Abstract

The incoherent emission of periodically structured Light Emitting Diodes (LEDs) can be computed at relatively low computational cost by applying the reciprocity method. We show that by another application of the reciprocity principle, the structure of the LED can be optimized to obtain a high emission. We demonstrate the method by optimizing one-dimensional grating structures. The optimized structures have twice the extraction efficiency of an optimized flat structure.

© 2010 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. T. Fujii, Y. Gao, R. Sharma, E. L. Hu, S. P. DenBaars, and S. Nakamura, "Increase in the extraction efficiency of GaN-based light-emitting diodes via surface roughening," Appl. Phys. Lett. 84(6), 855-857 (2004).
    [CrossRef]
  2. M. Boroditsky, T. F. Krauss, R. Coccioli, R. Vrijen, R. Bhat, and E. Yablonovitch, "Light extraction from optically pumped light-emitting diode by thin-slab photonic crystals," Appl. Phys. Lett. 75(8), 1036-1038 (1999).
    [CrossRef]
  3. W. J. Choi, Q. H. Park, D. Kim, H. Jeon, C. Sone, and Y. Park, "FDTD Simulation for Light Extraction in a GaN-Based LED," J. Korean Phys. Soc. 49, 877-880 (2006).
  4. Y. J. Lee, S. H. Kim, G. H. Kim, Y. H. Lee, S. H. Cho, Y. W. Song, Y. C. Kim, and Y. R. Do, "Far-field radiation of photonic crystal organic light-emitting diode," Opt. Express 13(15), 5864-5870 (2005).
    [CrossRef] [PubMed]
  5. H. Rigneault, F. Lemarchand, and A. Sentenac, "Dipole radiation into grating structures," J. Opt. Soc. Am. A 17(6), 1048-1058 (2000).
    [CrossRef]
  6. A. L. Fehrembach, S. Enoch, and A. Sentenac, "Highly directive light sources using two-dimensional photonic crystal slabs," Appl. Phys. Lett. 79(26), 4280-4282 (2001).
    [CrossRef]
  7. A. David, H. Benisty, and C. Weisbuch, "Optimization of Light-Diffracting Photonic-Crystals for High Extraction Efficiency LEDs," J. Disp. Technol. 3(2), 133-148 (2007).
    [CrossRef]
  8. A. Roger, and D. Maystre, "Inverse scattering method in electromagnetic optics- Application to diffraction gratings," J. Opt. Soc. Am. 70(12), 1483-1495 (1980).
    [CrossRef]
  9. S. J. Norton, "Iterative inverse scattering algorithms: Methods of computing Frechet derivatives," J. Acoust. Soc. Am. 106(5), 2653-2660 (1999).
    [CrossRef]
  10. L. D. Landau, and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamom Press, 1960).
  11. R. J. Potton, "Reciprocity in Optics," Rep. Prog. Phys. 67(5), 717-754 (2004).
    [CrossRef]
  12. J.-J. Greffet, and R. Carminati, "Image formation in near-field optics," Prog. Surf. Sci. 56(3), 133-237 (1997).
    [CrossRef]
  13. C. E. Reed, J. Giergiel, J. C. Hemminger, and S. Ushioda, "Dipole radiation in a multilayer geometry," Phys. Rev. B 36(9), 4990-5000 (1987).
    [CrossRef]
  14. A. Roger, "Reciprocity theorem applied to the computation of functional derivatives of the scattering matrix," Electromagnetics 2(1), 69-83 (1982).
    [CrossRef]
  15. S. Bonnard, P. Vincent, and M. Saillard, "Cross-borehole inverse scattering using a boundary finite-element method," Inverse Probl. 14(3), 521-534 (1998).
    [CrossRef]
  16. A. Litman, and K. Belkebir, "Two-dimensional inverse profiling problem using phaseless data," J. Opt. Soc. Am. A 23(11), 2737-2746 (2006).
    [CrossRef]
  17. E. F. Schubert, Light-Emitting Diodes (Cambridge University Press, 2006).
    [CrossRef]
  18. M. Yamanishi, and I. Suemune, "Comment on polarization dependent momentum matrix elements in quantum well lasers," Jpn. J. Appl. Phys. 23(Part 2, No. 1), L35-L36 (1984).
    [CrossRef]
  19. M. Cui, H. P. Urbach, and D. K. G. de Boer, "Optimization of light extraction from OLEDs," Opt. Express 15(8), 4398-4409 (2007).
    [CrossRef] [PubMed]
  20. D. G. Luenberger, Optimization by Vector Space Methods (Wiley-Interscience, 1967).
  21. X. Wei, A. J. Wachters, and H. P. Urbach, "Finite-element model for three-dimensional optical scattering problems," J. Opt. Soc. Am. A 24(3), 866-881 (2007).
    [CrossRef]
  22. J. J. Wierer, A. David, and M. M. Megens, "III-nitride photonic-crystal light-emitting diodes with high extraction efficiency," Nat. Photonics 3(3), 163-169 (2009).
    [CrossRef]
  23. P. B. Johnson, and R. W. Christy, "Optical constants of the noble metals," Phys. Rev. B 6(12), 4370-4379 (1972).
    [CrossRef]
  24. S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, "Guided modes in photonic crystal slabs," Phys. Rev. B 60(8), 5751-5758 (1999).
    [CrossRef]

