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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 16 — Aug. 11, 2014
  • pp: 19423–19439
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Adjoint-enabled optimization of optical devices based on coupled-mode equations

Yannick Lefevre, Pierre Wahl, Nathalie Vermeulen, and Hugo Thienpont  »View Author Affiliations


Optics Express, Vol. 22, Issue 16, pp. 19423-19439 (2014)
http://dx.doi.org/10.1364/OE.22.019423


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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

Optimization algorithms have proven to be a powerful tool for designing optical devices. For instance, in the field of nanophotonics, topology optimization has recently enabled the design of components with no constraints on their geometrical shapes [1

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

]. This approach treats the material distribution as a design parameter, and employs repeated finite-element or -difference analyses and gradient-based optimization updates in combination with an adjoint approach for efficient gradient calculations. It has been successfully applied to improve a large variety of devices, including photonic crystal waveguides with tailored dispersion characteristics [2

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]

], broadband photonic-crystal waveguide bends [3

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]

], and 90° nanowaveguide bends and splitters [4

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]

].

In this paper, we propose an optimization scheme for designing coupled-mode-based optical devices that depend on a large number of design parameters. Similar to how topology optimization combines gradient-based updates with an adjoint approach, our scheme combines gradient-based optimization algorithms with an adjoint sensitivity analysis of the coupled-mode equations describing the light propagation. In Sections 2 and 3, we describe this adjoint-enabled optimization and its mathematical aspects in detail. In Section 4, we illustrate the potential of the design method by considering a non-trivial design problem, namely the design of a silicon-on-insulator (SOI) Raman wavelength converter, in which multiple modes couple through a variety of interactions. We compare a converter designed by the conventional phase-matching design rule, with one designed through adjoint-enabled optimization. For the optimization, we consider a waveguide constructed from a series of interconnected linear tapers, and treat the waveguide width at each interconnection point, as well as the waveguide length and the pump-Stokes frequency difference, as independent design parameters. By investigating the conversion process occurring in the optimized design, we identify an alternative and more efficient method than conventional phase-matching for achieving efficient Raman conversion. We also show that adjoint-enabled optimization significantly improves the robustness with respect to the variations in the design parameters. Finally, we give our conclusions in Section 5.

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

Solving Eq. (3) for a given set of design parameters θ allows us to determine the corresponding output amplitude vector Af at the waveguide end position zf [see Fig. 1]. Based on Af, we can determine the performance corresponding to the parameter values θ, which is typically measured by a performance figure G=G(Af,Af*). Note that possibly G could also depend on other variables such as the input amplitude A0 or a subset of the design parameters θ, 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 θ̄ yielding an optimal G.

Fig. 1 A coupled-mode-based device is simulated by solving the coupled-mode equations for a given input vector A0, yielding the output vector Af. The input A0 as well as the coupled-mode equations depend on a set of design parameters θ1,...,θM, whereas the device’s performance figure G is calculated from the output Af. Designing the device consists of determining the parameter values θ̄ that optimize G.

For simple problems, the set of coupled equations of Eq. (3) can be solved analytically. In such cases, the parameter values θ̄ 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

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]

8

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

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

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]

], and perturbed waveguides with complex non-periodic spatial variations [12

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]

] require Eq. (3) to be solved numerically. The optimal parameter values θ̄ 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 M rises.

In case of a large number of parameters, gradient-based optimization algorithms are typically combined with an adjoint approach [1

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

]. To compute the gradient, such methods introduce a set of additional variables, called adjoint variables, as Lagrange multipliers to efficiently calculate the gradient (sensitivity) of the system. For systems described by a set of ordinary differential equations, the adjoint sensitivity analysis is typically formulated in terms of purely real variables [18

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]

, 19

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).

], in which case the number of adjoint variables introduced equals the number of independent differential equations of the problem under study.

