## Objective-first design of high-efficiency, small-footprint couplers between arbitrary nanophotonic waveguide modes |

Optics Express, Vol. 20, Issue 7, pp. 7221-7236 (2012)

http://dx.doi.org/10.1364/OE.20.007221

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### Abstract

We present an algorithm for designing high efficiency (∼98%), small-footprint (1.5–4 square vacuum wavelengths) couplers between arbitrary nanophotonic waveguide modes in two dimensions. Our “objective-first” method is computationally fast (15 minutes on a single-core personal computer), requires no trial-and-error, and does not require guessing a good starting design. We demonstrate designs for various coupling problems which suggest that our method allows for the design of any single-mode, linear optical device.

© 2012 OSA

## 1. Motivation

- Coupling between various nanophotonic waveguides, since different waveguides are best suited for different applications. For example, ridge waveguides seem ideal for low-loss transport [2], but other waveguides, such as photonic crystal waveguides or slot waveguides, may be better suited for slow-light [3] or nonlinear optical devices based on localized field intensities [4].
3. Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature

**438**, 65–69 (2005). [PubMed] - Coupling between different materials systems such as passive, active [5], and metallic [6] devices.
5. J. Van Campenhout, P. Rojo Romeo, P. Regreny, C. Seassal, D. Van Thourhout, S. Verstuyft, L. Di Cioccio, J.-M. Fedeli, C. Lagahe, and R. Baets, “Electrically pumped InP-based microdisk lasers integrated with a nanophotonic silicon-on-insulator waveguide circuit,” Opt. Express

**15**, 6744–6749 (2007). [PubMed]

## 2. Objective-first approach

*x*is the field variable and

*p*is the structure variable. Here,

*f*(

*x*), the

*design objective*, calculates the performance of the device (e.g. amount of power lost); while

*g*(

*x*,

*p*) is the underlying physical equation for the system (e.g. the electromagnetic wave equation).

*g*(

*x*,

*p*)||

^{2}is the

*physics residual*. We term this formulation “objective-first” because the design objective is prioritized even above satisfying physics; specifically, we force our design to always exhibit the desired performance (

*f*(

*x*) = 0), even at the expense of not perfectly satisfying the underlying physics which governs its operation.

*x*and

*p*to vary independently, as opposed to Eq. 1 where the value of

*x*is completely dependent on

*p*. Second, it enables an increase in the likelihood of arriving at a high efficiency design by forcibly imposing

*f*(

*x*) = 0, and thereby circumvents any local optima consisting of low-performance devices.

## 3. Objective-first design of waveguide couplers

*H*as the field variable

_{z}*x*, and

*ε*

^{−1}(the inverse of the permittivity) as the structure variable

*p*. This results in the following form of the physics residual, where

*ω*is the angular frequency, and

*μ*

_{0}is the permeability of free-space.

*∂H*/

_{z}*∂n*denotes the spatial derivative in the direction normal to the boundary.

*x*and

*p*are solved iteratively [7

7. J. Lu, S. Boyd, and J. Vuckovic, “Inverse design of a three-dimensional nanophotonic resonator,” Opt. Express **19**, 10563–10750 (2011). [PubMed]

*ε*to be between the permittivity of vacuum and of silicon, Note that a completely binary structure, where

*ε*= {

*ε*

_{0},

*ε*

_{silicon}}, is needed for manufacturing purposes; this will be pursued in a future work along with an analysis of robustness to fabrication imperfections. That said, the final designs presented here all have significant portions which are already binary.

### 3.1. Coupler designs

- coupling between waveguides of different refractive index and width (Fig. 2);
- coupling between waveguide modes of different order and symmetry (Fig. 3);
- coupling between waveguides that confine light using different principles (index guided vs. distributed Bragg reflection guided), i.e., between a slab waveguide and a photonic crystal fiber (Fig. 4)
- coupling from a dielectric to a plasmonic metal-insulator-metal waveguide (Fig. 5); and
- coupling from a dielectric waveguide to a (plasmonic) metal wire (Fig. 6).

8. G. Veronis and S. Fan, “Theoretical investigations of compact couplers between dielectric slab waveguides and two-dimensional metal-dielectric-metal plasmonic waveguides,” Opt. Express **15**, 1211–1221 (2007). [PubMed]

9. R. Yang, R. A. Wahsheh, Z. Lu, and M. A. G. Abushagur, “Efficient light coupling between dielectric slot waveguide and plasmonic slot waveguide,” Opt. Lett. **35**, 649–651 (2010). [PubMed]

*ε*= 9 everywhere (a somewhat arbitrary guess, other values work as well). Finally, only 15 minutes on a single-core personal computer were needed to obtain each design.

