Abstract

This paper employs topology optimization to systematically design free-topology loss-engineered slow-light waveguides with enlarged group index bandwidth product (GBP). The propagation losses of guided modes are evaluated by the imaginary part of eigenvalues in complex band structure calculations, where the scattering losses due to manufacturing imperfections are represented by an edge-related effective dissipation. The loss engineering of slow-light waveguides is realized by minimizing the propagation losses of design modes. Numerical examples illustrate that the propagation losses of free-topology dispersion-engineered waveguides can be significantly suppressed by loss engineering. Comparisons between fixed- and free-topology loss-engineered waveguides demonstrate that the GBP can be enhanced significantly by the free-topology loss-engineered waveguides with a small increase of the propagation losses.

© 2012 Optical Society of America

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    [CrossRef]

2012 (3)

J. Grgić, J. R. Ott, F. Wang, O. Sigmund, A.-P. Jauho, J. Mørk, and N. A. Mortensen, “Fundamental limitations to gain enhancement in periodic media and waveguides,” Phys. Rev. Lett. 108, 183903 (2012).
[CrossRef]

F. Wang, J. S. Jensen, and O. Sigmund, “High-performance slow light photonic crystal waveguides with topology optimized or circular-hole based material layouts,” Photon. Nanostruct. Fundam. Applic. 10, 378–388 (2012).
[CrossRef]

P. Colman, S. Combrié, G. Lehoucq, and A. De Rossi, “Control of dispersion in photonic crystal waveguides using group symmetry theory,” Opt. Express 20, 13108–13114 (2012).
[CrossRef]

2011 (3)

2010 (4)

M. Patterson and S. Hughes, “Theory of disorder-induced coherent scattering and light localization in slow-light photonic crystal waveguides,” J. Opt. 12, 104013 (2010).
[CrossRef]

S. A. Schulz, L. O’Faolain, D. M. Beggs, T. P. White, A. Melloni, and T. F. Krauss, “Dispersion engineered slow light in photonic crystals: a comparison,” J. Opt. 12, 104004 (2010).
[CrossRef]

M. Heuck, S. Blaaberg, and J. Mork, “Theory of passively mode-locked photonic crystal semiconductor lasers,” Opt. Express 18, 18003–18014 (2010).
[CrossRef]

L. O’Faolain, S. A. Schulz, D. M. Beggs, T. P. White, M. Spasenovic, L. Kuipers, F. Morichetti, A. Melloni, S. Mazoyer, J. P. Hugonin, P. Lalanne, and T. F. Krauss, “Loss engineered slow light waveguides,” Opt. Express 18, 27627–27638 (2010).
[CrossRef]

2009 (3)

A. Petrov, M. Krause, and M. Eich, “Backscattering and disorder limits in slow light photonic crystal waveguides,” Opt. Express 17, 8676–8684 (2009).
[CrossRef]

B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides,” Nat. Photonics 3, 206–210 (2009).
[CrossRef]

M. Patterson, S. Hughes, S. Schulz, D. M. Beggs, T. P. White, L. O’Faolain, and T. F. Krauss, “Disorder-induced incoherent scattering losses in photonic crystal waveguides: Bloch mode reshaping, multiple scattering, and breakdown of the Beer-Lambert law,” Phys. Rev. B 80, 195305 (2009).
[CrossRef]

2008 (6)

T. F. Krauss, “Why do we need slow light?” Nat. Photonics 2, 448–450 (2008).
[CrossRef]

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

J. Ma and C. Jiang, “Demonstration of ultraslow modes in asymmetric line-defect photonic crystal waveguides,” IEEE Photon. Technol. Lett. 20, 1237–1239 (2008).
[CrossRef]

J. G. Pedersen, S. S. Xiao, and N. A. Mortensen, “Limits of slow light in photonic crystals,” Phys. Rev. B 78, 153101 (2008).
[CrossRef]

D. M. Beggs, T. P. White, L. O’Faolain, and T. F. Krauss, “Ultracompact and low-power optical switch based on silicon photonic crystals,” Opt. Lett. 33, 147–149 (2008).
[CrossRef]

J. Li, T. P. White, L. O’Faolain, A. Gomez-Iglesias, and T. F. Krauss, “Systematic design of flat band slow light in photonic crystal waveguides,” Opt. Express 16, 6227–6232 (2008).
[CrossRef]

2007 (6)

D. Mori, S. Kubo, H. Sasaki, and T. Baba, “Experimental demonstration of wideband dispersion-compensated slow light by a chirped photonic crystal directional coupler,” Opt. Express 15, 5264–5270 (2007).
[CrossRef]

T. Kawasaki, D. Mori, and T. Baba, “Experimental observation of slow light in photonic crystal coupled waveguides,” Opt. Express 15, 10274–10281 (2007).
[CrossRef]

T. F. Krauss, “Slow light in photonic crystal waveguides,” J. Phys. D 40, 2666–2670 (2007).
[CrossRef]

O. Sigmund, “Morphology-based black and white filters for topology optimization,” Struct. Multidisc. Optim. 33, 401–424 (2007).
[CrossRef]

L. C. Andreani and D. Gerace, “Light-matter interaction in photonic crystal slabs,” Phys. Status Solidi B 244, 3528–3539 (2007).
[CrossRef]

R. Stainko and O. Sigmund, “Tailoring dispersion properties of photonic crystal waveguides by topology optimization,” Waves Random Complex Media 17, 477–489 (2007).
[CrossRef]

2006 (4)

2005 (3)

J. S. Jensen and O. Sigmund, “Topology optimization of photonic crystal structures: a high-bandwidth low-loss T-junction waveguide,” J. Opt. Soc. Am. B 22, 1191–1198 (2005).
[CrossRef]

S. Hughes, L. Ramunno, J. F. Young, and J. E. Sipe, “Extrinsic optical scattering loss in photonic crystal waveguides: role of fabrication disorder and photon group velocity,” Phys. Rev. Lett. 94, 033903 (2005).
[CrossRef]