We point out that the adjoint method can be simplified considerably in a special class of problems. For a performance function G for which the partial derivatives in Eq. (4b) satisfy G/Af*=(G/Af)*, we found that the second adjoint vector equals λ = μ* according to Eq. (4). As a consequence, only N independent adjoint variables remain, and the adjoint system of Eq. (4) simplifies to:
μz=(FA)μ(F*A)μ*,
(7a)
μ(zf)=(GA),
(7b)
and the adjoint sensitivity equation of Eq. (6) becomes:
dGdθk=Gθk+2Re[μ(z0)A0θk]+2z0zfRe[μFθk]dz,
(8)
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 Pj (zf) = |Aj (zf)|2 satisfies the condition G/Af*=(G/Af)*. Hence, any device for which the performance is measured only in terms of output powers can be treated by the simplified adjoint analysis of Eqs. (7)(8), enabling a much faster gradient computation for these devices than Eqs. (4)(6).

Fig. 2 Adjoint-enabled optimization optimizes the performance function G with respect to the design parameters θ by employing an iterative optimization algorithm consisting of five steps during each iteration. Note that, if G satisfies G/Af*=(G/Af)*, then λ = μ* so that only one adjoint vector has to be propagated in Step 4.

3. Opportunities and advantages of adjoint-enabled optimization

To overcome these limitations, our method starts from the coupled-mode equations of Eq. (3) that are approximations to Maxwell’s equation. The coupled-mode equations and the derived adjoint system of Eq. (4a) or (7a) are sets of first-order ordinary differential equations that can be solved by a simple 1-D numerical integration. Hence, for optical components that are accurately described by coupled-mode equations like Eq. (3), our adjoint-enabled optimization method is much more computationally efficient than topology optimization, and thus enables the design of components that are too long to be designed by topology optimization. In addition, nonlinear optical devices based on waveguides are commonly investigated by means of coupled-mode equations [14

14. G. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, 2001).

,15

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]

], and are thus in particular suited for our optimization method. Hence, in contrast to topology optimization, our optimization scheme can be implemented in a straightforward manner for complex nonlinear propagation equations, as we illustrate in Section 4.

These benefits are only available for geometries for which the approximations leading to the coupled-mode equations remain valid. Hence, a drawback of our adjoint-enabled optimization method is that it offers less design freedom and less geometrical flexibility than topology optimization which has no such constraints. Nevertheless, for devices that are accurately modeled by coupled-mode equations, the benefits of our method are substantial.

Generally optimization algorithms converge to local extrema and not necessarily to the sought-after global extremum. Hence, choosing a proper starting point is still essential for the design method described, and should be done with much care based on physical insight into the problem considered. By repeating the design method with multiple starting points, one can obtain a variety of locally-optimized designs that give insight into the different modes of operation improving the device’s performance.

An additional advantage of employing optimization algorithms to design optical components is that the sensitivity with respect to variations in the design parameters can be greatly reduced. Indeed, at an optimum for the performance G, the gradient (sensitivity) dG/dθ ≈ 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.

It should be noted that, although we focus in this paper on waveguide devices operating in a continuous-wave regime, the described design method can also be used for designing optical components operating in a pulsed regime. Indeed, the spectral components Aj in Eq. (1) could represent the spectrum of a pulse (or of multiple pulses). Alternatively, one could also define the amplitude vector A of Eq. (2) by sampling the amplitude in time rather than in frequency.

In Section 4 we illustrate the full potential of the design scheme by considering a non-trivial design problem of a component in which multiple modes couple with each other through a variety of interactions. A component that illustrates this well is the SOI-based Raman wavelength converter.

4. Design of a Raman wavelength converter

Raman wavelength converters employ the third-order nonlinear Raman effect to convert light from one frequency to another [23

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]

]. convert a low-frequency Stokes wave into a high frequency anti-Stokes wave by interacting with a strong pump wave through the process of coherent anti-Stokes Raman scattering (CARS). The conversion is such that the Stokes and anti-Stokes frequencies ωs and ωa are located symmetrically around the pump frequency ωp, i.e., ωpωs = ωaωp. Due to the resonant nature of the Raman effect, the Raman interactions are only significant if the frequency detuning ΔΩ = ωpωs is close to the Raman shift ΔΩR.