2009 (1)

J. J. Wierer, A. David, and M. M. Megens, "III-nitride photonic-crystal light-emitting diodes with high extraction efficiency," Nat. Photonics 3(3), 163-169 (2009).
[CrossRef]

2007 (3)

2006 (2)

A. Litman, and K. Belkebir, "Two-dimensional inverse profiling problem using phaseless data," J. Opt. Soc. Am. A 23(11), 2737-2746 (2006).
[CrossRef]

W. J. Choi, Q. H. Park, D. Kim, H. Jeon, C. Sone, and Y. Park, "FDTD Simulation for Light Extraction in a GaN-Based LED," J. Korean Phys. Soc. 49, 877-880 (2006).

2005 (1)

2004 (2)

T. Fujii, Y. Gao, R. Sharma, E. L. Hu, S. P. DenBaars, and S. Nakamura, "Increase in the extraction efficiency of GaN-based light-emitting diodes via surface roughening," Appl. Phys. Lett. 84(6), 855-857 (2004).
[CrossRef]

R. J. Potton, "Reciprocity in Optics," Rep. Prog. Phys. 67(5), 717-754 (2004).
[CrossRef]

2001 (1)

A. L. Fehrembach, S. Enoch, and A. Sentenac, "Highly directive light sources using two-dimensional photonic crystal slabs," Appl. Phys. Lett. 79(26), 4280-4282 (2001).
[CrossRef]

2000 (1)

1999 (3)

M. Boroditsky, T. F. Krauss, R. Coccioli, R. Vrijen, R. Bhat, and E. Yablonovitch, "Light extraction from optically pumped light-emitting diode by thin-slab photonic crystals," Appl. Phys. Lett. 75(8), 1036-1038 (1999).
[CrossRef]

S. J. Norton, "Iterative inverse scattering algorithms: Methods of computing Frechet derivatives," J. Acoust. Soc. Am. 106(5), 2653-2660 (1999).
[CrossRef]

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, "Guided modes in photonic crystal slabs," Phys. Rev. B 60(8), 5751-5758 (1999).
[CrossRef]

1998 (1)

S. Bonnard, P. Vincent, and M. Saillard, "Cross-borehole inverse scattering using a boundary finite-element method," Inverse Probl. 14(3), 521-534 (1998).
[CrossRef]

1997 (1)

J.-J. Greffet, and R. Carminati, "Image formation in near-field optics," Prog. Surf. Sci. 56(3), 133-237 (1997).
[CrossRef]

1987 (1)

C. E. Reed, J. Giergiel, J. C. Hemminger, and S. Ushioda, "Dipole radiation in a multilayer geometry," Phys. Rev. B 36(9), 4990-5000 (1987).
[CrossRef]

1984 (1)

M. Yamanishi, and I. Suemune, "Comment on polarization dependent momentum matrix elements in quantum well lasers," Jpn. J. Appl. Phys. 23(Part 2, No. 1), L35-L36 (1984).
[CrossRef]

1982 (1)

A. Roger, "Reciprocity theorem applied to the computation of functional derivatives of the scattering matrix," Electromagnetics 2(1), 69-83 (1982).
[CrossRef]

1980 (1)

1972 (1)

P. B. Johnson, and R. W. Christy, "Optical constants of the noble metals," Phys. Rev. B 6(12), 4370-4379 (1972).
[CrossRef]

Belkebir, K.

Benisty, H.

A. David, H. Benisty, and C. Weisbuch, "Optimization of Light-Diffracting Photonic-Crystals for High Extraction Efficiency LEDs," J. Disp. Technol. 3(2), 133-148 (2007).
[CrossRef]

Bhat, R.

