Abstract

The spatial distribution of the local density of states (LDOS) at band edges on large-sized two-dimensional Thue–Morse dielectric systems has been investigated by means of a parallel implementation of the multipole expansion method. From the study of the LDOS participation number, the optical modes have been classified as critical, namely as intermediate between uniformly extended and exponentially localized. The exponent of power-law decay of their envelopes has been determined. Furthermore, multifractal analysis by a standard box-counting method has been performed. Almost all of the investigated modes exhibit a strong multifractal character. These results can be exploited for the designing of novel light source devices based on aperiodic order.

© 2012 Optical Society of America

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    [CrossRef]

2012 (3)

L. Dal Negro and S. V. Boriskina, “Deterministic aperiodic nanostructures for photonics and plasmonics applications,” Laser Photon. Rev. 6, 178–218 (2012).
[CrossRef]

J. Trevino, S. F. Liew, H. Noh, H. Cao, and L. D. Negro, “Geometrical structure, multifractal spectra and localized optical modes of aperiodic Vogel spirals,” Opt. Express 20, 3015–3033 (2012).
[CrossRef]

E. Maciá “Exploiting aperiodic designs in nanophotonic devices,” Rep. Prog. Phys. 75, 036502 (2012).
[CrossRef]

2011 (2)

H. Noh, J.-K. Yang, S. V. Boriskina, M. J. Rooks, G. S. Solomon, L. Dal Negro, and H. Cao, “Lasing in Thue–Morse structures with optimized aperiodicity,” Appl. Phys. Lett. 98, 201109 (2011).
[CrossRef]

J. Andreasen, A. A. Asatryan, L. C. Botten, M. A. Bryne, H. Cao, L. Ge, L. Labonté, P. Sebbah, A. D. Stone, H. E. Tureci, and C. Vanneste, “Mode of random lasers,” Adv. Opt. Photon. 3, 88–127 (2011).
[CrossRef]

2010 (3)

J.-K. Yang, S. V. Boriskina, H. Noh, M. J. Rooks, G. S. Solomon, L. Dal Negro, and H. Cao, “Demonstration of laser action in a pseudorandom medium,” Appl. Phys. Lett. 97, 223101(2010).
[CrossRef]

S. V. Boriskina, S. Y. K. Lee, J. J. Amsden, F. G. Omenetto, and L. Dal Negro “Formation of colorimetric fingerprints on nano-patterned deterministic aperiodic surfaces” Opt. Express 18, 14568–14576 (2010).
[CrossRef]

V. Krachmalnicoff, E. Castanié, Y. De Wilde, and R. Carminati, “Fluctuations of the local density of states probe localized surface plasmons on disordered metal films,” Phys. Rev. Lett. 105, 183901 (2010).
[CrossRef]

2009 (2)

A. Rodriguez, L. J. Vasquez, and R. A. Roemer, “Optimisation of multifractal analysis using box-size scaling,” Eur. Phys. J. B 67, 77–82 (2009).
[CrossRef]

C. Vanneste and P. Sebbah, “Complexity of two-dimensional quasimodes at the transition from weak scattering to Anderson localization,” Phys. Rev. A 79, 041802 (2009).
[CrossRef]

2008 (2)

2007 (4)

L. Moretti, I. Rea, L. De Stefano, and I. Rendina, “Periodic versus aperiodic: enhancing the sensitivity of porous silicon based optical sensors,” Appl. Phys. Lett. 90, 191112 (2007).
[CrossRef]

Y. Lai, Z.-Q. Zhang, C.-H. Chan, and L. Tsang, “Anomalous properties of the band-edge states in large two-dimensional photonic quasicrystals,” Phys. Rev. B 76, 1–5 (2007).
[CrossRef]

R. Lifshitz, “What is a crystal?” Z. Kristallogr. 222, 313–317 (2007).
[CrossRef]

L. Moretti and V. Mocella, “Two-dimensional photonic aperiodic crystals based on Thue–Morse sequence,” Opt. Express 15, 15314–15323 (2007).
[CrossRef]

2006 (3)

C. Rockstuhl, U. Peschel, and F. Lederer, “Correlation between single-cylinder properties and bandgap formation in photonic structures,” Opt. Lett. 31, 1741–1743 (2006).
[CrossRef]

E. Maciá, “The role of aperiodic order in science and technology,” Rep. Prog. Phys. 69, 397–441 (2006).
[CrossRef]

Y. Lai, Z. Zhang, C. Chan, and L. Tsang, “Gap structures and wave functions of classical waves in large-sized two-dimensional quasiperiodic structures,” Phys. Rev. B 74, 0543051 (2006).
[CrossRef]

2005 (3)

X. Jiang, Y. Zhang, S. Feng, K. C. Huang, Y. Yi, and J. D. Joannopoulos, “Photonic band gaps and localization in the Thue–Morse structures,” Appl. Phys. Lett. 86, 201110(2005).
[CrossRef]

R. Lifshitz, A. Arie, and A. Bahabad, “Photonic quasicrystals for nonlinear optical frequency conversion,” Phys. Rev. Lett. 95, 133901 (2005).
[CrossRef]

L. Dal Negro, J. H. Yi, V. Nguyen, Y. Yi, J. Michel, and L. C. Kimerling, “Spectrally enhanced light emission from aperiodic photonic structures,” Appl. Phys. Lett. 86, 261905 (2005).
[CrossRef]

2004 (1)

L. D. Negro, M. Stolfi, Y. Yi, J. Michel, X. Duan, L. C. Kimerling, J. LeBlanc, and J. Haavisto, “Photon band gap properties and omnidirectional reflectance in Si/SiO2 Thue–Morse quasicrystals,” Appl. Phys. Lett. 84, 5186–5188 (2004).
[CrossRef]

2003 (1)

L. Dal Negro, C. Oton, Z. Gaburro, L. Pavesi, P. Johnson, A. Lagendijk, R. Righini, M. Colocci, and D. Wiersma, “Light transport through the band-edge states of Fibonacci quasicrystals,” Phys. Rev. Lett. 90, 055501 (2003).
[CrossRef]

2001 (2)

A. A. Asatryan, S. Fabre, K. Busch, R. McPhedran, L. C. Botten, M. de Sterke, and N. A. Nicorovici, “Two-dimensional local density of states in two-dimensional photonic crystals,” Opt. Express 8, 191–196 (2001).
[CrossRef]