In the terminology of Section 2, the SOI-based Raman converter is thus described by an amplitude vector A consisting of three elements, namely Ap, As, and Aa. Equations (9)(11) compose the corresponding elements of the function vector F(z, A, A*, θ) [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 G = Pa(zf)/Ps,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

The coupled-mode equations of Eqs. (9)(11) can be solved analytically if three assumptions are made: (1) the strong-pump assumption, which assumes that the pump power is much stronger than the Stokes and anti-Stokes powers throughout the waveguide (PpPs, Pa); (2) the undepleted-pump approximation, which assumes that the pump power remains approximately undepleted throughout the waveguide (|Ap (z)|2Pp,0); (3) and the assumption that the waveguide’s characteristics are uniform along the waveguide length, i.e., that all parameters other than Ap,s,a and 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

14. G. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, 2001).

, 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]

].

These solutions indicate that the conversion process strongly depends on the so-called phase-mismatch between the waves [8

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

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

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]

]. A small phase-mismatch results in an efficient anti-Stokes generation, whereas a large phase-mismatch suppresses the anti-Stokes generation so that amplification of the Stokes wave through stimulated Stokes Raman scattering is favoured instead [24

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]

]. Hence, according to the solutions to the simplified equations, designing a Raman wavelength converter simply corresponds to satisfying the phase-matching condition, i.e., to realizing a small phase-mismatch value. As the phase-mismatch depends on the waveguide’s dispersion characteristics, such phase-matching is typically achieved by engineering the waveguide geometry [7

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]

].

The phase-matching rule-of-thumb outlined above is commonly employed to design Raman wavelength converters in SOI nanowaveguides [8

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

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

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]

], even though the analytical solutions from which the rule was derived are actually not valid for these devices. Indeed, any near-infrared pump wave experiences severe losses in a SOI waveguide due to TPA and the associated FCA, resulting in a strong pump depletion [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]

], so that Eqs. (9)(11) should be solved numerically to simulate the light propagation.

Nevertheless, the phase-matching condition remains an efficient design rule, even for these devices. To illustrate this, we consider a rectangular, air-cladded SOI waveguide [see inset Fig. 3(a)] with a non-varying waveguide geometry. We assume a waveguide height 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/.

]. The phase-mismatch of this waveguide can be tailored by tuning the waveguide width w. For a waveguide length zf = 3 cm (we define z0 = 0), input pump and Stokes powers of Pp,0 = 300 mW and Ps,0 = 100 μW, a pump wavelength of λp = 1550 nm, and a pump-Stokes frequency difference at Raman resonance ΔΩ = ΩR, solving Eqs. (9)(11) over a range of w values between 700–800 nm yields the conversion efficiency Pa(zf)/Ps,0 and the Stokes amplification Ps(zf)/Ps,0 shown in Fig. 3. As expected, a high conversion efficiency Pa(zf)/Ps,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]

]. The peak in conversion efficiency is also accompanied by a dip in the Stokes amplification Ps(zf)/Ps,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.

Fig. 3 (a) The conversion efficiency Pa/Ps,0 and (b) Stokes amplification Ps/Ps,0 of that can be achieved with a 3 cm-long, non-varying, rectangular SOI waveguide [see inset (a)] with fixed height h = 220 nm depends strongly on its width w.

The phase-matching design rule allows us to easily estimate the width 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 θ1 = w. Specifically, the thus optimized Raman converter design yields G = 0.22 dB for w = 755 nm.

