Abstract

We investigate the design of taper structures for coupling to slow-light modes of various photonic-crystal waveguides while taking into account parameter uncertainties inherent in practical fabrication. Our short-length (11 periods) robust tapers designed for λ = 1.55μm and a slow-light group velocity of c/34 have a total loss of < 20dB even in the presence of nanometer-scale surface roughness, which outperform the corresponding non-robust designs by an order of magnitude. We discover a posteriori that the robust designs have smooth profiles that can be parameterized by a few-term (intrinsically smooth) sine series which helps the optimization to further boost the performance slightly. We ground these numerical results in an analytical foundation by deriving the scaling relationships between taper length, taper smoothness, and group velocity with the help of an exact equivalence with Fourier analysis.

© 2012 OSA

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2012 (2)

Y. Elesin, B. Lazarov, J. Jensen, and O. Sigmund, “Design of robust and efficient photonic switches using topology optimization,” Photon. and Nanostruc. 10, 153–165 (2012).
[CrossRef]

J. Lu and J. Vuckovic, “Objective-first design of high-efficiency, small-footprint couplers between arbitrary nanophotonic waveguide modes,” Opt. Express 20, 7221–7236 (2012).
[CrossRef] [PubMed]

2011 (5)

2010 (5)

J. Ma and M. Povinelli, “Effect of periodicity on optical forces between a one-dimensional periodic photonic crystal waveguide and an underlying substrate,” Appl. Phys. Letters 97(151102) (2010).
[CrossRef]

D. Bertsimas, O. Nohedani, and K. Teo, “Robust optimization for unconstrained simulation-based problems,” Oper. Research 58, 161–178 (2010).
[CrossRef]

M. Strain and M. Sorel, “Design and fabrication of integrated chirped bragg gratings for on-chip dispersion control,” IEEE J. Quant. Elec. 46, 774–782 (2010).
[CrossRef]

P. Bermel, M. Ghebrebrhan, W. Chan, Y. X. Yeng, M. Araghchini, R. Hamam, C. H. Marton, K. F. Jensen, M. Soljačić, J. D. Joannopoulos, S. G. Johnson, and I. Celanovic, “Design and global optimization of high-efficiency thermophotovoltaic systems,” Opt. Express 18, A314–A334 (2010).
[CrossRef] [PubMed]

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comp. Phys. Communications 181, 687–702 (2010).
[CrossRef]

2009 (3)

M. Ghebrebrhan, P. Bermel, Y. Avniel, J. D. Joannopoulos, and S. G. Johnson, “Global optimization of silicon photovoltaic cell front coatings,” Opt. Express 17, 7505–7518 (2009).
[CrossRef] [PubMed]

A. Mutapcic, S. Boyd, A. Farjadpour, S. Johnson, and Y. Avniel, “Robust design of slow-light tapers in periodic waveguides,” Engineering Optimization 41, 365–384 (2009).
[CrossRef]

D. Dobson and L. Simeonova, “Optimization of periodic composite structures for sub-wavelength focusing,” Appl. Math. Optim. 60, 133–150 (2009).
[CrossRef]

2008 (6)

J. Riishede and O. Sigmund, “Inverse design of dispersion compensating optical fiber using topology optimization,” J. Opt. Soc. Am. B 25, 88–97 (2008).
[CrossRef]

O. Sigmund and K. Hougaard, “Geometric properties of optimal photonic crystals,” Phys. Rev. Letters 100(153904) (2008).
[CrossRef]

T. Baba, “Slow light in photonic crystals,” Nat. Photonics 2, 465–473 (2008).
[CrossRef]

T. White, L. Botten, C. de Sterke, K. Dossou, and R. McPhedran, “Efficient slow-light coupling in a photonic crystal waveguide without transition region,” Opt. Letters 33, 2644–2646 (2008).
[CrossRef]

S. Anderson, A. Shroff, and P. Fauchet, “Slow light with photonic crystals for on-chip optical interconnects,” Adv. Opt. Technology 2008(293531) (2008).

A. Oskooi, L. Zhang, Y. Avniel, and S. Johnson, “The failure of perfectly matched layers, and towards their redemption by adiabatic absorbers,” Opt. Express 16(15), 11376–11392 (2008).
[CrossRef]

2007 (6)

W. Frei, D. Tortorelli, and H. Johnson, “Geometry projection method for optimizing photonic nanostructures,” Opt. Letters 32, 77–79 (2007).
[CrossRef]

L. He, C.-Y. Kao, and S. Osher, “Incorporating topological derivatives into shape derivatives based level set methods,” J. Comp. Physics 225, 891–909 (2007).
[CrossRef]

J. Hugonin, P. Lalanne, T. White, and T. Krauss, “Coupling into slow-mode photonic crystal waveguides,” Opt. Letters 32, 2638–2640 (2007).
[CrossRef]

P. Pottier, M. Gnan, and R. D. L. Rue, “Efficient coupling into slow-light photonic crystal channel guides using photonic crystal tapers,” Opt. Express 15, 6569–6575 (2007).
[CrossRef] [PubMed]

C. de Sterke, J. Walker, K. Dossou, and L. Botten, “Efficient slow light coupling into photonic crystals,” Opt. Express 15, 10984–10990 (2007).
[CrossRef]

P. Velha, J. Hugonin, and P. Lalanne, “Compact and efficient injection of light into band-edge slow-modes,” Opt. Express 15, 6102–6112 (2007).
[CrossRef] [PubMed]

2006 (3)

K. Dossou, L. Botten, C. de Sterke, R. McPhedran, A. Asatryan, S. Chen, and J. Brnovic, “Efficient couplers for photonic crystal waveguides,” Opt. Commun. 265, 207–219 (2006).
[CrossRef]

Y. Vlasov and S. McNab, “Coupling into the slow light mode in slab-type photonic crystal waveguides,” Opt. Letters 31, 50–52 (2006).
[CrossRef]

N. Ikeda, Y. Sugimoto, Y. Watanabe, N. Ozaki, A. Mizutani, Y. Takata, J. Jensen, O. Sigmund, P. Borel, M. Kristensen, and K. Asakawa, “Topology optimised photonic crystal waveguide intersections with high-transmittance and low crosstalk,” Elec. Letters 42, 1031–1033 (2006).
[CrossRef]

2005 (6)