A. A. Asatryan, K. Busch, R. McPhedran, L. Botten, C. Martijn de Sterke, and N. Nicorovici, “Two-dimensional Green’s function and local density of states in photonic crystals consisting of a finite number of cylinders of infinite length,” Phys. Rev. E 63, 046612 (2001).
[CrossRef]

2000 (4)

1998 (1)

T. Rieth and M. Schreiber, “Numerical investigation of electronic wave functions in quasiperiodic lattices,” J. Phys. Condens. Matter 10, 783–800 (1998).
[CrossRef]

1997 (1)

1994 (1)

M. Janssen, “Multifractal analysis of broadly distributed observables at criticality,” Int. J. Mod. Phys. B 8, 943–984(1994).
[CrossRef]

1992 (1)

M. Dulea, M. Johansson, and R. Riklund, “Localization of electrons and electromagnetic waves in a deterministic aperiodic system,” Phys. Rev. B 45, 105–114 (1992).
[CrossRef]

1991 (1)

M. Schreiber and H. Grussbach, “Multifractal wave function at Anderson transition,” Phys. Rev. Lett. 67, 607–610 (1991).
[CrossRef]

1989 (1)

A. Chhabra and R. V. Jensen, “Direct determination of f(α) singularity spectrum,” Phys. Rev. Lett. 62, 1327–1330 (1989).
[CrossRef]

1988 (1)

Z. Cheng, R. Savit, and R. Merlin, “Structure and electronic properties of Thue–Morse lattices,” Phys. Rev. B 37, 4375–4382 (1988).
[CrossRef]

1985 (1)

J. L. Pichard and G. Sarma, “Power-law decay and fractal character of eigenstates in two-dimensional disordered systems,” J. Phys. C 18, 3457–3466 (1985).
[CrossRef]

1962 (1)

1951 (1)

M. Lax, “Multiple scattering of waves,” Rev. Mod. Phys. 23, 287–310 (1951).
[CrossRef]

Allouche, J.-P.

J.-P. Allouche and J. Shallit, Automatic Sequences: Theory, Applications, Generalizations (Cambridge University, 2003).

Amsden, J. J.

Andreasen, J.

Arie, A.

R. Lifshitz, A. Arie, and A. Bahabad, “Photonic quasicrystals for nonlinear optical frequency conversion,” Phys. Rev. Lett. 95, 133901 (2005).
[CrossRef]

Asatryan, A. A.

Bahabad, A.

R. Lifshitz, A. Arie, and A. Bahabad, “Photonic quasicrystals for nonlinear optical frequency conversion,” Phys. Rev. Lett. 95, 133901 (2005).
[CrossRef]

Blackford, L. S.

L. S. Blackford, J. Choi, A. Cleary, E. D’Azevedo, J. Demmel, I. Dhillon, J. Dongarra, S. Hammarling, G. Henry, A. Petitet, K. Stanley, D. Walker, and R. C. Whaley, ScaLAPACK Users’ Guide (SIAM, 1997).

Boriskina, S. V.

L. Dal Negro and S. V. Boriskina, “Deterministic aperiodic nanostructures for photonics and plasmonics applications,” Laser Photon. Rev. 6, 178–218 (2012).
[CrossRef]

H. Noh, J.-K. Yang, S. V. Boriskina, M. J. Rooks, G. S. Solomon, L. Dal Negro, and H. Cao, “Lasing in Thue–Morse structures with optimized aperiodicity,” Appl. Phys. Lett. 98, 201109 (2011).
[CrossRef]

J.-K. Yang, S. V. Boriskina, H. Noh, M. J. Rooks, G. S. Solomon, L. Dal Negro, and H. Cao, “Demonstration of laser action in a pseudorandom medium,” Appl. Phys. Lett. 97, 223101(2010).
[CrossRef]

S. V. Boriskina, S. Y. K. Lee, J. J. Amsden, F. G. Omenetto, and L. Dal Negro “Formation of colorimetric fingerprints on nano-patterned deterministic aperiodic surfaces” Opt. Express 18, 14568–14576 (2010).
[CrossRef]

S. V. Boriskina, A. Gopinath, and L. D. Negro, “Optical gap formation and localization properties of optical modes in deterministic aperiodic photonic structures,” Opt. Express 16, 18813–18826 (2008).
[CrossRef]

Botten, L.

A. A. Asatryan, K. Busch, R. McPhedran, L. Botten, C. Martijn de Sterke, and N. Nicorovici, “Two-dimensional Green’s function and local density of states in photonic crystals consisting of a finite number of cylinders of infinite length,” Phys. Rev. E 63, 046612 (2001).
[CrossRef]

Botten, L. C.

Bryne, M. A.

Busch, K.

A. A. Asatryan, S. Fabre, K. Busch, R. McPhedran, L. C. Botten, M. de Sterke, and N. A. Nicorovici, “Two-dimensional local density of states in two-dimensional photonic crystals,” Opt. Express 8, 191–196 (2001).
[CrossRef]

A. A. Asatryan, K. Busch, R. McPhedran, L. Botten, C. Martijn de Sterke, and N. Nicorovici, “Two-dimensional Green’s function and local density of states in photonic crystals consisting of a finite number of cylinders of infinite length,” Phys. Rev. E 63, 046612 (2001).
[CrossRef]

Cao, H.

J. Trevino, S. F. Liew, H. Noh, H. Cao, and L. D. Negro, “Geometrical structure, multifractal spectra and localized optical modes of aperiodic Vogel spirals,” Opt. Express 20, 3015–3033 (2012).
[CrossRef]

H. Noh, J.-K. Yang, S. V. Boriskina, M. J. Rooks, G. S. Solomon, L. Dal Negro, and H. Cao, “Lasing in Thue–Morse structures with optimized aperiodicity,” Appl. Phys. Lett. 98, 201109 (2011).
[CrossRef]

J. Andreasen, A. A. Asatryan, L. C. Botten, M. A. Bryne, H. Cao, L. Ge, L. Labonté, P. Sebbah, A. D. Stone, H. E. Tureci, and C. Vanneste, “Mode of random lasers,” Adv. Opt. Photon. 3, 88–127 (2011).
[CrossRef]

J.-K. Yang, S. V. Boriskina, H. Noh, M. J. Rooks, G. S. Solomon, L. Dal Negro, and H. Cao, “Demonstration of laser action in a pseudorandom medium,” Appl. Phys. Lett. 97, 223101(2010).
[CrossRef]

Carminati, R.