E. Kuramochi, M. Notomi, S. Hughes, A. Shinya, T. Watanabe, and L. Ramunno, “Disorder-induced scattering loss of line-defect waveguides in photonic crystal slabs,” Phys. Rev. B 72, 161318 (2005).
[CrossRef]

2004 (2)

D. Mori and T. Baba, “Dispersion-controlled optical group delay device by chirped photonic crystal waveguides,” Appl. Phys. Lett. 85, 1101–1103 (2004).
[CrossRef]

K. C. Huang, E. Lidorikis, X. Y. Jiang, J. D. Joannopoulos, K. A. Nelson, P. Bienstman, and S. H. Fan, “Nature of lossy Bloch states in polaritonic photonic crystals,” Phys. Rev. B 69, 195111 (2004).
[CrossRef]

2003 (1)

2002 (1)

K. Svanberg, “A class of globally convergent optimization methods based on conservative convex separable approximations,” SIAM J. Optim. 12, 555–573 (2002).
[CrossRef]

2000 (1)

H. Benisty, D. Labilloy, C. Weisbuch, C. J. M. Smith, T. F. Krauss, D. Cassagne, A. Beraud, and C. Jouanin, “Radiation losses of waveguide-based two-dimensional photonic crystals: positive role of the substrate,” Appl. Phys. Lett. 76, 532–534 (2000).
[CrossRef]

1994 (1)

A. P. Seyranian, E. Lund, and N. Olhoff, “Multiple eigenvalues in structural optimization problems,” Struct. Optim. 8, 207–227 (1994).
[CrossRef]

Andreani, L. C.

L. C. Andreani and D. Gerace, “Light-matter interaction in photonic crystal slabs,” Phys. Status Solidi B 244, 3528–3539 (2007).
[CrossRef]

Asakawa, K.

Baba, T.

Beggs, D. M.

S. A. Schulz, L. O’Faolain, D. M. Beggs, T. P. White, A. Melloni, and T. F. Krauss, “Dispersion engineered slow light in photonic crystals: a comparison,” J. Opt. 12, 104004 (2010).
[CrossRef]

L. O’Faolain, S. A. Schulz, D. M. Beggs, T. P. White, M. Spasenovic, L. Kuipers, F. Morichetti, A. Melloni, S. Mazoyer, J. P. Hugonin, P. Lalanne, and T. F. Krauss, “Loss engineered slow light waveguides,” Opt. Express 18, 27627–27638 (2010).
[CrossRef]

M. Patterson, S. Hughes, S. Schulz, D. M. Beggs, T. P. White, L. O’Faolain, and T. F. Krauss, “Disorder-induced incoherent scattering losses in photonic crystal waveguides: Bloch mode reshaping, multiple scattering, and breakdown of the Beer-Lambert law,” Phys. Rev. B 80, 195305 (2009).
[CrossRef]

D. M. Beggs, T. P. White, L. O’Faolain, and T. F. Krauss, “Ultracompact and low-power optical switch based on silicon photonic crystals,” Opt. Lett. 33, 147–149 (2008).
[CrossRef]

Benisty, H.

Beraud, A.

H. Benisty, D. Labilloy, C. Weisbuch, C. J. M. Smith, T. F. Krauss, D. Cassagne, A. Beraud, and C. Jouanin, “Radiation losses of waveguide-based two-dimensional photonic crystals: positive role of the substrate,” Appl. Phys. Lett. 76, 532–534 (2000).
[CrossRef]

Bezerra, M.

Bienstman, P.

K. C. Huang, E. Lidorikis, X. Y. Jiang, J. D. Joannopoulos, K. A. Nelson, P. Bienstman, and S. H. Fan, “Nature of lossy Bloch states in polaritonic photonic crystals,” Phys. Rev. B 69, 195111 (2004).
[CrossRef]

Blaaberg, S.

Borel, P. I.

Cassagne, D.

H. Benisty, D. Labilloy, C. Weisbuch, C. J. M. Smith, T. F. Krauss, D. Cassagne, A. Beraud, and C. Jouanin, “Radiation losses of waveguide-based two-dimensional photonic crystals: positive role of the substrate,” Appl. Phys. Lett. 76, 532–534 (2000).
[CrossRef]

Colman, P.

Combrié, S.

Corcoran, B.

B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides,” Nat. Photonics 3, 206–210 (2009).
[CrossRef]

De Rossi, A.

Eggleton, B. J.

B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides,” Nat. Photonics 3, 206–210 (2009).
[CrossRef]

Eich, M.

Engelen, R. J. P.

Fage-Pedersen, J.

Fan, S. H.

K. C. Huang, E. Lidorikis, X. Y. Jiang, J. D. Joannopoulos, K. A. Nelson, P. Bienstman, and S. H. Fan, “Nature of lossy Bloch states in polaritonic photonic crystals,” Phys. Rev. B 69, 195111 (2004).
[CrossRef]

Ferrini, R.

Forchel, A.

Frandsen, L. H.

Gerace, D.

L. C. Andreani and D. Gerace, “Light-matter interaction in photonic crystal slabs,” Phys. Status Solidi B 244, 3528–3539 (2007).
[CrossRef]

Gomez-Iglesias, A.

Grgic, J.

J. Grgić, J. R. Ott, F. Wang, O. Sigmund, A.-P. Jauho, J. Mørk, and N. A. Mortensen, “Fundamental limitations to gain enhancement in periodic media and waveguides,” Phys. Rev. Lett. 108, 183903 (2012).
[CrossRef]

Grillet, C.

B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides,” Nat. Photonics 3, 206–210 (2009).
[CrossRef]

Heuck, M.

Houdre, R.

Huang, K. C.

K. C. Huang, E. Lidorikis, X. Y. Jiang, J. D. Joannopoulos, K. A. Nelson, P. Bienstman, and S. H. Fan, “Nature of lossy Bloch states in polaritonic photonic crystals,” Phys. Rev. B 69, 195111 (2004).
[CrossRef]

Hughes, S.