M. Boroditsky, T. F. Krauss, R. Coccioli, R. Vrijen, R. Bhat, and E. Yablonovitch, "Light extraction from optically pumped light-emitting diode by thin-slab photonic crystals," Appl. Phys. Lett. 75(8), 1036-1038 (1999).
[CrossRef]

Bonnard, S.

S. Bonnard, P. Vincent, and M. Saillard, "Cross-borehole inverse scattering using a boundary finite-element method," Inverse Probl. 14(3), 521-534 (1998).
[CrossRef]

Boroditsky, M.

M. Boroditsky, T. F. Krauss, R. Coccioli, R. Vrijen, R. Bhat, and E. Yablonovitch, "Light extraction from optically pumped light-emitting diode by thin-slab photonic crystals," Appl. Phys. Lett. 75(8), 1036-1038 (1999).
[CrossRef]

Carminati, R.

J.-J. Greffet, and R. Carminati, "Image formation in near-field optics," Prog. Surf. Sci. 56(3), 133-237 (1997).
[CrossRef]

Cho, S. H.

Choi, W. J.

W. J. Choi, Q. H. Park, D. Kim, H. Jeon, C. Sone, and Y. Park, "FDTD Simulation for Light Extraction in a GaN-Based LED," J. Korean Phys. Soc. 49, 877-880 (2006).

Christy, R. W.

P. B. Johnson, and R. W. Christy, "Optical constants of the noble metals," Phys. Rev. B 6(12), 4370-4379 (1972).
[CrossRef]

Coccioli, R.

M. Boroditsky, T. F. Krauss, R. Coccioli, R. Vrijen, R. Bhat, and E. Yablonovitch, "Light extraction from optically pumped light-emitting diode by thin-slab photonic crystals," Appl. Phys. Lett. 75(8), 1036-1038 (1999).
[CrossRef]

Cui, M.

David, A.

J. J. Wierer, A. David, and M. M. Megens, "III-nitride photonic-crystal light-emitting diodes with high extraction efficiency," Nat. Photonics 3(3), 163-169 (2009).
[CrossRef]

A. David, H. Benisty, and C. Weisbuch, "Optimization of Light-Diffracting Photonic-Crystals for High Extraction Efficiency LEDs," J. Disp. Technol. 3(2), 133-148 (2007).
[CrossRef]

de Boer, D. K. G.

DenBaars, S. P.

T. Fujii, Y. Gao, R. Sharma, E. L. Hu, S. P. DenBaars, and S. Nakamura, "Increase in the extraction efficiency of GaN-based light-emitting diodes via surface roughening," Appl. Phys. Lett. 84(6), 855-857 (2004).
[CrossRef]

Do, Y. R.

Enoch, S.

A. L. Fehrembach, S. Enoch, and A. Sentenac, "Highly directive light sources using two-dimensional photonic crystal slabs," Appl. Phys. Lett. 79(26), 4280-4282 (2001).
[CrossRef]

Fan, S.

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, "Guided modes in photonic crystal slabs," Phys. Rev. B 60(8), 5751-5758 (1999).
[CrossRef]

Fehrembach, A. L.

A. L. Fehrembach, S. Enoch, and A. Sentenac, "Highly directive light sources using two-dimensional photonic crystal slabs," Appl. Phys. Lett. 79(26), 4280-4282 (2001).
[CrossRef]

Fujii, T.

T. Fujii, Y. Gao, R. Sharma, E. L. Hu, S. P. DenBaars, and S. Nakamura, "Increase in the extraction efficiency of GaN-based light-emitting diodes via surface roughening," Appl. Phys. Lett. 84(6), 855-857 (2004).
[CrossRef]

Gao, Y.

T. Fujii, Y. Gao, R. Sharma, E. L. Hu, S. P. DenBaars, and S. Nakamura, "Increase in the extraction efficiency of GaN-based light-emitting diodes via surface roughening," Appl. Phys. Lett. 84(6), 855-857 (2004).
[CrossRef]

Giergiel, J.

C. E. Reed, J. Giergiel, J. C. Hemminger, and S. Ushioda, "Dipole radiation in a multilayer geometry," Phys. Rev. B 36(9), 4990-5000 (1987).
[CrossRef]

Greffet, J.-J.

J.-J. Greffet, and R. Carminati, "Image formation in near-field optics," Prog. Surf. Sci. 56(3), 133-237 (1997).
[CrossRef]

Hemminger, J. C.

C. E. Reed, J. Giergiel, J. C. Hemminger, and S. Ushioda, "Dipole radiation in a multilayer geometry," Phys. Rev. B 36(9), 4990-5000 (1987).
[CrossRef]

Hu, E. L.

