Abstract

Single-mode waveguide designs frequently support higher order transverse modes, usually as a consequence of process limitations such as lithography. In these systems, it is important to minimize coupling to higher-order modes so that the system nonetheless behaves single mode. We propose a variational approach to design adiabatic waveguide connections with minimal intermodal coupling. An application of this algorithm in designing the “S-bend” of a whispering-gallery spiral waveguide is demonstrated with approximately 0.05dB insertion loss. Compared to other approaches, our algorithm requires less fabrication resolution and is able to minimize the transition loss over a broadband spectrum. The method can be applied to a wide range of turns and connections and has the advantage of handling connections with arbitrary boundary conditions.

© 2012 OSA

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  1. F. Ladouceur and P. Labeye, “A new gerenal approach to optical waveguide path design,” J. Lightwave Tech. 13, 481–491 (1995).
    [CrossRef]
  2. R. Adar, M. Serbin, and V. Mizrahi, “Less than 1dB per meter propagation loss of silica waveguides measured using a ring resonator,” J. Lightwave Tech. 12, 1369–1372 (1994).
    [CrossRef]
  3. K. Takada, H. Yamada, Y. Hida, Y. Ohmori, and S. Mitachi, “Rayleigh backscattering measurement of 10m long silica-based waveguides,” Electron. Lett. 32, 1665–1667 (1996).
    [CrossRef]
  4. J. F. Bauters, M. Heck, D. John, D. Dai, M. Tien, J. S. Barton, A. Leinse, R. G. Heideman, D. J. Blumenthal, and J. E. Bowers, “Ultra-low-loss high-aspect-ratio Si3N4 waveguides,” Opt. Express 19, 3163–3174 (2011).
    [CrossRef] [PubMed]
  5. H. Lee, T. Chen, J. Li, O. Painter, and K. Vahala, “Ultra-low-loss optical delay line on a silicon chip,” Nat. Commun. 3, doi: (2012).
    [CrossRef] [PubMed]
  6. J. F. Bauters, M. Heck, D. D. John, J. S. Barton, C. M. Bruinink, A. Leinse, R. Heideman, D. J. Blumenthal, and J. E. Bowers, “Planar waveguides with less than 0.1dB/m propagation loss fabricated with wafer bonding,” Opt. Express 19, 24090–24101 (2011).
    [CrossRef] [PubMed]
  7. E. Marcatilli, “Bends in optical dielectric guides,” Bell Syst. Tech. J. 48, 2103–2132 (1969).
  8. R. Baets and P. Lagasse, “Loss calculation and design of arbitrarily curved integrated-optic waveguides,” J. Opt. Soc. Am. 73, 177–182 (1983).
    [CrossRef]
  9. V. Subramaniam, G. De Brabander, D. Naghski, and J. Boyd, “Measurement of mode field profiles and bending and transition losses in curved optical channel waveguides,” J. Lightwave Tech. 15, 990–997 (1997).
    [CrossRef]
  10. W. Gambling, H. Matsumura, and C. Ragdale, “Field deformation in a curved single-mode fiber,” Electron. Lett. 14, 130–132 (1978).
    [CrossRef]
  11. T. Kitoh, N. Takato, M. Yasu, and M. Kawachi, “Bending loss reduction in silica-based waveugide by using lateral offests,” J. Lightwave Tech. 13, 555–562 (1995).
    [CrossRef]
  12. A. Melloni, P. Monguzzi, R. Costa, and M. Martinelli, “Design of curved waveugide: the matched bend,” J. Opt. Soc. Am. A 20, 130–137 (2003).
    [CrossRef]
  13. T. Kominato, Y. Hida, M. Itoh, H. Takahashi, S. Sohma, T. Kitoh, and Y. Hibino, “Extremely low-loss (0.3 dB/m) and long silica-based waveguides with large width and clothoid curve connection,” in Proceedings of ECOC TuI.4.3 (2004).
  14. D. Meek and J. Harris, “Clothoid spline transition spirals,” Math. Comp. 59, 117–133 (1992).
    [CrossRef]
  15. D. J. Walton, “Spiral spline curves for highway design,” Microcomputers in Civil Engineering 4, 99–106 (1989).
    [CrossRef]
  16. K. G. Bass, “The use of clothoid templates in highway design,” Transportation Forum 1, 47–52 (1984).
  17. S. Fleury, P. Soueres, J. P. Laumond, and R. Chatila, “Primitives for smoothing mobile robot trajectories,” IEEE Trans. Robot. Autom. 11, 441–448 (1995).
    [CrossRef]
  18. J. McCrae and K. Singh, “Sketching piecewise clothoid curves,” Computers & Graphics 33, 452–461 (2008).
    [CrossRef] [PubMed]
  19. K. Takada, H. Yamada, Y. Hida, Y. Ohmori, and S. Mitachi, “New waveguide fabrication techniques for next-generation plcs,” NTT Technical Review 3, 37–41 (2005).
  20. A. W. Snyder, “Radiation losses due to variations of radius on dielectric or optical fibers,” IEEE Trans. Microwave Theory Tech. 18, 608–615 (1970).
    [CrossRef]
  21. A. W. Snyder, “Excitation and scattering of modes on a dielectic or optical fiber,” IEEE Trans. Microwave Theory Tech. 17, 1138–1144 (1969).
    [CrossRef]
  22. M. Heiblum and J. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quantum Electron. 11, 75–83 (1975).
    [CrossRef]
  23. R. Ulrich, “Fiber-optic rotation sensing with low drift,” Opt. Express 5, 173–175 (1980).
  24. C. Ciminelli, F. Dell’Olio, C. Campanella, and M. Armenise, “Photonic technologies for angular velocity sensing,” Adv. Opt. Photon. 2, 370–404 (2010).
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  26. X. Yao and L. Maleki, “Optoelectronic microwave oscillator,” J. Opt. Soc. Am. B 13, 1725–1735 (1996).
    [CrossRef]
  27. C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 3, 216–219 (2009).
    [CrossRef]
  28. R. L. Levien, “From spiral to spline: Optimal techniques in interactive curve design,” Ph.D. thesis, UC Berkeley (2009).
  29. S. Ohlin, Splines for Engineers (Eurographics Association, 1987).
  30. B. Soller, D. Gifford, M. Wolfe, and M. Froggatt, “High resolution optical frequency domain reflectometry for characterization of components and assemblies,” Opt. Express 13, 666–674 (2005).
    [CrossRef] [PubMed]
  31. H. Lee, T. Chen, J. Li, O. Painter, and K. Vahala, “Chemically etched ultrahigh-Q wedge-resonator on a silicon chip,” Nat. Photonics 6, 369–373 (2012).
    [CrossRef]
  32. M. Cai, O. Painter, and K. J. Vahala, “Observation of critical coupling in a fiber taper to silica-microsphere whispering gallery mode system,” Phys. Rev. Lett. 85, 1430–1432 (2000).
    [CrossRef]
  33. H. Rokhsari and K. J. Vahala, “Ultralow loss, high q, four port resonant couplers for quantum optics and photonics,” Phys. Rev. Lett. 92, 253905 (2004).
    [CrossRef] [PubMed]