M. Patterson and S. Hughes, “Theory of disorder-induced coherent scattering and light localization in slow-light photonic crystal waveguides,” J. Opt. 12, 104013 (2010).
[CrossRef]

M. Patterson, S. Hughes, S. Schulz, D. M. Beggs, T. P. White, L. O’Faolain, and T. F. Krauss, “Disorder-induced incoherent scattering losses in photonic crystal waveguides: Bloch mode reshaping, multiple scattering, and breakdown of the Beer-Lambert law,” Phys. Rev. B 80, 195305 (2009).
[CrossRef]

S. Hughes, L. Ramunno, J. F. Young, and J. E. Sipe, “Extrinsic optical scattering loss in photonic crystal waveguides: role of fabrication disorder and photon group velocity,” Phys. Rev. Lett. 94, 033903 (2005).
[CrossRef]

E. Kuramochi, M. Notomi, S. Hughes, A. Shinya, T. Watanabe, and L. Ramunno, “Disorder-induced scattering loss of line-defect waveguides in photonic crystal slabs,” Phys. Rev. B 72, 161318 (2005).
[CrossRef]

Hugonin, J. P.

Ikeda, N.

Jauho, A.-P.

J. Grgić, J. R. Ott, F. Wang, O. Sigmund, A.-P. Jauho, J. Mørk, and N. A. Mortensen, “Fundamental limitations to gain enhancement in periodic media and waveguides,” Phys. Rev. Lett. 108, 183903 (2012).
[CrossRef]

Jensen, J. S.

Jiang, C.

J. Ma and C. Jiang, “Demonstration of ultraslow modes in asymmetric line-defect photonic crystal waveguides,” IEEE Photon. Technol. Lett. 20, 1237–1239 (2008).
[CrossRef]

Jiang, X. Y.

K. C. Huang, E. Lidorikis, X. Y. Jiang, J. D. Joannopoulos, K. A. Nelson, P. Bienstman, and S. H. Fan, “Nature of lossy Bloch states in polaritonic photonic crystals,” Phys. Rev. B 69, 195111 (2004).
[CrossRef]

Joannopoulos, J. D.

K. C. Huang, E. Lidorikis, X. Y. Jiang, J. D. Joannopoulos, K. A. Nelson, P. Bienstman, and S. H. Fan, “Nature of lossy Bloch states in polaritonic photonic crystals,” Phys. Rev. B 69, 195111 (2004).
[CrossRef]

Jouanin, C.

H. Benisty, D. Labilloy, C. Weisbuch, C. J. M. Smith, T. F. Krauss, D. Cassagne, A. Beraud, and C. Jouanin, “Radiation losses of waveguide-based two-dimensional photonic crystals: positive role of the substrate,” Appl. Phys. Lett. 76, 532–534 (2000).
[CrossRef]

Kamp, M.

Kawasaki, T.

Korterik, J. P.

Krause, M.

Krauss, T. F.

S. A. Schulz, L. O’Faolain, D. M. Beggs, T. P. White, A. Melloni, and T. F. Krauss, “Dispersion engineered slow light in photonic crystals: a comparison,” J. Opt. 12, 104004 (2010).
[CrossRef]

L. O’Faolain, S. A. Schulz, D. M. Beggs, T. P. White, M. Spasenovic, L. Kuipers, F. Morichetti, A. Melloni, S. Mazoyer, J. P. Hugonin, P. Lalanne, and T. F. Krauss, “Loss engineered slow light waveguides,” Opt. Express 18, 27627–27638 (2010).
[CrossRef]

M. Patterson, S. Hughes, S. Schulz, D. M. Beggs, T. P. White, L. O’Faolain, and T. F. Krauss, “Disorder-induced incoherent scattering losses in photonic crystal waveguides: Bloch mode reshaping, multiple scattering, and breakdown of the Beer-Lambert law,” Phys. Rev. B 80, 195305 (2009).
[CrossRef]

B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides,” Nat. Photonics 3, 206–210 (2009).
[CrossRef]

T. F. Krauss, “Why do we need slow light?” Nat. Photonics 2, 448–450 (2008).
[CrossRef]

J. Li, T. P. White, L. O’Faolain, A. Gomez-Iglesias, and T. F. Krauss, “Systematic design of flat band slow light in photonic crystal waveguides,” Opt. Express 16, 6227–6232 (2008).
[CrossRef]

D. M. Beggs, T. P. White, L. O’Faolain, and T. F. Krauss, “Ultracompact and low-power optical switch based on silicon photonic crystals,” Opt. Lett. 33, 147–149 (2008).
[CrossRef]

T. F. Krauss, “Slow light in photonic crystal waveguides,” J. Phys. D 40, 2666–2670 (2007).
[CrossRef]

H. Benisty, D. Labilloy, C. Weisbuch, C. J. M. Smith, T. F. Krauss, D. Cassagne, A. Beraud, and C. Jouanin, “Radiation losses of waveguide-based two-dimensional photonic crystals: positive role of the substrate,” Appl. Phys. Lett. 76, 532–534 (2000).
[CrossRef]

Kubo, S.

Kuipers, L.

Kuramochi, E.

E. Kuramochi, M. Notomi, S. Hughes, A. Shinya, T. Watanabe, and L. Ramunno, “Disorder-induced scattering loss of line-defect waveguides in photonic crystal slabs,” Phys. Rev. B 72, 161318 (2005).
[CrossRef]

Labilloy, D.

H. Benisty, D. Labilloy, C. Weisbuch, C. J. M. Smith, T. F. Krauss, D. Cassagne, A. Beraud, and C. Jouanin, “Radiation losses of waveguide-based two-dimensional photonic crystals: positive role of the substrate,” Appl. Phys. Lett. 76, 532–534 (2000).
[CrossRef]

Lalanne, P.

Lavrinenko, A. V.