T. Fujii, Y. Gao, R. Sharma, E. L. Hu, S. P. DenBaars, and S. Nakamura, "Increase in the extraction efficiency of GaN-based light-emitting diodes via surface roughening," Appl. Phys. Lett. 84(6), 855-857 (2004).
[CrossRef]

Jeon, H.

W. J. Choi, Q. H. Park, D. Kim, H. Jeon, C. Sone, and Y. Park, "FDTD Simulation for Light Extraction in a GaN-Based LED," J. Korean Phys. Soc. 49, 877-880 (2006).

Joannopoulos, J. D.

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, "Guided modes in photonic crystal slabs," Phys. Rev. B 60(8), 5751-5758 (1999).
[CrossRef]

Johnson, P. B.

P. B. Johnson, and R. W. Christy, "Optical constants of the noble metals," Phys. Rev. B 6(12), 4370-4379 (1972).
[CrossRef]

Johnson, S. G.

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, "Guided modes in photonic crystal slabs," Phys. Rev. B 60(8), 5751-5758 (1999).
[CrossRef]

Kim, D.

W. J. Choi, Q. H. Park, D. Kim, H. Jeon, C. Sone, and Y. Park, "FDTD Simulation for Light Extraction in a GaN-Based LED," J. Korean Phys. Soc. 49, 877-880 (2006).

Kim, G. H.

Kim, S. H.

Kim, Y. C.

Kolodziejski, L. A.

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, "Guided modes in photonic crystal slabs," Phys. Rev. B 60(8), 5751-5758 (1999).
[CrossRef]

Krauss, T. F.

M. Boroditsky, T. F. Krauss, R. Coccioli, R. Vrijen, R. Bhat, and E. Yablonovitch, "Light extraction from optically pumped light-emitting diode by thin-slab photonic crystals," Appl. Phys. Lett. 75(8), 1036-1038 (1999).
[CrossRef]

Lee, Y. H.

Lee, Y. J.

Lemarchand, F.

Litman, A.

Maystre, D.

Megens, M. M.

J. J. Wierer, A. David, and M. M. Megens, "III-nitride photonic-crystal light-emitting diodes with high extraction efficiency," Nat. Photonics 3(3), 163-169 (2009).
[CrossRef]

Nakamura, S.

T. Fujii, Y. Gao, R. Sharma, E. L. Hu, S. P. DenBaars, and S. Nakamura, "Increase in the extraction efficiency of GaN-based light-emitting diodes via surface roughening," Appl. Phys. Lett. 84(6), 855-857 (2004).
[CrossRef]

Norton, S. J.

S. J. Norton, "Iterative inverse scattering algorithms: Methods of computing Frechet derivatives," J. Acoust. Soc. Am. 106(5), 2653-2660 (1999).
[CrossRef]

Park, Q. H.

W. J. Choi, Q. H. Park, D. Kim, H. Jeon, C. Sone, and Y. Park, "FDTD Simulation for Light Extraction in a GaN-Based LED," J. Korean Phys. Soc. 49, 877-880 (2006).

Park, Y.

W. J. Choi, Q. H. Park, D. Kim, H. Jeon, C. Sone, and Y. Park, "FDTD Simulation for Light Extraction in a GaN-Based LED," J. Korean Phys. Soc. 49, 877-880 (2006).

Potton, R. J.

R. J. Potton, "Reciprocity in Optics," Rep. Prog. Phys. 67(5), 717-754 (2004).
[CrossRef]

Reed, C. E.

C. E. Reed, J. Giergiel, J. C. Hemminger, and S. Ushioda, "Dipole radiation in a multilayer geometry," Phys. Rev. B 36(9), 4990-5000 (1987).
[CrossRef]

Rigneault, H.

Roger, A.

A. Roger, "Reciprocity theorem applied to the computation of functional derivatives of the scattering matrix," Electromagnetics 2(1), 69-83 (1982).
[CrossRef]

A. Roger, and D. Maystre, "Inverse scattering method in electromagnetic optics- Application to diffraction gratings," J. Opt. Soc. Am. 70(12), 1483-1495 (1980).
[CrossRef]

Saillard, M.

S. Bonnard, P. Vincent, and M. Saillard, "Cross-borehole inverse scattering using a boundary finite-element method," Inverse Probl. 14(3), 521-534 (1998).
[CrossRef]

Sentenac, A.

A. L. Fehrembach, S. Enoch, and A. Sentenac, "Highly directive light sources using two-dimensional photonic crystal slabs," Appl. Phys. Lett. 79(26), 4280-4282 (2001).
[CrossRef]

H. Rigneault, F. Lemarchand, and A. Sentenac, "Dipole radiation into grating structures," J. Opt. Soc. Am. A 17(6), 1048-1058 (2000).
[CrossRef]

Sharma, R.

