Abstract

We present an investigation of disordered photonic crystals (PhCs) based on the combination of photonic Wannier functions with the concept of the coherent potential approximation (CPA). In particular, we provide the theoretical foundation of a real-space cluster CPA that is causal, enforces the proper symmetries of the effective medium, and includes effects of multiple scattering of the same and nearby defects, which is essential for strong defects. Based on this, we present results for the density of states of disordered PhCs for different types of disorder. Our results are thus relevant to such diverse areas as random lasing and the analysis of fabricational imperfections in PhCs.

© 2012 Optical Society of America

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  1. E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987).
    [CrossRef]
  2. S. John, Phys. Rev. Lett. 58, 2486 (1987).
    [CrossRef]
  3. K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, Phys. Rep. 444, 101 (2007).
    [CrossRef]
  4. L. Sapienza, H. Thyrrestrup, S. Stobbe, P. D. Garcia, S. Smolka, and P. Lodahl, Science 327, 1352 (2010).
    [CrossRef]
  5. V. V. Zalamai, V. V. Ursaki, C. Klingshirn, H. Kalt, G. A. Emelchenko, and A. N. Redkin, Appl. Phys. B 97, 817 (2009).
    [CrossRef]
  6. E. Kuramochi, M. Notomi, S. Hughes, A. Shinya, T. Watanabe, and L. Ramunno, Phys. Rev. B 72, 161318(R) (2005).
    [CrossRef]
  7. T. Kaplan and M. Mostoller, Phys. Rev. B 9, 1783 (1974).
    [CrossRef]
  8. D. M. Esterling, Phys. Rev. B 12, 1596 (1975).
    [CrossRef]
  9. K. Busch, C. Blum, A. M. Graham, D. Hermann, M. Köhl, P. Mack, and C. Wolff, J. Mod. Opt. 58, 365 (2011).
    [CrossRef]
  10. K. Busch, S. F. Mingaleev, M. Schillinger, and D. Hermann, J. Phys. Condens. Matter 15, R1233 (2003).
    [CrossRef]
  11. A. Gonis, Green Functions for Ordered and Disordered Systems (North-Holland, 1992).
  12. H. Zhou, J. Fallert, J. Sartor, R. J. B. Dietz, C. Klingshirn, H. Kalt, D. Weissenberger, D. Gerthsen, H. Zeng, and W. Cai, Appl. Phys. Lett. 92, 132112 (2008).
    [CrossRef]

2011 (1)

K. Busch, C. Blum, A. M. Graham, D. Hermann, M. Köhl, P. Mack, and C. Wolff, J. Mod. Opt. 58, 365 (2011).
[CrossRef]

2010 (1)

L. Sapienza, H. Thyrrestrup, S. Stobbe, P. D. Garcia, S. Smolka, and P. Lodahl, Science 327, 1352 (2010).
[CrossRef]

2009 (1)

V. V. Zalamai, V. V. Ursaki, C. Klingshirn, H. Kalt, G. A. Emelchenko, and A. N. Redkin, Appl. Phys. B 97, 817 (2009).
[CrossRef]

2008 (1)

H. Zhou, J. Fallert, J. Sartor, R. J. B. Dietz, C. Klingshirn, H. Kalt, D. Weissenberger, D. Gerthsen, H. Zeng, and W. Cai, Appl. Phys. Lett. 92, 132112 (2008).
[CrossRef]

2007 (1)

K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, Phys. Rep. 444, 101 (2007).
[CrossRef]

2005 (1)

E. Kuramochi, M. Notomi, S. Hughes, A. Shinya, T. Watanabe, and L. Ramunno, Phys. Rev. B 72, 161318(R) (2005).
[CrossRef]

2003 (1)

K. Busch, S. F. Mingaleev, M. Schillinger, and D. Hermann, J. Phys. Condens. Matter 15, R1233 (2003).
[CrossRef]

1987 (2)

E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987).
[CrossRef]

S. John, Phys. Rev. Lett. 58, 2486 (1987).
[CrossRef]

1975 (1)

D. M. Esterling, Phys. Rev. B 12, 1596 (1975).
[CrossRef]

1974 (1)

T. Kaplan and M. Mostoller, Phys. Rev. B 9, 1783 (1974).
[CrossRef]

Blum, C.

K. Busch, C. Blum, A. M. Graham, D. Hermann, M. Köhl, P. Mack, and C. Wolff, J. Mod. Opt. 58, 365 (2011).
[CrossRef]

Busch, K.

