Abstract

A geometry projection method for the design of photonic nanostructures is demonstrated and compared with a topology optimization method. By using a higher-dimensional surface to delineate the material interfaces, the projection method restricts the regions of intermediate dielectric properties to a narrow user-defined band and gives some implicit control over feature size. Topology optimization addresses these issues by using a penalization to avoid areas of intermediate dielectric and a filter to obtain implicit control over feature size. The directional emission from a photonic crystal waveguide termination is improved by both methods by generating a series of irregularly shaped dielectric posts. Results are presented, and the relative merits of each method are discussed.

© 2006 Optical Society of America

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References

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  1. M. Burger, S. J. Osher, and E. Yablonovitch, IEICE Trans. Electron. E87-C, 258 (2004).
  2. M. P. Bendsøe and O. Sigmund, Topology Optimization (Springer, 2003).
  3. J. S. Jensen and O. Sigmund, J. Opt. Soc. Am. B 22, 1191 (2005).
    [Crossref]
  4. L. H. Frandsen, A. Harpøth, P. I. Borel, M. Kristensen, J. S. Jensen, and O. Sigmund, Opt. Express 12, 5916 (2004).
    [Crossref] [PubMed]
  5. W. R. Frei, D. A. Tortorelli, and H. T. Johnson, Appl. Phys. Lett. 86, 111114 (2005).
    [Crossref]
  6. O. Sigmund and J. Petersson, Struct. Optim. 16, 68 (1998).
    [Crossref]
  7. J. Norato, R. Haber, D. A. Tortorelli, and M. P. Bendsøe, Int. J. Numer. Methods Eng. 60, 2289 (2004).
    [Crossref]
  8. J. Jin, The Finite Element Method in Electromagnetics (Wiley, 2002).
  9. R. Franke, Math. Comput. 38, 181 (1982).
  10. D. A. Tortorelli and P. Michaleris, Inverse Probl. Eng. 1, 71 (1994).
    [Crossref]
  11. K. Svanberg, Int. J. Numer. Methods Eng. 24, 359 (1987).
    [Crossref]
  12. http://www.comsol.com.
  13. W. R. Frei, D. A. Tortorelli, and H. T. Johnson, in Proc. SPIE 5931 (2005).
  14. S. Assefa, G. S. Petrich, L. A. Kolodziejski, M. K. Mondol, and H. I. Smith, J. Vac. Sci. Technol. B 22, 3363 (2004).
    [Crossref]

2005 (2)

J. S. Jensen and O. Sigmund, J. Opt. Soc. Am. B 22, 1191 (2005).
[Crossref]

W. R. Frei, D. A. Tortorelli, and H. T. Johnson, Appl. Phys. Lett. 86, 111114 (2005).
[Crossref]

2004 (4)

L. H. Frandsen, A. Harpøth, P. I. Borel, M. Kristensen, J. S. Jensen, and O. Sigmund, Opt. Express 12, 5916 (2004).
[Crossref] [PubMed]

J. Norato, R. Haber, D. A. Tortorelli, and M. P. Bendsøe, Int. J. Numer. Methods Eng. 60, 2289 (2004).
[Crossref]

M. Burger, S. J. Osher, and E. Yablonovitch, IEICE Trans. Electron. E87-C, 258 (2004).

S. Assefa, G. S. Petrich, L. A. Kolodziejski, M. K. Mondol, and H. I. Smith, J. Vac. Sci. Technol. B 22, 3363 (2004).
[Crossref]

1998 (1)

O. Sigmund and J. Petersson, Struct. Optim. 16, 68 (1998).
[Crossref]

1994 (1)

D. A. Tortorelli and P. Michaleris, Inverse Probl. Eng. 1, 71 (1994).
[Crossref]

1987 (1)

K. Svanberg, Int. J. Numer. Methods Eng. 24, 359 (1987).
[Crossref]

1982 (1)

R. Franke, Math. Comput. 38, 181 (1982).

Assefa, S.

S. Assefa, G. S. Petrich, L. A. Kolodziejski, M. K. Mondol, and H. I. Smith, J. Vac. Sci. Technol. B 22, 3363 (2004).
[Crossref]

Bendsøe, M. P.

J. Norato, R. Haber, D. A. Tortorelli, and M. P. Bendsøe, Int. J. Numer. Methods Eng. 60, 2289 (2004).
[Crossref]

M. P. Bendsøe and O. Sigmund, Topology Optimization (Springer, 2003).

Borel, P. I.

Burger, M.

M. Burger, S. J. Osher, and E. Yablonovitch, IEICE Trans. Electron. E87-C, 258 (2004).

Frandsen, L. H.

Franke, R.

R. Franke, Math. Comput. 38, 181 (1982).

Frei, W. R.

W. R. Frei, D. A. Tortorelli, and H. T. Johnson, Appl. Phys. Lett. 86, 111114 (2005).
[Crossref]

W. R. Frei, D. A. Tortorelli, and H. T. Johnson, in Proc. SPIE 5931 (2005).

Haber, R.

J. Norato, R. Haber, D. A. Tortorelli, and M. P. Bendsøe, Int. J. Numer. Methods Eng. 60, 2289 (2004).
[Crossref]

Harpøth, A.

Jensen, J. S.

Jin, J.

J. Jin, The Finite Element Method in Electromagnetics (Wiley, 2002).

Johnson, H. T.