2012

H. Lee, T. Chen, J. Li, O. Painter, and K. Vahala, “Ultra-low-loss optical delay line on a silicon chip,” Nat. Commun. 3, doi: (2012).
[CrossRef] [PubMed]

H. Lee, T. Chen, J. Li, O. Painter, and K. Vahala, “Chemically etched ultrahigh-Q wedge-resonator on a silicon chip,” Nat. Photonics 6, 369–373 (2012).
[CrossRef]

2011

2010

2009

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 3, 216–219 (2009).
[CrossRef]

2008

J. McCrae and K. Singh, “Sketching piecewise clothoid curves,” Computers & Graphics 33, 452–461 (2008).
[CrossRef] [PubMed]

2005

K. Takada, H. Yamada, Y. Hida, Y. Ohmori, and S. Mitachi, “New waveguide fabrication techniques for next-generation plcs,” NTT Technical Review 3, 37–41 (2005).

B. Soller, D. Gifford, M. Wolfe, and M. Froggatt, “High resolution optical frequency domain reflectometry for characterization of components and assemblies,” Opt. Express 13, 666–674 (2005).
[CrossRef] [PubMed]

2004

H. Rokhsari and K. J. Vahala, “Ultralow loss, high q, four port resonant couplers for quantum optics and photonics,” Phys. Rev. Lett. 92, 253905 (2004).
[CrossRef] [PubMed]

T. Kominato, Y. Hida, M. Itoh, H. Takahashi, S. Sohma, T. Kitoh, and Y. Hibino, “Extremely low-loss (0.3 dB/m) and long silica-based waveguides with large width and clothoid curve connection,” in Proceedings of ECOC TuI.4.3 (2004).

