Abstract

We have studied the effect of the boundaries on the local density of states (LDOS) in one-, two-, and three-dimensional finite size photonic structures. The LDOS was calculated with the help of the local perturbation method (LPM) and a new LPM-Bloch method using periodicity of the system. The methods are applicable for the clusters made of small (relative to the incident wavelength) particles or for the clusters which can be considered as made of such particles. It was demonstrated that the LPM-Bloch method is an accurate numerical tool for calculation of the LDOS in the finite size photonic structures with weak interference.

© 2008 Optical Society of America

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  1. A. A. Asatryan, L. C. Botten, N. A. Nicorovici, R. C. McPhedran, and C. Martijn de Sterke, "Frequency shift of sources embedded in finite two-dimensional photonic clusters," Waves Random Complex Media 16, 151-165 (2006).
    [CrossRef]
  2. A. F. Koenderink, A. Lagendijk, and W. L. Vos, "Optical extinction due to intrinsic structural variations of photonic crystals," Phys. Rev. B 72, 153102 (2005).
    [CrossRef]
  3. F. G. Bass, V. D. Freilikher, and V. V. Prosentsov, "Electromagnetic wave scattering from small scatterers of arbitrary shape," J. Elm. Wav. Appl. 14, 269-283 (2000).
  4. V. Prosyentsov and A. Lagendijk, "The local density of states in finite size photonic structures, small particles approach," Photonics Nanostruct. 5, 189-199 (2007).
    [CrossRef]
  5. S. Kole, New methods for the numerical solution of Maxwell�??s equations, Ph. D. thesis, University of Groningen (2003).
  6. M. Wubs and A. Lagendijk, "Local optical density of states in finite crystals of plane scatterers," Phys. Rev. E 65, 046612 (2002).
    [CrossRef]
  7. R. C. McPhedran, L. C. Botten, J. McOrist, A. A. Asatryan, C. M. de Sterke, and N. A. Nicorovici, Density of States Functions for Photonic Crystals, Phys. Rev. E 69, 016609 (2004).
    [CrossRef]

2007

V. Prosyentsov and A. Lagendijk, "The local density of states in finite size photonic structures, small particles approach," Photonics Nanostruct. 5, 189-199 (2007).
[CrossRef]

2006

A. A. Asatryan, L. C. Botten, N. A. Nicorovici, R. C. McPhedran, and C. Martijn de Sterke, "Frequency shift of sources embedded in finite two-dimensional photonic clusters," Waves Random Complex Media 16, 151-165 (2006).
[CrossRef]

2005

A. F. Koenderink, A. Lagendijk, and W. L. Vos, "Optical extinction due to intrinsic structural variations of photonic crystals," Phys. Rev. B 72, 153102 (2005).
[CrossRef]

2004

R. C. McPhedran, L. C. Botten, J. McOrist, A. A. Asatryan, C. M. de Sterke, and N. A. Nicorovici, Density of States Functions for Photonic Crystals, Phys. Rev. E 69, 016609 (2004).
[CrossRef]

2002

M. Wubs and A. Lagendijk, "Local optical density of states in finite crystals of plane scatterers," Phys. Rev. E 65, 046612 (2002).
[CrossRef]

2000

F. G. Bass, V. D. Freilikher, and V. V. Prosentsov, "Electromagnetic wave scattering from small scatterers of arbitrary shape," J. Elm. Wav. Appl. 14, 269-283 (2000).

Asatryan, A. A.

A. A. Asatryan, L. C. Botten, N. A. Nicorovici, R. C. McPhedran, and C. Martijn de Sterke, "Frequency shift of sources embedded in finite two-dimensional photonic clusters," Waves Random Complex Media 16, 151-165 (2006).
[CrossRef]

R. C. McPhedran, L. C. Botten, J. McOrist, A. A. Asatryan, C. M. de Sterke, and N. A. Nicorovici, Density of States Functions for Photonic Crystals, Phys. Rev. E 69, 016609 (2004).
[CrossRef]

Bass, F. G.

F. G. Bass, V. D. Freilikher, and V. V. Prosentsov, "Electromagnetic wave scattering from small scatterers of arbitrary shape," J. Elm. Wav. Appl. 14, 269-283 (2000).

Botten, L. C.