W. R. Frei, D. A. Tortorelli, and H. T. Johnson, Appl. Phys. Lett. 86, 111114 (2005).
[Crossref]

W. R. Frei, D. A. Tortorelli, and H. T. Johnson, in Proc. SPIE 5931 (2005).

Kolodziejski, L. A.

S. Assefa, G. S. Petrich, L. A. Kolodziejski, M. K. Mondol, and H. I. Smith, J. Vac. Sci. Technol. B 22, 3363 (2004).
[Crossref]

Kristensen, M.

Michaleris, P.

D. A. Tortorelli and P. Michaleris, Inverse Probl. Eng. 1, 71 (1994).
[Crossref]

Mondol, M. K.

S. Assefa, G. S. Petrich, L. A. Kolodziejski, M. K. Mondol, and H. I. Smith, J. Vac. Sci. Technol. B 22, 3363 (2004).
[Crossref]

Norato, J.

J. Norato, R. Haber, D. A. Tortorelli, and M. P. Bendsøe, Int. J. Numer. Methods Eng. 60, 2289 (2004).
[Crossref]

Osher, S. J.

M. Burger, S. J. Osher, and E. Yablonovitch, IEICE Trans. Electron. E87-C, 258 (2004).

Petersson, J.

O. Sigmund and J. Petersson, Struct. Optim. 16, 68 (1998).
[Crossref]

Petrich, G. S.

S. Assefa, G. S. Petrich, L. A. Kolodziejski, M. K. Mondol, and H. I. Smith, J. Vac. Sci. Technol. B 22, 3363 (2004).
[Crossref]

Sigmund, O.

Smith, H. I.

S. Assefa, G. S. Petrich, L. A. Kolodziejski, M. K. Mondol, and H. I. Smith, J. Vac. Sci. Technol. B 22, 3363 (2004).
[Crossref]

Svanberg, K.

K. Svanberg, Int. J. Numer. Methods Eng. 24, 359 (1987).
[Crossref]

Tortorelli, D. A.

W. R. Frei, D. A. Tortorelli, and H. T. Johnson, Appl. Phys. Lett. 86, 111114 (2005).
[Crossref]

J. Norato, R. Haber, D. A. Tortorelli, and M. P. Bendsøe, Int. J. Numer. Methods Eng. 60, 2289 (2004).
[Crossref]

D. A. Tortorelli and P. Michaleris, Inverse Probl. Eng. 1, 71 (1994).
[Crossref]

W. R. Frei, D. A. Tortorelli, and H. T. Johnson, in Proc. SPIE 5931 (2005).

Yablonovitch, E.

M. Burger, S. J. Osher, and E. Yablonovitch, IEICE Trans. Electron. E87-C, 258 (2004).

Appl. Phys. Lett. (1)

W. R. Frei, D. A. Tortorelli, and H. T. Johnson, Appl. Phys. Lett. 86, 111114 (2005).
[Crossref]

IEICE Trans. Electron. (1)

M. Burger, S. J. Osher, and E. Yablonovitch, IEICE Trans. Electron. E87-C, 258 (2004).

Int. J. Numer. Methods Eng. (2)

J. Norato, R. Haber, D. A. Tortorelli, and M. P. Bendsøe, Int. J. Numer. Methods Eng. 60, 2289 (2004).
[Crossref]

K. Svanberg, Int. J. Numer. Methods Eng. 24, 359 (1987).
[Crossref]

Inverse Probl. Eng. (1)

D. A. Tortorelli and P. Michaleris, Inverse Probl. Eng. 1, 71 (1994).
[Crossref]

J. Opt. Soc. Am. B (1)

J. Vac. Sci. Technol. B (1)

S. Assefa, G. S. Petrich, L. A. Kolodziejski, M. K. Mondol, and H. I. Smith, J. Vac. Sci. Technol. B 22, 3363 (2004).
[Crossref]

Math. Comput. (1)

R. Franke, Math. Comput. 38, 181 (1982).

Opt. Express (1)

Struct. Optim. (1)

O. Sigmund and J. Petersson, Struct. Optim. 16, 68 (1998).
[Crossref]

Other (4)

J. Jin, The Finite Element Method in Electromagnetics (Wiley, 2002).

M. P. Bendsøe and O. Sigmund, Topology Optimization (Springer, 2003).

http://www.comsol.com.

W. R. Frei, D. A. Tortorelli, and H. T. Johnson, in Proc. SPIE 5931 (2005).

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Figures (4)

Fig. 1
Fig. 1

Model of PC waveguide termination. The design domain is symmetric about the waveguide axis.

Fig. 2
Fig. 2

Level set definition of dielectric constant distribution for the two-dimensional domain of the design area shown in Fig. 1.

Fig. 3
Fig. 3

Magnitude of the electric field. Left, unoptimized waveguide termination; center, waveguide optimized by TO with filter radius a 22 ; right, waveguide termination optimized by the GPM with control point spacing a 5 .

Fig. 4
Fig. 4

Top half of the design domain. (a)–(c) Optimized by TO with filter radius of a 20 , a 22 , and a 24 , respectively. (d)–(f) Optimized by the GPM with control point spacing of a 4 , a 5 , and a 6 , respectively.

Equations (5)

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

z = s ( x , q ) = c ( q ) x + p λ p ( q ) φ ( x x p ) ,
φ ( r ) = φ ( r 2 log r ) ,
d ( x , q ) = min x 0 x 0 x such that : s ( x 0 , q ) = 0 ,
d S ( x , q ) q p = sign [ s ( x , q ) ] λ s ( x 0 , q ) q p ,
max q P O ( E [ ϵ ( x , q ) ] ) .

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