4.2. Design by adjoint-enabled optimization

Due to its additional design freedom, we here also optimize a variable-width waveguide rather than a non-varying waveguide. To ensure an adiabatic width variation, we construct the waveguide as a series of interconnected linear tapers [see Fig. 4]. All tapers have an equal length LTaper, so that the width evolution w(z) can be described by:
zk+1=zk+LTaper,
(18a)
w(zkzzk+1)=wk+wk+1wkLTaper(zzk).
(18b)
If the taper length LTaper is much larger than any variation wk+1wk 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

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]

, 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]

, 34

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

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

], so that a taper length LTaper of several tens of micrometers is sufficiently long [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]

]. For our waveguide design, we employ a taper length LTaper = 50 μm.

Fig. 4 The variable-width waveguide proposed consists of a series of interconnected linear tapers with equal lengths of LTaper. The widths w1,...,wk, wk+1 ... at each interconnection point is taken as an independent design parameter.

To perform an adjoint-enabled optimization, we first have to identify the problem’s design parameters. Adjoint-enabled optimization enables us to treat each width wk in Eq. (18) as a separate design parameter. Additionally, rather than optimizing a design with a fixed length, we include the waveguide length zf as a design parameter (we define z0 = 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 wk to Mw so that zf ≤ (Mw − 1)LTaper. 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]

], we also include the frequency difference ΔΩ as another design parameter. The set of design parameters θ thus consists of θ1 = ΔΩ, θ2 = zf, and θk+2 = wk for k = 1,...,Mw with (Mw − 1) the maximum number of interconnected tapers allowed. Here we take Mw = 1601, resulting in a waveguide length limited by zf ≤ 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., Pp,0 = 300 mW, Ps,0 = 100 μW, λp = 1550 nm, and h = 220 nm.

During each iteration of the optimization algorithm, we execute the five steps discussed in Section 2 and depicted in Fig. 2 in the following manner: (Step 1) we update the design parameters θ as indicated by the steepest descent algorithm. (Step 2) We solve Eqs. (9)(11) over the current device length θ2 = zf. To evaluate the w-dependent parameters in these equations, we employ the method outlined by Driscoll et al. [9

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]

] of fitting for each parameter a polynomial in w to a set of calculated values [38

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

]. Evaluating the obtained polynomials at each position then allows us to directly solve the propagation equations [9

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]

, 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]

]. (Step 3) Based on the output anti-Stokes amplitude Aa(zf) obtained, we update the performance G = |Aa(zf)|2/Ps,0 and calculate the non-zero derivatives G/Aa(zf)=Aa*(zf)/Ps,0 and G/Aa*(zf)=Aa(zf)/Ps,0. (Step 4) We solve the Raman converter’s adjoint system in a similar fashion as the coupled-mode equations in Step 3. Since the performance derivatives satisfy G/Aa*(zf)=[G/Aa(zf)]*, we employ the simplified adjoint system described by Eq. (7). The equations for the three elements of the adjoint vector 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 dG/dθk for each θk. First, we compute the derivative dG/dΔΩ by Eq. (8). To find the function derivatives F/ΔΩ of Eqs. (9)(11) we employ the formula:
FΔΩ=FωaFωs,
(19)
which follows from the identities ωa = ωp + ΔΩ and ωs = ωp − ΔΩ. Second, we compute the derivatives dG/dwk also by Eq. (8). The function derivatives F/∂wk are obtained by taking into account that any wk only affects the light propagation in the tapers just before and just after the corresponding position zk, and this according to the formulas:
F(zkzzk+1)wk+1=Fwwwk+1=FwzzkLTaper,
(20)
F(zkzzk+1)wk=Fwwwk=FwLTaper(zzk)LTaper.
(21)
Third, for the derivative ∂G/∂zf, we do not use Eq. (8), but instead employ the simple formula directly derived from ∂G/∂zf itself:
Gzf=2Re[GAfAfzf]=2Re[μAz]z=zf.
(22)
Here the last equality follows from Eq. (7b).