A. A. Asatryan, L. C. Botten, N. A. Nicorovici, R. C. McPhedran, and C. Martijn de Sterke, "Frequency shift of sources embedded in finite two-dimensional photonic clusters," Waves Random Complex Media 16, 151-165 (2006).
[CrossRef]

R. C. McPhedran, L. C. Botten, J. McOrist, A. A. Asatryan, C. M. de Sterke, and N. A. Nicorovici, Density of States Functions for Photonic Crystals, Phys. Rev. E 69, 016609 (2004).
[CrossRef]

de Sterke, C. M.

R. C. McPhedran, L. C. Botten, J. McOrist, A. A. Asatryan, C. M. de Sterke, and N. A. Nicorovici, Density of States Functions for Photonic Crystals, Phys. Rev. E 69, 016609 (2004).
[CrossRef]

Freilikher, V. D.

F. G. Bass, V. D. Freilikher, and V. V. Prosentsov, "Electromagnetic wave scattering from small scatterers of arbitrary shape," J. Elm. Wav. Appl. 14, 269-283 (2000).

Koenderink, A. F.

A. F. Koenderink, A. Lagendijk, and W. L. Vos, "Optical extinction due to intrinsic structural variations of photonic crystals," Phys. Rev. B 72, 153102 (2005).
[CrossRef]

Lagendijk, A.

V. Prosyentsov and A. Lagendijk, "The local density of states in finite size photonic structures, small particles approach," Photonics Nanostruct. 5, 189-199 (2007).
[CrossRef]

A. F. Koenderink, A. Lagendijk, and W. L. Vos, "Optical extinction due to intrinsic structural variations of photonic crystals," Phys. Rev. B 72, 153102 (2005).
[CrossRef]

M. Wubs and A. Lagendijk, "Local optical density of states in finite crystals of plane scatterers," Phys. Rev. E 65, 046612 (2002).
[CrossRef]

Martijn de Sterke, C.

A. A. Asatryan, L. C. Botten, N. A. Nicorovici, R. C. McPhedran, and C. Martijn de Sterke, "Frequency shift of sources embedded in finite two-dimensional photonic clusters," Waves Random Complex Media 16, 151-165 (2006).
[CrossRef]

McOrist, J.

R. C. McPhedran, L. C. Botten, J. McOrist, A. A. Asatryan, C. M. de Sterke, and N. A. Nicorovici, Density of States Functions for Photonic Crystals, Phys. Rev. E 69, 016609 (2004).
[CrossRef]

McPhedran, R. C.

A. A. Asatryan, L. C. Botten, N. A. Nicorovici, R. C. McPhedran, and C. Martijn de Sterke, "Frequency shift of sources embedded in finite two-dimensional photonic clusters," Waves Random Complex Media 16, 151-165 (2006).
[CrossRef]

R. C. McPhedran, L. C. Botten, J. McOrist, A. A. Asatryan, C. M. de Sterke, and N. A. Nicorovici, Density of States Functions for Photonic Crystals, Phys. Rev. E 69, 016609 (2004).
[CrossRef]

Nicorovici, N. A.

A. A. Asatryan, L. C. Botten, N. A. Nicorovici, R. C. McPhedran, and C. Martijn de Sterke, "Frequency shift of sources embedded in finite two-dimensional photonic clusters," Waves Random Complex Media 16, 151-165 (2006).
[CrossRef]

R. C. McPhedran, L. C. Botten, J. McOrist, A. A. Asatryan, C. M. de Sterke, and N. A. Nicorovici, Density of States Functions for Photonic Crystals, Phys. Rev. E 69, 016609 (2004).
[CrossRef]

Prosentsov, V. V.

F. G. Bass, V. D. Freilikher, and V. V. Prosentsov, "Electromagnetic wave scattering from small scatterers of arbitrary shape," J. Elm. Wav. Appl. 14, 269-283 (2000).

Prosyentsov, V.

V. Prosyentsov and A. Lagendijk, "The local density of states in finite size photonic structures, small particles approach," Photonics Nanostruct. 5, 189-199 (2007).
[CrossRef]

Vos, W. L.

A. F. Koenderink, A. Lagendijk, and W. L. Vos, "Optical extinction due to intrinsic structural variations of photonic crystals," Phys. Rev. B 72, 153102 (2005).
[CrossRef]

Wubs, M.

M. Wubs and A. Lagendijk, "Local optical density of states in finite crystals of plane scatterers," Phys. Rev. E 65, 046612 (2002).
[CrossRef]

J. Elm. Wav. Appl.

F. G. Bass, V. D. Freilikher, and V. V. Prosentsov, "Electromagnetic wave scattering from small scatterers of arbitrary shape," J. Elm. Wav. Appl. 14, 269-283 (2000).

