## Adjoint shape optimization applied to electromagnetic design |

Optics Express, Vol. 21, Issue 18, pp. 21693-21701 (2013)

http://dx.doi.org/10.1364/OE.21.021693

Acrobat PDF (1083 KB)

### Abstract

We present an adjoint-based optimization for electromagnetic design. It embeds commercial Maxwell solvers within a steepest-descent inverse-design optimization algorithm. The adjoint approach calculates shape derivatives at *all* points in space, but requires only two “forward” simulations. Geometrical shape parameterization is by the level set method. Our adjoint design optimization is applied to a Silicon photonics Y-junction splitter that had previously been investigated by stochastic methods. Owing to the speed of calculating shape derivatives within the adjoint method, convergence is much faster, within a larger design space. This is an extremely efficient method for the design of complex electromagnetic components.

© 2013 OSA

## 1. Introduction and motivations

3. Y. Zhang, S. Yang, A. E. J. Lim, G. Q. Lo, C. Galland, T. Baehr-Jones, and M. Hochberg, “A compact and low loss Y-junction for submicron silicon waveguide,” Opt. Express **21**(1), 1310–1316 (2013). [CrossRef] [PubMed]

3. Y. Zhang, S. Yang, A. E. J. Lim, G. Q. Lo, C. Galland, T. Baehr-Jones, and M. Hochberg, “A compact and low loss Y-junction for submicron silicon waveguide,” Opt. Express **21**(1), 1310–1316 (2013). [CrossRef] [PubMed]

4. P. Sanchis, P. Villalba, F. Cuesta, A. Håkansson, A. Griol, J. V. Galán, A. Brimont, and J. Martí, “Highly efficient crossing structure for silicon-on-insulator waveguides,” Opt. Lett. **34**(18), 2760–2762 (2009). [CrossRef] [PubMed]

4. P. Sanchis, P. Villalba, F. Cuesta, A. Håkansson, A. Griol, J. V. Galán, A. Brimont, and J. Martí, “Highly efficient crossing structure for silicon-on-insulator waveguides,” Opt. Lett. **34**(18), 2760–2762 (2009). [CrossRef] [PubMed]

4. P. Sanchis, P. Villalba, F. Cuesta, A. Håkansson, A. Griol, J. V. Galán, A. Brimont, and J. Martí, “Highly efficient crossing structure for silicon-on-insulator waveguides,” Opt. Lett. **34**(18), 2760–2762 (2009). [CrossRef] [PubMed]

3. Y. Zhang, S. Yang, A. E. J. Lim, G. Q. Lo, C. Galland, T. Baehr-Jones, and M. Hochberg, “A compact and low loss Y-junction for submicron silicon waveguide,” Opt. Express **21**(1), 1310–1316 (2013). [CrossRef] [PubMed]

5. Y. Zhang, S. Yang, E. J. Lim, G. Lo, T. Baehr-Jones, and M. Hochberg, “A CMOS-compatible, low-loss and low-crosstalk silicon waveguide crossing,” IEEE Photon. Technol. Lett. **25**(5), 422–425 (2013). [CrossRef]

6. T. Tanemura, K. C. Balram, D. S. Ly-Gagnon, P. Wahl, J. S. White, M. L. Brongersma, and D. A. B. Miller, “Multiple-wavelength focusing of surface plasmons with a nonperiodic nanoslit coupler,” Nano Lett. **11**(7), 2693–2698 (2011). [CrossRef] [PubMed]

## 2. Presentation of the adjoint method for electromagnetic problems

9. T. Borrvall and J. Petersson, “Topology optimization of fluids in Stokes flow,” Int. J. Numer. Methods Fluids **41**(1), 77–107 (2003). [CrossRef]

10. J. S. Jensen and O. Sigmund, “Topology optimization for nano-photonics,” Las. Photon. Rev. **5**(2), 308–321 (2011). [CrossRef]

14. G. Veronis, R. W. Dutton, and S. Fan, “Method for sensitivity analysis of photonic crystal devices,” Opt. Lett. **29**(19), 2288–2290 (2004). [CrossRef] [PubMed]

15. A. F. J. Levi and I. G. Rosen, “A novel formulation of the adjoint method in the optimal design of quantum electronic devices,” SIAM J. Contr. Optim. **48**(5), 3191–3223 (2010). [CrossRef]

**E**and

**D**are continuous across the boundary [14

14. G. Veronis, R. W. Dutton, and S. Fan, “Method for sensitivity analysis of photonic crystal devices,” Opt. Lett. **29**(19), 2288–2290 (2004). [CrossRef] [PubMed]

17. O. D. Miller, PhD thesis (2012), University of California at Berkeley, http://optoelectronics.eecs.berkeley.edu/ThesisOwenMiller.pdf

18. S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **65**(6), 066611 (2002). [CrossRef] [PubMed]

*single*simulation, even though it provides the derivative with respect to permittivity at

*every*point in the computational region

17. O. D. Miller, PhD thesis (2012), University of California at Berkeley, http://optoelectronics.eecs.berkeley.edu/ThesisOwenMiller.pdf