W. Frei, D. Tortorelli, and H. Johnson, “Topology optimization of a photonic crystal waveguide termination to maximize directional emission,” Appl. Phys. Letters 86(111114) (2005).
[CrossRef]

A. Håkansson, J. Sánchez-Dehesa, and L. Sanchis, “Inverse design of photonic crystal devices,” IEEE J. Selected Areas in Communications 23, 1365–1371 (2005).
[CrossRef]

C. Kao, S. Osher, and E. Yablonovitch, “Maximizing band gaps in two-dimensional photonic crystals,” Appl. Phys. B 81, 235–244 (2005).
[CrossRef]

E. Khoo, A. Liu, and J. Wu, “Nonuniform photonic crystal taper for high-efficiency mode coupling,” Opt. Express 13, 7748–7759 (2005).
[CrossRef] [PubMed]

P. Sanchis, J. Marti, W. Bogaerts, P. Dumon, D. V. Thourhout, and R. Baets, “Experimental results on adiabatic coupling into SOI photonic crystal coupled-cavity waveguides,” IEEE Photon. Tech. Letters 17, 1199–1201 (2005).
[CrossRef]

M. Povinelli, S. Johnson, and J. Joannopoulos, “Slow-light, band-edge waveguides for tunable time delays,” Opt. Express 13(8), 7145–7159 (2005).
[CrossRef] [PubMed]

2004 (9)

M. L. Povinelli, M. Ibanescu, S. G. Johnson, and J. D. Joannopoulos, “Slow-light enhancement of radiation pressure in an omnidirectional-reflector waveguide,” Appl. Phys. Letters 85, 1466–1468 (2004).
[CrossRef]

M. L. Povinelli, S. G. Johnson, E. Lidorikis, J. D. Joannopoulos, and M. Soljačić, “Effect of a photonic band gap on scattering from waveguide disorder,” Appl. Phys. Letters 84, 3639–3641 (2004).
[CrossRef]

S. Assefa, P. T. Rakich, P. Bienstman, S. G. Johnson, G. S. Petrich, J. D. Joannopoulos, L. A. Kolodziejski, E. P. Ippen, and H. I. Smith, “Guiding 1.5μm light in photonic crystals based on dielectric rods,” Appl. Phys. Letters 85, 6110–6112 (2004).
[CrossRef]

T. Segawa, S. Matsuo, Y. Ohiso, T. Ishii, and H. Suzuki, “Apodised sampled grating using InGaAsP/InP deep-ridge waveguide with vertical-groove surface grating,” Elec. Letters 40, 804–805 (2004).
[CrossRef]

E. Miyai and S. Noda, “Structural dependence of coupling between a two-dimensional photonic crystal waveguide and a wire waveguide,” J. Opt. Soc. Am. B 21, 67–72 (2004).
[CrossRef]

P. Sanchis, P. Bienstman, B. Luyssaert, R. Baets, and J. Marti, “Analysis of butt coupling in photonic crystals,” IEEE J. Quant. Elec. 40, 541–550 (2004).
[CrossRef]

L. Frandsen, A. Harpoth, P. Borel, M. Kristensen, and J. Jensen, “Broadband photonic crystal waveguide 60° bend obtained utilizing topology optimization,” Opt. Express 12, 5916–5921 (2004).
[CrossRef] [PubMed]

P. Borel, A. Harpoth, 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]

D. Dobson and F. Santosa, “Optimal localization of eigenfunctions in an inhomogeneous medium,” SIAM J. Appl. Math 64, 762–774 (2004).
[CrossRef]

2003 (4)

P. Bienstman, S. Assefa, S. Johnson, J. Joannopoulos, G. Petrich, and L. Kolodziejski, “Taper structures for coupling into photonic crystal slab waveguides,” J. Opt. Soc. Am. B 20, 1817–1821 (2003).
[CrossRef]

N. Moll and G.-L. Bona, “Comparison of three-dimensional photonic crystal slab waveguides with two-dimensional photonic crystal waveguides: efficient butt coupling into these photonic crystal waveguides,” J. Appl. Physics 93, 4986–4991 (2003).
[CrossRef]

P. Sanchis, J. Garcia, A. Martinez, F. Cuesta, A. Griol, and J. Marti, “Analysis of adiabatic coupling between photonic crystal single-line-defect and coupled-resonator optical waveguides,” Opt. Letters 28, 1903–1905 (2003).
[CrossRef]

S. G. Johnson, M. L. Povinelli, P. Bienstman, M. Skorobogatiy, M. Soljačić, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos, “Coupling, scattering and perturbation theory: semi-analytical analyses of photonic-crystal waveguides,” in Proc. 2003 5th Intl. Conf. on Transparent Optical Networks and 2nd European Symp. on Photonic Crystals, vol. 1, pp. 103–109 (2003).
[CrossRef]

2002 (5)

M. Soljačić, S. G. Johnson, S. Fan, M. Ibanescu, E. Ippen, and J. D. Joannopoulos, “Photonic-crystal slow-light enhancement of non-linear phase sensitivity,” J. Opt. Soc. Am. B 19, 2052–2059 (2002).
[CrossRef]

J. Hastings, M. Lim, J. Goodberlet, and H. Smith, “Optical waveguides with apodized sidewall gratings via spatial-phase- locked electron-beam lithography,” J. Vac. Sci. Tech. B 20, 2753–2757 (2002).
[CrossRef]

S. Johnson, P. Bienstman, M. Skorobogatiy, M. Ibanescu, E. Lidorikis, and J. Joannopoulos, “Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals,” Phys. Rev. E 66(066608) (2002).
[CrossRef]

P. Sanchis, J. Marti, J. Blasco, A. Martinez, and A. Garcia, “Mode matching technique for highly efficient coupling between dielectric waveguides and planar photonic crystal circuits,” Opt. Express 10, 1391–1397 (2002).
[PubMed]

A. Talneau, P. Lalanne, M. Agio, and C. Soukoulis, “Low-reflection photonic-crystal taper for efficient coupling between guide sections of arbitrary widths,” Opt. Letters 27, 1522–1524 (2002).
[CrossRef]

2001 (5)

T. Happ, M. Kamp, and A. Forchel, “Photonic crystal tapers for ultracompact mode conversion,” Opt. Letters 26, 1102–1104 (2001).
[CrossRef]