K. Busch, C. Blum, A. M. Graham, D. Hermann, M. Köhl, P. Mack, and C. Wolff, J. Mod. Opt. 58, 365 (2011).
[CrossRef]

K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, Phys. Rep. 444, 101 (2007).
[CrossRef]

K. Busch, S. F. Mingaleev, M. Schillinger, and D. Hermann, J. Phys. Condens. Matter 15, R1233 (2003).
[CrossRef]

Cai, W.

H. Zhou, J. Fallert, J. Sartor, R. J. B. Dietz, C. Klingshirn, H. Kalt, D. Weissenberger, D. Gerthsen, H. Zeng, and W. Cai, Appl. Phys. Lett. 92, 132112 (2008).
[CrossRef]

Dietz, R. J. B.

H. Zhou, J. Fallert, J. Sartor, R. J. B. Dietz, C. Klingshirn, H. Kalt, D. Weissenberger, D. Gerthsen, H. Zeng, and W. Cai, Appl. Phys. Lett. 92, 132112 (2008).
[CrossRef]

Emelchenko, G. A.

V. V. Zalamai, V. V. Ursaki, C. Klingshirn, H. Kalt, G. A. Emelchenko, and A. N. Redkin, Appl. Phys. B 97, 817 (2009).
[CrossRef]

Esterling, D. M.

D. M. Esterling, Phys. Rev. B 12, 1596 (1975).
[CrossRef]

Fallert, J.

H. Zhou, J. Fallert, J. Sartor, R. J. B. Dietz, C. Klingshirn, H. Kalt, D. Weissenberger, D. Gerthsen, H. Zeng, and W. Cai, Appl. Phys. Lett. 92, 132112 (2008).
[CrossRef]

Garcia, P. D.

L. Sapienza, H. Thyrrestrup, S. Stobbe, P. D. Garcia, S. Smolka, and P. Lodahl, Science 327, 1352 (2010).
[CrossRef]

Gerthsen, D.

H. Zhou, J. Fallert, J. Sartor, R. J. B. Dietz, C. Klingshirn, H. Kalt, D. Weissenberger, D. Gerthsen, H. Zeng, and W. Cai, Appl. Phys. Lett. 92, 132112 (2008).
[CrossRef]

Gonis, A.

A. Gonis, Green Functions for Ordered and Disordered Systems (North-Holland, 1992).

Graham, A. M.

K. Busch, C. Blum, A. M. Graham, D. Hermann, M. Köhl, P. Mack, and C. Wolff, J. Mod. Opt. 58, 365 (2011).
[CrossRef]

Hermann, D.

K. Busch, C. Blum, A. M. Graham, D. Hermann, M. Köhl, P. Mack, and C. Wolff, J. Mod. Opt. 58, 365 (2011).
[CrossRef]

K. Busch, S. F. Mingaleev, M. Schillinger, and D. Hermann, J. Phys. Condens. Matter 15, R1233 (2003).
[CrossRef]

Hughes, S.

E. Kuramochi, M. Notomi, S. Hughes, A. Shinya, T. Watanabe, and L. Ramunno, Phys. Rev. B 72, 161318(R) (2005).
[CrossRef]

John, S.

S. John, Phys. Rev. Lett. 58, 2486 (1987).
[CrossRef]

Kalt, H.

V. V. Zalamai, V. V. Ursaki, C. Klingshirn, H. Kalt, G. A. Emelchenko, and A. N. Redkin, Appl. Phys. B 97, 817 (2009).
[CrossRef]

H. Zhou, J. Fallert, J. Sartor, R. J. B. Dietz, C. Klingshirn, H. Kalt, D. Weissenberger, D. Gerthsen, H. Zeng, and W. Cai, Appl. Phys. Lett. 92, 132112 (2008).
[CrossRef]

Kaplan, T.

T. Kaplan and M. Mostoller, Phys. Rev. B 9, 1783 (1974).
[CrossRef]

Klingshirn, C.

V. V. Zalamai, V. V. Ursaki, C. Klingshirn, H. Kalt, G. A. Emelchenko, and A. N. Redkin, Appl. Phys. B 97, 817 (2009).
[CrossRef]

H. Zhou, J. Fallert, J. Sartor, R. J. B. Dietz, C. Klingshirn, H. Kalt, D. Weissenberger, D. Gerthsen, H. Zeng, and W. Cai, Appl. Phys. Lett. 92, 132112 (2008).
[CrossRef]

Köhl, M.

