## 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

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

## 1. Introduction

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

2. F. Wang, J. S. Jensen, and O. Sigmund, “Robust topology optimization of photonic crystal waveguides with tailored dispersion properties,” J. Opt. Soc. Am. B **28**, 387–397 (2011). [CrossRef]

3. P. Borel, A. Harpøth, L. Frandsen, M. Kristensen, P. Shi, J. Jensen, and O. Sigmund, “Topology optimization and fabrication of photonic crystal structures,” Opt. Express **12**, 1996–2001 (2004). [CrossRef] [PubMed]

4. 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]

5. 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]

6. H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave raman silicon laser,” Nature **433**, 725–728 (2005). [CrossRef] [PubMed]

5. 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]

7. 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. Express **15**, 12949–12958 (2007). [CrossRef] [PubMed]

9. 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. Express **20**, 9227–9242 (2012). [CrossRef] [PubMed]

5. 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]

10. D. T. Tan, P. C. Sun, and Y. Fainman, “Monolithic nonlinear pulse compressor on a silicon chip,” Nat. Commun. **1**, 116 (2010). [CrossRef] [PubMed]

**1**, 162–235 (2009). [CrossRef]

13. W.-P. Huang and J. Mu, “Complex coupled-mode theory for optical waveguides,” Opt. Express **17**, 19134–19152 (2009). [CrossRef]

**1**, 162–235 (2009). [CrossRef]

## 2. Adjoint-enabled optimization of coupled-mode-based devices

*ω*

_{1},

*ω*

_{2},··· ,

*ω*: Here the amplitudes

_{N}*A*vary along

_{j}*z*, the propagation direction of the waveguide, and the mode profiles

**e**capture the transverse variation as well as the polarization of each mode. The mode profiles are conventionally normalized such that |

_{j}*A*|

_{j}^{2}equals the optical power flow

*P*of each mode along

_{j}*z*. For notational simplicity, we denote by

**A**the column vector composed of the amplitudes

*A*:

_{j}**A**along

*z*. In general, this evolution is described by Maxwell’s equations, which for a wide variety of applications can be reduced to a set of coupled-mode equations, i.e., a set of coupled, first-order ordinary differential equations [11, 13

13. W.-P. Huang and J. Mu, “Complex coupled-mode theory for optical waveguides,” Opt. Express **17**, 19134–19152 (2009). [CrossRef]

15. Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: Modeling and applications,” Opt. Express **15**, 16604–16644 (2007). [CrossRef] [PubMed]

**A**indicates the value of the amplitude vector

_{0}**A**at the initial position

*z*

_{0}, and the elements of

**F**represent the coupled-mode equations of the corresponding elements of

**A**. Generally, each coupled-mode equation depends on the position

*z*, on the full amplitude vector

**A**and its complex conjugate as denoted by the superscript *, and on

**, the set of design parameters**

*θ**θ*

_{1},

*θ*

_{2},··· ,

*θ*that have an impact on the light propagation. The latter include both parameters that affect the input amplitude vector

_{M}**A**, such as the power, phase, and frequency of the different modal amplitudes, and parameters that affect the coupled-mode equations

_{0}**F**themselves, such as structural parameters of the (local) waveguide geometry or the waveguide length. Note that we explicitly included the dependency of

**F**on both

**A**and

**A**

^{*}to emphasize that they have to be treated as two distinct variables in the context of partial derivatives since

**allows us to determine the corresponding output amplitude vector**

*θ***A**at the waveguide end position

_{f}*z*[see Fig. 1]. Based on

_{f}**A**, we can determine the performance corresponding to the parameter values

_{f}**, which is typically measured by a performance figure**

*θ**G*could also depend on other variables such as the input amplitude

**A**or a subset of the design parameters

_{0}**, but we do not explicitly describe these dependencies here since they are not relevant for the current discussion. In these terms, designing a device corresponds to determining the parameter values**

*θ***yielding an optimal**

*θ̄**G*.