Lazarov, B. S.

F. Wang, B. S. Lazarov, and O. Sigmund, “On projection methods, convergence and robust formulations in topology optimization,” Struct. Multidisc. Optim. 43, 767–784 (2011).
[CrossRef]

Lehoucq, G.

Leuenberger, D.

Li, J.

Lidorikis, E.

K. C. Huang, E. Lidorikis, X. Y. Jiang, J. D. Joannopoulos, K. A. Nelson, P. Bienstman, and S. H. Fan, “Nature of lossy Bloch states in polaritonic photonic crystals,” Phys. Rev. B 69, 195111 (2004).
[CrossRef]

Lund, E.

A. P. Seyranian, E. Lund, and N. Olhoff, “Multiple eigenvalues in structural optimization problems,” Struct. Optim. 8, 207–227 (1994).
[CrossRef]

Ma, J.

J. Ma and C. Jiang, “Demonstration of ultraslow modes in asymmetric line-defect photonic crystal waveguides,” IEEE Photon. Technol. Lett. 20, 1237–1239 (2008).
[CrossRef]

Mazoyer, S.

McMillan, J. E.

Melloni, A.

L. O’Faolain, S. A. Schulz, D. M. Beggs, T. P. White, M. Spasenovic, L. Kuipers, F. Morichetti, A. Melloni, S. Mazoyer, J. P. Hugonin, P. Lalanne, and T. F. Krauss, “Loss engineered slow light waveguides,” Opt. Express 18, 27627–27638 (2010).
[CrossRef]

S. A. Schulz, L. O’Faolain, D. M. Beggs, T. P. White, A. Melloni, and T. F. Krauss, “Dispersion engineered slow light in photonic crystals: a comparison,” J. Opt. 12, 104004 (2010).
[CrossRef]

Monat, C.

B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides,” Nat. Photonics 3, 206–210 (2009).
[CrossRef]

Moosburger, J.

Mori, D.

Morichetti, F.

Mork, J.

Mørk, J.

J. Grgić, J. R. Ott, F. Wang, O. Sigmund, A.-P. Jauho, J. Mørk, and N. A. Mortensen, “Fundamental limitations to gain enhancement in periodic media and waveguides,” Phys. Rev. Lett. 108, 183903 (2012).
[CrossRef]

Mortensen, N. A.

J. Grgić, J. R. Ott, F. Wang, O. Sigmund, A.-P. Jauho, J. Mørk, and N. A. Mortensen, “Fundamental limitations to gain enhancement in periodic media and waveguides,” Phys. Rev. Lett. 108, 183903 (2012).
[CrossRef]

J. G. Pedersen, S. S. Xiao, and N. A. Mortensen, “Limits of slow light in photonic crystals,” Phys. Rev. B 78, 153101 (2008).
[CrossRef]

Moss, D. J.

B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides,” Nat. Photonics 3, 206–210 (2009).
[CrossRef]

Nelson, K. A.

K. C. Huang, E. Lidorikis, X. Y. Jiang, J. D. Joannopoulos, K. A. Nelson, P. Bienstman, and S. H. Fan, “Nature of lossy Bloch states in polaritonic photonic crystals,” Phys. Rev. B 69, 195111 (2004).
[CrossRef]

Notomi, M.

E. Kuramochi, M. Notomi, S. Hughes, A. Shinya, T. Watanabe, and L. Ramunno, “Disorder-induced scattering loss of line-defect waveguides in photonic crystal slabs,” Phys. Rev. B 72, 161318 (2005).
[CrossRef]

Nunes, F. D.

O’Faolain, L.

L. O’Faolain, S. A. Schulz, D. M. Beggs, T. P. White, M. Spasenovic, L. Kuipers, F. Morichetti, A. Melloni, S. Mazoyer, J. P. Hugonin, P. Lalanne, and T. F. Krauss, “Loss engineered slow light waveguides,” Opt. Express 18, 27627–27638 (2010).
[CrossRef]

S. A. Schulz, L. O’Faolain, D. M. Beggs, T. P. White, A. Melloni, and T. F. Krauss, “Dispersion engineered slow light in photonic crystals: a comparison,” J. Opt. 12, 104004 (2010).
[CrossRef]

M. Patterson, S. Hughes, S. Schulz, D. M. Beggs, T. P. White, L. O’Faolain, and T. F. Krauss, “Disorder-induced incoherent scattering losses in photonic crystal waveguides: Bloch mode reshaping, multiple scattering, and breakdown of the Beer-Lambert law,” Phys. Rev. B 80, 195305 (2009).
[CrossRef]

B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides,” Nat. Photonics 3, 206–210 (2009).
[CrossRef]

J. Li, T. P. White, L. O’Faolain, A. Gomez-Iglesias, and T. F. Krauss, “Systematic design of flat band slow light in photonic crystal waveguides,” Opt. Express 16, 6227–6232 (2008).
[CrossRef]

D. M. Beggs, T. P. White, L. O’Faolain, and T. F. Krauss, “Ultracompact and low-power optical switch based on silicon photonic crystals,” Opt. Lett. 33, 147–149 (2008).
[CrossRef]

Olhoff, N.

A. P. Seyranian, E. Lund, and N. Olhoff, “Multiple eigenvalues in structural optimization problems,” Struct. Optim. 8, 207–227 (1994).
[CrossRef]

Osgood, R. M.

Ott, J. R.

J. Grgić, J. R. Ott, F. Wang, O. Sigmund, A.-P. Jauho, J. Mørk, and N. A. Mortensen, “Fundamental limitations to gain enhancement in periodic media and waveguides,” Phys. Rev. Lett. 108, 183903 (2012).
[CrossRef]

Panoiu, N. C.

Patterson, M.