A. Mekis and J. Joannopoulos, “Tapered couplers for efficient interfacing between dielectric and photonic crystal waveguides,” J. Lightwave Tech. 19(6), 861–865 (2001).
[CrossRef]

M. Palamaru and P. Lalanne, “Photonic crystal waveguides: out of plane losses and adiabatic modal conversion,” Appl. Phys. Letters 78, 1466–1468 (2001).
[CrossRef]

P. Bienstman and R. Baets, “Optical modelling of photonic crystals and VCSELs using eigenmode expansion and perfectly matched layers,” Opt. and Quant. Electronics 33, 327–341 (2001).
[CrossRef]

S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8(3), 173–190 (2001).
[CrossRef] [PubMed]

2000 (2)

Y. Xu, R. Lee, and A. Yariv, “Propagation and second-harmonic generation of electromagnetic waves in a coupled-resonator optical waveguide,” JOSA-B 17(387–400) (2000).
[CrossRef]

S. Cox and D. Dobson, “Band structure optimization of two-dimensional photonic crystals in H-polarization,” J. Comp. Physics 158, 214–224 (2000).
[CrossRef]

1999 (2)

S. Cox and D. Dobson, “Maximizing band gaps in two-dimensional photonic crystals,” SIAM J. Appl. Math 59, 2108–2120 (1999).
[CrossRef]

J. E. Avron and A. Elgart, “Adiabatic theorem without a gap condition,” Commun. Math. Physics 203, 445–463 (1999).
[CrossRef]

1997 (1)

J. Foresi, P. Villeneuve, J. Ferrera, E. Thoen, G. Steinmeyer, S. Fan, J. Joannopoulos, L. Kimerling, H. Smith, and E. P. Ippen, “Photonic-bandgap microcavities in optical waveguides,” Nature 390, 143–145 (1997).
[CrossRef]

1995 (1)

S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Theoretical investigation of fabrication-related disorder on the properties of photonic crystals,” J. Appl. Physics 78, 1415–1418 (1995).
[CrossRef]

1973 (1)

K. O. Mead and L. M. Delves, “On the convergence rate of generalized Fourier expansions,” IMA J. Appl. Math. 12(3), 247–259 (1973).
[CrossRef]

Agio, M.

A. Talneau, P. Lalanne, M. Agio, and C. Soukoulis, “Low-reflection photonic-crystal taper for efficient coupling between guide sections of arbitrary widths,” Opt. Letters 27, 1522–1524 (2002).
[CrossRef]

Anderson, S.

S. Anderson, A. Shroff, and P. Fauchet, “Slow light with photonic crystals for on-chip optical interconnects,” Adv. Opt. Technology 2008(293531) (2008).

Araghchini, M.

Asakawa, K.

N. Ikeda, Y. Sugimoto, Y. Watanabe, N. Ozaki, A. Mizutani, Y. Takata, J. Jensen, O. Sigmund, P. Borel, M. Kristensen, and K. Asakawa, “Topology optimised photonic crystal waveguide intersections with high-transmittance and low crosstalk,” Elec. Letters 42, 1031–1033 (2006).
[CrossRef]

Asatryan, A.

K. Dossou, L. Botten, C. de Sterke, R. McPhedran, A. Asatryan, S. Chen, and J. Brnovic, “Efficient couplers for photonic crystal waveguides,” Opt. Commun. 265, 207–219 (2006).
[CrossRef]

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S. Assefa, P. T. Rakich, P. Bienstman, S. G. Johnson, G. S. Petrich, J. D. Joannopoulos, L. A. Kolodziejski, E. P. Ippen, and H. I. Smith, “Guiding 1.5μm light in photonic crystals based on dielectric rods,” Appl. Phys. Letters 85, 6110–6112 (2004).
[CrossRef]

P. Bienstman, S. Assefa, S. Johnson, J. Joannopoulos, G. Petrich, and L. Kolodziejski, “Taper structures for coupling into photonic crystal slab waveguides,” J. Opt. Soc. Am. B 20, 1817–1821 (2003).
[CrossRef]

Avniel, Y.

Avron, J. E.

J. E. Avron and A. Elgart, “Adiabatic theorem without a gap condition,” Commun. Math. Physics 203, 445–463 (1999).
[CrossRef]

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T. Baba, “Slow light in photonic crystals,” Nat. Photonics 2, 465–473 (2008).
[CrossRef]

Baets, R.

P. Sanchis, J. Marti, W. Bogaerts, P. Dumon, D. V. Thourhout, and R. Baets, “Experimental results on adiabatic coupling into SOI photonic crystal coupled-cavity waveguides,” IEEE Photon. Tech. Letters 17, 1199–1201 (2005).
[CrossRef]

P. Sanchis, P. Bienstman, B. Luyssaert, R. Baets, and J. Marti, “Analysis of butt coupling in photonic crystals,” IEEE J. Quant. Elec. 40, 541–550 (2004).
[CrossRef]

P. Bienstman and R. Baets, “Optical modelling of photonic crystals and VCSELs using eigenmode expansion and perfectly matched layers,” Opt. and Quant. Electronics 33, 327–341 (2001).
[CrossRef]

Bental, A.

A. Bental, L. E. Ghaoui, and A. Nemirovski, Robust Optimization (Princeton University Press, 2009).

Bermel, P.

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Bienstman, P.

P. Sanchis, P. Bienstman, B. Luyssaert, R. Baets, and J. Marti, “Analysis of butt coupling in photonic crystals,” IEEE J. Quant. Elec. 40, 541–550 (2004).
[CrossRef]

S. Assefa, P. T. Rakich, P. Bienstman, S. G. Johnson, G. S. Petrich, J. D. Joannopoulos, L. A. Kolodziejski, E. P. Ippen, and H. I. Smith, “Guiding 1.5μm light in photonic crystals based on dielectric rods,” Appl. Phys. Letters 85, 6110–6112 (2004).
[CrossRef]

P. Bienstman, S. Assefa, S. Johnson, J. Joannopoulos, G. Petrich, and L. Kolodziejski, “Taper structures for coupling into photonic crystal slab waveguides,” J. Opt. Soc. Am. B 20, 1817–1821 (2003).
[CrossRef]

S. G. Johnson, M. L. Povinelli, P. Bienstman, M. Skorobogatiy, M. Soljačić, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos, “Coupling, scattering and perturbation theory: semi-analytical analyses of photonic-crystal waveguides,” in Proc. 2003 5th Intl. Conf. on Transparent Optical Networks and 2nd European Symp. on Photonic Crystals, vol. 1, pp. 103–109 (2003).
[CrossRef]

S. Johnson, P. Bienstman, M. Skorobogatiy, M. Ibanescu, E. Lidorikis, and J. Joannopoulos, “Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals,” Phys. Rev. E 66(066608) (2002).
[CrossRef]

P. Bienstman and R. Baets, “Optical modelling of photonic crystals and VCSELs using eigenmode expansion and perfectly matched layers,” Opt. and Quant. Electronics 33, 327–341 (2001).
[CrossRef]

Blasco, J.