**are easily derived from these solutions. More complex problems, including waveguides based on materials that display a variety of nonlinear interactions, such as for instance silicon [5**

*θ̄***1**, 162–235 (2009). [CrossRef]

8. 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]

15. Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: Modeling and applications,” Opt. Express **15**, 16604–16644 (2007). [CrossRef] [PubMed]

16. Q. Lin, J. Zhang, P. M. Fauchet, and G. P. Agrawal, “Ultrabroadband parametric generation and wavelength conversion in silicon waveguides,” Opt. Express **14**, 4786–4799 (2006). [CrossRef] [PubMed]

12. 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]

**then also have to be determined numerically. Often this is done by solving Eq. (3) for a wide range of parameter values and subsequently comparing the performance of each configuration. Such a scheme is however computationally very inefficient, and becomes quickly impractical as the number of parameters**

*θ̄**M*rises.

17. J. Nocedal and S. J. Wright, *Numerical Optimization*, 2nd ed. (Springer, 1999). [CrossRef]

*G*at each estimate. Faster gradient-based algorithms such as the steepest descent method, the conjugate gradient method, or quasi-Newton methods [17

17. J. Nocedal and S. J. Wright, *Numerical Optimization*, 2nd ed. (Springer, 1999). [CrossRef]

*G*/d

*θ*, i.e., the first-order derivatives of the performance with respect to each

_{k}*θ*. Note that these derivatives are equivalent to the sensitivity of the performance with respect to each parameter

_{k}*θ*. Computing the gradient of the performance hence requires a sensitivity analysis of the set of ordinary differential equations of Eq. (3).

_{k}1. J. Jensen and O. Sigmund, “Topology optimization for nano-photonics,” Laser Photon. Rev. **5**, 308–321 (2011). [CrossRef]

18. 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]

*G*for which the partial derivatives in Eq. (4b) satisfy

**=**

*λ*

*μ*^{*}according to Eq. (4). As a consequence, only

*N*independent adjoint variables remain, and the adjoint system of Eq. (4) simplifies to: and the adjoint sensitivity equation of Eq. (6) becomes: The adjoint sensitivity analysis now requires only a set of

*N*coupled differential equations to be solved. Any performance function

*G*that is purely a function of the output powers

*P*(

_{j}*z*) = |

_{f}*A*(

_{j}*z*)|

_{f}^{2}satisfies the condition

*adjoint-enabled optimization*consists of five steps per iteration [see Fig. 2]: (Step 1) update the parameters

**based on the data obtained in the previous iterations. (Step 2) For the current parameter values**

*θ***, simulate the forward propagation of the amplitude vector**

*θ***A**by solving the coupled-mode equations of Eq. (3). This yields the output amplitude vector

**A**. (Step 3) Based on

_{f}**A**, calculate the current performance

_{f}*G*and its derivatives

*∂G/∂*

**A**and

_{f}**and**

*μ***by solving the adjoint system of Eq. (4). This requires the knowledge of the amplitude vector**

*λ***A**across the whole waveguide length. To reduce memory usage, the evolution of

**A**can be recalculated segment-wise by employing a check-pointing algorithm [19]. (Step 5) Combine the amplitude and adjoint vectors calculated to compute the gradient of the performance function d

*G*/d

**by evaluating Eq. (6) for each**

*θ**θ*. If the performance function

_{k}*G*satisfies

*G*converges towards an optimum value.

## 3. Opportunities and advantages of adjoint-enabled optimization

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

20. 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]

*λ*)

^{3}on supercomputers. Additionally, there exists no general framework to design nonlinear optical devices displaying intricate nonlinear interactions with topology optimization. Indeed, only nonlinear optical devices based on a Kerr-type nonlinear refractive index in a 1-D [21

21. 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]

22. J. S. Jensen, “Topology optimization of nonlinear optical devices,” Struct. Multidisc. Optim. **43**, 731–743 (2011). [CrossRef]

15. Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: Modeling and applications,” Opt. Express **15**, 16604–16644 (2007). [CrossRef] [PubMed]

*G*, the gradient (sensitivity) d

*G*/d

**≈ 0 by definition vanishes. Hence, adjoint-enabled optimization can greatly reduce the sensitivity with respect to a large number of design parameters, and allows the design of more robust devices.**

*θ**A*in Eq. (1) could represent the spectrum of a pulse (or of multiple pulses). Alternatively, one could also define the amplitude vector

_{j}**A**of Eq. (2) by sampling the amplitude in time rather than in frequency.