2003

2000

M. Cai, O. Painter, and K. J. Vahala, “Observation of critical coupling in a fiber taper to silica-microsphere whispering gallery mode system,” Phys. Rev. Lett. 85, 1430–1432 (2000).
[CrossRef]

1997

V. Subramaniam, G. De Brabander, D. Naghski, and J. Boyd, “Measurement of mode field profiles and bending and transition losses in curved optical channel waveguides,” J. Lightwave Tech. 15, 990–997 (1997).
[CrossRef]

1996

X. Yao and L. Maleki, “Optoelectronic microwave oscillator,” J. Opt. Soc. Am. B 13, 1725–1735 (1996).
[CrossRef]

K. Takada, H. Yamada, Y. Hida, Y. Ohmori, and S. Mitachi, “Rayleigh backscattering measurement of 10m long silica-based waveguides,” Electron. Lett. 32, 1665–1667 (1996).
[CrossRef]

1995

S. Fleury, P. Soueres, J. P. Laumond, and R. Chatila, “Primitives for smoothing mobile robot trajectories,” IEEE Trans. Robot. Autom. 11, 441–448 (1995).
[CrossRef]

F. Ladouceur and P. Labeye, “A new gerenal approach to optical waveguide path design,” J. Lightwave Tech. 13, 481–491 (1995).
[CrossRef]

T. Kitoh, N. Takato, M. Yasu, and M. Kawachi, “Bending loss reduction in silica-based waveugide by using lateral offests,” J. Lightwave Tech. 13, 555–562 (1995).
[CrossRef]

1994

R. Adar, M. Serbin, and V. Mizrahi, “Less than 1dB per meter propagation loss of silica waveguides measured using a ring resonator,” J. Lightwave Tech. 12, 1369–1372 (1994).
[CrossRef]

1992

D. Meek and J. Harris, “Clothoid spline transition spirals,” Math. Comp. 59, 117–133 (1992).
[CrossRef]

1989

D. J. Walton, “Spiral spline curves for highway design,” Microcomputers in Civil Engineering 4, 99–106 (1989).
[CrossRef]

1984

K. G. Bass, “The use of clothoid templates in highway design,” Transportation Forum 1, 47–52 (1984).

1983

1980

R. Ulrich, “Fiber-optic rotation sensing with low drift,” Opt. Express 5, 173–175 (1980).

1978

W. Gambling, H. Matsumura, and C. Ragdale, “Field deformation in a curved single-mode fiber,” Electron. Lett. 14, 130–132 (1978).
[CrossRef]

1975

M. Heiblum and J. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quantum Electron. 11, 75–83 (1975).
[CrossRef]

1970

A. W. Snyder, “Radiation losses due to variations of radius on dielectric or optical fibers,” IEEE Trans. Microwave Theory Tech. 18, 608–615 (1970).
[CrossRef]

1969

A. W. Snyder, “Excitation and scattering of modes on a dielectic or optical fiber,” IEEE Trans. Microwave Theory Tech. 17, 1138–1144 (1969).
[CrossRef]

E. Marcatilli, “Bends in optical dielectric guides,” Bell Syst. Tech. J. 48, 2103–2132 (1969).

Adar, R.

R. Adar, M. Serbin, and V. Mizrahi, “Less than 1dB per meter propagation loss of silica waveguides measured using a ring resonator,” J. Lightwave Tech. 12, 1369–1372 (1994).
[CrossRef]

Armenise, M.

Baets, R.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 3, 216–219 (2009).
[CrossRef]

R. Baets and P. Lagasse, “Loss calculation and design of arbitrarily curved integrated-optic waveguides,” J. Opt. Soc. Am. 73, 177–182 (1983).
[CrossRef]

Barton, J. S.

Bass, K. G.

K. G. Bass, “The use of clothoid templates in highway design,” Transportation Forum 1, 47–52 (1984).

Bauters, J. F.

Biaggio, I.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 3, 216–219 (2009).
[CrossRef]

Blumenthal, D. J.

Bogaerts, W.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 3, 216–219 (2009).
[CrossRef]

Bowers, J. E.