T. Fujii, Y. Gao, R. Sharma, E. L. Hu, S. P. DenBaars, and S. Nakamura, "Increase in the extraction efficiency of GaN-based light-emitting diodes via surface roughening," Appl. Phys. Lett. 84(6), 855-857 (2004).
[CrossRef]

Sone, C.

W. J. Choi, Q. H. Park, D. Kim, H. Jeon, C. Sone, and Y. Park, "FDTD Simulation for Light Extraction in a GaN-Based LED," J. Korean Phys. Soc. 49, 877-880 (2006).

Song, Y. W.

Suemune, I.

M. Yamanishi, and I. Suemune, "Comment on polarization dependent momentum matrix elements in quantum well lasers," Jpn. J. Appl. Phys. 23(Part 2, No. 1), L35-L36 (1984).
[CrossRef]

Urbach, H. P.

Ushioda, S.

C. E. Reed, J. Giergiel, J. C. Hemminger, and S. Ushioda, "Dipole radiation in a multilayer geometry," Phys. Rev. B 36(9), 4990-5000 (1987).
[CrossRef]

Villeneuve, P. R.

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, "Guided modes in photonic crystal slabs," Phys. Rev. B 60(8), 5751-5758 (1999).
[CrossRef]

Vincent, P.

S. Bonnard, P. Vincent, and M. Saillard, "Cross-borehole inverse scattering using a boundary finite-element method," Inverse Probl. 14(3), 521-534 (1998).
[CrossRef]

Vrijen, R.

M. Boroditsky, T. F. Krauss, R. Coccioli, R. Vrijen, R. Bhat, and E. Yablonovitch, "Light extraction from optically pumped light-emitting diode by thin-slab photonic crystals," Appl. Phys. Lett. 75(8), 1036-1038 (1999).
[CrossRef]

Wachters, A. J.

Wei, X.

Weisbuch, C.

A. David, H. Benisty, and C. Weisbuch, "Optimization of Light-Diffracting Photonic-Crystals for High Extraction Efficiency LEDs," J. Disp. Technol. 3(2), 133-148 (2007).
[CrossRef]

Wierer, J. J.

J. J. Wierer, A. David, and M. M. Megens, "III-nitride photonic-crystal light-emitting diodes with high extraction efficiency," Nat. Photonics 3(3), 163-169 (2009).
[CrossRef]

Yablonovitch, E.

M. Boroditsky, T. F. Krauss, R. Coccioli, R. Vrijen, R. Bhat, and E. Yablonovitch, "Light extraction from optically pumped light-emitting diode by thin-slab photonic crystals," Appl. Phys. Lett. 75(8), 1036-1038 (1999).
[CrossRef]

Yamanishi, M.

M. Yamanishi, and I. Suemune, "Comment on polarization dependent momentum matrix elements in quantum well lasers," Jpn. J. Appl. Phys. 23(Part 2, No. 1), L35-L36 (1984).
[CrossRef]

Appl. Phys. Lett. (3)

T. Fujii, Y. Gao, R. Sharma, E. L. Hu, S. P. DenBaars, and S. Nakamura, "Increase in the extraction efficiency of GaN-based light-emitting diodes via surface roughening," Appl. Phys. Lett. 84(6), 855-857 (2004).
[CrossRef]

M. Boroditsky, T. F. Krauss, R. Coccioli, R. Vrijen, R. Bhat, and E. Yablonovitch, "Light extraction from optically pumped light-emitting diode by thin-slab photonic crystals," Appl. Phys. Lett. 75(8), 1036-1038 (1999).
[CrossRef]

A. L. Fehrembach, S. Enoch, and A. Sentenac, "Highly directive light sources using two-dimensional photonic crystal slabs," Appl. Phys. Lett. 79(26), 4280-4282 (2001).
[CrossRef]

Electromagnetics (1)

A. Roger, "Reciprocity theorem applied to the computation of functional derivatives of the scattering matrix," Electromagnetics 2(1), 69-83 (1982).
[CrossRef]

Inverse Probl. (1)

S. Bonnard, P. Vincent, and M. Saillard, "Cross-borehole inverse scattering using a boundary finite-element method," Inverse Probl. 14(3), 521-534 (1998).
[CrossRef]

J. Acoust. Soc. Am. (1)

S. J. Norton, "Iterative inverse scattering algorithms: Methods of computing Frechet derivatives," J. Acoust. Soc. Am. 106(5), 2653-2660 (1999).
[CrossRef]

J. Disp. Technol. (1)

A. David, H. Benisty, and C. Weisbuch, "Optimization of Light-Diffracting Photonic-Crystals for High Extraction Efficiency LEDs," J. Disp. Technol. 3(2), 133-148 (2007).
[CrossRef]

J. Korean Phys. Soc. (1)

W. J. Choi, Q. H. Park, D. Kim, H. Jeon, C. Sone, and Y. Park, "FDTD Simulation for Light Extraction in a GaN-Based LED," J. Korean Phys. Soc. 49, 877-880 (2006).