## 4. Design of a Raman wavelength converter

23. 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]

*ω*and

_{s}*ω*are located symmetrically around the pump frequency

_{a}*ω*, i.e.,

_{p}*ω*−

_{p}*ω*=

_{s}*ω*−

_{a}*ω*. Due to the resonant nature of the Raman effect, the Raman interactions are only significant if the frequency detuning ΔΩ =

_{p}*ω*−

_{p}*ω*is close to the Raman shift ΔΩ

_{s}*.*

_{R}**A**consisting of three elements, namely

*A*,

_{p}*A*, and

_{s}*A*. Equations (9)–(11) compose the corresponding elements of the function vector

_{a}**F**(

*z*,

**A**,

**A**

^{*},

**) [see Eq. (3a)], in which the set**

*θ***consists of any design parameters of choice (see further). The performance of a Raman converter is measured by the conversion efficiency**

*θ**G*=

*P*(

_{a}*z*)/

_{f}*P*

_{s,0}, defined as the output anti-Stokes power divided by the input Stokes power. In the following sections, we compare the performance of a conventional Raman converter designed based on the phase-matching design rule, with that of one designed by adjoint-enabled optimization.

### 4.1. Conventional design based on phase matching

*P*≫

_{p}*P*,

_{s}*P*); (2) the undepleted-pump approximation, which assumes that the pump power remains approximately undepleted throughout the waveguide (|

_{a}*A*(

_{p}*z*)|

^{2}≈

*P*

_{p,0}); (3) and the assumption that the waveguide’s characteristics are uniform along the waveguide length, i.e., that all parameters other than

*A*and

_{p,s,a}*N*in Eqs. (9)–(17) do not vary in function of

*z*. The resulting equations can be solved analytically, and the solutions thus obtained are often employed to directly describe FWM and wavelength conversion in optical fibers [14, 32

32. 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]

8. 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]

16. Q. Lin, J. Zhang, P. M. Fauchet, and G. P. Agrawal, “Ultrabroadband parametric generation and wavelength conversion in silicon waveguides,” Opt. Express **14**, 4786–4799 (2006). [CrossRef] [PubMed]

27. 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. Express **19**, 18810–18826 (2011). [CrossRef] [PubMed]

24. R. Claps, V. Raghunathan, D. Dimitropoulos, and B. Jalali, “Anti-Stokes Raman conversion in silicon waveguides,” Opt. Express **11**, 2862–2872 (2003). [CrossRef] [PubMed]

7. 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. Express **15**, 12949–12958 (2007). [CrossRef] [PubMed]

8. 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]

16. Q. Lin, J. Zhang, P. M. Fauchet, and G. P. Agrawal, “Ultrabroadband parametric generation and wavelength conversion in silicon waveguides,” Opt. Express **14**, 4786–4799 (2006). [CrossRef] [PubMed]

24. R. Claps, V. Raghunathan, D. Dimitropoulos, and B. Jalali, “Anti-Stokes Raman conversion in silicon waveguides,” Opt. Express **11**, 2862–2872 (2003). [CrossRef] [PubMed]

27. 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. Express **19**, 18810–18826 (2011). [CrossRef] [PubMed]

*h*with a fixed value of

*h*= 220 nm, as is common for silicon photonics foundries [33

33. ePIXfab, The silicon photonics website, http://www.epixfab.eu/.