V. Krachmalnicoff, E. Castanié, Y. De Wilde, and R. Carminati, “Fluctuations of the local density of states probe localized surface plasmons on disordered metal films,” Phys. Rev. Lett. 105, 183901 (2010).
[CrossRef]

Castanié, E.

V. Krachmalnicoff, E. Castanié, Y. De Wilde, and R. Carminati, “Fluctuations of the local density of states probe localized surface plasmons on disordered metal films,” Phys. Rev. Lett. 105, 183901 (2010).
[CrossRef]

Chan, C.

Y. Lai, Z. Zhang, C. Chan, and L. Tsang, “Gap structures and wave functions of classical waves in large-sized two-dimensional quasiperiodic structures,” Phys. Rev. B 74, 0543051 (2006).
[CrossRef]

Chan, C.-H.

Y. Lai, Z.-Q. Zhang, C.-H. Chan, and L. Tsang, “Anomalous properties of the band-edge states in large two-dimensional photonic quasicrystals,” Phys. Rev. B 76, 1–5 (2007).
[CrossRef]

Cheng, Z.

Z. Cheng, R. Savit, and R. Merlin, “Structure and electronic properties of Thue–Morse lattices,” Phys. Rev. B 37, 4375–4382 (1988).
[CrossRef]

Chhabra, A.

A. Chhabra and R. V. Jensen, “Direct determination of f(α) singularity spectrum,” Phys. Rev. Lett. 62, 1327–1330 (1989).
[CrossRef]

Choi, J.

L. S. Blackford, J. Choi, A. Cleary, E. D’Azevedo, J. Demmel, I. Dhillon, J. Dongarra, S. Hammarling, G. Henry, A. Petitet, K. Stanley, D. Walker, and R. C. Whaley, ScaLAPACK Users’ Guide (SIAM, 1997).

Cleary, A.

L. S. Blackford, J. Choi, A. Cleary, E. D’Azevedo, J. Demmel, I. Dhillon, J. Dongarra, S. Hammarling, G. Henry, A. Petitet, K. Stanley, D. Walker, and R. C. Whaley, ScaLAPACK Users’ Guide (SIAM, 1997).

Colocci, M.

L. Dal Negro, C. Oton, Z. Gaburro, L. Pavesi, P. Johnson, A. Lagendijk, R. Righini, M. Colocci, and D. Wiersma, “Light transport through the band-edge states of Fibonacci quasicrystals,” Phys. Rev. Lett. 90, 055501 (2003).
[CrossRef]

D’Azevedo, E.

L. S. Blackford, J. Choi, A. Cleary, E. D’Azevedo, J. Demmel, I. Dhillon, J. Dongarra, S. Hammarling, G. Henry, A. Petitet, K. Stanley, D. Walker, and R. C. Whaley, ScaLAPACK Users’ Guide (SIAM, 1997).

Dal Negro, L.

L. Dal Negro and S. V. Boriskina, “Deterministic aperiodic nanostructures for photonics and plasmonics applications,” Laser Photon. Rev. 6, 178–218 (2012).
[CrossRef]

H. Noh, J.-K. Yang, S. V. Boriskina, M. J. Rooks, G. S. Solomon, L. Dal Negro, and H. Cao, “Lasing in Thue–Morse structures with optimized aperiodicity,” Appl. Phys. Lett. 98, 201109 (2011).
[CrossRef]

J.-K. Yang, S. V. Boriskina, H. Noh, M. J. Rooks, G. S. Solomon, L. Dal Negro, and H. Cao, “Demonstration of laser action in a pseudorandom medium,” Appl. Phys. Lett. 97, 223101(2010).
[CrossRef]

S. V. Boriskina, S. Y. K. Lee, J. J. Amsden, F. G. Omenetto, and L. Dal Negro “Formation of colorimetric fingerprints on nano-patterned deterministic aperiodic surfaces” Opt. Express 18, 14568–14576 (2010).
[CrossRef]

L. Dal Negro, J. H. Yi, V. Nguyen, Y. Yi, J. Michel, and L. C. Kimerling, “Spectrally enhanced light emission from aperiodic photonic structures,” Appl. Phys. Lett. 86, 261905 (2005).
[CrossRef]

L. Dal Negro, C. Oton, Z. Gaburro, L. Pavesi, P. Johnson, A. Lagendijk, R. Righini, M. Colocci, and D. Wiersma, “Light transport through the band-edge states of Fibonacci quasicrystals,” Phys. Rev. Lett. 90, 055501 (2003).
[CrossRef]

De Stefano, L.

L. Moretti, I. Rea, L. De Stefano, and I. Rendina, “Periodic versus aperiodic: enhancing the sensitivity of porous silicon based optical sensors,” Appl. Phys. Lett. 90, 191112 (2007).
[CrossRef]

de Sterke, M.

De Wilde, Y.

V. Krachmalnicoff, E. Castanié, Y. De Wilde, and R. Carminati, “Fluctuations of the local density of states probe localized surface plasmons on disordered metal films,” Phys. Rev. Lett. 105, 183901 (2010).
[CrossRef]

Demmel, J.

L. S. Blackford, J. Choi, A. Cleary, E. D’Azevedo, J. Demmel, I. Dhillon, J. Dongarra, S. Hammarling, G. Henry, A. Petitet, K. Stanley, D. Walker, and R. C. Whaley, ScaLAPACK Users’ Guide (SIAM, 1997).

Dhillon, I.

L. S. Blackford, J. Choi, A. Cleary, E. D’Azevedo, J. Demmel, I. Dhillon, J. Dongarra, S. Hammarling, G. Henry, A. Petitet, K. Stanley, D. Walker, and R. C. Whaley, ScaLAPACK Users’ Guide (SIAM, 1997).

Dongarra, J.

L. S. Blackford, J. Choi, A. Cleary, E. D’Azevedo, J. Demmel, I. Dhillon, J. Dongarra, S. Hammarling, G. Henry, A. Petitet, K. Stanley, D. Walker, and R. C. Whaley, ScaLAPACK Users’ Guide (SIAM, 1997).