Bogaerts, W.

P. Sanchis, J. Marti, W. Bogaerts, P. Dumon, D. V. Thourhout, and R. Baets, “Experimental results on adiabatic coupling into SOI photonic crystal coupled-cavity waveguides,” IEEE Photon. Tech. Letters 17, 1199–1201 (2005).
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N. Moll and G.-L. Bona, “Comparison of three-dimensional photonic crystal slab waveguides with two-dimensional photonic crystal waveguides: efficient butt coupling into these photonic crystal waveguides,” J. Appl. Physics 93, 4986–4991 (2003).
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Botten, L.

T. White, L. Botten, C. de Sterke, K. Dossou, and R. McPhedran, “Efficient slow-light coupling in a photonic crystal waveguide without transition region,” Opt. Letters 33, 2644–2646 (2008).
[CrossRef]

C. de Sterke, J. Walker, K. Dossou, and L. Botten, “Efficient slow light coupling into photonic crystals,” Opt. Express 15, 10984–10990 (2007).
[CrossRef]

K. Dossou, L. Botten, C. de Sterke, R. McPhedran, A. Asatryan, S. Chen, and J. Brnovic, “Efficient couplers for photonic crystal waveguides,” Opt. Commun. 265, 207–219 (2006).
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J. P. Boyd, Chebyshev And Fourier Spectral Methods, 2nd ed. (Springer, 1989).
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Boyd, S.

J. Lu, S. Boyd, and J. Vuckovic, “Inverse design of a three-dimensional nanophotonic resonator,” Opt. Express 19, 10563–10570 (2011).
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A. Mutapcic, S. Boyd, A. Farjadpour, S. Johnson, and Y. Avniel, “Robust design of slow-light tapers in periodic waveguides,” Engineering Optimization 41, 365–384 (2009).
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S. Boyd and L. Vandenberghe, Convex Optimization (Cambridge University Press, Cambridge, UK, 2004).

Brnovic, J.

K. Dossou, L. Botten, C. de Sterke, R. McPhedran, A. Asatryan, S. Chen, and J. Brnovic, “Efficient couplers for photonic crystal waveguides,” Opt. Commun. 265, 207–219 (2006).
[CrossRef]

Celanovic, I.

Chan, W.

Chen, S.

K. Dossou, L. Botten, C. de Sterke, R. McPhedran, A. Asatryan, S. Chen, and J. Brnovic, “Efficient couplers for photonic crystal waveguides,” Opt. Commun. 265, 207–219 (2006).
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A. Conn, N. Gould, and P. Toint, Trust Region Methods (SIAM, Philadelphia, PA, 2000).
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Cox, S.

S. Cox and D. Dobson, “Band structure optimization of two-dimensional photonic crystals in H-polarization,” J. Comp. Physics 158, 214–224 (2000).
[CrossRef]

S. Cox and D. Dobson, “Maximizing band gaps in two-dimensional photonic crystals,” SIAM J. Appl. Math 59, 2108–2120 (1999).
[CrossRef]

Cuesta, F.

P. Sanchis, J. Garcia, A. Martinez, F. Cuesta, A. Griol, and J. Marti, “Analysis of adiabatic coupling between photonic crystal single-line-defect and coupled-resonator optical waveguides,” Opt. Letters 28, 1903–1905 (2003).
[CrossRef]

de Sterke, C.

T. White, L. Botten, C. de Sterke, K. Dossou, and R. McPhedran, “Efficient slow-light coupling in a photonic crystal waveguide without transition region,” Opt. Letters 33, 2644–2646 (2008).
[CrossRef]

C. de Sterke, J. Walker, K. Dossou, and L. Botten, “Efficient slow light coupling into photonic crystals,” Opt. Express 15, 10984–10990 (2007).
[CrossRef]

K. Dossou, L. Botten, C. de Sterke, R. McPhedran, A. Asatryan, S. Chen, and J. Brnovic, “Efficient couplers for photonic crystal waveguides,” Opt. Commun. 265, 207–219 (2006).
[CrossRef]

Delves, L. M.

K. O. Mead and L. M. Delves, “On the convergence rate of generalized Fourier expansions,” IMA J. Appl. Math. 12(3), 247–259 (1973).
[CrossRef]

Dobson, D.

D. Dobson and L. Simeonova, “Optimization of periodic composite structures for sub-wavelength focusing,” Appl. Math. Optim. 60, 133–150 (2009).
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D. Dobson and F. Santosa, “Optimal localization of eigenfunctions in an inhomogeneous medium,” SIAM J. Appl. Math 64, 762–774 (2004).
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S. Cox and D. Dobson, “Band structure optimization of two-dimensional photonic crystals in H-polarization,” J. Comp. Physics 158, 214–224 (2000).
[CrossRef]

S. Cox and D. Dobson, “Maximizing band gaps in two-dimensional photonic crystals,” SIAM J. Appl. Math 59, 2108–2120 (1999).
[CrossRef]

Dossou, K.

T. White, L. Botten, C. de Sterke, K. Dossou, and R. McPhedran, “Efficient slow-light coupling in a photonic crystal waveguide without transition region,” Opt. Letters 33, 2644–2646 (2008).
[CrossRef]

C. de Sterke, J. Walker, K. Dossou, and L. Botten, “Efficient slow light coupling into photonic crystals,” Opt. Express 15, 10984–10990 (2007).
[CrossRef]

K. Dossou, L. Botten, C. de Sterke, R. McPhedran, A. Asatryan, S. Chen, and J. Brnovic, “Efficient couplers for photonic crystal waveguides,” Opt. Commun. 265, 207–219 (2006).
[CrossRef]

Dumon, P.