Boyd, J.

V. Subramaniam, G. De Brabander, D. Naghski, and J. Boyd, “Measurement of mode field profiles and bending and transition losses in curved optical channel waveguides,” J. Lightwave Tech. 15, 990–997 (1997).
[CrossRef]

Bruinink, C. M.

Cai, M.

M. Cai, O. Painter, and K. J. Vahala, “Observation of critical coupling in a fiber taper to silica-microsphere whispering gallery mode system,” Phys. Rev. Lett. 85, 1430–1432 (2000).
[CrossRef]

Campanella, C.

Chatila, R.

S. Fleury, P. Soueres, J. P. Laumond, and R. Chatila, “Primitives for smoothing mobile robot trajectories,” IEEE Trans. Robot. Autom. 11, 441–448 (1995).
[CrossRef]

Chen, T.

H. Lee, T. Chen, J. Li, O. Painter, and K. Vahala, “Chemically etched ultrahigh-Q wedge-resonator on a silicon chip,” Nat. Photonics 6, 369–373 (2012).
[CrossRef]

H. Lee, T. Chen, J. Li, O. Painter, and K. Vahala, “Ultra-low-loss optical delay line on a silicon chip,” Nat. Commun. 3, doi: (2012).
[CrossRef] [PubMed]

Ciminelli, C.

Costa, R.

Dai, D.

De Brabander, G.

V. Subramaniam, G. De Brabander, D. Naghski, and J. Boyd, “Measurement of mode field profiles and bending and transition losses in curved optical channel waveguides,” J. Lightwave Tech. 15, 990–997 (1997).
[CrossRef]

Dell’Olio, F.

Diederich, F.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 3, 216–219 (2009).
[CrossRef]

Dumon, P.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 3, 216–219 (2009).
[CrossRef]

Esembeson, B.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 3, 216–219 (2009).
[CrossRef]

Fleury, S.

S. Fleury, P. Soueres, J. P. Laumond, and R. Chatila, “Primitives for smoothing mobile robot trajectories,” IEEE Trans. Robot. Autom. 11, 441–448 (1995).
[CrossRef]

Freude, W.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 3, 216–219 (2009).
[CrossRef]

Froggatt, M.

Gambling, W.

W. Gambling, H. Matsumura, and C. Ragdale, “Field deformation in a curved single-mode fiber,” Electron. Lett. 14, 130–132 (1978).
[CrossRef]

Gifford, D.

Harris, J.

D. Meek and J. Harris, “Clothoid spline transition spirals,” Math. Comp. 59, 117–133 (1992).
[CrossRef]

M. Heiblum and J. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quantum Electron. 11, 75–83 (1975).
[CrossRef]

Heck, M.

Heiblum, M.

M. Heiblum and J. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quantum Electron. 11, 75–83 (1975).
[CrossRef]

Heideman, R.

Heideman, R. G.

Hibino, Y.

T. Kominato, Y. Hida, M. Itoh, H. Takahashi, S. Sohma, T. Kitoh, and Y. Hibino, “Extremely low-loss (0.3 dB/m) and long silica-based waveguides with large width and clothoid curve connection,” in Proceedings of ECOC TuI.4.3 (2004).

Hida, Y.

K. Takada, H. Yamada, Y. Hida, Y. Ohmori, and S. Mitachi, “New waveguide fabrication techniques for next-generation plcs,” NTT Technical Review 3, 37–41 (2005).

T. Kominato, Y. Hida, M. Itoh, H. Takahashi, S. Sohma, T. Kitoh, and Y. Hibino, “Extremely low-loss (0.3 dB/m) and long silica-based waveguides with large width and clothoid curve connection,” in Proceedings of ECOC TuI.4.3 (2004).

K. Takada, H. Yamada, Y. Hida, Y. Ohmori, and S. Mitachi, “Rayleigh backscattering measurement of 10m long silica-based waveguides,” Electron. Lett. 32, 1665–1667 (1996).
[CrossRef]

Itoh, M.

T. Kominato, Y. Hida, M. Itoh, H. Takahashi, S. Sohma, T. Kitoh, and Y. Hibino, “Extremely low-loss (0.3 dB/m) and long silica-based waveguides with large width and clothoid curve connection,” in Proceedings of ECOC TuI.4.3 (2004).

John, D.

John, D. D.