Duan, X.

L. D. Negro, M. Stolfi, Y. Yi, J. Michel, X. Duan, L. C. Kimerling, J. LeBlanc, and J. Haavisto, “Photon band gap properties and omnidirectional reflectance in Si/SiO2 Thue–Morse quasicrystals,” Appl. Phys. Lett. 84, 5186–5188 (2004).
[CrossRef]

Dulea, M.

M. Dulea, M. Johansson, and R. Riklund, “Localization of electrons and electromagnetic waves in a deterministic aperiodic system,” Phys. Rev. B 45, 105–114 (1992).
[CrossRef]

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Feng, S.

X. Jiang, Y. Zhang, S. Feng, K. C. Huang, Y. Yi, and J. D. Joannopoulos, “Photonic band gaps and localization in the Thue–Morse structures,” Appl. Phys. Lett. 86, 201110(2005).
[CrossRef]

Gaburro, Z.

L. Dal Negro, C. Oton, Z. Gaburro, L. Pavesi, P. Johnson, A. Lagendijk, R. Righini, M. Colocci, and D. Wiersma, “Light transport through the band-edge states of Fibonacci quasicrystals,” Phys. Rev. Lett. 90, 055501 (2003).
[CrossRef]

Ge, L.

Gopinath, A.

Grimm, U.

H. Q. Yuan, U. Grimm, P. Repetowicz, and M. Schreiber, “Energy spectra, wavefunctions and quantum diffusion for quasiperiodic systems,” Phys. Rev. B 62, 15569–15578 (2000).
[CrossRef]

Grussbach, H.

M. Schreiber and H. Grussbach, “Multifractal wave function at Anderson transition,” Phys. Rev. Lett. 67, 607–610 (1991).
[CrossRef]

Haavisto, J.

L. D. Negro, M. Stolfi, Y. Yi, J. Michel, X. Duan, L. C. Kimerling, J. LeBlanc, and J. Haavisto, “Photon band gap properties and omnidirectional reflectance in Si/SiO2 Thue–Morse quasicrystals,” Appl. Phys. Lett. 84, 5186–5188 (2004).
[CrossRef]

Hammarling, S.

L. S. Blackford, J. Choi, A. Cleary, E. D’Azevedo, J. Demmel, I. Dhillon, J. Dongarra, S. Hammarling, G. Henry, A. Petitet, K. Stanley, D. Walker, and R. C. Whaley, ScaLAPACK Users’ Guide (SIAM, 1997).

Henry, G.

L. S. Blackford, J. Choi, A. Cleary, E. D’Azevedo, J. Demmel, I. Dhillon, J. Dongarra, S. Hammarling, G. Henry, A. Petitet, K. Stanley, D. Walker, and R. C. Whaley, ScaLAPACK Users’ Guide (SIAM, 1997).

Huang, K. C.

X. Jiang, Y. Zhang, S. Feng, K. C. Huang, Y. Yi, and J. D. Joannopoulos, “Photonic band gaps and localization in the Thue–Morse structures,” Appl. Phys. Lett. 86, 201110(2005).
[CrossRef]

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M. Janssen, “Multifractal analysis of broadly distributed observables at criticality,” Int. J. Mod. Phys. B 8, 943–984(1994).
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A. Chhabra and R. V. Jensen, “Direct determination of f(α) singularity spectrum,” Phys. Rev. Lett. 62, 1327–1330 (1989).
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X. Jiang, Y. Zhang, S. Feng, K. C. Huang, Y. Yi, and J. D. Joannopoulos, “Photonic band gaps and localization in the Thue–Morse structures,” Appl. Phys. Lett. 86, 201110(2005).
[CrossRef]

Joannopoulos, J. D.

X. Jiang, Y. Zhang, S. Feng, K. C. Huang, Y. Yi, and J. D. Joannopoulos, “Photonic band gaps and localization in the Thue–Morse structures,” Appl. Phys. Lett. 86, 201110(2005).
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Johansson, M.

M. Dulea, M. Johansson, and R. Riklund, “Localization of electrons and electromagnetic waves in a deterministic aperiodic system,” Phys. Rev. B 45, 105–114 (1992).
[CrossRef]

Johnson, P.

L. Dal Negro, C. Oton, Z. Gaburro, L. Pavesi, P. Johnson, A. Lagendijk, R. Righini, M. Colocci, and D. Wiersma, “Light transport through the band-edge states of Fibonacci quasicrystals,” Phys. Rev. Lett. 90, 055501 (2003).
[CrossRef]

Kimerling, L. C.

L. Dal Negro, J. H. Yi, V. Nguyen, Y. Yi, J. Michel, and L. C. Kimerling, “Spectrally enhanced light emission from aperiodic photonic structures,” Appl. Phys. Lett. 86, 261905 (2005).
[CrossRef]

L. D. Negro, M. Stolfi, Y. Yi, J. Michel, X. Duan, L. C. Kimerling, J. LeBlanc, and J. Haavisto, “Photon band gap properties and omnidirectional reflectance in Si/SiO2 Thue–Morse quasicrystals,” Appl. Phys. Lett. 84, 5186–5188 (2004).
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Lagendijk, A.

L. Dal Negro, C. Oton, Z. Gaburro, L. Pavesi, P. Johnson, A. Lagendijk, R. Righini, M. Colocci, and D. Wiersma, “Light transport through the band-edge states of Fibonacci quasicrystals,” Phys. Rev. Lett. 90, 055501 (2003).
[CrossRef]

Lai, Y.

Y. Lai, Z.-Q. Zhang, C.-H. Chan, and L. Tsang, “Anomalous properties of the band-edge states in large two-dimensional photonic quasicrystals,” Phys. Rev. B 76, 1–5 (2007).
[CrossRef]

Y. Lai, Z. Zhang, C. Chan, and L. Tsang, “Gap structures and wave functions of classical waves in large-sized two-dimensional quasiperiodic structures,” Phys. Rev. B 74, 0543051 (2006).
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L. D. Negro, M. Stolfi, Y. Yi, J. Michel, X. Duan, L. C. Kimerling, J. LeBlanc, and J. Haavisto, “Photon band gap properties and omnidirectional reflectance in Si/SiO2 Thue–Morse quasicrystals,” Appl. Phys. Lett. 84, 5186–5188 (2004).
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Maystre, D.