P. Sanchis, J. Marti, W. Bogaerts, P. Dumon, D. V. Thourhout, and R. Baets, “Experimental results on adiabatic coupling into SOI photonic crystal coupled-cavity waveguides,” IEEE Photon. Tech. Letters 17, 1199–1201 (2005).
[CrossRef]

Elesin, Y.

Y. Elesin, B. Lazarov, J. Jensen, and O. Sigmund, “Design of robust and efficient photonic switches using topology optimization,” Photon. and Nanostruc. 10, 153–165 (2012).
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Elgart, A.

J. E. Avron and A. Elgart, “Adiabatic theorem without a gap condition,” Commun. Math. Physics 203, 445–463 (1999).
[CrossRef]

Fan, S.

M. Soljačić, S. G. Johnson, S. Fan, M. Ibanescu, E. Ippen, and J. D. Joannopoulos, “Photonic-crystal slow-light enhancement of non-linear phase sensitivity,” J. Opt. Soc. Am. B 19, 2052–2059 (2002).
[CrossRef]

J. Foresi, P. Villeneuve, J. Ferrera, E. Thoen, G. Steinmeyer, S. Fan, J. Joannopoulos, L. Kimerling, H. Smith, and E. P. Ippen, “Photonic-bandgap microcavities in optical waveguides,” Nature 390, 143–145 (1997).
[CrossRef]

S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Theoretical investigation of fabrication-related disorder on the properties of photonic crystals,” J. Appl. Physics 78, 1415–1418 (1995).
[CrossRef]

Farjadpour, A.

A. Mutapcic, S. Boyd, A. Farjadpour, S. Johnson, and Y. Avniel, “Robust design of slow-light tapers in periodic waveguides,” Engineering Optimization 41, 365–384 (2009).
[CrossRef]

Fauchet, P.

S. Anderson, A. Shroff, and P. Fauchet, “Slow light with photonic crystals for on-chip optical interconnects,” Adv. Opt. Technology 2008(293531) (2008).

Favuzzi, P.

A. Oskooi, P. Favuzzi, Y. Kawakami, and S. Noda, “Tailoring repulsive optical forces in nanophotonic waveguides,” Opt. Letters 36, 4638–4640 (2011).
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Ferrera, J.

J. Foresi, P. Villeneuve, J. Ferrera, E. Thoen, G. Steinmeyer, S. Fan, J. Joannopoulos, L. Kimerling, H. Smith, and E. P. Ippen, “Photonic-bandgap microcavities in optical waveguides,” Nature 390, 143–145 (1997).
[CrossRef]

Forchel, A.

T. Happ, M. Kamp, and A. Forchel, “Photonic crystal tapers for ultracompact mode conversion,” Opt. Letters 26, 1102–1104 (2001).
[CrossRef]

Foresi, J.

J. Foresi, P. Villeneuve, J. Ferrera, E. Thoen, G. Steinmeyer, S. Fan, J. Joannopoulos, L. Kimerling, H. Smith, and E. P. Ippen, “Photonic-bandgap microcavities in optical waveguides,” Nature 390, 143–145 (1997).
[CrossRef]

Frandsen, L.

Frei, W.

W. Frei, D. Tortorelli, and H. Johnson, “Geometry projection method for optimizing photonic nanostructures,” Opt. Letters 32, 77–79 (2007).
[CrossRef]

W. Frei, D. Tortorelli, and H. Johnson, “Topology optimization of a photonic crystal waveguide termination to maximize directional emission,” Appl. Phys. Letters 86(111114) (2005).
[CrossRef]

Garcia, A.

Garcia, J.

P. Sanchis, J. Garcia, A. Martinez, F. Cuesta, A. Griol, and J. Marti, “Analysis of adiabatic coupling between photonic crystal single-line-defect and coupled-resonator optical waveguides,” Opt. Letters 28, 1903–1905 (2003).
[CrossRef]

Ghaoui, L. E.

A. Bental, L. E. Ghaoui, and A. Nemirovski, Robust Optimization (Princeton University Press, 2009).

Ghebrebrhan, M.

Gnan, M.

Goodberlet, J.

J. Hastings, M. Lim, J. Goodberlet, and H. Smith, “Optical waveguides with apodized sidewall gratings via spatial-phase- locked electron-beam lithography,” J. Vac. Sci. Tech. B 20, 2753–2757 (2002).
[CrossRef]

Gould, N.

A. Conn, N. Gould, and P. Toint, Trust Region Methods (SIAM, Philadelphia, PA, 2000).
[CrossRef]

Griol, A.

P. Sanchis, J. Garcia, A. Martinez, F. Cuesta, A. Griol, and J. Marti, “Analysis of adiabatic coupling between photonic crystal single-line-defect and coupled-resonator optical waveguides,” Opt. Letters 28, 1903–1905 (2003).
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A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech, Norwood, MA, 2005).

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A. Håkansson, J. Sánchez-Dehesa, and L. Sanchis, “Inverse design of photonic crystal devices,” IEEE J. Selected Areas in Communications 23, 1365–1371 (2005).
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Hamam, R.

Happ, T.

T. Happ, M. Kamp, and A. Forchel, “Photonic crystal tapers for ultracompact mode conversion,” Opt. Letters 26, 1102–1104 (2001).
[CrossRef]

Harpoth, A.

Hastings, J.

J. Hastings, M. Lim, J. Goodberlet, and H. Smith, “Optical waveguides with apodized sidewall gratings via spatial-phase- locked electron-beam lithography,” J. Vac. Sci. Tech. B 20, 2753–2757 (2002).
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L. He, C.-Y. Kao, and S. Osher, “Incorporating topological derivatives into shape derivatives based level set methods,” J. Comp. Physics 225, 891–909 (2007).
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O. Sigmund and K. Hougaard, “Geometric properties of optimal photonic crystals,” Phys. Rev. Letters 100(153904) (2008).
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J. Hugonin, P. Lalanne, T. White, and T. Krauss, “Coupling into slow-mode photonic crystal waveguides,” Opt. Letters 32, 2638–2640 (2007).
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P. Velha, J. Hugonin, and P. Lalanne, “Compact and efficient injection of light into band-edge slow-modes,” Opt. Express 15, 6102–6112 (2007).
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Ibanescu, M.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comp. Phys. Communications 181, 687–702 (2010).
[CrossRef]