K. Busch, C. Blum, A. M. Graham, D. Hermann, M. Köhl, P. Mack, and C. Wolff, J. Mod. Opt. 58, 365 (2011).
[CrossRef]

Kuramochi, E.

E. Kuramochi, M. Notomi, S. Hughes, A. Shinya, T. Watanabe, and L. Ramunno, Phys. Rev. B 72, 161318(R) (2005).
[CrossRef]

Linden, S.

K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, Phys. Rep. 444, 101 (2007).
[CrossRef]

Lodahl, P.

L. Sapienza, H. Thyrrestrup, S. Stobbe, P. D. Garcia, S. Smolka, and P. Lodahl, Science 327, 1352 (2010).
[CrossRef]

Mack, P.

K. Busch, C. Blum, A. M. Graham, D. Hermann, M. Köhl, P. Mack, and C. Wolff, J. Mod. Opt. 58, 365 (2011).
[CrossRef]

Mingaleev, S. F.

K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, Phys. Rep. 444, 101 (2007).
[CrossRef]

K. Busch, S. F. Mingaleev, M. Schillinger, and D. Hermann, J. Phys. Condens. Matter 15, R1233 (2003).
[CrossRef]

Mostoller, M.

T. Kaplan and M. Mostoller, Phys. Rev. B 9, 1783 (1974).
[CrossRef]

Notomi, M.

E. Kuramochi, M. Notomi, S. Hughes, A. Shinya, T. Watanabe, and L. Ramunno, Phys. Rev. B 72, 161318(R) (2005).
[CrossRef]

Ramunno, L.

E. Kuramochi, M. Notomi, S. Hughes, A. Shinya, T. Watanabe, and L. Ramunno, Phys. Rev. B 72, 161318(R) (2005).
[CrossRef]

Redkin, A. N.

V. V. Zalamai, V. V. Ursaki, C. Klingshirn, H. Kalt, G. A. Emelchenko, and A. N. Redkin, Appl. Phys. B 97, 817 (2009).
[CrossRef]

Sapienza, L.

L. Sapienza, H. Thyrrestrup, S. Stobbe, P. D. Garcia, S. Smolka, and P. Lodahl, Science 327, 1352 (2010).
[CrossRef]

Sartor, J.

H. Zhou, J. Fallert, J. Sartor, R. J. B. Dietz, C. Klingshirn, H. Kalt, D. Weissenberger, D. Gerthsen, H. Zeng, and W. Cai, Appl. Phys. Lett. 92, 132112 (2008).
[CrossRef]

Schillinger, M.

K. Busch, S. F. Mingaleev, M. Schillinger, and D. Hermann, J. Phys. Condens. Matter 15, R1233 (2003).
[CrossRef]

Shinya, A.

E. Kuramochi, M. Notomi, S. Hughes, A. Shinya, T. Watanabe, and L. Ramunno, Phys. Rev. B 72, 161318(R) (2005).
[CrossRef]

Smolka, S.

L. Sapienza, H. Thyrrestrup, S. Stobbe, P. D. Garcia, S. Smolka, and P. Lodahl, Science 327, 1352 (2010).
[CrossRef]

Stobbe, S.

L. Sapienza, H. Thyrrestrup, S. Stobbe, P. D. Garcia, S. Smolka, and P. Lodahl, Science 327, 1352 (2010).
[CrossRef]

Thyrrestrup, H.

L. Sapienza, H. Thyrrestrup, S. Stobbe, P. D. Garcia, S. Smolka, and P. Lodahl, Science 327, 1352 (2010).
[CrossRef]

Tkeshelashvili, L.

K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, Phys. Rep. 444, 101 (2007).
[CrossRef]

Ursaki, V. V.

V. V. Zalamai, V. V. Ursaki, C. Klingshirn, H. Kalt, G. A. Emelchenko, and A. N. Redkin, Appl. Phys. B 97, 817 (2009).
[CrossRef]

von Freymann, G.

K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, Phys. Rep. 444, 101 (2007).
[CrossRef]

Watanabe, T.

E. Kuramochi, M. Notomi, S. Hughes, A. Shinya, T. Watanabe, and L. Ramunno, Phys. Rev. B 72, 161318(R) (2005).
[CrossRef]

Wegener, M.

K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, Phys. Rep. 444, 101 (2007).
[CrossRef]

Weissenberger, D.

H. Zhou, J. Fallert, J. Sartor, R. J. B. Dietz, C. Klingshirn, H. Kalt, D. Weissenberger, D. Gerthsen, H. Zeng, and W. Cai, Appl. Phys. Lett. 92, 132112 (2008).
[CrossRef]

Wolff, C.