McPhedran, R.

A. A. Asatryan, S. Fabre, K. Busch, R. McPhedran, L. C. Botten, M. de Sterke, and N. A. Nicorovici, “Two-dimensional local density of states in two-dimensional photonic crystals,” Opt. Express 8, 191–196 (2001).
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A. A. Asatryan, K. Busch, R. McPhedran, L. Botten, C. Martijn de Sterke, and N. Nicorovici, “Two-dimensional Green’s function and local density of states in photonic crystals consisting of a finite number of cylinders of infinite length,” Phys. Rev. E 63, 046612 (2001).
[CrossRef]

McPhedran, R. C.

Merlin, R.

Z. Cheng, R. Savit, and R. Merlin, “Structure and electronic properties of Thue–Morse lattices,” Phys. Rev. B 37, 4375–4382 (1988).
[CrossRef]

Michel, J.

L. Dal Negro, J. H. Yi, V. Nguyen, Y. Yi, J. Michel, and L. C. Kimerling, “Spectrally enhanced light emission from aperiodic photonic structures,” Appl. Phys. Lett. 86, 261905 (2005).
[CrossRef]

L. D. Negro, M. Stolfi, Y. Yi, J. Michel, X. Duan, L. C. Kimerling, J. LeBlanc, and J. Haavisto, “Photon band gap properties and omnidirectional reflectance in Si/SiO2 Thue–Morse quasicrystals,” Appl. Phys. Lett. 84, 5186–5188 (2004).
[CrossRef]

Mirlin, A. D.

A. D. Mirlin, “Statistics of energy levels and eigenfunctions in disordered systems,” Phys. Rep. 326, 259–382 (2000).
[CrossRef]

Mocella, V.

L. Moretti and V. Mocella, “The square Thue–Morse tiling for photonic application,” Philos. Mag. B 88, 2275–2284(2008).
[CrossRef]

L. Moretti and V. Mocella, “Two-dimensional photonic aperiodic crystals based on Thue–Morse sequence,” Opt. Express 15, 15314–15323 (2007).
[CrossRef]

Moretti, L.

L. Moretti and V. Mocella, “The square Thue–Morse tiling for photonic application,” Philos. Mag. B 88, 2275–2284(2008).
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L. Moretti and V. Mocella, “Two-dimensional photonic aperiodic crystals based on Thue–Morse sequence,” Opt. Express 15, 15314–15323 (2007).
[CrossRef]

L. Moretti, I. Rea, L. De Stefano, and I. Rendina, “Periodic versus aperiodic: enhancing the sensitivity of porous silicon based optical sensors,” Appl. Phys. Lett. 90, 191112 (2007).
[CrossRef]

Negro, L. D.

Nguyen, V.

L. Dal Negro, J. H. Yi, V. Nguyen, Y. Yi, J. Michel, and L. C. Kimerling, “Spectrally enhanced light emission from aperiodic photonic structures,” Appl. Phys. Lett. 86, 261905 (2005).
[CrossRef]

Nicorovici, N.

A. A. Asatryan, K. Busch, R. McPhedran, L. Botten, C. Martijn de Sterke, and N. Nicorovici, “Two-dimensional Green’s function and local density of states in photonic crystals consisting of a finite number of cylinders of infinite length,” Phys. Rev. E 63, 046612 (2001).
[CrossRef]

Nicorovici, N. A.

Nicorovici, N.-A. P.

Noh, H.

J. Trevino, S. F. Liew, H. Noh, H. Cao, and L. D. Negro, “Geometrical structure, multifractal spectra and localized optical modes of aperiodic Vogel spirals,” Opt. Express 20, 3015–3033 (2012).
[CrossRef]

H. Noh, J.-K. Yang, S. V. Boriskina, M. J. Rooks, G. S. Solomon, L. Dal Negro, and H. Cao, “Lasing in Thue–Morse structures with optimized aperiodicity,” Appl. Phys. Lett. 98, 201109 (2011).
[CrossRef]

J.-K. Yang, S. V. Boriskina, H. Noh, M. J. Rooks, G. S. Solomon, L. Dal Negro, and H. Cao, “Demonstration of laser action in a pseudorandom medium,” Appl. Phys. Lett. 97, 223101(2010).
[CrossRef]

Omenetto, F. G.

Oton, C.

L. Dal Negro, C. Oton, Z. Gaburro, L. Pavesi, P. Johnson, A. Lagendijk, R. Righini, M. Colocci, and D. Wiersma, “Light transport through the band-edge states of Fibonacci quasicrystals,” Phys. Rev. Lett. 90, 055501 (2003).
[CrossRef]

Pavesi, L.

L. Dal Negro, C. Oton, Z. Gaburro, L. Pavesi, P. Johnson, A. Lagendijk, R. Righini, M. Colocci, and D. Wiersma, “Light transport through the band-edge states of Fibonacci quasicrystals,” Phys. Rev. Lett. 90, 055501 (2003).
[CrossRef]

Peschel, U.

Petitet, A.

L. S. Blackford, J. Choi, A. Cleary, E. D’Azevedo, J. Demmel, I. Dhillon, J. Dongarra, S. Hammarling, G. Henry, A. Petitet, K. Stanley, D. Walker, and R. C. Whaley, ScaLAPACK Users’ Guide (SIAM, 1997).

Pichard, J. L.

J. L. Pichard and G. Sarma, “Power-law decay and fractal character of eigenstates in two-dimensional disordered systems,” J. Phys. C 18, 3457–3466 (1985).
[CrossRef]

Rea, I.

L. Moretti, I. Rea, L. De Stefano, and I. Rendina, “Periodic versus aperiodic: enhancing the sensitivity of porous silicon based optical sensors,” Appl. Phys. Lett. 90, 191112 (2007).
[CrossRef]

Rendina, I.

L. Moretti, I. Rea, L. De Stefano, and I. Rendina, “Periodic versus aperiodic: enhancing the sensitivity of porous silicon based optical sensors,” Appl. Phys. Lett. 90, 191112 (2007).
[CrossRef]

Repetowicz, P.