19. Lumerical FDTD Solutions, www.lumerical.com

## 3. Y-Splitter optimization example using the level set method for shape representation

**21**(1), 1310–1316 (2013). [CrossRef] [PubMed]

**21**(1), 1310–1316 (2013). [CrossRef] [PubMed]

**21**(1), 1310–1316 (2013). [CrossRef] [PubMed]

**21**(1), 1310–1316 (2013). [CrossRef] [PubMed]

20. S. Osher and J. A. Sethian, “Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations,” J. Comput. Phys. **79**(1), 12–49 (1988). [CrossRef]

**21**(1), 1310–1316 (2013). [CrossRef] [PubMed]

5. Y. Zhang, S. Yang, E. J. Lim, G. Lo, T. Baehr-Jones, and M. Hochberg, “A CMOS-compatible, low-loss and low-crosstalk silicon waveguide crossing,” IEEE Photon. Technol. Lett. **25**(5), 422–425 (2013). [CrossRef]

10. J. S. Jensen and O. Sigmund, “Topology optimization for nano-photonics,” Las. Photon. Rev. **5**(2), 308–321 (2011). [CrossRef]

17. O. D. Miller, PhD thesis (2012), University of California at Berkeley, http://optoelectronics.eecs.berkeley.edu/ThesisOwenMiller.pdf

*every*point in the design region is calculated by combining the forward and adjoint simulations results into Eq. (5). FDTD is perfectly suited to solve the direct and adjoint problem, which consists of propagating waves in a dielectric.

**21**(1), 1310–1316 (2013). [CrossRef] [PubMed]

**21**(1), 1310–1316 (2013). [CrossRef] [PubMed]

## 4. Conclusion

## Acknowledgments

## References and links

1. | P. Sandborn, N. Quack, N. Hoghooghi, J. B. Chou, J. Ferrara, S. Gambini, B. Behroozpour, L. Zhu, B. Boser, C. Chang-Hasnain, and M. C. Wu, “Linear frequency chirp generation employing opto-electronic feedback loop and integrated Silicon photonics,” in |

2. | A. Sakai, T. Fukazawa, and T. Baba, “Low loss ultra-small branches in a silicon photonic wire waveguide,” IEICE Trans. Electron. |

3. | Y. Zhang, S. Yang, A. E. J. Lim, G. Q. Lo, C. Galland, T. Baehr-Jones, and M. Hochberg, “A compact and low loss Y-junction for submicron silicon waveguide,” Opt. Express |

4. | P. Sanchis, P. Villalba, F. Cuesta, A. Håkansson, A. Griol, J. V. Galán, A. Brimont, and J. Martí, “Highly efficient crossing structure for silicon-on-insulator waveguides,” Opt. Lett. |

5. | Y. Zhang, S. Yang, E. J. Lim, G. Lo, T. Baehr-Jones, and M. Hochberg, “A CMOS-compatible, low-loss and low-crosstalk silicon waveguide crossing,” IEEE Photon. Technol. Lett. |

6. | T. Tanemura, K. C. Balram, D. S. Ly-Gagnon, P. Wahl, J. S. White, M. L. Brongersma, and D. A. B. Miller, “Multiple-wavelength focusing of surface plasmons with a nonperiodic nanoslit coupler,” Nano Lett. |

7. | M. P. Bendsøe and N. Kikuchi, “Generating optimal topologies in structural design using a homogenization method,” Comput. Meth. In Appl. M. |

8. | M. P. Bendsoe and O. Sigmund, |

9. | T. Borrvall and J. Petersson, “Topology optimization of fluids in Stokes flow,” Int. J. Numer. Methods Fluids |

10. | J. S. Jensen and O. Sigmund, “Topology optimization for nano-photonics,” Las. Photon. Rev. |

11. | P. Seliger, M. Mahvash, C. Wang, and A. F. J. Levi, “Optimization of aperiodic dielectric structures,” J. Appl. Phys. |

12. | W. R. Frei, D. A. Tortorelli, and H. T. Johnson, “Geometry projection method for optimizing photonic nanostructures,” Opt. Lett. |

13. | V. Liu and S. Fan, “Compact bends for multi-mode photonic crystal waveguides with high transmission and suppressed modal crosstalk,” Opt. Express |

14. | G. Veronis, R. W. Dutton, and S. Fan, “Method for sensitivity analysis of photonic crystal devices,” Opt. Lett. |

15. | A. F. J. Levi and I. G. Rosen, “A novel formulation of the adjoint method in the optimal design of quantum electronic devices,” SIAM J. Contr. Optim. |

16. | G. Strang, Computational Science and Engineering , (Wellesley-Cambridge, 2007). |

17. | O. D. Miller, PhD thesis (2012), University of California at Berkeley, http://optoelectronics.eecs.berkeley.edu/ThesisOwenMiller.pdf |

18. | S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

19. | Lumerical FDTD Solutions, www.lumerical.com |

20. | S. Osher and J. A. Sethian, “Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations,” J. Comput. Phys. |