*w*. For a waveguide length

*z*= 3 cm (we define

_{f}*z*

_{0}= 0), input pump and Stokes powers of

*P*

_{p,0}= 300 mW and

*P*

_{s,0}= 100 μW, a pump wavelength of

*λ*= 1550 nm, and a pump-Stokes frequency difference at Raman resonance ΔΩ = Ω

_{p}*, solving Eqs. (9)–(11) over a range of*

_{R}*w*values between 700–800 nm yields the conversion efficiency

*P*(

_{a}*z*)/

_{f}*P*

_{s,0}and the Stokes amplification

*P*(

_{s}*z*)/

_{f}*P*

_{s,0}shown in Fig. 3. As expected, a high conversion efficiency

*P*(

_{a}*z*)/

_{f}*P*

_{s,0}is only achieved near

*w*= 755 nm, which corresponds to a small phase-mismatch [26

26. 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]

*P*(

_{s}*z*)/

_{f}*P*

_{s,0}.For

*w*values away from the phase-matching condition, the conversion efficiency quickly decreases, whereas the Stokes amplification first increases before flattening at

*w*-values far away from phase-matching.

*w*of a non-varying waveguide yielding optimal conversion efficiency. In other words, it allows us to optimize the performance

*G*with respect to but a single design parameter

*θ*

_{1}=

*w*. Specifically, the thus optimized Raman converter design yields

*G*= 0.22 dB for

*w*= 755 nm.

### 4.2. Design by adjoint-enabled optimization

34. 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]

9. 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. Express **20**, 9227–9242 (2012). [CrossRef] [PubMed]

35. V. R. Almeida, R. R. Panepucci, and M. Lipson, “Nanotaper for compact mode conversion,” Opt. Lett. **28**, 1302–1304 (2003). [CrossRef] [PubMed]

9. 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. Express **20**, 9227–9242 (2012). [CrossRef] [PubMed]

36. D. Zografopoulos, R. Beccherelli, and E. Kriezis, “Quasi-soliton propagation in dispersion-engineered silicon nanowires,” Opt. Commun. **285**, 3306–3311 (2012). [CrossRef]

37. 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]

**20**, 9227–9242 (2012). [CrossRef] [PubMed]

26. 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]

*L*

_{Taper}, so that the width evolution

*w*(

*z*) can be described by: If the taper length

*L*

_{Taper}is much larger than any variation

*w*

_{k+1}−

*w*in width, then a waveguiding functionality is indeed ensured. For SOI nanowaveguides, typical width variations range from several tens to several hundreds of nanometers [9

_{k}**20**, 9227–9242 (2012). [CrossRef] [PubMed]

26. 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]

34. 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]

35. V. R. Almeida, R. R. Panepucci, and M. Lipson, “Nanotaper for compact mode conversion,” Opt. Lett. **28**, 1302–1304 (2003). [CrossRef] [PubMed]

*L*

_{Taper}of several tens of micrometers is sufficiently long [26

**31**, 2113–2121 (2013). [CrossRef]

*L*

_{Taper}= 50 μm.

*w*in Eq. (18) as a separate design parameter. Additionally, rather than optimizing a design with a fixed length, we include the waveguide length

_{k}*z*as a design parameter (we define

_{f}*z*

_{0}= 0), and impose an upper limit for this parameter. This upper limit should exceed the device length expected, and can be specified by limiting the number of independent parameters

*w*to

_{k}*M*so that

_{w}*z*≤ (

_{f}*M*− 1)

_{w}*L*

_{Taper}. Moreover, since numerical simulations indicate that the Raman conversion efficiency can be improved by operating slightly off Raman resonance [27

27. 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. Express **19**, 18810–18826 (2011). [CrossRef] [PubMed]

**thus consists of**

*θ**θ*

_{1}= ΔΩ,

*θ*

_{2}=

*z*, and

_{f}*θ*

_{k+2}=

*w*for

_{k}*k*= 1,...,

*M*with (

_{w}*M*− 1) the maximum number of interconnected tapers allowed. Here we take

_{w}*M*= 1601, resulting in a waveguide length limited by

_{w}*z*≤ 8 cm and a total of 1603 independent design parameters. All other parameters, including the input pump and Stokes powers, the pump wavelength, and the waveguide height, are taken as fixed values identical to those in Section 4.1, i.e.,

_{f}*P*

_{p,0}= 300 mW,

*P*

_{s,0}= 100 μW,

*λ*= 1550 nm, and

_{p}*h*= 220 nm.