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Jpn. J. Appl. Phys. (1)

M. Yamanishi, and I. Suemune, "Comment on polarization dependent momentum matrix elements in quantum well lasers," Jpn. J. Appl. Phys. 23(Part 2, No. 1), L35-L36 (1984).
[CrossRef]

Nat. Photonics (1)

J. J. Wierer, A. David, and M. M. Megens, "III-nitride photonic-crystal light-emitting diodes with high extraction efficiency," Nat. Photonics 3(3), 163-169 (2009).
[CrossRef]

Opt. Express (2)

Phys. Rev. B (3)

P. B. Johnson, and R. W. Christy, "Optical constants of the noble metals," Phys. Rev. B 6(12), 4370-4379 (1972).
[CrossRef]

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, "Guided modes in photonic crystal slabs," Phys. Rev. B 60(8), 5751-5758 (1999).
[CrossRef]

C. E. Reed, J. Giergiel, J. C. Hemminger, and S. Ushioda, "Dipole radiation in a multilayer geometry," Phys. Rev. B 36(9), 4990-5000 (1987).
[CrossRef]

Prog. Surf. Sci. (1)

J.-J. Greffet, and R. Carminati, "Image formation in near-field optics," Prog. Surf. Sci. 56(3), 133-237 (1997).
[CrossRef]

Rep. Prog. Phys. (1)

R. J. Potton, "Reciprocity in Optics," Rep. Prog. Phys. 67(5), 717-754 (2004).
[CrossRef]

Other (3)

L. D. Landau, and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamom Press, 1960).

D. G. Luenberger, Optimization by Vector Space Methods (Wiley-Interscience, 1967).

E. F. Schubert, Light-Emitting Diodes (Cambridge University Press, 2006).
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

(left) A typical photonic crystal LED geometry. (right) Schematic of one periodic cell with pitch Λ of a periodic LED. Two different dipole sources p�� and p are shown, one in the active layer �� and the other in the half-space above interface Γ.

Fig. 2
Fig. 2

Angle dependent intensity at the wavelength of 450nm of isotropically radiating dipoles in a geometry with 700nm thick GaN, an active layer 100nm above the metal contact, and a grating with a pitch of 250nm and a depth of 5nm. Values are normalized by the intensity of the flat geometry I flat 0 at kx = 0. The solid and dashed vertical lines depict, respectively, the location of the S and P waveguide modes of the flat geometry. The insets show the corresponding radiation patterns. (a) The metal contact is considered a perfect metallic conductor. (b) The metal contact is considered to be silver (Ag) with complex permittivity.

Fig. 3
Fig. 3

Gateaux derivative at the surface Γ as defined for Fig. 2. The x-axis is cut as to emphasize the derivative close to the resonances. (a) Gateaux derivative with respect to the average layer thickness t. (b) Gateaux derivative with respect to the grating depth d.

Fig. 4
Fig. 4

Radiated intensity of an LED with a block grating and pitch Λ = 450nm, normalized to the maximum intensity obtainable with an unpatterned geometry. The indicated paths are optimization paths from arbitrary starting points leading to several local maxima.

Fig. 5
Fig. 5

Plot of the radiated intensity in a cone with 60° opening angle (x < 0.5) resulting from a local incoherent dipole as a function of its position within the grating. The plot on the right is the radiated intensity of incoherent dipoles in a thin flat active layer at the corresponding z-coordinate. (a) After optimizing with surface 2 parameters. (b) After optimizing a surface parameterized by 10 parameters.

Fig. 6
Fig. 6

Radiated intensity of a grating with t = 700nm and d = 300nm as a function of the normalized pitch Λ and angle x. Red shades indicate contributions from p-polarized emission, while blue shades indicate the contribution by s-polarized emission. The plot on the right gives the total radiated intensity in a cone with a 60° opening angle (x < 0.5).