Kawachi, M.

T. Kitoh, N. Takato, M. Yasu, and M. Kawachi, “Bending loss reduction in silica-based waveugide by using lateral offests,” J. Lightwave Tech. 13, 555–562 (1995).
[CrossRef]

Kitoh, T.

T. Kominato, Y. Hida, M. Itoh, H. Takahashi, S. Sohma, T. Kitoh, and Y. Hibino, “Extremely low-loss (0.3 dB/m) and long silica-based waveguides with large width and clothoid curve connection,” in Proceedings of ECOC TuI.4.3 (2004).

T. Kitoh, N. Takato, M. Yasu, and M. Kawachi, “Bending loss reduction in silica-based waveugide by using lateral offests,” J. Lightwave Tech. 13, 555–562 (1995).
[CrossRef]

Kominato, T.

T. Kominato, Y. Hida, M. Itoh, H. Takahashi, S. Sohma, T. Kitoh, and Y. Hibino, “Extremely low-loss (0.3 dB/m) and long silica-based waveguides with large width and clothoid curve connection,” in Proceedings of ECOC TuI.4.3 (2004).

Koos, C.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 3, 216–219 (2009).
[CrossRef]

Labeye, P.

F. Ladouceur and P. Labeye, “A new gerenal approach to optical waveguide path design,” J. Lightwave Tech. 13, 481–491 (1995).
[CrossRef]

Ladouceur, F.

F. Ladouceur and P. Labeye, “A new gerenal approach to optical waveguide path design,” J. Lightwave Tech. 13, 481–491 (1995).
[CrossRef]

Lagasse, P.

Laumond, J. P.

S. Fleury, P. Soueres, J. P. Laumond, and R. Chatila, “Primitives for smoothing mobile robot trajectories,” IEEE Trans. Robot. Autom. 11, 441–448 (1995).
[CrossRef]

Lee, H.

H. Lee, T. Chen, J. Li, O. Painter, and K. Vahala, “Chemically etched ultrahigh-Q wedge-resonator on a silicon chip,” Nat. Photonics 6, 369–373 (2012).
[CrossRef]

H. Lee, T. Chen, J. Li, O. Painter, and K. Vahala, “Ultra-low-loss optical delay line on a silicon chip,” Nat. Commun. 3, doi: (2012).
[CrossRef] [PubMed]

Leinse, A.

Leuthold, J.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 3, 216–219 (2009).
[CrossRef]

Levien, R. L.

R. L. Levien, “From spiral to spline: Optimal techniques in interactive curve design,” Ph.D. thesis, UC Berkeley (2009).

Li, J.

H. Lee, T. Chen, J. Li, O. Painter, and K. Vahala, “Chemically etched ultrahigh-Q wedge-resonator on a silicon chip,” Nat. Photonics 6, 369–373 (2012).
[CrossRef]

H. Lee, T. Chen, J. Li, O. Painter, and K. Vahala, “Ultra-low-loss optical delay line on a silicon chip,” Nat. Commun. 3, doi: (2012).
[CrossRef] [PubMed]

Maleki, L.

Marcatilli, E.

E. Marcatilli, “Bends in optical dielectric guides,” Bell Syst. Tech. J. 48, 2103–2132 (1969).

Martinelli, M.

Matsumura, H.

W. Gambling, H. Matsumura, and C. Ragdale, “Field deformation in a curved single-mode fiber,” Electron. Lett. 14, 130–132 (1978).
[CrossRef]

McCrae, J.

J. McCrae and K. Singh, “Sketching piecewise clothoid curves,” Computers & Graphics 33, 452–461 (2008).
[CrossRef] [PubMed]

Meek, D.

D. Meek and J. Harris, “Clothoid spline transition spirals,” Math. Comp. 59, 117–133 (1992).
[CrossRef]

Melloni, A.

Michinobu, T.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 3, 216–219 (2009).
[CrossRef]

Mitachi, S.

K. Takada, H. Yamada, Y. Hida, Y. Ohmori, and S. Mitachi, “New waveguide fabrication techniques for next-generation plcs,” NTT Technical Review 3, 37–41 (2005).

K. Takada, H. Yamada, Y. Hida, Y. Ohmori, and S. Mitachi, “Rayleigh backscattering measurement of 10m long silica-based waveguides,” Electron. Lett. 32, 1665–1667 (1996).
[CrossRef]

Mizrahi, V.