17. J. Nocedal and S. J. Wright, *Numerical Optimization*, 2nd ed. (Springer, 1999). [CrossRef]

*θ*

_{1}= ΔΩ = Ω

*,*

_{R}*θ*

_{2}=

*z*= 4 cm, and

_{f}*θ*

_{k+2}=

*w*= 755 nm for

_{k}*k*= 1,...,

*M*.

_{w}**as indicated by the steepest descent algorithm. (Step 2) We solve Eqs. (9)–(11) over the current device length**

*θ**θ*

_{2}=

*z*. To evaluate the

_{f}*w*-dependent parameters in these equations, we employ the method outlined by Driscoll

*et al.*[9

**20**, 9227–9242 (2012). [CrossRef] [PubMed]

*w*to a set of calculated values [38]. Evaluating the obtained polynomials at each position then allows us to directly solve the propagation equations [9

**20**, 9227–9242 (2012). [CrossRef] [PubMed]

**31**, 2113–2121 (2013). [CrossRef]

*A*(

_{a}*z*) obtained, we update the performance

_{f}*G*= |

*A*(

_{a}*z*)|

_{f}^{2}/

*P*

_{s,0}and calculate the non-zero derivatives

*μ*are derived in a straightforward manner by applying Eq. (7) to the pump, Stokes, and anti-Stokes equations of Eqs. (9)–(11). However, as these adjoint equations are rather lengthy, we do not give them here explicitly. (Step 5) Based on the found

**A**and

*μ*evolutions, we calculate the gradient d

*G*/d

*θ*for each

_{k}*θ*. First, we compute the derivative d

_{k}*G*/dΔΩ by Eq. (8). To find the function derivatives

*∂*

**F**/

*∂*ΔΩ of Eqs. (9)–(11) we employ the formula: which follows from the identities

*ω*=

_{a}*ω*+ ΔΩ and

_{p}*ω*=

_{s}*ω*− ΔΩ. Second, we compute the derivatives d

_{p}*G*/d

*w*also by Eq. (8). The function derivatives

_{k}*∂*

**F**/

*∂w*are obtained by taking into account that any

_{k}*w*only affects the light propagation in the tapers just before and just after the corresponding position

_{k}*z*, and this according to the formulas: Third, for the derivative

_{k}*∂G/∂z*, we do not use Eq. (8), but instead employ the simple formula directly derived from

_{f}*∂G/∂z*itself: Here the last equality follows from Eq. (7b).

_{f}*w*of the optimized design varies over a range of more than 25 nm, whereas its length and frequency difference equal

*z*= 6.28 cm and ΔΩ = Ω

_{f}*− 187 GHz respectively. The variations in the width remain adiabatic, as the maximal relative change in width max(|*

_{R}*w*

_{k+1}+

*w*|/

_{k}*L*

_{Taper})= 4.9 nm/50 μm is smaller than the variation 60 nm/500 μm of an experimentally demonstrated variable-width waveguide [9

**20**, 9227–9242 (2012). [CrossRef] [PubMed]

*P*(

_{a}*z*)/

_{f}*P*

_{s,0}= 10.8 dB, corresponding to a more than 10 dB improvement with respect to the initial phase-matched design. In addition, the optimized design results in an output Stokes amplification of

*P*(

_{s}*z*)/

_{f}*P*

_{s,0}= 14.5 dB, which is more than 7 dB higher than for the initial design. Note that the output Stokes amplification of the initial phase-matched design could also be enhanced by employing a longer waveguide, but this would be accompanied by a reduction in the conversion efficiency as the anti-Stokes power

*P*experiences no longer gain but loss after 4 cm in the phase-matched converter [see Fig. 5(b)]. Hence, the optimized design does not only yield a much higher conversion efficiency than can be achieved with the phase-matched design, but also leads to a Stokes amplification of the same level as that of a conventional Raman amplifier operating far from phase-matching [see Fig. 3]. In other words, the design combines the optimized Raman wavelength conversion with the functionality of a conventional Raman amplifier operating away from phase-matching.