The resulting optimized Raman converter design is compared with the initial phase-matched design in Fig. 5. The width profile w of the optimized design varies over a range of more than 25 nm, whereas its length and frequency difference equal zf = 6.28 cm and ΔΩ = ΩR − 187 GHz respectively. The variations in the width remain adiabatic, as the maximal relative change in width max(|wk+1 + wk|/LTaper)= 4.9 nm/50 μm is smaller than the variation 60 nm/500 μm of an experimentally demonstrated variable-width waveguide [9

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]

]. The optimized design’s performance is Pa(zf)/Ps,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 Ps(zf)/Ps,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 Pa 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.

Fig. 5 Comparison between the evolutions in (a) waveguide width w, (b) conversion efficiency Pa/Ps,0, (c) Stokes amplification Ps/Ps,0, and (d) phase difference Δϕ for the initial Raman converter design derived from the phase-matching rule (dashed lines) and the design optimized by adjoint-enabled optimization (full lines). In (d), also the phases −ΔϕFWM,a and −ΔϕFWM,s of the anti-Stokes and Stokes FWM gains are shown, both for the initial frequency difference ΔΩ = ΩR (dash-dotted lines) and for the optimized one ΔΩ = ΩR − 187.0 GHz (dotted lines), to indicate at which Δϕ values the waves experience maximal gain.

To investigate the physical origins of these characteristics, we consider the evolution of the phase difference Δϕ along the initial and final waveguides [see Fig. 5(d)]. The phase difference, defined as Δϕ = 2ϕpϕsϕa with ϕj the phase of Aj, is an essential parameter in the conversion process [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]

, 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]

]. Its value determines whether the anti-Stokes and Stokes waves experience gain or loss due to the FWM processes, which consist of both CARS and Kerr-based FWM. As explained in reference [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]

], there is anti-Stokes (Stokes) gain as long as Δϕ is within a range π around the value −ΔϕFWM,a (−ΔϕFWM,s), which is the negative of the phase of the total complex FWM anti-Stokes (Stokes) gain GFWM,a (GFWM,s):
GFWM,j=i(γK,jΓspapK+γR,jΓspapRHR(ωjωp))SpSsSa
(23)
In Fig. 5(d), we depict −ΔϕFWM,a and −ΔϕFWM,s both for the phase-matched design with ΔΩ = ΩR (dash-dotted lines) and for the optimized design with ΔΩ = ΩR − 187 GHz (dotted lines). Conventional phase-matched operation corresponds to maintaining Δϕ as close as possible to −ΔϕFWM,a 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, Δϕ (full black line) is not maintained at the value −ΔϕFWM,a, but rather halfway between −ΔϕFWM,a and −ΔϕFWM,s (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 As [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]

], the increased Stokes power enhances the FWM interactions further down the waveguide resulting eventually also in an increase of the conversion efficiency.

Our optimized design reveals 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.

As discussed towards the end of Section 2, an additional advantage of a design through optimization is a reduced sensitivity with respect to the design parameters. In case of the optimized Raman converter, this translates to a reduction of the relative sensitivity Pa(zf)−1∂Pa(zf)/∂wk 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.

Fig. 6 The sensitivity of the Raman converter performance Pa(zf)−1∂Pa(zf)/∂wk with respect to local variations in waveguide width wk is several orders magnitude smaller for the design optimized by adjoint-enabled optimization (full line) than for the initial design derived from the phase-matching rule (dashed line). The inset shows a close-up of the sensitivity Pa(zf)−1∂Pa(zf)/∂wk between −1·10−5 nm−1 and 1·10−5 nm−1.

5. Conclusions

We proposed a design method for optical components that are based on coupled-mode equations. The method combines gradient-based optimization algorithms with an adjoint sensitivity analysis of the coupled-mode equations describing the light propagation to efficiently handle a large number of design parameters.

