## Adjoint-enabled optimization of optical devices based on coupled-mode equations |

Optics Express, Vol. 22, Issue 16, pp. 19423-19439 (2014)

http://dx.doi.org/10.1364/OE.22.019423

Enhanced HTML Acrobat PDF (826 KB)

### Abstract

In this work, we propose a method for designing optical devices described by coupled-mode equations. Following a commonly applied optimization strategy, we combine gradient-based optimization algorithms with an adjoint sensitivity analysis of the coupled-mode equations to obtain an optimization scheme that can handle a large number of design parameters. To demonstrate this adjoint-enabled optimization method, we design a silicon-on-insulator Raman wavelength converter. As structure, we consider a waveguide constructed from a series of interconnected and adiabatically-varying linear tapers, and treat the width at each interconnection point, the waveguide length, and the pump-Stokes frequency difference as independent design parameters. Optimizing with respect to these 1603 parameters results in an improvement of more than 10 dB in the conversion efficiency for a waveguide length of 6.28 cm and frequency difference 187 GHz below the Raman shift as compared to a converter designed by the conventional phase-matching design rule and operating at perfect Raman resonance. The increase in conversion efficiency is also accompanied by a more than 7 dB-improvement in the Stokes amplification. Hence, the adjoint-enabled optimization allows us to identify a more efficient method for achieving Raman conversion than conventional phase-matching. We also show that adjoint-enabled optimization significantly improves design robustness. In case of the Raman converter example, this leads to a sensitivity with respect to local variations in waveguide width that is several orders of magnitude smaller for the optimized design than for the phase-matched one.

© 2014 Optical Society of America

**OCIS Codes**

(000.3860) General : Mathematical methods in physics

(000.4430) General : Numerical approximation and analysis

(190.4360) Nonlinear optics : Nonlinear optics, devices

(190.5650) Nonlinear optics : Raman effect

(230.7370) Optical devices : Waveguides

**ToC Category:**

Optoelectronics

**History**

Original Manuscript: May 13, 2014

Revised Manuscript: June 27, 2014

Manuscript Accepted: June 30, 2014

Published: August 4, 2014

**Citation**

Yannick Lefevre, Pierre Wahl, Nathalie Vermeulen, and Hugo Thienpont, "Adjoint-enabled optimization of optical devices based on coupled-mode equations," Opt. Express **22**, 19423-19439 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-16-19423

