Abstract

In this paper, we propose a method for designing Photonic Crystal (PhC) devices that consist of dielectric rods with varying size. In the proposed design method, PhC devices are modeled with the Transformation Optics (TO) approach, and then they are optimized using the gradient method. By applying the TO technique, the original device model is transformed into an equivalent model that consists of uniform and fixed-sized rods, with parameterized permittivity and permeability distributions. Therefore, mesh refinement around small rods can be avoided, and PhC devices can be simulated more efficiently. In addition, gradient of the optimization object function is calculated with the Adjoint-Variable Method (AVM), which is very efficient for optimizing devices subject to multiple design variables. The proposed method opens up a new avenue to design and optimize a variety of photonic devices for optical computing and information processing.

© 2014 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2013 (3)

2012 (4)

M. Minkov and V. Savona, “Effect of hole-shape irregularities on photonic crystal waveguides,” Opt. Lett. 37(15), 3108–3110 (2012).
[CrossRef] [PubMed]

Y. M. Liu and X. Zhang, “Recent advances in transformation optics,” Nanoscale 4(17), 5277–5292 (2012).
[CrossRef] [PubMed]

L. H. Gabrielli, D. Liu, S. G. Johnson, and M. Lipson, “On-chip transformation optics for multimode waveguide bends,” Nat. Commun. 3:1217(2012).
[PubMed]

Q. Wu, J. P. Turpin, and D. H. Werner, “Integrated photonic systems based on transformation optics enabled gradient index devices,” Light: Science & Applications 1, e38; (2012).
[CrossRef]

2011 (1)

V. Savona, “Electromagnetic modes of a disordered photonic crystal,” Phys. Rev. B 83, 085301 (2011).
[CrossRef]

2009 (1)

2006 (2)

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[CrossRef] [PubMed]

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

2005 (1)

2004 (1)

2003 (1)

2002 (1)

N. K. Georgieva, S. Glavic, M. H. Bakr, and J. W. Bandler, “Feasible Adjoint Sensitivity Technique for EM Design Optimization,” IEEE Tans. Microwave Theory Tech. 50(12), 2751–2758 (2002).
[CrossRef]

2001 (3)

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

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

A. Adibi, Y. Xu, R. K. Lee, A. Yariv, and A. Scherer, “Guiding mechanisms in dielectric-core photonic-crystal optical waveguides,” Phys. Rev. B 64, 033308 (2001).
[CrossRef]

1996 (1)

A. J. Ward and J. B. Pendry, “Refraction and geometry in Maxwell’s equation,” J. Mod. Opt. 43(4), 773–793 (1996).
[CrossRef]

Adibi, A.

A. Adibi, Y. Xu, R. K. Lee, A. Yariv, and A. Scherer, “Guiding mechanisms in dielectric-core photonic-crystal optical waveguides,” Phys. Rev. B 64, 033308 (2001).
[CrossRef]

Asatryan, A. A.

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

Assefa, S.

Baehr-Jones, T.

Bakr, M. H.

N. K. Georgieva, S. Glavic, M. H. Bakr, and J. W. Bandler, “Feasible Adjoint Sensitivity Technique for EM Design Optimization,” IEEE Tans. Microwave Theory Tech. 50(12), 2751–2758 (2002).
[CrossRef]

Bandler, J. W.

N. K. Georgieva, S. Glavic, M. H. Bakr, and J. W. Bandler, “Feasible Adjoint Sensitivity Technique for EM Design Optimization,” IEEE Tans. Microwave Theory Tech. 50(12), 2751–2758 (2002).
[CrossRef]

Bhargava, S.

Bienstman, P.

Botten, L. C.

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

Brimont, A.

Brnovic, J.

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

Chen, S.

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

Cuesta, F.

Dossou, K.

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

Dutton, R. W.

Fan, S.

Gabrielli, L. H.

L. H. Gabrielli, D. Liu, S. G. Johnson, and M. Lipson, “On-chip transformation optics for multimode waveguide bends,” Nat. Commun. 3:1217(2012).
[PubMed]

Galán, J. V.