We illustrated the potential of our design method by considering the non-trivial problem of a SOI-based Raman wavelength converter that is constructed from a series of interconnected linear tapers. Optimizing with respect to 1603 design parameters, including the width at the connection points of the different tapers, the waveguide length, and the pump-Stokes frequency difference, resulted in an optimal conversion efficiency of 10.8 dB for a length of 6.28 cm and a frequency difference 187 GHz below the Raman shift. This corresponds to a more than 10 dB improvement in performance compared to a design derived from the conventional phase-matching design rule and that operates at perfect Raman resonance. Additionally, the optimized design also achieved a 14.5 dB Stokes amplification, which is more than 7 dB higher than for the phase-matched design. The adjoint-enabled optimization also allowed us to identify an alternative and more efficient method for achieving efficient Raman wavelength conversion than conventional phase-matching. By introducing a strong initial amplification of the Stokes wave, the conversion process is enhanced further down the waveguide, resulting in an overall improvement of the conversion efficiency in the optimized design. Finally, we showed that the adjoint-enabled optimization also considerably improves the design’s robustness towards parameter variations. Specifically, the optimized Raman converter displays a sensitivity with respect to local variations in the waveguide width that is several orders of magnitude smaller than for the phase-matched design.

Our results show that adjoint-enabled optimization is an efficient design tool for optical components based on coupled-mode equations. The method is especially suited for non-trivial design problems that cannot be solved analytically and in which multiple modes couple through a variety of interactions. It does not only allow to improve the performance and robustness of such optical devices, but also to gain better physical insight in the mechanisms that lead to optimal performance, and even to novel classes of optical devices.

Acknowledgments

This work was supported by FWO-Vlaanderen, which provides an Aspirant grant for Y. Lefevre and a Postdoctoraal Onderzoeker grant for N. Vermeulen, VUB-Methusalem, VUB-OZR, IAPBELSPO under grant IAP P7-35, and the European Research Council ( ERC-FP7/2007-2013) under grant 336940.

References and links

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]

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]

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]

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]

11.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).

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]

13.

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

14.

G. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, 2001).

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]

17.

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

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. 138, 467–478 (2013). [CrossRef]

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]

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]

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]

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. 31, 2113–2121 (2013). [CrossRef]

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]

28.

R. Claps, D. Dimitropoulos, V. Raghunathan, Y. Han, and B. Jalali, “Observation of stimulated Raman amplification in silicon waveguides,” Opt. Express 11, 1731–1739 (2003). [CrossRef] [PubMed]

29.

R. Soref and B. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. 23, 123–129 (1987). [CrossRef]

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. 86, 071115 (2005). [CrossRef]

31.

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]

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]

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. 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]

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]

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


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References

  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. B28, 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. Express12, 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,” Nature433, 725–728 (2005). [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. Express15, 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]
  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. Express20, 9227–9242 (2012). [CrossRef] [PubMed]
  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]
  11. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).
  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]
  13. W.-P. Huang and J. Mu, “Complex coupled-mode theory for optical waveguides,” Opt. Express17, 19134–19152 (2009). [CrossRef]
  14. G. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, 2001).
  15. 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]
  16. 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]
  17. J. Nocedal and S. J. Wright, Numerical Optimization, 2nd ed. (Springer, 1999). [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]
  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.138, 467–478 (2013). [CrossRef]
  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]
  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]
  24. R. Claps, V. Raghunathan, D. Dimitropoulos, and B. Jalali, “Anti-Stokes Raman conversion in silicon waveguides,” Opt. Express11, 2862–2872 (2003). [CrossRef] [PubMed]
  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.31, 2113–2121 (2013). [CrossRef]
  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. Express19, 18810–18826 (2011). [CrossRef] [PubMed]
  28. 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]
  29. R. Soref and B. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron.23, 123–129 (1987). [CrossRef]
  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.86, 071115 (2005). [CrossRef]
  31. 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]
  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]
  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.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]
  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]
  38. 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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