical coupling (r=a), the finesse drops by a factor of 2 and, more important, the transmission drops to zero.

S. Pereira, J. E. Sipe, J. E. Heebner, and R. W. Boyd, “Gap solitons in a two-channel side-coupled, integrated, space sequence of resonator structure,” Opt. Lett (to be published).

A. Taflove and S. C. Hagness, Computational Electrodynamics, the Finite-Difference Time-Domain Method (Artech House, Boston, 2000).

S. Blair, J. E. Heebner, and R. W. Boyd, “Beyond the absorption-limited nonlinear phase shift with micro-ring resonators,” Opt. Lett. (to be published).

For the purpose of quoting the material nonlinearity of standard silica fiber, we have defaulted to a more intuitive definition of the nonlinear coefficient γ such that γPL is the nonlinear phase shift acquired for a power level of P over a distance L.

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Figures (11)

Fig. 1
Fig. 1

Structured, fully transmissive waveguide and resonator configuration, forming a SCISSOR. E1 is the incident field, E4 is the field injected into the disk, E3 is the field after one pass around the resonator, and E2 is the transmitted field.

Fig. 2
Fig. 2

(a) Transmission, (b) build-up factor, (c) effective phase shift acquired on transmission, and (d) phase sensitivity [derivative of (c)] plotted versus the internal phase shift for a waveguide-coupled resonator with a finesse of 10π.

Fig. 3
Fig. 3

Dispersion relation (propagation constant versus frequency) for light propagation in a SCISSOR with differing values of the self-coupling coefficient r. For generality, the waveguide contribution of constant slope k0 has been subtracted from the effective propagation constant keff.

Fig. 4
Fig. 4

A weak pulse tuned to the dispersion maxima disperses while propagating in a SCISSOR. A 10-ps FWHM hyperbolic secant pulse tuned for maximum anomalous GVD (B=0.13) enters the system consisting of 100 resonators each with a 5-µm diameter and finesse of 10π, spaced by 10 µm. Note that the peak power is reduced by a factor of ∼4 after propagating only 1 mm as a consequence of the strong induced dispersion.

Fig. 5
Fig. 5

A pulse with amplitude corresponding to the fundamental soliton propagates in a SCISSOR without dispersing. The same parameters were used as in Fig. 4, but with a peak power of 125 mW (Γ=0.0196) in a chalcogenide-glass-based system.

Fig. 6
Fig. 6

A negative pulse in a uniform intensity background with parameters corresponding to the fundamental dark soliton propagates in a SCISSOR without dispersing. The incident field distribution was a hyperbolic tangent with twice the pulse width of the bright soliton and a background power that was one fourth that of its peak power in Fig. 5.

Fig. 7
Fig. 7

A higher-order breathing soliton is unstable under the influence of the resonator-induced intensity-dependent group velocity (SS). Here a second-order soliton splits into two stable fundamental solitons on propagation in a SCISSOR. The incident field distribution was the same as in Fig. 5 but with four times the peak power.

Fig. 8
Fig. 8

Demonstration of modulation instability in a SCISSOR. The input field consists of 800 mW of cw power with a 1% power ripple. The SCISSOR parameters are chosen such that the peak of the instability gain is at the input modulation frequency of 100 GHz. Note that the modulation frequency given by Eq. (25) need not be a resonance frequency of the structure.

Fig. 9
Fig. 9

Radiation-loss-limited finesse of the lowest-order radial TE and TM whispering-gallery modes of a dielectric cylinder of index n1 in a medium of index n2 plotted versus normalized radius (n1ω/c)R. The family of diagonal curves represents varying refractive-index contrasts (n1/n2). The family of nearly vertical curves corresponds to whispering-gallery mode resonances, each characterized by an azimuthal mode number m. The plots were obtained by numerical solution of the dispersion relation for whispering-gallery modes.