M. L. Povinelli, M. Ibanescu, S. G. Johnson, and J. D. Joannopoulos, “Slow-light enhancement of radiation pressure in an omnidirectional-reflector waveguide,” Appl. Phys. Letters 85, 1466–1468 (2004).
[CrossRef]

S. G. Johnson, M. L. Povinelli, P. Bienstman, M. Skorobogatiy, M. Soljačić, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos, “Coupling, scattering and perturbation theory: semi-analytical analyses of photonic-crystal waveguides,” in Proc. 2003 5th Intl. Conf. on Transparent Optical Networks and 2nd European Symp. on Photonic Crystals, vol. 1, pp. 103–109 (2003).
[CrossRef]

M. Soljačić, S. G. Johnson, S. Fan, M. Ibanescu, E. Ippen, and J. D. Joannopoulos, “Photonic-crystal slow-light enhancement of non-linear phase sensitivity,” J. Opt. Soc. Am. B 19, 2052–2059 (2002).
[CrossRef]

S. Johnson, P. Bienstman, M. Skorobogatiy, M. Ibanescu, E. Lidorikis, and J. Joannopoulos, “Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals,” Phys. Rev. E 66(066608) (2002).
[CrossRef]

Ikeda, N.

N. Ikeda, Y. Sugimoto, Y. Watanabe, N. Ozaki, A. Mizutani, Y. Takata, J. Jensen, O. Sigmund, P. Borel, M. Kristensen, and K. Asakawa, “Topology optimised photonic crystal waveguide intersections with high-transmittance and low crosstalk,” Elec. Letters 42, 1031–1033 (2006).
[CrossRef]

Ippen, E.

Ippen, E. P.

S. Assefa, P. T. Rakich, P. Bienstman, S. G. Johnson, G. S. Petrich, J. D. Joannopoulos, L. A. Kolodziejski, E. P. Ippen, and H. I. Smith, “Guiding 1.5μm light in photonic crystals based on dielectric rods,” Appl. Phys. Letters 85, 6110–6112 (2004).
[CrossRef]

J. Foresi, P. Villeneuve, J. Ferrera, E. Thoen, G. Steinmeyer, S. Fan, J. Joannopoulos, L. Kimerling, H. Smith, and E. P. Ippen, “Photonic-bandgap microcavities in optical waveguides,” Nature 390, 143–145 (1997).
[CrossRef]

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T. Segawa, S. Matsuo, Y. Ohiso, T. Ishii, and H. Suzuki, “Apodised sampled grating using InGaAsP/InP deep-ridge waveguide with vertical-groove surface grating,” Elec. Letters 40, 804–805 (2004).
[CrossRef]

Jensen, J.

Jensen, K. F.

Joannopoulos, J.

M. Povinelli, S. Johnson, and J. Joannopoulos, “Slow-light, band-edge waveguides for tunable time delays,” Opt. Express 13(8), 7145–7159 (2005).
[CrossRef] [PubMed]

P. Bienstman, S. Assefa, S. Johnson, J. Joannopoulos, G. Petrich, and L. Kolodziejski, “Taper structures for coupling into photonic crystal slab waveguides,” J. Opt. Soc. Am. B 20, 1817–1821 (2003).
[CrossRef]

S. Johnson, P. Bienstman, M. Skorobogatiy, M. Ibanescu, E. Lidorikis, and J. Joannopoulos, “Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals,” Phys. Rev. E 66(066608) (2002).
[CrossRef]

A. Mekis and J. Joannopoulos, “Tapered couplers for efficient interfacing between dielectric and photonic crystal waveguides,” J. Lightwave Tech. 19(6), 861–865 (2001).
[CrossRef]

J. Foresi, P. Villeneuve, J. Ferrera, E. Thoen, G. Steinmeyer, S. Fan, J. Joannopoulos, L. Kimerling, H. Smith, and E. P. Ippen, “Photonic-bandgap microcavities in optical waveguides,” Nature 390, 143–145 (1997).
[CrossRef]

Joannopoulos, J. D.

A. Kurs, J. D. Joannopoulos, M. Soljačić, and S. G. Johnson, “Abrupt coupling between strongly dissimilar waveguides with 100% transmission,” Opt. Express 19, 13714–13721 (2011).
[CrossRef]

P. Bermel, M. Ghebrebrhan, W. Chan, Y. X. Yeng, M. Araghchini, R. Hamam, C. H. Marton, K. F. Jensen, M. Soljačić, J. D. Joannopoulos, S. G. Johnson, and I. Celanovic, “Design and global optimization of high-efficiency thermophotovoltaic systems,” Opt. Express 18, A314–A334 (2010).
[CrossRef] [PubMed]

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comp. Phys. Communications 181, 687–702 (2010).
[CrossRef]

M. Ghebrebrhan, P. Bermel, Y. Avniel, J. D. Joannopoulos, and S. G. Johnson, “Global optimization of silicon photovoltaic cell front coatings,” Opt. Express 17, 7505–7518 (2009).
[CrossRef] [PubMed]

S. Assefa, P. T. Rakich, P. Bienstman, S. G. Johnson, G. S. Petrich, J. D. Joannopoulos, L. A. Kolodziejski, E. P. Ippen, and H. I. Smith, “Guiding 1.5μm light in photonic crystals based on dielectric rods,” Appl. Phys. Letters 85, 6110–6112 (2004).
[CrossRef]

M. L. Povinelli, S. G. Johnson, E. Lidorikis, J. D. Joannopoulos, and M. Soljačić, “Effect of a photonic band gap on scattering from waveguide disorder,” Appl. Phys. Letters 84, 3639–3641 (2004).
[CrossRef]

M. L. Povinelli, M. Ibanescu, S. G. Johnson, and J. D. Joannopoulos, “Slow-light enhancement of radiation pressure in an omnidirectional-reflector waveguide,” Appl. Phys. Letters 85, 1466–1468 (2004).
[CrossRef]

S. G. Johnson, M. L. Povinelli, P. Bienstman, M. Skorobogatiy, M. Soljačić, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos, “Coupling, scattering and perturbation theory: semi-analytical analyses of photonic-crystal waveguides,” in Proc. 2003 5th Intl. Conf. on Transparent Optical Networks and 2nd European Symp. on Photonic Crystals, vol. 1, pp. 103–109 (2003).
[CrossRef]