Galland, C.

Georgieva, N. K.

N. K. Georgieva, S. Glavic, M. H. Bakr, and J. W. Bandler, “Feasible Adjoint Sensitivity Technique for EM Design Optimization,” IEEE Tans. Microwave Theory Tech. 50(12), 2751–2758 (2002).
[CrossRef]

Glavic, S.

N. K. Georgieva, S. Glavic, M. H. Bakr, and J. W. Bandler, “Feasible Adjoint Sensitivity Technique for EM Design Optimization,” IEEE Tans. Microwave Theory Tech. 50(12), 2751–2758 (2002).
[CrossRef]

Griol, A.

Guryev, I. V.

I. A. Sukhoivanov and I. V. Guryev, Photonic Crystals: Physics and Practical Modeling (Springer-Verlag, Berlin, 2009).
[CrossRef]

Håkansson, A.

Hochberg, M.

Jensen, J. S.

Joannopoulos, J. D.

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

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

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2 (Princeton University, 2008).

Joanopoulos, J. D.

Johnson, S. G.

L. H. Gabrielli, D. Liu, S. G. Johnson, and M. Lipson, “On-chip transformation optics for multimode waveguide bends,” Nat. Commun. 3:1217(2012).
[PubMed]

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

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

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2 (Princeton University, 2008).

Kolodziejski, L. A.

Kuzuoglu, M.

O. Ozgun and M. Kuzuoglu, “Software metamaterials: Transformation media based multiscale techniques for computational electromagnetics”, J. Comput. Phys. 236, 203–219 (2013).
[CrossRef]

Lalau-Keraly, C. M.

Lee, R. K.

A. Adibi, Y. Xu, R. K. Lee, A. Yariv, and A. Scherer, “Guiding mechanisms in dielectric-core photonic-crystal optical waveguides,” Phys. Rev. B 64, 033308 (2001).
[CrossRef]

Lim, A. E. J.

Lipson, M.

L. H. Gabrielli, D. Liu, S. G. Johnson, and M. Lipson, “On-chip transformation optics for multimode waveguide bends,” Nat. Commun. 3:1217(2012).
[PubMed]

Liu, D.

L. H. Gabrielli, D. Liu, S. G. Johnson, and M. Lipson, “On-chip transformation optics for multimode waveguide bends,” Nat. Commun. 3:1217(2012).
[PubMed]

Liu, Y. M.

Y. M. Liu and X. Zhang, “Recent advances in transformation optics,” Nanoscale 4(17), 5277–5292 (2012).
[CrossRef] [PubMed]

Lo, G. Q.

Marcuse, D.

D. Marcuse, Theory of Dielectric Optical Waveguides(Academic Press, 1974).

Marti, J.

McPhedran, R. C.

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

Meade, R. D.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2 (Princeton University, 2008).

Mekis, A.

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

Miller, O. D.

Minkov, M.

Ozgun, O.

O. Ozgun and M. Kuzuoglu, “Software metamaterials: Transformation media based multiscale techniques for computational electromagnetics”, J. Comput. Phys. 236, 203–219 (2013).
[CrossRef]

Pendry, J. B.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[CrossRef] [PubMed]

A. J. Ward and J. B. Pendry, “Refraction and geometry in Maxwell’s equation,” J. Mod. Opt. 43(4), 773–793 (1996).
[CrossRef]

Petrich, G. S.

Sanchis, P.

Savona, V.

Scherer, A.

A. Adibi, Y. Xu, R. K. Lee, A. Yariv, and A. Scherer, “Guiding mechanisms in dielectric-core photonic-crystal optical waveguides,” Phys. Rev. B 64, 033308 (2001).
[CrossRef]

Schurig, D.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[CrossRef] [PubMed]

Sigmund, O.

Smith, D. R.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[CrossRef] [PubMed]

Sterke, C. M.

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

Sukhoivanov, I. A.

I. A. Sukhoivanov and I. V. Guryev, Photonic Crystals: Physics and Practical Modeling (Springer-Verlag, Berlin, 2009).
[CrossRef]

Turpin, J. P.