## 4. Conclusion

## A. Appendix A

## B. Appendix B

## C. Appendix C

## Acknowledgments

## References and links

1. | Y. Tang, Z. Wang, L. Wosinski, U. Westergren, and S. He, “Highly efficient nonuniform grating coupler for silicon-on-insulator nanophotonic circuits,” Opt. Lett. |

2. | K. K. Lee, D. R. Lim, L.C. Kimerling, J. Shin, and F. Cerrina, “Fabrication of ultralow-loss Si/SiO2 waveguides by roughness reduction,” Opt. Lett. |

3. | Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature |

4. | M. Lipson, “Guiding, modulating, and emitting light on silicon-challenges and opportunities,” J. Lightwave Technol. |

5. | J. Van Campenhout, P. Rojo Romeo, P. Regreny, C. Seassal, D. Van Thourhout, S. Verstuyft, L. Di Cioccio, J.-M. Fedeli, C. Lagahe, and R. Baets, “Electrically pumped InP-based microdisk lasers integrated with a nanophotonic silicon-on-insulator waveguide circuit,” Opt. Express |

6. | L. Tang, S. E. Kocabas, S. Latif, A. K. Okyay, D. S. Ly-Gagnon, K. C. Saraswat, and D. A. B. Miller, “Nanometre-scale germanium photodetector enhanced by a near-infrared dipole antenna,” Nat. Photonics |

7. | J. Lu, S. Boyd, and J. Vuckovic, “Inverse design of a three-dimensional nanophotonic resonator,” Opt. Express |

8. | G. Veronis and S. Fan, “Theoretical investigations of compact couplers between dielectric slab waveguides and two-dimensional metal-dielectric-metal plasmonic waveguides,” Opt. Express |

9. | R. Yang, R. A. Wahsheh, Z. Lu, and M. A. G. Abushagur, “Efficient light coupling between dielectric slot waveguide and plasmonic slot waveguide,” Opt. Lett. |

10. |

**OCIS Codes**

(230.7370) Optical devices : Waveguides

(130.3990) Integrated optics : Micro-optical devices

**ToC Category:**

Integrated Optics

**History**

Original Manuscript: January 11, 2012

Revised Manuscript: March 7, 2012

Manuscript Accepted: March 7, 2012

Published: March 14, 2012

**Citation**

Jesse Lu and Jelena Vučković, "Objective-first design of high-efficiency, small-footprint couplers between arbitrary nanophotonic waveguide modes," Opt. Express **20**, 7221-7236 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-7-7221

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### References

- Y. Tang, Z. Wang, L. Wosinski, U. Westergren, S. He, “Highly efficient nonuniform grating coupler for silicon-on-insulator nanophotonic circuits,” Opt. Lett. 35, 1290–1292 (2010). [PubMed]
- K. K. Lee, D. R. Lim, L.C. Kimerling, J. Shin, F. Cerrina, “Fabrication of ultralow-loss Si/SiO2 waveguides by roughness reduction,” Opt. Lett. 26, 1888–1890 (2001).
- Y. A. Vlasov, M. O’Boyle, H. F. Hamann, S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature 438, 65–69 (2005). [PubMed]
- M. Lipson, “Guiding, modulating, and emitting light on silicon-challenges and opportunities,” J. Lightwave Technol. 23, 4222–4238 (2005).
- J. Van Campenhout, P. Rojo Romeo, P. Regreny, C. Seassal, D. Van Thourhout, S. Verstuyft, L. Di Cioccio, J.-M. Fedeli, C. Lagahe, R. Baets, “Electrically pumped InP-based microdisk lasers integrated with a nanophotonic silicon-on-insulator waveguide circuit,” Opt. Express 15, 6744–6749 (2007). [PubMed]
- L. Tang, S. E. Kocabas, S. Latif, A. K. Okyay, D. S. Ly-Gagnon, K. C. Saraswat, D. A. B. Miller, “Nanometre-scale germanium photodetector enhanced by a near-infrared dipole antenna,” Nat. Photonics 2, 226–229 (2008).
- J. Lu, S. Boyd, J. Vuckovic, “Inverse design of a three-dimensional nanophotonic resonator,” Opt. Express 19, 10563–10750 (2011). [PubMed]
- G. Veronis, S. Fan, “Theoretical investigations of compact couplers between dielectric slab waveguides and two-dimensional metal-dielectric-metal plasmonic waveguides,” Opt. Express 15, 1211–1221 (2007). [PubMed]
- R. Yang, R. A. Wahsheh, Z. Lu, M. A. G. Abushagur, “Efficient light coupling between dielectric slot waveguide and plasmonic slot waveguide,” Opt. Lett. 35, 649–651 (2010). [PubMed]
- www.github.com/JesseLu/objective-first

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