## A general design algorithm for low optical loss adiabatic connections in waveguides |

Optics Express, Vol. 20, Issue 20, pp. 22819-22829 (2012)

http://dx.doi.org/10.1364/OE.20.022819

Acrobat PDF (2030 KB)

### 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

## 1. Introduction

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]

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]

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: [CrossRef] (2012). [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]

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

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]

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]

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]

## 2. Design Algorithm

### 2.1. Overview

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]

*n*(

*x*,

*y*) denotes the transverse refractive index of the waveguide,

*κ*(

*z*) is the curvature, (

*x*,

*y*) are the transverse coordinates (

*x*=

*R*

_{1}and

*x*=

*R*

_{2}at the inner and outer boundary) and z is the coordinate along the direction of propagation, then using the conformal transformation (see Fig. 2), one can show that a bent waveguide with curvature

*κ*(

*z*) behaves like a straight waveguide with a refractive index profile [22

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

*ε*(

*r⃗*) and constant permeability

*μ*. The fields are expressed by the superposition of local normal modes, {

*e⃗*(

_{p}*x*,

*y*),

*⃗h*(

_{p}*x*,

*y*)}, having propagation constants {

*β*} with orthonormal condition where

_{p}*z*̂ is the unit vector along the axis of propagation and “

*S*” represents the local surface area that is normal to the axis of propagation. For slow variation in

*ε*, the coupling coefficient

*C*(

*β*,

_{p}*β*) between two modes

_{q}*p*and

*q*is given by (

*q*≠

*p*) where

*ω*is the optical frequency in radians/sec. We see that the coupling is directly proportional to

*z*

_{0}to

*z*

_{1}and suppose that at

*z*=

*z*

_{0}the only non-zero modal amplitude is

*A*(

*β*,

_{p}*z*

_{0}). To leading order, the propagation solution is given by: Thus, power transfer from mode

*p*to

*q*is found to be proportional to

*ε*(

_{eq}*x*,

*y*,

*z*) =

*ε*(

*x,y*)

*e*

^{2uκ(z)}and the width of waveguide much smaller than its curvature radius (

*i.e. w*=

*R*

_{2}−

*R*

_{1}≪

*R*

_{1}), we have Herein, it is apparent that a narrower waveguide is desirable to minimize the power transfer. Assuming the width of the waveguide to be unchanged, a tempting objective functional to be minimized is given by: where, for mathematical simplicity, we have replaced the

*z*with the arc-length parameter

*s*. This equation is the working equation used here to minimize the power transfer in a connection waveguide having evolving curvature along the connection path.

### 2.2. S-Bend Design

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: [CrossRef] (2012). [PubMed]

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]

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]

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 Si_{3}N_{4} 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: [CrossRef] (2012). [PubMed]

**3**, doi: [CrossRef] (2012). [PubMed]

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 Si_{3}N_{4} waveguides,” Opt. Express **19**, 3163–3174 (2011). [CrossRef] [PubMed]

*E*is the

*L*

^{2}norm of the variation of curvature along the arc. This problem is therefore similar to the “minimization of variation of curvature (MVC)” problem in computer graphics and free-way design [15

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

*E*leads to the following corresponding Euler-Lagrange equation, The solution family is

*κ*=

*k*

_{0}+

*k*

_{1}

*s*, which are similarity transformations of the basic Euler spiral (

*i.e.*clothoid)

*κ*=

*s*[14

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

*κ*rather than curve (

*x*,

*y*); also, the boundary conditions are expressed in terms of curvature rather than

*z*. Similarly, positional endpoint constraints are missing. The most general endpoint constraints are the specification of curve length, end point positions, and end point tangents. Of these, the end point position constraints require Lagrange multipliers. The end point constraints (

*x*

_{0},

*y*

_{0}) and (

*x*

_{1},

*y*

_{1}) for connection to waveguides A and B are expressed as the integral of the unit tangent vector of direction

*θ*(

*s*): Therefore, by adding the Lagrange multipliers (

*λ*

_{1}and

*λ*

_{2}) and eliminating

*κ*in favor of

*θ*, the following the functional must be minimized over the length

*l*[28] The corresponding Euler-Poisson equation is then given by: Without loss of generality we may assume

*λ*

_{1}= 0 and observe that

*κ*(

*s*) =

*θ*′(

*s*) and the constant of integration has been set at zero by a translation of the curve. As an aside, it is interesting to note that for

*λ*

_{2}= 0, this equation is equivalent to that of the Euler spiral. To approximately solve this ODE (Eq. (16)), we may consider a family of curves with curvature given in terms of a cubic polynomial of arc length “

*s*” [28, 29]. This curve family provides a very good approximation to the original variational problem and provides an analytical expression of the connection path [28]. To see this, assuming small turning angles,

*y*is almost proportional to

*s*plus a constant offset, so substituting into Eq. (16),

*κ*″ ≈

*c*

_{1}

*s*+

*c*

_{2}, and then integrating twice yields Eq. (17).