Q. Wu, J. P. Turpin, and D. H. Werner, “Integrated photonic systems based on transformation optics enabled gradient index devices,” Light: Science & Applications 1, e38; (2012).
[CrossRef]

Veronis, G.

Villalba, P.

Ward, A. J.

A. J. Ward and J. B. Pendry, “Refraction and geometry in Maxwell’s equation,” J. Mod. Opt. 43(4), 773–793 (1996).
[CrossRef]

Werner, D. H.

Q. Wu, J. P. Turpin, and D. H. Werner, “Integrated photonic systems based on transformation optics enabled gradient index devices,” Light: Science & Applications 1, e38; (2012).
[CrossRef]

Winn, J. N.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2 (Princeton University, 2008).

Wu, Q.

Q. Wu, J. P. Turpin, and D. H. Werner, “Integrated photonic systems based on transformation optics enabled gradient index devices,” Light: Science & Applications 1, e38; (2012).
[CrossRef]

Xu, Y.

A. Adibi, Y. Xu, R. K. Lee, A. Yariv, and A. Scherer, “Guiding mechanisms in dielectric-core photonic-crystal optical waveguides,” Phys. Rev. B 64, 033308 (2001).
[CrossRef]

Yablonovitch, E.

Yang, S.

Yariv, A.

A. Adibi, Y. Xu, R. K. Lee, A. Yariv, and A. Scherer, “Guiding mechanisms in dielectric-core photonic-crystal optical waveguides,” Phys. Rev. B 64, 033308 (2001).
[CrossRef]

Zhang, X.

Y. M. Liu and X. Zhang, “Recent advances in transformation optics,” Nanoscale 4(17), 5277–5292 (2012).
[CrossRef] [PubMed]

Zhang, Y.

IEEE Tans. Microwave Theory Tech. (1)

N. K. Georgieva, S. Glavic, M. H. Bakr, and J. W. Bandler, “Feasible Adjoint Sensitivity Technique for EM Design Optimization,” IEEE Tans. Microwave Theory Tech. 50(12), 2751–2758 (2002).
[CrossRef]

J. Comput. Phys. (1)

O. Ozgun and M. Kuzuoglu, “Software metamaterials: Transformation media based multiscale techniques for computational electromagnetics”, J. Comput. Phys. 236, 203–219 (2013).
[CrossRef]

J. Lightw. Technol. (1)

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

J. Mod. Opt. (1)

A. J. Ward and J. B. Pendry, “Refraction and geometry in Maxwell’s equation,” J. Mod. Opt. 43(4), 773–793 (1996).
[CrossRef]

J. Opt. Soc. Am. B (2)

Light: Science & Applications (1)

Q. Wu, J. P. Turpin, and D. H. Werner, “Integrated photonic systems based on transformation optics enabled gradient index devices,” Light: Science & Applications 1, e38; (2012).
[CrossRef]

Nanoscale (1)

Y. M. Liu and X. Zhang, “Recent advances in transformation optics,” Nanoscale 4(17), 5277–5292 (2012).
[CrossRef] [PubMed]

Nat. Commun. (1)

L. H. Gabrielli, D. Liu, S. G. Johnson, and M. Lipson, “On-chip transformation optics for multimode waveguide bends,” Nat. Commun. 3:1217(2012).
[PubMed]

Opt. Commun. (1)

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

Opt. Express (3)

Opt. Lett. (3)

Phys. Rev. B (2)

A. Adibi, Y. Xu, R. K. Lee, A. Yariv, and A. Scherer, “Guiding mechanisms in dielectric-core photonic-crystal optical waveguides,” Phys. Rev. B 64, 033308 (2001).
[CrossRef]

V. Savona, “Electromagnetic modes of a disordered photonic crystal,” Phys. Rev. B 83, 085301 (2011).
[CrossRef]

Science (1)

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[CrossRef] [PubMed]

Other (5)

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2 (Princeton University, 2008).