R. Adar, M. Serbin, and V. Mizrahi, “Less than 1dB per meter propagation loss of silica waveguides measured using a ring resonator,” J. Lightwave Tech. 12, 1369–1372 (1994).
[CrossRef]

Monguzzi, P.

Naghski, D.

V. Subramaniam, G. De Brabander, D. Naghski, and J. Boyd, “Measurement of mode field profiles and bending and transition losses in curved optical channel waveguides,” J. Lightwave Tech. 15, 990–997 (1997).
[CrossRef]

Ohlin, S.

S. Ohlin, Splines for Engineers (Eurographics Association, 1987).

Ohmori, Y.

K. Takada, H. Yamada, Y. Hida, Y. Ohmori, and S. Mitachi, “New waveguide fabrication techniques for next-generation plcs,” NTT Technical Review 3, 37–41 (2005).

K. Takada, H. Yamada, Y. Hida, Y. Ohmori, and S. Mitachi, “Rayleigh backscattering measurement of 10m long silica-based waveguides,” Electron. Lett. 32, 1665–1667 (1996).
[CrossRef]

Painter, O.

H. Lee, T. Chen, J. Li, O. Painter, and K. Vahala, “Ultra-low-loss optical delay line on a silicon chip,” Nat. Commun. 3, doi: (2012).
[CrossRef] [PubMed]

H. Lee, T. Chen, J. Li, O. Painter, and K. Vahala, “Chemically etched ultrahigh-Q wedge-resonator on a silicon chip,” Nat. Photonics 6, 369–373 (2012).
[CrossRef]

M. Cai, O. Painter, and K. J. Vahala, “Observation of critical coupling in a fiber taper to silica-microsphere whispering gallery mode system,” Phys. Rev. Lett. 85, 1430–1432 (2000).
[CrossRef]

Ragdale, C.

W. Gambling, H. Matsumura, and C. Ragdale, “Field deformation in a curved single-mode fiber,” Electron. Lett. 14, 130–132 (1978).
[CrossRef]

Rokhsari, H.

H. Rokhsari and K. J. Vahala, “Ultralow loss, high q, four port resonant couplers for quantum optics and photonics,” Phys. Rev. Lett. 92, 253905 (2004).
[CrossRef] [PubMed]

Serbin, M.

R. Adar, M. Serbin, and V. Mizrahi, “Less than 1dB per meter propagation loss of silica waveguides measured using a ring resonator,” J. Lightwave Tech. 12, 1369–1372 (1994).
[CrossRef]

Singh, K.

J. McCrae and K. Singh, “Sketching piecewise clothoid curves,” Computers & Graphics 33, 452–461 (2008).
[CrossRef] [PubMed]

Snyder, A. W.

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H. Rokhsari and K. J. Vahala, “Ultralow loss, high q, four port resonant couplers for quantum optics and photonics,” Phys. Rev. Lett. 92, 253905 (2004).
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A. W. Snyder, “Excitation and scattering of modes on a dielectic or optical fiber,” IEEE Trans. Microwave Theory Tech. 17, 1138–1144 (1969).
[CrossRef]

IEEE Trans. Robot. Autom.

S. Fleury, P. Soueres, J. P. Laumond, and R. Chatila, “Primitives for smoothing mobile robot trajectories,” IEEE Trans. Robot. Autom. 11, 441–448 (1995).
[CrossRef]

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[CrossRef]

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[CrossRef] [PubMed]

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Figures (6)

Fig. 1:
Fig. 1:

An illustration of the generic connection design problem. Waveguides A and B are shown linked by the connection waveguide (dashed). To create a low insertion loss, coupling to higher-order modes must be reduced at the connecting points as well as through out the transition. The inset shows the curvature (κ) versus path length, z.

Fig. 2:
Fig. 2:

Conformal mapping between a bent waveguide and a straight waveguide. The transverse refractive index, n(x, y), of the curved waveguide is mapped to n eq 2 ( u , v , z ) = n 2 ( u , v ) e 2 u κ ( z ) for a straight guide.

Fig. 3:
Fig. 3:

The optimal S-bend design that minimizes mode coupling between clockwise and counter-clockwise Archimedean spiral waveguides. A jump from 0.5π to −0.5π of the tangent vector in the upper right panel is due to the convention that the tangent vector is defined as [−0.5π, 0.5π] (for example, a tangent vector of 0.6π is considered as 0.6ππ = −0.4π). The geometry property is still continuous.