*a*) are determined by matching the endpoint positions, endpoint tangents and the curvature between the S-bend and Archimedean spiral. By symmetry, we only need to design the incoming arc of the S-bend. This curve starts at the origin (

_{i}*x*= 0,

*y*= 0) and has curvature

*κ*= 0 at

*x*= 0,

*y*= 0, which leads to the following formulation

*a*

_{0}= 0. By solving a set of unknowns (

*a*

_{1},

*a*

_{2},

*a*

_{3},

*θ*

_{0},

*s*

_{1}) to match the input parameters (

*θ*

_{1},

*κ*

_{1},

*κ*′

_{1}, (

*x*

_{1},

*iy*

_{1})) from the end point of the Archimedean spiral, a curve for the adiabatic coupler is successfully defined as shown is Fig. 3.

## 3. Experimental Verification

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]

**3**, doi: [CrossRef] (2012). [PubMed]

_{2}. Further details on the processing are given in Ref. [5

**3**, doi: [CrossRef] (2012). [PubMed]

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]

*z*;

*α*

_{1}is the waveguide loss per unit length and

*α*

_{0}is the insertion loss of the S-bend connection. Such a plot is shown in Fig. 4(d). The slope of the linear fit gives approximately 0.35dB/m loss for the waveguide, while the intercept gives an estimated insertion loss for the adiabatic coupler of 0.05dB. This insertion loss and the indicated confidence interval result from linear regression on all of the points. Data points within 0.25 meters of the S-bend have been omitted in this estimate as there is a large increase in the variance on account of the steep slope associated with the backscatter singularity (see Fig. 4(c)). As an aside, the spiral device of this measurement was fabricated using a contact aligner and therefore features a higher waveguide loss as compared to that reported in Ref. [5

**3**, doi: [CrossRef] (2012). [PubMed]

*μm*). We compared measured insertion loss to attenuation calculations based upon atomic force microscope (AFM) roughness data [5

**3**, doi: [CrossRef] (2012). [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]

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]

**3**, doi: [CrossRef] (2012). [PubMed]

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]

## 4. Conclusion

## Acknowledgments

## References and links

1. | F. Ladouceur and P. Labeye, “A new gerenal approach to optical waveguide path design,” J. Lightwave Tech. |

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. |

3. | K. Takada, H. Yamada, Y. Hida, Y. Ohmori, and S. Mitachi, “Rayleigh backscattering measurement of 10m long silica-based waveguides,” Electron. Lett. |

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 Si |

5. | H. Lee, T. Chen, J. Li, O. Painter, and K. Vahala, “Ultra-low-loss optical delay line on a silicon chip,” Nat. Commun. |

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 |

7. | E. Marcatilli, “Bends in optical dielectric guides,” Bell Syst. Tech. J. |

8. | R. Baets and P. Lagasse, “Loss calculation and design of arbitrarily curved integrated-optic waveguides,” J. Opt. Soc. Am. |

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. |

10. | W. Gambling, H. Matsumura, and C. Ragdale, “Field deformation in a curved single-mode fiber,” Electron. Lett. |

11. | T. Kitoh, N. Takato, M. Yasu, and M. Kawachi, “Bending loss reduction in silica-based waveugide by using lateral offests,” J. Lightwave Tech. |

12. | A. Melloni, P. Monguzzi, R. Costa, and M. Martinelli, “Design of curved waveugide: the matched bend,” J. Opt. Soc. Am. A |

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. |

15. | D. J. Walton, “Spiral spline curves for highway design,” Microcomputers in Civil Engineering |

16. | K. G. Bass, “The use of clothoid templates in highway design,” Transportation Forum |

17. | S. Fleury, P. Soueres, J. P. Laumond, and R. Chatila, “Primitives for smoothing mobile robot trajectories,” IEEE Trans. Robot. Autom. |

18. | J. McCrae and K. Singh, “Sketching piecewise clothoid curves,” Computers & Graphics |

19. | K. Takada, H. Yamada, Y. Hida, Y. Ohmori, and S. Mitachi, “New waveguide fabrication techniques for next-generation plcs,” NTT Technical Review |

20. | A. W. Snyder, “Radiation losses due to variations of radius on dielectric or optical fibers,” IEEE Trans. Microwave Theory Tech. |

21. | A. W. Snyder, “Excitation and scattering of modes on a dielectic or optical fiber,” IEEE Trans. Microwave Theory Tech. |

22. | M. Heiblum and J. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quantum Electron. |

23. | R. Ulrich, “Fiber-optic rotation sensing with low drift,” Opt. Express |

24. | C. Ciminelli, F. Dell’Olio, C. Campanella, and M. Armenise, “Photonic technologies for angular velocity sensing,” Adv. Opt. Photon. |