Sort: Year | Journal | Reset

### References

- J. Jensen and O. Sigmund, “Topology optimization for nano-photonics,” Laser Photon. Rev.5, 308–321 (2011). [CrossRef]
- F. Wang, J. S. Jensen, and O. Sigmund, “Robust topology optimization of photonic crystal waveguides with tailored dispersion properties,” J. Opt. Soc. Am. B28, 387–397 (2011). [CrossRef]
- P. Borel, A. Harpøth, L. Frandsen, M. Kristensen, P. Shi, J. Jensen, and O. Sigmund, “Topology optimization and fabrication of photonic crystal structures,” Opt. Express12, 1996–2001 (2004). [CrossRef] [PubMed]
- Y. Tsuji, K. Hirayama, T. Nomura, K. Sato, and S. Nishiwaki, “Design of optical circuit devices based on topology optimization,” IEEE Photon. Technol. Lett.18, 850–852 (2006). [CrossRef]
- J. Osgood, N. C. Panoiu, J. I. Dadap, X. Liu, X. Chen, I.-W. Hsieh, E. Dulkeith, W. M. Green, and Y. A. Vlasov, “Engineering nonlinearities in nanoscale optical systems: physics and applications in dispersion-engineered silicon nanophotonic wires,” Adv. Opt. Photon.1, 162–235 (2009). [CrossRef]
- H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave raman silicon laser,” Nature433, 725–728 (2005). [CrossRef] [PubMed]
- M. A. Foster, A. C. Turner, R. Salem, M. Lipson, and A. L. Gaeta, “Broad-band continuous-wave parametric wavelength conversion in silicon nanowaveguides,” Opt. Express15, 12949–12958 (2007). [CrossRef] [PubMed]
- V. Raghunathan, R. Claps, D. Dimitropoulos, and B. Jalali, “Parametric Raman wavelength conversion in scaled silicon waveguides,” J. Lightwave Technol.23, 2094–2102 (2005). [CrossRef]
- J. B. Driscoll, N. Ophir, R. R. Grote, J. I. Dadap, N. C. Panoiu, K. Bergman, and R. M. Osgood, “Width-modulation of Si photonic wires for quasi-phase-matching of four-wave-mixing: experimental and theoretical demonstration,” Opt. Express20, 9227–9242 (2012). [CrossRef] [PubMed]
- D. T. Tan, P. C. Sun, and Y. Fainman, “Monolithic nonlinear pulse compressor on a silicon chip,” Nat. Commun.1, 116 (2010). [CrossRef] [PubMed]
- A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).
- L. Jin, W. Jin, J. Ju, and Y. Wang, “Coupled local-mode theory for strongly modulated long period gratings,” J. Lightwave Technol.28, 1745–1751 (2010). [CrossRef]
- W.-P. Huang and J. Mu, “Complex coupled-mode theory for optical waveguides,” Opt. Express17, 19134–19152 (2009). [CrossRef]
- G. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, 2001).
- Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: Modeling and applications,” Opt. Express15, 16604–16644 (2007). [CrossRef] [PubMed]
- Q. Lin, J. Zhang, P. M. Fauchet, and G. P. Agrawal, “Ultrabroadband parametric generation and wavelength conversion in silicon waveguides,” Opt. Express14, 4786–4799 (2006). [CrossRef] [PubMed]
- J. Nocedal and S. J. Wright, Numerical Optimization, 2nd ed. (Springer, 1999). [CrossRef]
- Y. Cao, S. Li, L. Petzold, and R. Serban, “Adjoint sensitivity analysis for differential-algebraic equations: the adjoint DAE system and its numerical solution,” SIAM J. Sci. Comput.24, 1076–1089 (2003). [CrossRef]
- R. Serban and A. C. Hindmarsh, “CVODES, the sensitivity-enabled ODE solver in SUNDIALS,” in “Proceedings of the 5th International Conference on Multibody Systems, Nonlinear Dynamics and Control, Long Beach, CA” (2005).
- P. Wahl, D. S. Ly Gagnon, C. Debaes, J. Van Erps, N. Vermeulen, D. A. B. Miller, and H. Thienpont, “B-CALM: an open-source multi-GPU-based 3D-FDTD with multi-pole dispersion for plasmonics,” Prog. Electromagn. Res.138, 467–478 (2013). [CrossRef]
- Y. Elesin, B. Lazarov, J. Jensen, and O. Sigmund, “Design of robust and efficient photonic switches using topology optimization,” Phot. Nano. Fund. Appl.10, 153–165 (2012). [CrossRef]
- J. S. Jensen, “Topology optimization of nonlinear optical devices,” Struct. Multidisc. Optim.43, 731–743 (2011). [CrossRef]
- N. Vermeulen, C. Debaes, and H. Thienpont, “Coherent anti-Stokes Raman scattering in Raman lasers and Raman wavelength converters,” Laser Photon. Rev.4, 656–670 (2010). [CrossRef]
- R. Claps, V. Raghunathan, D. Dimitropoulos, and B. Jalali, “Anti-Stokes Raman conversion in silicon waveguides,” Opt. Express11, 2862–2872 (2003). [CrossRef] [PubMed]
- P. Koonath, D. R. Solli, and B. Jalali, “High efficiency CARS conversion in silicon,” in “Conference on Lasers and Electro-Optics and on Quantum Electronics and Laser Science” (2008), pp. 1–2.
- Y. Lefevre, N. Vermeulen, and H. Thienpont, “Quasi-phase-matching of four-wave-mixing-based wavelength conversion by phase-mismatch switching,” J. Lightwave Technol.31, 2113–2121 (2013). [CrossRef]
- Y. Lefevre, N. Vermeulen, C. Debaes, and H. Thienpont, “Optimized wavelength conversion in silicon waveguides based on “off-Raman-resonance” operation: extending the phase mismatch formalism,” Opt. Express19, 18810–18826 (2011). [CrossRef] [PubMed]
- R. Claps, D. Dimitropoulos, V. Raghunathan, Y. Han, and B. Jalali, “Observation of stimulated Raman amplification in silicon waveguides,” Opt. Express11, 1731–1739 (2003). [CrossRef] [PubMed]
- R. Soref and B. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron.23, 123–129 (1987). [CrossRef]
- D. Dimitropoulos, R. Jhaveri, R. Claps, J. C. S. Woo, and B. Jalali, “Lifetime of photogenerated carriers in silicon-on-insulator rib waveguides,” Appl. Phys. Lett.86, 071115 (2005). [CrossRef]
- X. Chen, N. Panoiu, and R. Osgood, “Theory of Raman-mediated pulsed amplification in silicon-wire waveguides,” IEEE J. Quantum Electron.42, 160–170 (2006). [CrossRef]
- E. Golovchenko, P. Mamyshev, A. Pilipetskii, and E. Dianov, “Mutual influence of the parametric effects and stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron.26, 1815–1820 (1990). [CrossRef]
- ePIXfab, The silicon photonics website, http://www.epixfab.eu/ .
- T. Shoji, T. Tsuchizawa, T. Watanabe, K. Yamada, and H. Morita, “Low loss mode size converter from 0.3 μm square Si wire waveguides to singlemode fibres,” Electron. Lett.38, 1669–1670 (2002). [CrossRef]
- V. R. Almeida, R. R. Panepucci, and M. Lipson, “Nanotaper for compact mode conversion,” Opt. Lett.28, 1302–1304 (2003). [CrossRef] [PubMed]
- D. Zografopoulos, R. Beccherelli, and E. Kriezis, “Quasi-soliton propagation in dispersion-engineered silicon nanowires,” Opt. Commun.285, 3306–3311 (2012). [CrossRef]
- O. Tsilipakos, D. C. Zografopoulos, and E. E. Kriezis, “Quasi-soliton pulse-train propagation in dispersion-managed silicon rib waveguides,” IEEE Photon. Technol. Lett.25, 724–727 (2013). [CrossRef]
- We employed the commercial software package MODE Solutions by Lumerical to calculate the dispersion characteristics and mode profiles of SOI waveguides.

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.