K. Busch, C. Blum, A. M. Graham, D. Hermann, M. Köhl, P. Mack, and C. Wolff, J. Mod. Opt. 58, 365 (2011).
[CrossRef]

Yablonovitch, E.

E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987).
[CrossRef]

Zalamai, V. V.

V. V. Zalamai, V. V. Ursaki, C. Klingshirn, H. Kalt, G. A. Emelchenko, and A. N. Redkin, Appl. Phys. B 97, 817 (2009).
[CrossRef]

Zeng, H.

H. Zhou, J. Fallert, J. Sartor, R. J. B. Dietz, C. Klingshirn, H. Kalt, D. Weissenberger, D. Gerthsen, H. Zeng, and W. Cai, Appl. Phys. Lett. 92, 132112 (2008).
[CrossRef]

Zhou, H.

H. Zhou, J. Fallert, J. Sartor, R. J. B. Dietz, C. Klingshirn, H. Kalt, D. Weissenberger, D. Gerthsen, H. Zeng, and W. Cai, Appl. Phys. Lett. 92, 132112 (2008).
[CrossRef]

Appl. Phys. B (1)

V. V. Zalamai, V. V. Ursaki, C. Klingshirn, H. Kalt, G. A. Emelchenko, and A. N. Redkin, Appl. Phys. B 97, 817 (2009).
[CrossRef]

Appl. Phys. Lett. (1)

H. Zhou, J. Fallert, J. Sartor, R. J. B. Dietz, C. Klingshirn, H. Kalt, D. Weissenberger, D. Gerthsen, H. Zeng, and W. Cai, Appl. Phys. Lett. 92, 132112 (2008).
[CrossRef]

J. Mod. Opt. (1)

K. Busch, C. Blum, A. M. Graham, D. Hermann, M. Köhl, P. Mack, and C. Wolff, J. Mod. Opt. 58, 365 (2011).
[CrossRef]

J. Phys. Condens. Matter (1)

K. Busch, S. F. Mingaleev, M. Schillinger, and D. Hermann, J. Phys. Condens. Matter 15, R1233 (2003).
[CrossRef]

Phys. Rep. (1)

K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, Phys. Rep. 444, 101 (2007).
[CrossRef]

Phys. Rev. B (3)

E. Kuramochi, M. Notomi, S. Hughes, A. Shinya, T. Watanabe, and L. Ramunno, Phys. Rev. B 72, 161318(R) (2005).
[CrossRef]

T. Kaplan and M. Mostoller, Phys. Rev. B 9, 1783 (1974).
[CrossRef]

D. M. Esterling, Phys. Rev. B 12, 1596 (1975).
[CrossRef]

Phys. Rev. Lett. (2)

E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987).
[CrossRef]

S. John, Phys. Rev. Lett. 58, 2486 (1987).
[CrossRef]

Science (1)

L. Sapienza, H. Thyrrestrup, S. Stobbe, P. D. Garcia, S. Smolka, and P. Lodahl, Science 327, 1352 (2010).
[CrossRef]

Other (1)

A. Gonis, Green Functions for Ordered and Disordered Systems (North-Holland, 1992).

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Figures (3)

Fig. 1.
Fig. 1.

Illustration of theoretical aspects of the cluster coherent potential approximation (CCPA).

Fig. 2.
Fig. 2.

Disorder-averaged DOS for the model system computed within SSCPA using six bands.

Fig. 3.
Fig. 3.

Disorder-averaged DOS for the model system computed within CCPA of different cluster sizes and using six bands.

Equations (7)

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

DOS(ω)2ωπVWSCn=1Im((Geff)nn,0(ω)).
G(α)=Geff+GeffVeff(α)G(α)=Geff+GeffTeff(α)Geff
Teff(α)=C1[teff,C1(α)+C2C2C1teff,C1(α)Geffteff,C2(α)+C2C2C1C3C3C2teff,C1(α)Geffteff,C2(α)Geffteff,C3(α)+],
teff,C1(α)Geffteff,C2(α)Geffteff,CN(α)=teff,C1(α)Geffteff,C2(α)Geffteff,CN(α)
teff,Ci(α)=!0.
Ltotal(α)G(r,rsrc;ω)=δ(rrsrc),
Ltotal(α)=1ϵper(r)Δ2D+ω2c2(1+δϵ(α)(r)ϵper(r)).

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