H. Q. Yuan, U. Grimm, P. Repetowicz, and M. Schreiber, “Energy spectra, wavefunctions and quantum diffusion for quasiperiodic systems,” Phys. Rev. B 62, 15569–15578 (2000).
[CrossRef]

Rieth, T.

T. Rieth and M. Schreiber, “Numerical investigation of electronic wave functions in quasiperiodic lattices,” J. Phys. Condens. Matter 10, 783–800 (1998).
[CrossRef]

Righini, R.

L. Dal Negro, C. Oton, Z. Gaburro, L. Pavesi, P. Johnson, A. Lagendijk, R. Righini, M. Colocci, and D. Wiersma, “Light transport through the band-edge states of Fibonacci quasicrystals,” Phys. Rev. Lett. 90, 055501 (2003).
[CrossRef]

Riklund, R.

M. Dulea, M. Johansson, and R. Riklund, “Localization of electrons and electromagnetic waves in a deterministic aperiodic system,” Phys. Rev. B 45, 105–114 (1992).
[CrossRef]

Robinson, P. A.

Rockstuhl, C.

Rodriguez, A.

A. Rodriguez, L. J. Vasquez, and R. A. Roemer, “Optimisation of multifractal analysis using box-size scaling,” Eur. Phys. J. B 67, 77–82 (2009).
[CrossRef]

Roemer, R. A.

A. Rodriguez, L. J. Vasquez, and R. A. Roemer, “Optimisation of multifractal analysis using box-size scaling,” Eur. Phys. J. B 67, 77–82 (2009).
[CrossRef]

Rooks, M. J.

H. Noh, J.-K. Yang, S. V. Boriskina, M. J. Rooks, G. S. Solomon, L. Dal Negro, and H. Cao, “Lasing in Thue–Morse structures with optimized aperiodicity,” Appl. Phys. Lett. 98, 201109 (2011).
[CrossRef]

J.-K. Yang, S. V. Boriskina, H. Noh, M. J. Rooks, G. S. Solomon, L. Dal Negro, and H. Cao, “Demonstration of laser action in a pseudorandom medium,” Appl. Phys. Lett. 97, 223101(2010).
[CrossRef]

Sarma, G.

J. L. Pichard and G. Sarma, “Power-law decay and fractal character of eigenstates in two-dimensional disordered systems,” J. Phys. C 18, 3457–3466 (1985).
[CrossRef]

Savit, R.

Z. Cheng, R. Savit, and R. Merlin, “Structure and electronic properties of Thue–Morse lattices,” Phys. Rev. B 37, 4375–4382 (1988).
[CrossRef]

Schreiber, M.

H. Q. Yuan, U. Grimm, P. Repetowicz, and M. Schreiber, “Energy spectra, wavefunctions and quantum diffusion for quasiperiodic systems,” Phys. Rev. B 62, 15569–15578 (2000).
[CrossRef]

T. Rieth and M. Schreiber, “Numerical investigation of electronic wave functions in quasiperiodic lattices,” J. Phys. Condens. Matter 10, 783–800 (1998).
[CrossRef]

M. Schreiber and H. Grussbach, “Multifractal wave function at Anderson transition,” Phys. Rev. Lett. 67, 607–610 (1991).
[CrossRef]

Sebbah, P.

J. Andreasen, A. A. Asatryan, L. C. Botten, M. A. Bryne, H. Cao, L. Ge, L. Labonté, P. Sebbah, A. D. Stone, H. E. Tureci, and C. Vanneste, “Mode of random lasers,” Adv. Opt. Photon. 3, 88–127 (2011).
[CrossRef]

C. Vanneste and P. Sebbah, “Complexity of two-dimensional quasimodes at the transition from weak scattering to Anderson localization,” Phys. Rev. A 79, 041802 (2009).
[CrossRef]

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J.-P. Allouche and J. Shallit, Automatic Sequences: Theory, Applications, Generalizations (Cambridge University, 2003).

Solomon, G. S.

H. Noh, J.-K. Yang, S. V. Boriskina, M. J. Rooks, G. S. Solomon, L. Dal Negro, and H. Cao, “Lasing in Thue–Morse structures with optimized aperiodicity,” Appl. Phys. Lett. 98, 201109 (2011).
[CrossRef]

J.-K. Yang, S. V. Boriskina, H. Noh, M. J. Rooks, G. S. Solomon, L. Dal Negro, and H. Cao, “Demonstration of laser action in a pseudorandom medium,” Appl. Phys. Lett. 97, 223101(2010).
[CrossRef]

Stanley, K.

L. S. Blackford, J. Choi, A. Cleary, E. D’Azevedo, J. Demmel, I. Dhillon, J. Dongarra, S. Hammarling, G. Henry, A. Petitet, K. Stanley, D. Walker, and R. C. Whaley, ScaLAPACK Users’ Guide (SIAM, 1997).

Sterke, C. M. D.

Stolfi, M.

L. D. Negro, M. Stolfi, Y. Yi, J. Michel, X. Duan, L. C. Kimerling, J. LeBlanc, and J. Haavisto, “Photon band gap properties and omnidirectional reflectance in Si/SiO2 Thue–Morse quasicrystals,” Appl. Phys. Lett. 84, 5186–5188 (2004).
[CrossRef]

Stone, A. D.

Tayeb, G.

Trevino, J.

Tsang, L.

Y. Lai, Z.-Q. Zhang, C.-H. Chan, and L. Tsang, “Anomalous properties of the band-edge states in large two-dimensional photonic quasicrystals,” Phys. Rev. B 76, 1–5 (2007).
[CrossRef]

Y. Lai, Z. Zhang, C. Chan, and L. Tsang, “Gap structures and wave functions of classical waves in large-sized two-dimensional quasiperiodic structures,” Phys. Rev. B 74, 0543051 (2006).
[CrossRef]

Tureci, H. E.

Twersky, V.

Vanneste, C.

J. Andreasen, A. A. Asatryan, L. C. Botten, M. A. Bryne, H. Cao, L. Ge, L. Labonté, P. Sebbah, A. D. Stone, H. E. Tureci, and C. Vanneste, “Mode of random lasers,” Adv. Opt. Photon. 3, 88–127 (2011).
[CrossRef]

C. Vanneste and P. Sebbah, “Complexity of two-dimensional quasimodes at the transition from weak scattering to Anderson localization,” Phys. Rev. A 79, 041802 (2009).
[CrossRef]

Vasquez, L. J.