M. Soljačić, S. G. Johnson, S. Fan, M. Ibanescu, E. Ippen, and J. D. Joannopoulos, “Photonic-crystal slow-light enhancement of non-linear phase sensitivity,” J. Opt. Soc. Am. B 19, 2052–2059 (2002).
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A. Mutapcic, S. Boyd, A. Farjadpour, S. Johnson, and Y. Avniel, “Robust design of slow-light tapers in periodic waveguides,” Engineering Optimization 41, 365–384 (2009).
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S. Johnson, P. Bienstman, M. Skorobogatiy, M. Ibanescu, E. Lidorikis, and J. Joannopoulos, “Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals,” Phys. Rev. E 66(066608) (2002).
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A. Kurs, J. D. Joannopoulos, M. Soljačić, and S. G. Johnson, “Abrupt coupling between strongly dissimilar waveguides with 100% transmission,” Opt. Express 19, 13714–13721 (2011).
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P. Bermel, M. Ghebrebrhan, W. Chan, Y. X. Yeng, M. Araghchini, R. Hamam, C. H. Marton, K. F. Jensen, M. Soljačić, J. D. Joannopoulos, S. G. Johnson, and I. Celanovic, “Design and global optimization of high-efficiency thermophotovoltaic systems,” Opt. Express 18, A314–A334 (2010).
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A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comp. Phys. Communications 181, 687–702 (2010).
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N. Ikeda, Y. Sugimoto, Y. Watanabe, N. Ozaki, A. Mizutani, Y. Takata, J. Jensen, O. Sigmund, P. Borel, M. Kristensen, and K. Asakawa, “Topology optimised photonic crystal waveguide intersections with high-transmittance and low crosstalk,” Elec. Letters 42, 1031–1033 (2006).
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Petrich, G. S.

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M. L. Povinelli, S. G. Johnson, E. Lidorikis, J. D. Joannopoulos, and M. Soljačić, “Effect of a photonic band gap on scattering from waveguide disorder,” Appl. Phys. Letters 84, 3639–3641 (2004).
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M. L. Povinelli, M. Ibanescu, S. G. Johnson, and J. D. Joannopoulos, “Slow-light enhancement of radiation pressure in an omnidirectional-reflector waveguide,” Appl. Phys. Letters 85, 1466–1468 (2004).
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[CrossRef]

S. Johnson, P. Bienstman, M. Skorobogatiy, M. Ibanescu, E. Lidorikis, and J. Joannopoulos, “Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals,” Phys. Rev. E 66(066608) (2002).
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J. Hastings, M. Lim, J. Goodberlet, and H. Smith, “Optical waveguides with apodized sidewall gratings via spatial-phase- locked electron-beam lithography,” J. Vac. Sci. Tech. B 20, 2753–2757 (2002).
[CrossRef]

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S. Assefa, P. T. Rakich, P. Bienstman, S. G. Johnson, G. S. Petrich, J. D. Joannopoulos, L. A. Kolodziejski, E. P. Ippen, and H. I. Smith, “Guiding 1.5μm light in photonic crystals based on dielectric rods,” Appl. Phys. Letters 85, 6110–6112 (2004).
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A. Kurs, J. D. Joannopoulos, M. Soljačić, and S. G. Johnson, “Abrupt coupling between strongly dissimilar waveguides with 100% transmission,” Opt. Express 19, 13714–13721 (2011).
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M. L. Povinelli, S. G. Johnson, E. Lidorikis, J. D. Joannopoulos, and M. Soljačić, “Effect of a photonic band gap on scattering from waveguide disorder,” Appl. Phys. Letters 84, 3639–3641 (2004).
[CrossRef]

S. G. Johnson, M. L. Povinelli, P. Bienstman, M. Skorobogatiy, M. Soljačić, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos, “Coupling, scattering and perturbation theory: semi-analytical analyses of photonic-crystal waveguides,” in Proc. 2003 5th Intl. Conf. on Transparent Optical Networks and 2nd European Symp. on Photonic Crystals, vol. 1, pp. 103–109 (2003).
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T. Segawa, S. Matsuo, Y. Ohiso, T. Ishii, and H. Suzuki, “Apodised sampled grating using InGaAsP/InP deep-ridge waveguide with vertical-groove surface grating,” Elec. Letters 40, 804–805 (2004).
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A. Kurs, J. D. Joannopoulos, M. Soljačić, and S. G. Johnson, “Abrupt coupling between strongly dissimilar waveguides with 100% transmission,” Opt. Express 19, 13714–13721 (2011).
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Opt. Letters (8)

T. White, L. Botten, C. de Sterke, K. Dossou, and R. McPhedran, “Efficient slow-light coupling in a photonic crystal waveguide without transition region,” Opt. Letters 33, 2644–2646 (2008).
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A. Oskooi, P. Favuzzi, Y. Kawakami, and S. Noda, “Tailoring repulsive optical forces in nanophotonic waveguides,” Opt. Letters 36, 4638–4640 (2011).
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W. Frei, D. Tortorelli, and H. Johnson, “Geometry projection method for optimizing photonic nanostructures,” Opt. Letters 32, 77–79 (2007).
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A. Talneau, P. Lalanne, M. Agio, and C. Soukoulis, “Low-reflection photonic-crystal taper for efficient coupling between guide sections of arbitrary widths,” Opt. Letters 27, 1522–1524 (2002).
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J. Hugonin, P. Lalanne, T. White, and T. Krauss, “Coupling into slow-mode photonic crystal waveguides,” Opt. Letters 32, 2638–2640 (2007).
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P. Sanchis, J. Garcia, A. Martinez, F. Cuesta, A. Griol, and J. Marti, “Analysis of adiabatic coupling between photonic crystal single-line-defect and coupled-resonator optical waveguides,” Opt. Letters 28, 1903–1905 (2003).
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Other (10)

G. Taguchi, R. Jugulum, and S. Taguchi, Computer-Based Robust Engineering: Essentials For DFSS (ASQ Quality Press, 2004).

A. Bental, L. E. Ghaoui, and A. Nemirovski, Robust Optimization (Princeton University Press, 2009).

J. D. Joannopoulos, S. G. Johnson, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding The Flow Of Light, 2nd ed. (Princeton Univ. Press, 2008).

P. Bienstman, “CAMFR: CAvity Modeling FRamework,” Software at http://camfr.sourceforge.net .

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech, Norwood, MA, 2005).