25. | W. Chang, ed., |

26. | X. Yao and L. Maleki, “Optoelectronic microwave oscillator,” J. Opt. Soc. Am. B |

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 |

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

29. | S. Ohlin, |

30. | B. Soller, D. Gifford, M. Wolfe, and M. Froggatt, “High resolution optical frequency domain reflectometry for characterization of components and assemblies,” Opt. Express |

31. | H. Lee, T. Chen, J. Li, O. Painter, and K. Vahala, “Chemically etched ultrahigh-Q wedge-resonator on a silicon chip,” Nat. Photonics |

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. |

33. | H. Rokhsari and K. J. Vahala, “Ultralow loss, high q, four port resonant couplers for quantum optics and photonics,” Phys. Rev. Lett. |

**OCIS Codes**

(220.0220) Optical design and fabrication : Optical design and fabrication

(230.7370) Optical devices : Waveguides

(230.7390) Optical devices : Waveguides, planar

**ToC Category:**

Optical Design and Fabrication

**History**

Original Manuscript: July 3, 2012

Revised Manuscript: September 6, 2012

Manuscript Accepted: September 7, 2012

Published: September 20, 2012

**Citation**

Tong Chen, Hansuek Lee, Jiang Li, and Kerry J. Vahala, "A general design algorithm for low optical loss adiabatic connections in waveguides," Opt. Express **20**, 22819-22829 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-20-22819

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### References

- F. Ladouceur and P. Labeye, “A new gerenal approach to optical waveguide path design,” J. Lightwave Tech.13, 481–491 (1995). [CrossRef]
- 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]
- 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]
- 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. Express19, 3163–3174 (2011). [CrossRef] [PubMed]
- 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]
- 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. Express19, 24090–24101 (2011). [CrossRef] [PubMed]
- E. Marcatilli, “Bends in optical dielectric guides,” Bell Syst. Tech. J.48, 2103–2132 (1969).
- R. Baets and P. Lagasse, “Loss calculation and design of arbitrarily curved integrated-optic waveguides,” J. Opt. Soc. Am.73, 177–182 (1983). [CrossRef]
- 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]
- W. Gambling, H. Matsumura, and C. Ragdale, “Field deformation in a curved single-mode fiber,” Electron. Lett.14, 130–132 (1978). [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]
- A. Melloni, P. Monguzzi, R. Costa, and M. Martinelli, “Design of curved waveugide: the matched bend,” J. Opt. Soc. Am. A20, 130–137 (2003). [CrossRef]
- 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).
- D. Meek and J. Harris, “Clothoid spline transition spirals,” Math. Comp.59, 117–133 (1992). [CrossRef]
- D. J. Walton, “Spiral spline curves for highway design,” Microcomputers in Civil Engineering4, 99–106 (1989). [CrossRef]
- K. G. Bass, “The use of clothoid templates in highway design,” Transportation Forum1, 47–52 (1984).
- 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]
- J. McCrae and K. Singh, “Sketching piecewise clothoid curves,” Computers & Graphics33, 452–461 (2008). [CrossRef] [PubMed]
- K. Takada, H. Yamada, Y. Hida, Y. Ohmori, and S. Mitachi, “New waveguide fabrication techniques for next-generation plcs,” NTT Technical Review3, 37–41 (2005).
- 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]
- A. W. Snyder, “Excitation and scattering of modes on a dielectic or optical fiber,” IEEE Trans. Microwave Theory Tech.17, 1138–1144 (1969). [CrossRef]
- M. Heiblum and J. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quantum Electron.11, 75–83 (1975). [CrossRef]
- R. Ulrich, “Fiber-optic rotation sensing with low drift,” Opt. Express5, 173–175 (1980).
- C. Ciminelli, F. Dell’Olio, C. Campanella, and M. Armenise, “Photonic technologies for angular velocity sensing,” Adv. Opt. Photon.2, 370–404 (2010). [CrossRef]
- W. Chang, ed., RF Photonic Technology in Optical Fiber Links (Cambridge University Press, 2002). [CrossRef]
- X. Yao and L. Maleki, “Optoelectronic microwave oscillator,” J. Opt. Soc. Am. B13, 1725–1735 (1996). [CrossRef]
- 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. Photonics3, 216–219 (2009). [CrossRef]
- R. L. Levien, “From spiral to spline: Optimal techniques in interactive curve design,” Ph.D. thesis, UC Berkeley (2009).
- S. Ohlin, Splines for Engineers (Eurographics Association, 1987).
- B. Soller, D. Gifford, M. Wolfe, and M. Froggatt, “High resolution optical frequency domain reflectometry for characterization of components and assemblies,” Opt. Express13, 666–674 (2005). [CrossRef] [PubMed]
- H. Lee, T. Chen, J. Li, O. Painter, and K. Vahala, “Chemically etched ultrahigh-Q wedge-resonator on a silicon chip,” Nat. Photonics6, 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]
- 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]

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