A. Rodriguez, L. J. Vasquez, and R. A. Roemer, “Optimisation of multifractal analysis using box-size scaling,” Eur. Phys. J. B 67, 77–82 (2009).
[CrossRef]

Walker, D.

L. S. Blackford, J. Choi, A. Cleary, E. D’Azevedo, J. Demmel, I. Dhillon, J. Dongarra, S. Hammarling, G. Henry, A. Petitet, K. Stanley, D. Walker, and R. C. Whaley, ScaLAPACK Users’ Guide (SIAM, 1997).

Whaley, R. C.

L. S. Blackford, J. Choi, A. Cleary, E. D’Azevedo, J. Demmel, I. Dhillon, J. Dongarra, S. Hammarling, G. Henry, A. Petitet, K. Stanley, D. Walker, and R. C. Whaley, ScaLAPACK Users’ Guide (SIAM, 1997).

Wiersma, D.

L. Dal Negro, C. Oton, Z. Gaburro, L. Pavesi, P. Johnson, A. Lagendijk, R. Righini, M. Colocci, and D. Wiersma, “Light transport through the band-edge states of Fibonacci quasicrystals,” Phys. Rev. Lett. 90, 055501 (2003).
[CrossRef]

Yang, J.-K.

H. Noh, J.-K. Yang, S. V. Boriskina, M. J. Rooks, G. S. Solomon, L. Dal Negro, and H. Cao, “Lasing in Thue–Morse structures with optimized aperiodicity,” Appl. Phys. Lett. 98, 201109 (2011).
[CrossRef]

J.-K. Yang, S. V. Boriskina, H. Noh, M. J. Rooks, G. S. Solomon, L. Dal Negro, and H. Cao, “Demonstration of laser action in a pseudorandom medium,” Appl. Phys. Lett. 97, 223101(2010).
[CrossRef]

Yi, J. H.

L. Dal Negro, J. H. Yi, V. Nguyen, Y. Yi, J. Michel, and L. C. Kimerling, “Spectrally enhanced light emission from aperiodic photonic structures,” Appl. Phys. Lett. 86, 261905 (2005).
[CrossRef]

Yi, Y.

L. Dal Negro, J. H. Yi, V. Nguyen, Y. Yi, J. Michel, and L. C. Kimerling, “Spectrally enhanced light emission from aperiodic photonic structures,” Appl. Phys. Lett. 86, 261905 (2005).
[CrossRef]

X. Jiang, Y. Zhang, S. Feng, K. C. Huang, Y. Yi, and J. D. Joannopoulos, “Photonic band gaps and localization in the Thue–Morse structures,” Appl. Phys. Lett. 86, 201110(2005).
[CrossRef]

L. D. Negro, M. Stolfi, Y. Yi, J. Michel, X. Duan, L. C. Kimerling, J. LeBlanc, and J. Haavisto, “Photon band gap properties and omnidirectional reflectance in Si/SiO2 Thue–Morse quasicrystals,” Appl. Phys. Lett. 84, 5186–5188 (2004).
[CrossRef]

Yuan, H. Q.

H. Q. Yuan, U. Grimm, P. Repetowicz, and M. Schreiber, “Energy spectra, wavefunctions and quantum diffusion for quasiperiodic systems,” Phys. Rev. B 62, 15569–15578 (2000).
[CrossRef]

Zhang, Y.

X. Jiang, Y. Zhang, S. Feng, K. C. Huang, Y. Yi, and J. D. Joannopoulos, “Photonic band gaps and localization in the Thue–Morse structures,” Appl. Phys. Lett. 86, 201110(2005).
[CrossRef]

Zhang, Z.

Y. Lai, Z. Zhang, C. Chan, and L. Tsang, “Gap structures and wave functions of classical waves in large-sized two-dimensional quasiperiodic structures,” Phys. Rev. B 74, 0543051 (2006).
[CrossRef]

Zhang, Z.-Q.

Y. Lai, Z.-Q. Zhang, C.-H. Chan, and L. Tsang, “Anomalous properties of the band-edge states in large two-dimensional photonic quasicrystals,” Phys. Rev. B 76, 1–5 (2007).
[CrossRef]

Adv. Opt. Photon. (1)

Appl. Phys. Lett. (6)

X. Jiang, Y. Zhang, S. Feng, K. C. Huang, Y. Yi, and J. D. Joannopoulos, “Photonic band gaps and localization in the Thue–Morse structures,” Appl. Phys. Lett. 86, 201110(2005).
[CrossRef]

L. D. Negro, M. Stolfi, Y. Yi, J. Michel, X. Duan, L. C. Kimerling, J. LeBlanc, and J. Haavisto, “Photon band gap properties and omnidirectional reflectance in Si/SiO2 Thue–Morse quasicrystals,” Appl. Phys. Lett. 84, 5186–5188 (2004).
[CrossRef]

L. Dal Negro, J. H. Yi, V. Nguyen, Y. Yi, J. Michel, and L. C. Kimerling, “Spectrally enhanced light emission from aperiodic photonic structures,” Appl. Phys. Lett. 86, 261905 (2005).
[CrossRef]

L. Moretti, I. Rea, L. De Stefano, and I. Rendina, “Periodic versus aperiodic: enhancing the sensitivity of porous silicon based optical sensors,” Appl. Phys. Lett. 90, 191112 (2007).
[CrossRef]

J.-K. Yang, S. V. Boriskina, H. Noh, M. J. Rooks, G. S. Solomon, L. Dal Negro, and H. Cao, “Demonstration of laser action in a pseudorandom medium,” Appl. Phys. Lett. 97, 223101(2010).
[CrossRef]

H. Noh, J.-K. Yang, S. V. Boriskina, M. J. Rooks, G. S. Solomon, L. Dal Negro, and H. Cao, “Lasing in Thue–Morse structures with optimized aperiodicity,” Appl. Phys. Lett. 98, 201109 (2011).
[CrossRef]

Eur. Phys. J. B (1)

A. Rodriguez, L. J. Vasquez, and R. A. Roemer, “Optimisation of multifractal analysis using box-size scaling,” Eur. Phys. J. B 67, 77–82 (2009).
[CrossRef]