21. | http://optoelectronics.eecs.berkeley.edu/PhotonicInverseDesign/ |

**OCIS Codes**

(130.3120) Integrated optics : Integrated optics devices

(230.1360) Optical devices : Beam splitters

(230.7370) Optical devices : Waveguides

**ToC Category:**

Integrated Optics

**History**

Original Manuscript: July 26, 2013

Revised Manuscript: August 19, 2013

Manuscript Accepted: August 20, 2013

Published: September 6, 2013

**Citation**

Christopher M. Lalau-Keraly, Samarth Bhargava, Owen D. Miller, and Eli Yablonovitch, "Adjoint shape optimization applied to electromagnetic design," Opt. Express **21**, 21693-21701 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-18-21693

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### References

- P. Sandborn, N. Quack, N. Hoghooghi, J. B. Chou, J. Ferrara, S. Gambini, B. Behroozpour, L. Zhu, B. Boser, C. Chang-Hasnain, and M. C. Wu, “Linear frequency chirp generation employing opto-electronic feedback loop and integrated Silicon photonics,” in CLEO, (2013) pp. 5–6.
- A. Sakai, T. Fukazawa, and T. Baba, “Low loss ultra-small branches in a silicon photonic wire waveguide,” IEICE Trans. Electron.E85, 1033–1038 (2002).
- Y. Zhang, S. Yang, A. E. J. Lim, G. Q. Lo, C. Galland, T. Baehr-Jones, and M. Hochberg, “A compact and low loss Y-junction for submicron silicon waveguide,” Opt. Express21(1), 1310–1316 (2013). [CrossRef] [PubMed]
- P. Sanchis, P. Villalba, F. Cuesta, A. Håkansson, A. Griol, J. V. Galán, A. Brimont, and J. Martí, “Highly efficient crossing structure for silicon-on-insulator waveguides,” Opt. Lett.34(18), 2760–2762 (2009). [CrossRef] [PubMed]
- Y. Zhang, S. Yang, E. J. Lim, G. Lo, T. Baehr-Jones, and M. Hochberg, “A CMOS-compatible, low-loss and low-crosstalk silicon waveguide crossing,” IEEE Photon. Technol. Lett.25(5), 422–425 (2013). [CrossRef]
- T. Tanemura, K. C. Balram, D. S. Ly-Gagnon, P. Wahl, J. S. White, M. L. Brongersma, and D. A. B. Miller, “Multiple-wavelength focusing of surface plasmons with a nonperiodic nanoslit coupler,” Nano Lett.11(7), 2693–2698 (2011). [CrossRef] [PubMed]
- M. P. Bendsøe and N. Kikuchi, “Generating optimal topologies in structural design using a homogenization method,” Comput. Meth. In Appl. M. 71, 197–224 (1988).
- M. P. Bendsoe and O. Sigmund, Topology Optimization Theory, Methods and Applications. (Springer, 2003)
- T. Borrvall and J. Petersson, “Topology optimization of fluids in Stokes flow,” Int. J. Numer. Methods Fluids41(1), 77–107 (2003). [CrossRef]
- J. S. Jensen and O. Sigmund, “Topology optimization for nano-photonics,” Las. Photon. Rev.5(2), 308–321 (2011). [CrossRef]
- P. Seliger, M. Mahvash, C. Wang, and A. F. J. Levi, “Optimization of aperiodic dielectric structures,” J. Appl. Phys.100(3), 034310 (2006). [CrossRef]
- W. R. Frei, D. A. Tortorelli, and H. T. Johnson, “Geometry projection method for optimizing photonic nanostructures,” Opt. Lett.32(1), 77–79 (2007). [CrossRef] [PubMed]
- V. Liu and S. Fan, “Compact bends for multi-mode photonic crystal waveguides with high transmission and suppressed modal crosstalk,” Opt. Express21(7), 8069–8075 (2013). [CrossRef] [PubMed]
- G. Veronis, R. W. Dutton, and S. Fan, “Method for sensitivity analysis of photonic crystal devices,” Opt. Lett.29(19), 2288–2290 (2004). [CrossRef] [PubMed]
- A. F. J. Levi and I. G. Rosen, “A novel formulation of the adjoint method in the optimal design of quantum electronic devices,” SIAM J. Contr. Optim.48(5), 3191–3223 (2010). [CrossRef]
- G. Strang, Computational Science and Engineering, (Wellesley-Cambridge, 2007).
- O. D. Miller, PhD thesis (2012), University of California at Berkeley, http://optoelectronics.eecs.berkeley.edu/ThesisOwenMiller.pdf
- S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.65(6), 066611 (2002). [CrossRef] [PubMed]
- Lumerical FDTD Solutions, www.lumerical.com
- S. Osher and J. A. Sethian, “Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations,” J. Comput. Phys.79(1), 12–49 (1988). [CrossRef]
- http://optoelectronics.eecs.berkeley.edu/PhotonicInverseDesign/

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