M. Patterson and S. Hughes, “Theory of disorder-induced coherent scattering and light localization in slow-light photonic crystal waveguides,” J. Opt. 12, 104013 (2010).
[CrossRef]

M. Patterson, S. Hughes, S. Schulz, D. M. Beggs, T. P. White, L. O’Faolain, and T. F. Krauss, “Disorder-induced incoherent scattering losses in photonic crystal waveguides: Bloch mode reshaping, multiple scattering, and breakdown of the Beer-Lambert law,” Phys. Rev. B 80, 195305 (2009).
[CrossRef]

Pedersen, J. G.

J. G. Pedersen, S. S. Xiao, and N. A. Mortensen, “Limits of slow light in photonic crystals,” Phys. Rev. B 78, 153101 (2008).
[CrossRef]

Petrov, A.

Qiu, M.

Ramunno, L.

E. Kuramochi, M. Notomi, S. Hughes, A. Shinya, T. Watanabe, and L. Ramunno, “Disorder-induced scattering loss of line-defect waveguides in photonic crystal slabs,” Phys. Rev. B 72, 161318 (2005).
[CrossRef]

S. Hughes, L. Ramunno, J. F. Young, and J. E. Sipe, “Extrinsic optical scattering loss in photonic crystal waveguides: role of fabrication disorder and photon group velocity,” Phys. Rev. Lett. 94, 033903 (2005).
[CrossRef]

Sasaki, H.

Schulz, S.

M. Patterson, S. Hughes, S. Schulz, D. M. Beggs, T. P. White, L. O’Faolain, and T. F. Krauss, “Disorder-induced incoherent scattering losses in photonic crystal waveguides: Bloch mode reshaping, multiple scattering, and breakdown of the Beer-Lambert law,” Phys. Rev. B 80, 195305 (2009).
[CrossRef]

Schulz, S. A.

S. A. Schulz, L. O’Faolain, D. M. Beggs, T. P. White, A. Melloni, and T. F. Krauss, “Dispersion engineered slow light in photonic crystals: a comparison,” J. Opt. 12, 104004 (2010).
[CrossRef]

L. O’Faolain, S. A. Schulz, D. M. Beggs, T. P. White, M. Spasenovic, L. Kuipers, F. Morichetti, A. Melloni, S. Mazoyer, J. P. Hugonin, P. Lalanne, and T. F. Krauss, “Loss engineered slow light waveguides,” Opt. Express 18, 27627–27638 (2010).
[CrossRef]

Seyranian, A. P.

A. P. Seyranian, E. Lund, and N. Olhoff, “Multiple eigenvalues in structural optimization problems,” Struct. Optim. 8, 207–227 (1994).
[CrossRef]

Shinya, A.

E. Kuramochi, M. Notomi, S. Hughes, A. Shinya, T. Watanabe, and L. Ramunno, “Disorder-induced scattering loss of line-defect waveguides in photonic crystal slabs,” Phys. Rev. B 72, 161318 (2005).
[CrossRef]

Sigmund, O.

J. Grgić, J. R. Ott, F. Wang, O. Sigmund, A.-P. Jauho, J. Mørk, and N. A. Mortensen, “Fundamental limitations to gain enhancement in periodic media and waveguides,” Phys. Rev. Lett. 108, 183903 (2012).
[CrossRef]

F. Wang, J. S. Jensen, and O. Sigmund, “High-performance slow light photonic crystal waveguides with topology optimized or circular-hole based material layouts,” Photon. Nanostruct. Fundam. Applic. 10, 378–388 (2012).
[CrossRef]

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]

F. Wang, B. S. Lazarov, and O. Sigmund, “On projection methods, convergence and robust formulations in topology optimization,” Struct. Multidisc. Optim. 43, 767–784 (2011).
[CrossRef]

O. Sigmund, “Morphology-based black and white filters for topology optimization,” Struct. Multidisc. Optim. 33, 401–424 (2007).
[CrossRef]

R. Stainko and O. Sigmund, “Tailoring dispersion properties of photonic crystal waveguides by topology optimization,” Waves Random Complex Media 17, 477–489 (2007).
[CrossRef]

J. S. Jensen and O. Sigmund, “Topology optimization of photonic crystal structures: a high-bandwidth low-loss T-junction waveguide,” J. Opt. Soc. Am. B 22, 1191–1198 (2005).
[CrossRef]

Sipe, J. E.

S. Hughes, L. Ramunno, J. F. Young, and J. E. Sipe, “Extrinsic optical scattering loss in photonic crystal waveguides: role of fabrication disorder and photon group velocity,” Phys. Rev. Lett. 94, 033903 (2005).
[CrossRef]

Smith, C. J. M.

H. Benisty, D. Labilloy, C. Weisbuch, C. J. M. Smith, T. F. Krauss, D. Cassagne, A. Beraud, and C. Jouanin, “Radiation losses of waveguide-based two-dimensional photonic crystals: positive role of the substrate,” Appl. Phys. Lett. 76, 532–534 (2000).
[CrossRef]

Spasenovic, M.

Stainko, R.

R. Stainko and O. Sigmund, “Tailoring dispersion properties of photonic crystal waveguides by topology optimization,” Waves Random Complex Media 17, 477–489 (2007).
[CrossRef]

Sugimoto, Y.

Svanberg, K.

K. Svanberg, “A class of globally convergent optimization methods based on conservative convex separable approximations,” SIAM J. Optim. 12, 555–573 (2002).
[CrossRef]

van Hulst, N. F.

Vasconcelos, T. C.

Wang, F.

J. Grgić, J. R. Ott, F. Wang, O. Sigmund, A.-P. Jauho, J. Mørk, and N. A. Mortensen, “Fundamental limitations to gain enhancement in periodic media and waveguides,” Phys. Rev. Lett. 108, 183903 (2012).
[CrossRef]

F. Wang, J. S. Jensen, and O. Sigmund, “High-performance slow light photonic crystal waveguides with topology optimized or circular-hole based material layouts,” Photon. Nanostruct. Fundam. Applic. 10, 378–388 (2012).
[CrossRef]

F. Wang, B. S. Lazarov, and O. Sigmund, “On projection methods, convergence and robust formulations in topology optimization,” Struct. Multidisc. Optim. 43, 767–784 (2011).
[CrossRef]

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

Watanabe, T.