Equations (38)

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

E J 1 J 2 d 3 r = E J 2 J 1 d 3 r ,
E p 𝒪 ( k ^ r p ) p = E k ( r p 𝒪 ) p 𝒪 .
E k inc ( r ) = k 2 4 π ɛ 0 r e ikr e ik ( k ^ r ) ( k ^ × p ) × k ^ ,
E k inc ( r ) = k 2 4 π ɛ 0 r e ikr e i k inc r p .
ω 2 μ 0 ɛ ( r ) E k ν ( r ) × × E k ν ( r ) = 0 , E k ν ( r ) E k inc , ν ( r ) satisfies the o . r . c ,
lim r p E p 𝒪 ( r p k ^ ) ν ^ = E k ν ( r P 𝒪 ) p 𝒪 .
lim r p | E p 𝒪 ( r p k ^ ) | 2 = lim r p ν = S , P | E p 𝒪 ( r p k ^ ) ν ^ | 2 = ν = S , P | E k ν ( r p 𝒪 ) p 𝒪 | 2 .
I ( k , r p 𝒪 ) = 1 2 ɛ 0 ɛ 2 μ 0 1 4 π 0 2 π 0 π ν = S , P | E k ν ( r p 𝒪 ) p 𝒪 | 2 sin θ d θ d ϕ = p 𝒪 2 6 ɛ 0 ɛ 2 μ 0 ν = S , P | E k ν ( r p 𝒪 ) | 2 .
I ˜ ( k , r p 𝒪 ) = 1 2 ɛ 0 ɛ 2 μ 0 1 2 π 0 2 π ν = S , P | E k ν ( r p 𝒪 ) p 𝒪 | 2 d ϕ = p 𝒪 2 4 ɛ 0 ɛ 2 μ 0 ν = S , P ( | E k , x ν ( r p 𝒪 ) | 2 + | E k , y ν ( r p 𝒪 ) | 2 ) .
I ( k ^ ) = 𝒪 ν = S , P | E k ν ( r p 𝒪 ) | 2 d 3 r p 𝒪 .
I ( Ω ) = I ( k ^ ) d Ω ,
ɛ ( Γ ) ( x , z ) = { ɛ 0 ɛ 1 for z Γ ( x ) b ; ɛ 0 ɛ 1 for z Γ ( x ) + b ; smooth for Γ ( x ) b z Γ ( x ) + b .
S b Γ = { ( x , z ) , 0 x a , Γ ( x ) b z Γ ( x ) + b } .
I ( Γ ) = ν = S , P 𝒪 | E ν ( Γ ) | 2 d x d z ,
δ I ( Γ ; h ) = lim ξ 0 1 ξ [ I ( Γ + ξ h ) I ( Γ ) ] = 2 Re [ 𝒪 δ E ( Γ ; h ) E ( Γ ) * d x d z ] ,
ω 2 μ 0 ɛ ( Γ ) E ( Γ ) × × E ( Γ ) 0 , E ( Γ ) E inc satisfies the o . r . c ,
ω 2 μ 0 ɛ ( Γ + ξ h ) E ( Γ + ξ h ) × × E ( Γ + ξ h ) = 0 , E ( Γ + ξ h ) E inc satisfies the o . r . c ,
ω 2 μ 0 ɛ ( Γ ) [ E ( Γ ) E ( Γ + ξ h ) ] × × [ E ( Γ ) E ( Γ + ξ h ) ] = ω 2 μ 0 [ ɛ ( Γ + ξ h ) ɛ ( Γ ) ] E ( Γ + ξ h ) , E ( Γ ) E ( Γ + ξ h ) satisfies the o . r . c .
ɛ ( Γ + ξ h ) ( x , z ) = ɛ ( Γ ) ( x , z ξ h ( x ) ) for all x , z .
lim ξ 0 [ ɛ ( Γ + ξ h ) ( x , z ) ɛ ( Γ ) ( x , z ) ] = lim ξ 0 [ ɛ ( Γ ) ( x , z ξ h ( x ) ) ɛ ( Γ ) ( x , z ) ] = ɛ ( Γ ) z ( x , z ) h ( x ) 𝟙 S b Γ ( x , z ) ,
ω 2 μ 0 ɛ ( Γ ) δ E ( Γ ; h ) × × δ E ( Γ ; h ) = ω 2 μ 0 ɛ ( Γ ) z h E ( Γ ) 𝟙 S b Γ . δ E ( Γ ; h ) satisfies the o . r . c .