J. Nocedal and S. Wright, Numerical Optimization (Springer, New York, 1999).
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A. Conn, N. Gould, and P. Toint, Trust Region Methods (SIAM, Philadelphia, PA, 2000).
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Figures (6)

Fig. 1
Fig. 1

Schematic taper design problem, with a transition between two waveguides characterized by a dimensionless parameter s ∈ [0, 1]. The taper goes from a uniform dielectric waveguide (s = 0) to a periodic waveguide (s = 1), with 0 < s < 1 describing intermediate structures (e.g. s ∼ hole radius, flange width, or block spacing in the structures shown at right). A taper design (bottom) is a “shape” function s(z/L) (for z ∈ [0, L]) describing a continuous transition from s(0) = 0 to s(1) = 1. For optimization, we parameterize s by its values at uniformly spaced points and linearly interpolate in between (bottom left).

Fig. 2
Fig. 2

Comparison of coupled-mode theory (CMT) calculation of reflection coefficient from double taper (from uniform to periodic to uniform) structure, using a simple linear profile s(u) = u, with two other brute-force numerical methods, scattering matrix (CAMFR [46]) and finite-difference time-domain (FDTD [74]). (a) 2d period a 0.4a×0.4a blocks example (top left) with TM operating mode at vg = c/4 [40, 43]. CMT and CAMFR agree well. (b) 2d period-a flanged waveguide (inner width a, outer width 2a, 50% duty cycle) with TE operating mode at vg = c/34, adapted from Ref. 37 (top right). For such slow light, the scattering from a short linear taper is too large for CMT to be valid, but optimization of the taper shape will drive the reflections low enough for the CMT calculation to be suitable.

Fig. 3
Fig. 3

Optimization results for taper structure of square blocks (operating TM mode with vg = c/4) from Fig. 2(a), with block spacing as the taper variable (top: linear tapers). (a) Performance of three tapers (linear in green, nominal optimum in blue, and robust optimum in red, with the optimizations performed independently at each length L) computed using a brute-force scattering-matrix method (CAMFR [46]) (with no disorder introduced). (b) Performance of nominal taper (optimized independently at each taper length) computed using two different methods (CMT and CAMFR)—the large disparity is due to the nominal optimum relying on a delicate interference cancellation that is destroyed even by the slight numerical differences between the two techniques. (c) Performance of linear, nominal, and robust tapers designed at L = 50a and then rescaled to other taper lengths: the nominal optimum relies on a delicate cancellation that only works close to L = 50a. Inset compares CMT and CAMFR for the L = 20a design rescaled to different lengths, and illustrates that the two approaches agree everywhere except at L = 20a, where there is a delicate interference cancellation that is inherently irreproducible (non-robust), and at small L where the reflection exceeds 10% (causing CMT to break down). (d) Nominal and robust taper profiles s(u) designed at L = 50a: the nominal design is only slightly different from a linear taper, with the slight changes (inset) sufficient to create a delicate reflection cancellation.

Fig. 4
Fig. 4

Optimization results for taper from uniform to flanged waveguide (operating TE mode with vg = c/6) from Fig. 2(b), with flange (outer) width as the taper variable (top: linear tapers). (a) Performance of three tapers (linear in green, nominal optimum in blue, and robust optimum in red, with the optimizations performed independently at each length L) computed using a brute-force scattering-matrix method (CAMFR [46]), where surface roughness (±10−3a every 0.01a) was introduced, averaged over 50 structures. The robust optimum maintains good (∼ 30 dB) performance even in the presence of disorder, whereas the nominal optimum is greatly degraded. (b) Nominal and robust taper profiles designed at L = 13a (vertical axis exaggerated for clarity): the nominal design includes fine cusps (circled) absent from the robust design, whose apparent function is to introduce reflection cancellations (which are spoiled by disorder).

Fig. 5
Fig. 5

Optimization results for taper from uniform to flanged slow-light waveguide (operating TE mode with vg = c/34) with flange (outer) width as the taper variable (top: linear tapers). (a) Performance of three tapers (linear in green, nominal in blue, and robust in red, with the optimizations performed independently at each length) computed using a brute-force scattering-matrix method (CAMFR [46]), where surface roughness (±10−3a every 0.01a) was introduced, averaged over 50 structures. (b) Same as (a) but with 1-transmission: the inclusion of radiative scattering loss eliminates the optimization-created dip (cancellation) in the reflection losses from (a). As in Fig. 4, the robust optimum greatly outperforms the nominal optimum in the presence of disorder. (c) Even without disorder, the nominal optimum performs poorly in CAMFR compared to CMT: the slight differences in simulation accuracy are enough to spoil the nominal optimum. (d) Same as (a) but with robust optimization using the original (1024-point) piecewise-linear (“tent function”) parameterization (red squares) compared to robust optimization using a sine-series parameterization (cyan) with 4, 16, and 64 terms: the latter is converging towards similar performance as the former, and towards similar structures as seen in Fig. 6, so the robust optimum is not merely a local minimum obtained as a byproduct of the discretization choice.

Fig. 6
Fig. 6

Linear, nominal, and robust-optimum taper profiles from the slow-light (c/34) optimization of Fig. 5, optimized at a taper length L = 13a. Vertical axes are exaggerated for scale; topmost (black) figure shows the robust taper design to scale. Even the robust design shows complicated sub-flange oscillations, but similar oscillations are reproduced independent of the parameterization of the taper design during optimization. Top middle (red): robust design using 1024-point linear interpolation (tent functions). Cyan (bottom/right): robust optimization using sine series with successive increase of the number of series terms. 64-term sine series structure (top right) is similar to tent-function (top middle) structure, and is superimposed on the latter as a dotted cyan curve. Nominal optimum (lower left) has even more radical oscillations, designed to create delicate reflection cancellations (which are spoiled by disorder).

Equations (3)

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R min ( L ) = min s 𝒮 max v 𝒱 R [ s , v , L ] ,
c r = 0 1 d u s ( u ) k M k [ s ( u ) ] Δ β k [ s ( u ) ] e i L 0 u Δ β k [ s ( u ) ] d u .
c r = d x s ( u ( x ) ) M k [ s ( u ( x ) ) ] Δ β k [ s ( u ( x ) ) ] 2 e i L x ,

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