E. Kuramochi, M. Notomi, S. Hughes, A. Shinya, T. Watanabe, and L. Ramunno, “Disorder-induced scattering loss of line-defect waveguides in photonic crystal slabs,” Phys. Rev. B 72, 161318 (2005).
[CrossRef]

Watanabe, Y.

Weiner, J.

Weisbuch, C.

H. Benisty, D. Labilloy, C. Weisbuch, C. J. M. Smith, T. F. Krauss, D. Cassagne, A. Beraud, and C. Jouanin, “Radiation losses of waveguide-based two-dimensional photonic crystals: positive role of the substrate,” Appl. Phys. Lett. 76, 532–534 (2000).
[CrossRef]

White, T. P.

S. A. Schulz, L. O’Faolain, D. M. Beggs, T. P. White, A. Melloni, and T. F. Krauss, “Dispersion engineered slow light in photonic crystals: a comparison,” J. Opt. 12, 104004 (2010).
[CrossRef]

L. O’Faolain, S. A. Schulz, D. M. Beggs, T. P. White, M. Spasenovic, L. Kuipers, F. Morichetti, A. Melloni, S. Mazoyer, J. P. Hugonin, P. Lalanne, and T. F. Krauss, “Loss engineered slow light waveguides,” Opt. Express 18, 27627–27638 (2010).
[CrossRef]

M. Patterson, S. Hughes, S. Schulz, D. M. Beggs, T. P. White, L. O’Faolain, and T. F. Krauss, “Disorder-induced incoherent scattering losses in photonic crystal waveguides: Bloch mode reshaping, multiple scattering, and breakdown of the Beer-Lambert law,” Phys. Rev. B 80, 195305 (2009).
[CrossRef]

B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides,” Nat. Photonics 3, 206–210 (2009).
[CrossRef]

J. Li, T. P. White, L. O’Faolain, A. Gomez-Iglesias, and T. F. Krauss, “Systematic design of flat band slow light in photonic crystal waveguides,” Opt. Express 16, 6227–6232 (2008).
[CrossRef]

D. M. Beggs, T. P. White, L. O’Faolain, and T. F. Krauss, “Ultracompact and low-power optical switch based on silicon photonic crystals,” Opt. Lett. 33, 147–149 (2008).
[CrossRef]

Wong, C. W.

Xiao, S. S.

J. G. Pedersen, S. S. Xiao, and N. A. Mortensen, “Limits of slow light in photonic crystals,” Phys. Rev. B 78, 153101 (2008).
[CrossRef]

Yang, X. D.

Young, J. F.

S. Hughes, L. Ramunno, J. F. Young, and J. E. Sipe, “Extrinsic optical scattering loss in photonic crystal waveguides: role of fabrication disorder and photon group velocity,” Phys. Rev. Lett. 94, 033903 (2005).
[CrossRef]

Appl. Phys. Lett. (2)

D. Mori and T. Baba, “Dispersion-controlled optical group delay device by chirped photonic crystal waveguides,” Appl. Phys. Lett. 85, 1101–1103 (2004).
[CrossRef]

H. Benisty, D. Labilloy, C. Weisbuch, C. J. M. Smith, T. F. Krauss, D. Cassagne, A. Beraud, and C. Jouanin, “Radiation losses of waveguide-based two-dimensional photonic crystals: positive role of the substrate,” Appl. Phys. Lett. 76, 532–534 (2000).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

J. Ma and C. Jiang, “Demonstration of ultraslow modes in asymmetric line-defect photonic crystal waveguides,” IEEE Photon. Technol. Lett. 20, 1237–1239 (2008).
[CrossRef]

J. Opt. (2)

M. Patterson and S. Hughes, “Theory of disorder-induced coherent scattering and light localization in slow-light photonic crystal waveguides,” J. Opt. 12, 104013 (2010).
[CrossRef]

S. A. Schulz, L. O’Faolain, D. M. Beggs, T. P. White, A. Melloni, and T. F. Krauss, “Dispersion engineered slow light in photonic crystals: a comparison,” J. Opt. 12, 104004 (2010).
[CrossRef]

J. Opt. Soc. Am. B (4)

J. Phys. D (1)

T. F. Krauss, “Slow light in photonic crystal waveguides,” J. Phys. D 40, 2666–2670 (2007).
[CrossRef]

Nat. Photonics (3)

T. F. Krauss, “Why do we need slow light?” Nat. Photonics 2, 448–450 (2008).
[CrossRef]

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

B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides,” Nat. Photonics 3, 206–210 (2009).
[CrossRef]

Opt. Express (9)

P. Colman, S. Combrié, G. Lehoucq, and A. De Rossi, “Control of dispersion in photonic crystal waveguides using group symmetry theory,” Opt. Express 20, 13108–13114 (2012).
[CrossRef]

J. Li, T. P. White, L. O’Faolain, A. Gomez-Iglesias, and T. F. Krauss, “Systematic design of flat band slow light in photonic crystal waveguides,” Opt. Express 16, 6227–6232 (2008).
[CrossRef]

A. Petrov, M. Krause, and M. Eich, “Backscattering and disorder limits in slow light photonic crystal waveguides,” Opt. Express 17, 8676–8684 (2009).
[CrossRef]

M. Heuck, S. Blaaberg, and J. Mork, “Theory of passively mode-locked photonic crystal semiconductor lasers,” Opt. Express 18, 18003–18014 (2010).
[CrossRef]

L. O’Faolain, S. A. Schulz, D. M. Beggs, T. P. White, M. Spasenovic, L. Kuipers, F. Morichetti, A. Melloni, S. Mazoyer, J. P. Hugonin, P. Lalanne, and T. F. Krauss, “Loss engineered slow light waveguides,” Opt. Express 18, 27627–27638 (2010).
[CrossRef]