J Γ ( x , z ) = i ω ɛ ( Γ ) z ( x , z ) h ( x ) E ( Γ ) ( x , z ) 𝟙 S b Γ ( x , z ) ,
J 𝒪 = E ( Γ ) * 𝟙 𝒪 .
ω 2 μ 0 ɛ ( Γ ) E J 𝒪 ( Γ ) × × E J 𝒪 ( Γ ) = i ω μ 0 E ( Γ ) * 𝟙 𝒪 , E J 𝒪 ( Γ ) satisfies the o . r . c .
δ I ( Γ ; h ) = 2 Re [ 𝒪 δ E ( Γ ; h ) J 𝒪 d x d z ] = 2 Re [ J Γ E J 𝒪 ( Γ ) d x d z ] = 2 Im [ ω S b Γ ɛ ( Γ ) z h E ( Γ ) E J 𝒪 ( Γ ) d x d z ]
h ˜ ( x ) = C Im [ Γ ( x ) b Γ ( x ) + b ɛ ( Γ ) z ( x , z ) E ( Γ ) ( x , z ) E J 𝒪 ( x , z ) d z ] ,
h ˜ ( x ) = C Im [ ( ɛ 2 ɛ 1 ) E || ( Γ ) ( x , Γ ( x ) ) E J 𝒪 , || ( x , Γ ( x ) ) ɛ 2 ( 1 ɛ 2 ɛ 1 ) E ( Γ ) ( x , Γ ( x ) + 0 ) E J 𝒪 , ( x , Γ ( x ) + 0 ) ] .
0 Λ ɛ 0 ɛ 2 h ( x ) E ( Γ ) ( x , Γ ( x ) + b ) E J 𝒪 ( x , Γ ( x ) + b ) d x
0 Λ ɛ 0 ɛ 1 h ( x ) E ( Γ ) ( x , Γ ( x ) b ) E J 𝒪 ( x , Γ ( x ) b ) d x
0 Λ Γ ( x ) b Γ ( x ) + b ɛ ( Γ ) ( x , z ) h ( x ) E ( Γ ) z ( x , z ) E J 𝒪 ( x , z ) d x d z
0 Λ Γ ( x ) b Γ ( x ) + b ɛ ( Γ ) ( x , z ) h ( x ) E ( Γ ) ( x , z ) E J 𝒪 z ( x , z ) d x d z .
0 Λ ɛ 0 ɛ 2 h ( x ) E || ( Γ ) ( x , Γ ( x ) ) E J 𝒪 , || ( x , Γ ( x ) ) d x 0 Λ ɛ 0 ɛ 1 h ( x ) E || ( Γ ) ( x , Γ ( x ) ) E J 𝒪 , || ( x , Γ ( x ) ) d x = 0 Λ ɛ 0 ( ɛ 2 ɛ 1 ) h ( x ) E || ( Γ ) ( x , Γ ( x ) . E J 𝒪 , || ( x , Γ ( x ) ) d x .
E ( x , Γ ( x ) + 0 ) E ( x , Γ ( x ) 0 ) = ( 1 ɛ 2 ɛ 1 ) E ( x , Γ ( x ) + 0 ) .
0 Λ ɛ 0 ɛ 2 ( 1 ɛ 2 ɛ 1 ) h ( x ) E ( Γ ) ( x , Γ ( x ) + 0 ) E J 𝒪 , ( x , Γ ( x ) + 0 ) d x .
lim b 0 E ( Γ ) z = [ E ( Γ + 0 ) E ( Γ 0 ) ] δ Γ ,
0 Λ h ( x ) ɛ 0 ɛ 2 ( 1 ɛ 2 ɛ 1 ) E ( Γ ) ( x , Γ ( x ) + 0 ) E J 𝒪 , ( x , Γ ( x ) + 0 ) d x .
δ I ( Γ ; h ) = 2 ω ɛ 0 Im 0 Λ h ( x ) [ ( ɛ 2 ɛ 1 ) E || ( Γ ) ( x , Γ ( x ) ) E j 𝒪 , || ( x , Γ ( x ) ) ɛ 2 ( 1 ɛ 2 ɛ 1 ) E ( Γ ) ( x , Γ ( x ) + 0 ) E J 𝒪 , ( x , Γ ( x ) + 0 ) ] d x .
h ˜ ( x ) = C Im [ ( ɛ 2 ɛ 1 ) E || ( Γ ) ( x , Γ ( x ) ) E J 𝒪 , || ( x , Γ ( x ) ) ɛ 2 ( 1 ɛ 2 ɛ 1 ) E ( Γ ) ( x , Γ ( x ) + 0 ) E J 𝒪 , ( x , Γ ( x ) + 0 ) ] ,

Metrics