Int. J. Mod. Phys. B (1)

M. Janssen, “Multifractal analysis of broadly distributed observables at criticality,” Int. J. Mod. Phys. B 8, 943–984(1994).
[CrossRef]

J. Opt. Soc. Am. (1)

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

J. Phys. C (1)

J. L. Pichard and G. Sarma, “Power-law decay and fractal character of eigenstates in two-dimensional disordered systems,” J. Phys. C 18, 3457–3466 (1985).
[CrossRef]

J. Phys. Condens. Matter (1)

T. Rieth and M. Schreiber, “Numerical investigation of electronic wave functions in quasiperiodic lattices,” J. Phys. Condens. Matter 10, 783–800 (1998).
[CrossRef]

Laser Photon. Rev. (1)

L. Dal Negro and S. V. Boriskina, “Deterministic aperiodic nanostructures for photonics and plasmonics applications,” Laser Photon. Rev. 6, 178–218 (2012).
[CrossRef]

Opt. Express (5)

Opt. Lett. (1)

Philos. Mag. B (1)

L. Moretti and V. Mocella, “The square Thue–Morse tiling for photonic application,” Philos. Mag. B 88, 2275–2284(2008).
[CrossRef]

Phys. Rep. (1)

A. D. Mirlin, “Statistics of energy levels and eigenfunctions in disordered systems,” Phys. Rep. 326, 259–382 (2000).
[CrossRef]

Phys. Rev. A (1)

C. Vanneste and P. Sebbah, “Complexity of two-dimensional quasimodes at the transition from weak scattering to Anderson localization,” Phys. Rev. A 79, 041802 (2009).
[CrossRef]

Phys. Rev. B (5)

Z. Cheng, R. Savit, and R. Merlin, “Structure and electronic properties of Thue–Morse lattices,” Phys. Rev. B 37, 4375–4382 (1988).
[CrossRef]

Y. Lai, Z.-Q. Zhang, C.-H. Chan, and L. Tsang, “Anomalous properties of the band-edge states in large two-dimensional photonic quasicrystals,” Phys. Rev. B 76, 1–5 (2007).
[CrossRef]

Y. Lai, Z. Zhang, C. Chan, and L. Tsang, “Gap structures and wave functions of classical waves in large-sized two-dimensional quasiperiodic structures,” Phys. Rev. B 74, 0543051 (2006).
[CrossRef]

M. Dulea, M. Johansson, and R. Riklund, “Localization of electrons and electromagnetic waves in a deterministic aperiodic system,” Phys. Rev. B 45, 105–114 (1992).
[CrossRef]

H. Q. Yuan, U. Grimm, P. Repetowicz, and M. Schreiber, “Energy spectra, wavefunctions and quantum diffusion for quasiperiodic systems,” Phys. Rev. B 62, 15569–15578 (2000).
[CrossRef]

Phys. Rev. E (1)

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Figures (8)

Fig. 1.
Fig. 1.

Construction of 2D ThMo dielectric systems.

Fig. 2.
Fig. 2.

Normalized LDOS at the center of ThMo systems versus normalized frequency. The labels G1–G5 represent the gaps. The dashed line corresponds to the free space level of the LDOS.

Fig. 3.
Fig. 3.

Normalized LDOS at the center of ThMo systems as a function of system size at the frequencies corresponding to the minimum of the bandgaps. The dashed lines represent the exponential fits.

Fig. 4.
Fig. 4.

(a)–(c) Spatial LDOS for the optical modes; (d)–(f) contour plots of log(ρπc2/2ω); (g), (h), (i) LDOS along the x axis for the normalized frequency a/λ=0.184, 0.238, and 0.300, respectively. The dashed lines represent the LDOS of free space.

Fig. 5.
Fig. 5.

Participation number Pldos versus number of sites Ns for the modes. The uppermost solid line is the maximally possible value obtained for an extended mode (β=1).

Fig. 6.
Fig. 6.

(a) Calculated anomalous exponent Δq by means of the box-counting method. The slope of the gIPR plotted versus the box size in bilogarithmic scales yield Δq. One example is shown in the inset for q=[4,3,2,1,0,1,2,3,4] and a/λ=0.238. (b) Singularity spectrum calculated for the several modes under investigation. For comparison purposes, it also reports MFA of a square photonic crystal.

Fig. 7.
Fig. 7.

Histograms of the logarithm of the LDOS at 0.300 (passband) and 0.238 (band edge). The continuous curve is the distribution function following from the corresponding f(α) spectrum via Eq. (9).

Fig. 8.
Fig. 8.

(a) Behavior of α0 and D(1) for various modes. For all frequencies, α0>2 and D(1)<2. (b) Fractal dimension d*=dβ calculated from the Table 1 and the correlation dimension D(2). The relative discrepancy between the two quantities is smaller than 15% for almost all the modes.

Tables (1)

Tables Icon

Table 1. Normalized Frequency, β, and γ Values Carried out from Linear Fit of P versus Nsa

Equations (12)

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A[ABBA]andB[BAAB]
ρ(r,ω)=mδ(ωωm)|ψm(r)|2=2ωπc2Im[G(ω,r,r)],
|ψω(r)|{|rr0|γ(ω)for|rr0|ξ(ω)|rr0|γ(ω)exp(|rr0|/ξ(ω))for|rr0|>ξ(ω),
P1(Ns)=Ld|ψ(r)|4ddr(Ld|ψ(r)|2ddr)2LdβNsβ,
ρ(r,ω)1Γ(ω)|ψω(r)|2,
P1(ω,Ns)Pldos1(ω,Ns)=Ldρ2(r,ω)ddr(Ldρ(r,ω)ddr)2.
Pq(ω,l)=i=1Nlμiq(ω,l)withμi(ω,l)=Biρ(ω,r)d2rLdρ(ω,r)d2r,
Pq(l)(Ll)τ(q)withτ(q)=d(q1)+Δq,
Π(μ,L/l)dμ(Ll)f(α)ddα,
α(q)=liml01ln(l/L)(1Pq(l)i=1Nlμiq(l)lnμi(l)),
f(α(q))=liml01ln(l/L)(1Pq(l)i=1Nlμiq(l)lnμiq(l)lnPq(l)).
D(q)=liml0τ(q)(q1).

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