R. J. P. Engelen, Y. Sugimoto, Y. Watanabe, J. P. Korterik, N. Ikeda, N. F. van Hulst, K. Asakawa, and L. Kuipers, “The effect of higher-order dispersion on slow light propagation in photonic crystal waveguides,” Opt. Express 14, 1658–1672 (2006).
[CrossRef]

L. H. Frandsen, A. V. Lavrinenko, J. Fage-Pedersen, and P. I. Borel, “Photonic crystal waveguides with semi-slow light and tailored dispersion properties,” Opt. Express 14, 9444–9450 (2006).
[CrossRef]

D. Mori, S. Kubo, H. Sasaki, and T. Baba, “Experimental demonstration of wideband dispersion-compensated slow light by a chirped photonic crystal directional coupler,” Opt. Express 15, 5264–5270 (2007).
[CrossRef]

T. Kawasaki, D. Mori, and T. Baba, “Experimental observation of slow light in photonic crystal coupled waveguides,” Opt. Express 15, 10274–10281 (2007).
[CrossRef]

Opt. Lett. (3)

Photon. Nanostruct. Fundam. Applic. (1)

F. Wang, J. S. Jensen, and O. Sigmund, “High-performance slow light photonic crystal waveguides with topology optimized or circular-hole based material layouts,” Photon. Nanostruct. Fundam. Applic. 10, 378–388 (2012).
[CrossRef]

Phys. Rev. B (4)

J. G. Pedersen, S. S. Xiao, and N. A. Mortensen, “Limits of slow light in photonic crystals,” Phys. Rev. B 78, 153101 (2008).
[CrossRef]

K. C. Huang, E. Lidorikis, X. Y. Jiang, J. D. Joannopoulos, K. A. Nelson, P. Bienstman, and S. H. Fan, “Nature of lossy Bloch states in polaritonic photonic crystals,” Phys. Rev. B 69, 195111 (2004).
[CrossRef]

M. Patterson, S. Hughes, S. Schulz, D. M. Beggs, T. P. White, L. O’Faolain, and T. F. Krauss, “Disorder-induced incoherent scattering losses in photonic crystal waveguides: Bloch mode reshaping, multiple scattering, and breakdown of the Beer-Lambert law,” Phys. Rev. B 80, 195305 (2009).
[CrossRef]

E. Kuramochi, M. Notomi, S. Hughes, A. Shinya, T. Watanabe, and L. Ramunno, “Disorder-induced scattering loss of line-defect waveguides in photonic crystal slabs,” Phys. Rev. B 72, 161318 (2005).
[CrossRef]

Phys. Rev. Lett. (2)

J. Grgić, J. R. Ott, F. Wang, O. Sigmund, A.-P. Jauho, J. Mørk, and N. A. Mortensen, “Fundamental limitations to gain enhancement in periodic media and waveguides,” Phys. Rev. Lett. 108, 183903 (2012).
[CrossRef]

S. Hughes, L. Ramunno, J. F. Young, and J. E. Sipe, “Extrinsic optical scattering loss in photonic crystal waveguides: role of fabrication disorder and photon group velocity,” Phys. Rev. Lett. 94, 033903 (2005).
[CrossRef]

Phys. Status Solidi B (1)

L. C. Andreani and D. Gerace, “Light-matter interaction in photonic crystal slabs,” Phys. Status Solidi B 244, 3528–3539 (2007).
[CrossRef]

SIAM J. Optim. (1)

K. Svanberg, “A class of globally convergent optimization methods based on conservative convex separable approximations,” SIAM J. Optim. 12, 555–573 (2002).
[CrossRef]

Struct. Multidisc. Optim. (2)

O. Sigmund, “Morphology-based black and white filters for topology optimization,” Struct. Multidisc. Optim. 33, 401–424 (2007).
[CrossRef]

F. Wang, B. S. Lazarov, and O. Sigmund, “On projection methods, convergence and robust formulations in topology optimization,” Struct. Multidisc. Optim. 43, 767–784 (2011).
[CrossRef]

Struct. Optim. (1)

A. P. Seyranian, E. Lund, and N. Olhoff, “Multiple eigenvalues in structural optimization problems,” Struct. Optim. 8, 207–227 (1994).
[CrossRef]

Waves Random Complex Media (1)

R. Stainko and O. Sigmund, “Tailoring dispersion properties of photonic crystal waveguides by topology optimization,” Waves Random Complex Media 17, 477–489 (2007).
[CrossRef]

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Figures (14)

Fig. 1.
Fig. 1.

Band structure calculations. (a) Supercell. (b) Complex band structure. The solid curves represent the band structure calculated using the ω(k) formulation, and the symbols represent the band structure calculated using the k(ω) formulation, where the asterisks denote the guided modes and the circles denote the evanescent modes.

Fig. 2.
Fig. 2.

Influence of material loss/gain on the performance of waveguides. (a) Influence of material loss. The solid curve represents the guided band of the lossless waveguide, the dashed curve shows the guided band when considering material loss of n=103, and the dashed–dotted curve shows the guided band of n=102. (b) Influence of material gain. The solid curve shows the guided band of the lossless waveguide, the dashed curve represents the guided band of the waveguide with the material gain of n=103, and the dashed–dotted curve denotes the guided band when considering the material gain of n=102.

Fig. 3.
Fig. 3.

Characteristics of guided bands considering material loss or gain. The solid curves show the characteristics of the guided band in the waveguide with material loss, and the dashed curves show the characteristics of the guided band in the waveguide with the same amount of material gain.

Fig. 4.
Fig. 4.

Fixed-topology dispersion-engineered waveguides for ng*=50. (a) Longitudinal-location-tuned waveguide. (b) Lateral-location-tuned waveguide. (c) Radius-tuned waveguide. (d) Performance of different waveguides. Left panel: group index versus frequency. Right panel: propagation loss versus frequency. The dashed curves represent the performance of the longitudinal-location-tuned waveguide, the dashed–dotted curves represent the performance of the lateral-location-tuned waveguide, and the solid curvess represent the performance of the radius-tuned waveguide.