Fig. 4:
Fig. 4:

(a) Optical micrograph of a spiral waveguide having a physical path length of 7 meters. The input port is in the upper left of the image, and there are two small spirals at the input and output ports (not resolved in the backscatter trace of panel (c)). The entire chip is 4.5cm × 4.5 cm. (b) A magnified view of the adiabatic coupling section (approximately 1mm in diameter). Light brown regions are silicon (under oxide or exposed) while darker brown regions along the border of the light brown are silica that has been undercut by dry etching. Very-dark-brown border regions are also silica but having a wedge profile. For further details see Ref. [5]. (c) Optical backscatter reflectometer measurement of the spiral waveguide. Besides occasional random noise spikes that we believe are associated with small dust particles on the surface of the waveguide, the major singularities in the backscatter signal correspond to the input facet, the optical wave transiting the inner adiabatic coupling region of the spiral and the output facet. The inset shows a close-in view of the adiabatic coupler region (i.e., peak of the singularity). There is no apparent drop in signal within this region. (d) Analysis of the adiabatic coupler insertion loss using backscatter data. Data points are generated by taking the ratio of backscatter signals at symmetrically offset distances away from the adiabatic coupler in (a). The intercept reveals the insertion loss of the S connection as given by a range of possible values falling within a confidence interval determined by linear regression.

Fig. 5:
Fig. 5:

The S connection intercept (see Fig. 4(d)) measured with backscattering reflectometry over a wavelength range from 1536 to 1598 nm in a 1 meter long spiral waveguide. A measurement window of 10 nm is applied. The error bars are obtained from three independent measurements.

Fig. 6:
Fig. 6:

(a) A high-spatial-resolution, backscatter trace of an unsuccessful S-bend design based on a conventional clothoid curve design. About 2.5dB insertion loss was measured using the OBR measurement technique. (b) Micrograph of the clothoid curve S connection.

Equations (19)

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( x , y ) ( u , v ) : u = R 2 log ( x / R 2 ) , v = y
n eq 2 ( u , v , z ) = n 2 ( u , v ) e 2 u κ ( z )
E = p A ( β p , z ) e p ( x , y )
H = p A ( β p , z ) h p ( x , y )
S z ^ ( e p × h q * ) d S = δ p q
C ( β p , β q ) = 1 4 S z ^ ( e q × h p * z e p * × h q z ) d S = ω 4 ( β p β q ) S ( e q e p * ) ε z d S
A ( β q , z 1 ) ~ A ( β p , z 0 ) z 0 z 1 C ( β p , β q ) exp ( i ( β p β q ) z ) d z × exp ( i z 0 z 1 β q ( z ) d z )
| A ( β q , z 1 ) A ( β p , z 0 ) | 2 z 0 z 1 1 ( β p β q ) 2 ( S ( 1 ε ε z ) d S ) 2 d z
| 1 ε ε z | | 2 ( R 2 x ) κ ( z ) z |
E [ κ ( s ) ] = z 0 z 1 ( κ ( s ) s ) 2 d s
κ ( s ) = 0
{ x 0 + 0 s cos ( θ ( s ) ) d s = x 1 x 0 + 0 s sin ( θ ( s ) ) d s = y 1
E = 0 l [ ( θ ) 2 + λ 1 sin θ + λ 2 cos θ ] d s
2 θ + λ 1 cos θ λ 2 sin θ = 0 .
θ λ 2 y = 0 .
κ = λ 2 y
κ ( s ) = a 0 + a 1 s + a 2 s 2 + a 3 s 3 .
{ θ 1 = θ 0 + 0 s 1 κ ( s ) d s = θ 0 + 0 s 1 a 1 s + a 2 s 2 + a 3 s 3 d s κ 1 = a 1 s + a 2 s 2 + a 3 s 3 | s = s 1 κ 1 = d κ ( s ) d s | s = s 1 = d d s ( a 0 + a 1 s + a 2 s 2 + a 3 s 3 ) | s = s 1 ( x 1 , i y 1 ) = 0 s 1 exp ( i θ ( s ) ) d s
log ( P backscatter ( z 2 ) P backscatter ( z 2 ) ) = α 1 z α 0

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