_{a}*ϕ*along the initial and final waveguides [see Fig. 5(d)]. The phase difference, defined as Δ

*ϕ*= 2

*ϕ*−

_{p}*ϕ*−

_{s}*ϕ*with

_{a}*ϕ*the phase of

_{j}*A*, is an essential parameter in the conversion process [26

_{j}**31**, 2113–2121 (2013). [CrossRef]

**19**, 18810–18826 (2011). [CrossRef] [PubMed]

**19**, 18810–18826 (2011). [CrossRef] [PubMed]

*ϕ*is within a range

*π*around the value −Δ

*ϕ*(−Δ

_{FWM,a}*ϕ*), which is the negative of the phase of the total complex FWM anti-Stokes (Stokes) gain

_{FWM,s}*G*(

_{FWM,a}*G*): In Fig. 5(d), we depict −Δ

_{FWM,s}*ϕ*and −Δ

_{FWM,a}*ϕ*both for the phase-matched design with ΔΩ = Ω

_{FWM,s}*(dash-dotted lines) and for the optimized design with ΔΩ = Ω*

_{R}*− 187 GHz (dotted lines). Conventional phase-matched operation corresponds to maintaining Δ*

_{R}*ϕ*as close as possible to −Δ

*ϕ*so that the anti-Stokes gain is maximal throughout the waveguide. However, for the optimized waveguide design, efficient conversion is realized in a different manner entirely. Throughout the first half of the waveguide, Δ

_{FWM,a}*ϕ*(full black line) is not maintained at the value −Δ

*ϕ*, but rather halfway between −Δ

_{FWM,a}*ϕ*and −Δ

_{FWM,a}*ϕ*(blue and red dash-dotted lines respectively). As a consequence, the conversion efficiency is at the beginning of the waveguide reduced [see Fig. 5(b)] and the signal amplification increased [see Fig. 5(c)] as compared to the quantities in the initial phase-matched waveguide. However, since the anti-Stokes FWM interactions scale with

_{FWM,s}*A*[27

_{s}**19**, 18810–18826 (2011). [CrossRef] [PubMed]

*a posteriori*a more efficient scheme for achieving efficient Raman wavelength conversion than conventional phase-matching. Rather than maximizing the conversion locally throughout the waveguide conform the phase-matching method, the design first realizes a strong Stokes amplification. The enhanced Stokes power then enables a higher conversion efficiency towards the end of the waveguide, despite the depleted pump powers there. This scheme allows to improve the efficiency of Raman converters and even to combine conventional Raman converters and amplifiers in a single device. Additionally, it also suggests that the conversion efficiency of any phase-matched converter could potentially be improved by an initial amplification of the input signal without increasing the overall power consumption.

*P*(

_{a}*z*)

_{f}^{−1}

*∂P*(

_{a}*z*)/

_{f}*∂w*with respect to local variations in width by several orders of magnitude as compared to the sensitivity of the phase-matched design [see Fig. 6]. Hence, the optimized design is much more robust with respect to local fabrication errors of the waveguide width. This robustness with respect to local variations is only made possible by the adjoint-enabled optimization technique and the large number of design parameters it allows.