Fig. 5.
Fig. 5.

Energy density and dissipative energy density of slow-light modes with ng50 in fixed-topology dispersion-engineered waveguides. (a) Longitudinal-location-tuned waveguide. Top: energy density. Bottom: dissipative energy density. (b) Lateral-location-tuned waveguide. (c) Radius-tuned waveguide.

Fig. 6.
Fig. 6.

Initial design and the design domain. The gray dashed rectangle indicates the design domain with a symmetry along the waveguide direction.

Fig. 7.
Fig. 7.

Loss-engineered slow-light waveguides for ng*=50. (a) Free-topology loss-engineered waveguide for er=10%. (b) Free-topology loss-engineered waveguide with er=5%. (c) Fixed-topology loss-engineered waveguide for er=5%. (d) Performances of different waveguides. The dotted curves show the performance of the initial waveguide, the dashed–dotted curves show the performance of the free-topology loss-engineered waveguide for er=10%, the dashed curves show the performance of the free-topology loss-engineered waveguide er=5%, and the solid curves show the performances of the fixed-topology loss-engineered waveguides. The gray regions indicate the design range.

Fig. 8.
Fig. 8.

Energy density and dissipative energy density of slow-light modes with ng50 in the optimized waveguides. (a) Free-topology loss-engineered waveguide for er=10%. (b) Free-topology loss-engineered waveguide for er=5%. (c) Fixed-topology loss-engineered waveguide for er=5%.

Fig. 9.
Fig. 9.

Free-topology optimized slow-light waveguides with an enlarged GBP for ng*=50. (a) Free-topology dispersion-engineered waveguide. (b) Free-topology loss-engineered waveguide. (c) Group index and loss of different waveguides. The dashed–dotted curves represent the performance of the initial design, the dashed curves represent the performances of the dispersion-engineered waveguide, and the solid curves represent the performance of the loss-engineered waveguides. The gray regions indicate the design range.

Fig. 10.
Fig. 10.

Energy density and dissipative energy density of slow-light modes with ng50 in different waveguides. (a) Dispersion-engineered waveguide of ng*=50. (b) Loss-engineered waveguide of ng*=50.

Fig. 11.
Fig. 11.

Comparison between fixed- and free-topology loss-engineered waveguides for different group indices. Left panel: average group index versus average propagation loss. Right panel: average group index versus GBP. The dashed–dotted curves represent the performance of the fixed-topology loss-engineered waveguides with circles indicating the different designs considered. The solid curves represent the performance of the loss-engineered slow-light waveguides, with crosses indicating the different designs considered.

Fig. 12.
Fig. 12.

Loss-engineered waveguides for ng*=90. (a) Fixed-topology loss-engineered waveguide. (b) Free-topology loss-engineered waveguide. (c) Performance of different waveguides. The dashed curves represent the performance of the fixed-topology loss-engineered waveguide, the black solid curves represent the free-topology loss-engineered waveguides, and the bold red center parts of the solid curves represent the design ranges.

Fig. 13.
Fig. 13.

Energy density and dissipative energy density of slow-light modes with ng94 at different frequencies in the free-topology loss-engineered waveguide for ng*=90. (a) ωa/2π=0.21034. (b) ωa/2π=0.21123.

Fig. 14.
Fig. 14.

Transmission spectra of the loss-engineered waveguides of 20 periods in the presence of the edge-related loss. The dashed–dotted curve represents the transmission spectrum of the free-topology dispersion-engineered waveguide, the dashed curve represents the one of the fixed-topology loss-engineered waveguide, and the solid curve represents the one of the free-topology loss-engineered waveguide.

Tables (3)

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Table 1. Performance Comparison of Fixed-Topology Waveguidesa

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Table 2. Performance Comparison of Loss-Engineered Waveguides with ng*=50a

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Table 3. Performance Comparison of the Optimized Waveguides with ng*=50 and Enlarged GBPa

Equations (25)

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·(εr1h)+(ω/c)2μrh=0.
(+ik)·εr1(+ik)u+(ω/c)2μru=0,
(K0ω2M)u+K1ku+K2k2u=0.
K0=eεr1e(NTx1Nx1+NTx2Nx2)dV,M=eμr/c2e(NTN)dV,K1=eεr1ie(NTNx2NTx2N)dV,K2=eεr1e(NTN)dV,
(KlkMl)U=0,
Kl=[K1(K0ω2M)I0],Ml=[K200I],U=[kuu].
ng=cvg=(kω),
kω=(U0(KlωkMlω)U),
(KlkMl)U0=0.
U0MlU=1.
K0(ε¯r)=K0(εr),M(ε¯r)=M(εr),K1(ε¯r)=K1(εr),K2(ε¯r)=K2(εr).
εe=ε1+ρ¯e(x)(ε2ε1).
ρ¯e=tanh(βη)+tanh(β(ρ˜eη))tanh(βη)+tanh(β(1η)),
ρ˜e=jNew(rj)vjρjjNew(rj)vj,
Ne={j|rjrer},
εe=ε1+ρ¯e(ε2ε1)+iνepε0,
minxmaxωif(ρ¯,ωi)=(ng(ρ¯,ωi)/ng*1)2.
minxf(ρ¯)=ωi((k(ρ¯,ωi)))s.t.(ng(ρ¯,ωi)/ng*1)2<er2.
(ωn1(kii))a1min(ωi),(ωn+1(kii))a2max(ωi),
ng=ΔkΔω=(k(ωi))(k(ω))ωiω,
ngρ¯e=((k(ωi)ρ¯e)(k(ω)ρ¯e))/(ωiω).
(k)ρ¯e=(kρ¯e).
fxe=fρ¯eρ¯exe.
GBP=Δωng/ω0,
Is=12ε0εr|E|2,Id=12ε0εr|E|2.

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