Fig. 10
Fig. 10

Finite-difference time-domain method of solving Maxwell’s equations for a SCISSOR structure composed of 5 microresonators. A TE field of wavelength 1.55 µm is launched into the 0.4-µm-wide waveguide evanescently side coupled to a disk with a diameter of 5.1 µm. The refractive index of the air-clad disk and guide is 2. Exclusive coupling to the m=16 azimuthal whispering-gallery mode is achieved by careful selection of parameters. (a) Strong scattering losses result due to roughness associated with a 50 nm grid. (b) Scattering losses are made negligible by use of a 30-nm grid. Consequently, a buildup factor of 16 and finesse of 25 are achieved in this structure.

Fig. 11
Fig. 11

Functional dependence of the (a) group-velocity reduction, (b) GVD, (c) third-order dispersion, (d) SPM coefficient, and (e) SS coefficient on the normalized detuning ϕ for a SCISSOR. The parameters have been scaled such that the curves are universal and fit within the same plot limits.

Equations (32)

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E˜4(ω)E˜2(ω)=rititrE˜3(ω)E˜1(ω),
E˜3(ω)=exp[iϕ(ω)]E˜4(ω),
ϕ(ω)=2πRn0ω/c.
E˜2(ω)=r-exp[iϕ(ω)]1-r exp[iϕ(ω)]E˜1(ω),
E˜3(ω)=itexp[iϕ(ω)]1-r exp[iϕ(ω)]E˜1(ω).
E˜2(ω)=exp[iΦ(ω)]E˜1(ω),
Φ(ω)=π+ϕ(ω)+2 arctan  r sin ϕ(ω)1-r cos ϕ(ω).
B(ω)=E˜3(ω)E˜1(ω)2=1-r21-2r cos ϕ(ω)+r2
r1B01+B02  sin2[ϕ(ω)/2].
k0(ω)=n0ω/c+Φ(ω)/L.
H(ω)=exp(iΦ)=exp(iΦ0)1+n=1  inn! m=1  1m! dmΦdϕm ϕ0×(ϕ-ϕ0)mn.
E˜j+1(ω)=expin0ω0c+Φ0Lδz1+n=1  inn! n0cΔωδz+m=1  1m! δzL dmΦdϕmϕ0(ϕ-ϕ0)mnE˜j(ω).
Aj+1(t)=Aj(t)+n=1  inn! i n0cδz t+m=1  1m! δzL dmΦdωm ϕ0iTR  tmnAj(t).
dAdz=-n0c t+i m=1  1m! 1L dmΦdωmϕ0iTR  tmA.
keff=dkeffdω=n0c+1L dΦdω=n0c1+2πRL 1-r21-2r cos ϕ0+r2
ϕ0=0,r1n0c 1+4RL F.
keff=dkeff2dω2=1L d2Φdω2=TR2L -2r(1-r2)sin ϕ0(1-2r cos ϕ0+r2)2
ϕ0=±πF333F2TR24π2L.
keff1L d3Φdω3=TR3L -2r(1-r2)[(1+r2)cos ϕ0-3r+r cos 2ϕ0](1-2r cos ϕ0+r2)3
ϕ0=0,r1-4π3 F3TR3L.
γeff1L dΦd|E˜1|2=1L dΦdϕ dϕd|E˜3|2 d|E˜3|2d|E˜1|2=γ2πRL 1-r21-2r cos ϕ0+r22 ϕ0=0,r1γ 8RπL F2.
dAdz=-n0c t+i m=1  1m! 1L dmΦdϕm ϕ0×γ2πRB|A|2+iTR  tmA.
zA=-i 12keff  2τ2A+iγeff|A|2A.
A(z, τ)=A0  sech(τ/TP)exp(i½γeff|A0|2z),
Γ=π23  arcsech2(1/2)B2B2.
s=γeffγeff=2B dBdωϕ0=±π/F33FTRπ.
γeff(|A|2)r1  γ 2πRL Bϕ021+γ2πRBϕ022π-Φ0|A|2,
zA+keff  tA=-i 12keff  2t2A+16keff  3t3A+i1+is t  γeff|A|21+|A|2/|AS|2A.
Ωm=±2γeff|A0|2/|keff|,
F=FSRFWHD=2π2 arccos[2r/(1+r2)]r1π1-r.
F=2π2 arccos{2ra/[1+(ra)2]}ra1  π1-ra,
T=a2-2ra cos ϕ+r21-2ra cos ϕ+(ra)2,

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