_{k}## 5. Conclusions

## Acknowledgments

## References and links

1. | J. Jensen and O. Sigmund, “Topology optimization for nano-photonics,” Laser Photon. Rev. |

2. | F. Wang, J. S. Jensen, and O. Sigmund, “Robust topology optimization of photonic crystal waveguides with tailored dispersion properties,” J. Opt. Soc. Am. B |

3. | P. Borel, A. Harpøth, L. Frandsen, M. Kristensen, P. Shi, J. Jensen, and O. Sigmund, “Topology optimization and fabrication of photonic crystal structures,” Opt. Express |

4. | Y. Tsuji, K. Hirayama, T. Nomura, K. Sato, and S. Nishiwaki, “Design of optical circuit devices based on topology optimization,” IEEE Photon. Technol. Lett. |

5. | 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. |

6. | H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave raman silicon laser,” Nature |

7. | 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. Express |

8. | V. Raghunathan, R. Claps, D. Dimitropoulos, and B. Jalali, “Parametric Raman wavelength conversion in scaled silicon waveguides,” J. Lightwave Technol. |

9. | 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. Express |

10. | D. T. Tan, P. C. Sun, and Y. Fainman, “Monolithic nonlinear pulse compressor on a silicon chip,” Nat. Commun. |

11. | A. W. Snyder and J. D. Love, |

12. | L. Jin, W. Jin, J. Ju, and Y. Wang, “Coupled local-mode theory for strongly modulated long period gratings,” J. Lightwave Technol. |

13. | W.-P. Huang and J. Mu, “Complex coupled-mode theory for optical waveguides,” Opt. Express |

14. | G. Agrawal, |

15. | Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: Modeling and applications,” Opt. Express |

16. | Q. Lin, J. Zhang, P. M. Fauchet, and G. P. Agrawal, “Ultrabroadband parametric generation and wavelength conversion in silicon waveguides,” Opt. Express |

17. | J. Nocedal and S. J. Wright, |

18. | 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. |

19. | 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). |

20. | 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. |

21. | Y. Elesin, B. Lazarov, J. Jensen, and O. Sigmund, “Design of robust and efficient photonic switches using topology optimization,” Phot. Nano. Fund. Appl. |

22. | J. S. Jensen, “Topology optimization of nonlinear optical devices,” Struct. Multidisc. Optim. |

23. | N. Vermeulen, C. Debaes, and H. Thienpont, “Coherent anti-Stokes Raman scattering in Raman lasers and Raman wavelength converters,” Laser Photon. Rev. |

24. | R. Claps, V. Raghunathan, D. Dimitropoulos, and B. Jalali, “Anti-Stokes Raman conversion in silicon waveguides,” Opt. Express |

25. | 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. |

26. | Y. Lefevre, N. Vermeulen, and H. Thienpont, “Quasi-phase-matching of four-wave-mixing-based wavelength conversion by phase-mismatch switching,” J. Lightwave Technol. |

27. | 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. Express |

28. | R. Claps, D. Dimitropoulos, V. Raghunathan, Y. Han, and B. Jalali, “Observation of stimulated Raman amplification in silicon waveguides,” Opt. Express |

29. | R. Soref and B. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. |

30. | 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. |

31. | X. Chen, N. Panoiu, and R. Osgood, “Theory of Raman-mediated pulsed amplification in silicon-wire waveguides,” IEEE J. Quantum Electron. |

32. | 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. |

33. | ePIXfab, The silicon photonics website, http://www.epixfab.eu/. |

34. | 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. |

35. | V. R. Almeida, R. R. Panepucci, and M. Lipson, “Nanotaper for compact mode conversion,” Opt. Lett. |

36. | D. Zografopoulos, R. Beccherelli, and E. Kriezis, “Quasi-soliton propagation in dispersion-engineered silicon nanowires,” Opt. Commun. |

37. | O. Tsilipakos, D. C. Zografopoulos, and E. E. Kriezis, “Quasi-soliton pulse-train propagation in dispersion-managed silicon rib waveguides,” IEEE Photon. Technol. Lett. |

38. | We employed the commercial software package MODE Solutions by Lumerical to calculate the dispersion characteristics and mode profiles of SOI waveguides. |

**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

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- 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]
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- 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]
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- 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]
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- 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]
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- 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.

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