I. A. Sukhoivanov and I. V. Guryev, Photonic Crystals: Physics and Practical Modeling (Springer-Verlag, Berlin, 2009).
[CrossRef]

COMSOL Multiphysics, http://www.comsol.com/

D. Marcuse, Theory of Dielectric Optical Waveguides(Academic Press, 1974).

M. Schmidt, http://www.di.ens.fr/~Emschmidt/Software/minConf.html

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Figures (8)

Fig. 1
Fig. 1

(a) A PhC device consisting of varying-sized dielectric rods. (b) Transformed model of (a), in which the transformed rods have a uniform and fixed size. Gray color represents the permittivity distributions. (c) A single dielectric rod, surrounded by air. (d) Transformed model of (c).

Fig. 2
Fig. 2

(a) Light scattering of a single dielectric rod. (b) Light scattering of the same dielectric rod, using the transformed model. (c) The difference of the field distribution between (a) and (b) in the region of r > ra, showing almost negligible difference (in the order of 10−4).

Fig. 3
Fig. 3

(a) Meshes of the transformed rod. (b)–(e) Meshes of the original model, in region of rrb. Radius of the rod is rc = 0.2, 0.1, 0.05, 0.01μm, respectively.

Fig. 4
Fig. 4

(a) Schematic diagram for PhC coupler. PhC couplers, indicated by dashed lines, couple light between the PhC line defect waveguide and the dielectric slab waveguides. (b) Rods radii of the optimised coupler, which are optimized with sampling wavelengths obj1–obj3, as defined in Eq. (6).

Fig. 5
Fig. 5

(a) Transmission spectra of the PhC coupler for a straight PhC waveguide. Spectrum of the unoptimized coupler is indicated by dotted line (black), spectrum of the coupler that optimized with obj1, obj2, and obj3 is indicated by dashed line (green), solid line (blue), and dash-dotted line (red) respectively. (b) Electric field Ez of the unoptimized PhC coupler, calculated at the wavelength of 1.55μm. (c) Electric field Ez of the coupler optimized with obj3.

Fig. 6
Fig. 6

(a) Schematic diagram for PhC coupler and PhC waveguide bend. PhC couplers are indicated by dashed lines, and PhC waveguide bend is indicated by dotted line. (b) Rods radii of the optimised PhC coupler and PhC waveguide bend, which are optimized with sampling wavelengths obj1–obj3, as defined in Eq. (6).

Fig. 7
Fig. 7

(a) Transmission spectra of the PhC coupler and PhC waveguide bend. Spectrum of the unoptimized device is indicated by dotted line, spectra of the device that optimized with obj1, obj2, and obj3 is indicated by dashed line, solid line, and dash-dotted line respectively. (b) Electric field Ez of PhC device optimized with obj3, calculated at the wavelength of 1.55μm. The rod of No. 12 and its counterpart at symmetric position are removed.

Fig. 8
Fig. 8

Magnitude of the normalized sensitivity of (a) the optimized PhC coupler in Section 3, (b) the optimized PhC coupler and waveguide bend in Section 4. Symbols “+”, “○”, and “×” represent sensitivity for PhC coupler and waveguide bend that are optimized with obj1, obj2, and obj3 respectively.

Tables (1)

Tables Icon

Table 1 Statistic data of meshes of the original and transformed model, in the region of rrb. Unit for radius: μm.

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

r ˜ = r b r c r = r b r c r ( r Ω 1 , r ˜ Ω ˜ 1 )
r ˜ = r a r b r a r c ( r r c ) + r b = r a r b r a r c ( r r c ) + r b ( r Ω 2 , r ˜ Ω ˜ 2 )
ε ˜ r = 1 det J ε r , μ ˜ r = J T J det J μ r
J = ( x ˜ , y ˜ ) ( x , y ) = ( x ˜ / x x ˜ / y y ˜ / x y ˜ / y )
J = i ( 1 p i ) 2
obj 1 : λ i = 1.5 , 1.525 , 1.55 , 1.575 , 1.6 ( μ m ) obj 2 : λ i = 1.545 , 1.55 , 1.555 ( μ m ) obj 3 : λ i = 1.55 ( μ m )
J r j = i 2 ( 1 p i ) p i r j

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