Photonics Nanostruct.

V. Prosyentsov and A. Lagendijk, "The local density of states in finite size photonic structures, small particles approach," Photonics Nanostruct. 5, 189-199 (2007).
[CrossRef]

Phys. Rev. B

A. F. Koenderink, A. Lagendijk, and W. L. Vos, "Optical extinction due to intrinsic structural variations of photonic crystals," Phys. Rev. B 72, 153102 (2005).
[CrossRef]

Phys. Rev. E

M. Wubs and A. Lagendijk, "Local optical density of states in finite crystals of plane scatterers," Phys. Rev. E 65, 046612 (2002).
[CrossRef]

R. C. McPhedran, L. C. Botten, J. McOrist, A. A. Asatryan, C. M. de Sterke, and N. A. Nicorovici, Density of States Functions for Photonic Crystals, Phys. Rev. E 69, 016609 (2004).
[CrossRef]

Waves Random Complex Media

A. A. Asatryan, L. C. Botten, N. A. Nicorovici, R. C. McPhedran, and C. Martijn de Sterke, "Frequency shift of sources embedded in finite two-dimensional photonic clusters," Waves Random Complex Media 16, 151-165 (2006).
[CrossRef]

Other

S. Kole, New methods for the numerical solution of Maxwell�??s equations, Ph. D. thesis, University of Groningen (2003).

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Figures (5)

Fig. 1.
Fig. 1.

The normalized scalar LDOS of the finite 1D periodic system versus size parameter ωd/2πc for observation point x=0.1 µm. The system consists of 21 layers centered at x=0. The permittivities of the layers are εsc =1.5 (Fig. 1(a)) and εsc =3 (Fig. 1(b)) and their width is L=0.0125 µm. The period of the system is d=1 µm and the permittivity of the host medium is ε 0=1. The red curve shows the LPM result (“exact result”) and the blue dashed one represents the LPM-Bloch result calculated by using 201 subdivisions in k-space for numerical integration.

Fig. 2.
Fig. 2.

The normalized scalar LDOS of the finite 2D system versus size parameter ωd/2πc at the point x=0.01 µm, y=0. The system consists of 13×13 (Fig. 2(a)) and 21×21 (Fig. 2(b)) circles with radius L=0.01 µm and permittivity εsc =9. The period of the system is d=0.05 µm and the permittivity of the host medium is ε 0=1. The scatterers are arranged into square lattice centered at x=y=0. The red curve shows the LPM result (“exact result”). The blue dashed and solid lines represent the LPM-Bloch results calculated by using 11 and 81 subdivisions in k-space (along each axis) for numerical integration.

Fig. 3.
Fig. 3.

The normalized scalar LDOS of the finite 3D system versus size parameter ωd/2πc at the point x=0.01 µm, y=z=0. The system consists of 13×13×13 (Fig. 3(a)) and 21×21×21 (Fig. 3(b)) spheres with radius L=0.01 µm and permittivity εsc =9. The period of the system is d=0.05 µm and the permittivity of the host medium is ε 0=1. The particles are arranged into simple cubic lattice centered at r=0. The red curve shows the LPM result (“exact result”). The blue dashed and solid lines represent the LPM-Bloch results calculated by using 11 and 81 subdivisions in k-space (along each axis) for numerical integration.

Fig. 4.
Fig. 4.

The normalized vector LDOS of the finite 3D system versus size parameter ωd/2πc at the point x=0.01 µm, y=z=0. The system consists of 9×9×9 spheres with radius L=0.01 µm and permittivity εsc =9. The periods of the systems are d=0.025 µm (Fig. 4(a)) and d=0.05 µm (Fig. 4(b)) and the permittivity of the host medium is ε 0=1. The particles are arranged into simple cubic lattice centered at the point r=0. The red curve shows the LPM result (“exact result”). The blue dashed and solid lines represent the LPMBloch results calculated by using 11 and 81 subdivisions in k-space (along each axis) for numerical integration.

Fig. 5.
Fig. 5.

The relative error Δ versus normalized period of the cluster d/2L at wavelength λ=14.5 µm. The solid line shows Δ while the dashed one shows cubically decreasing function ~ (d/2L)-3 for comparison. The LDOS was calculated at the point x=0.01 µm, y=z=0 for the system consisting of 9×9×9 spheres with radius L=0.01 µm and permittivity εsc εsc =9. The permittivity of the host medium is ε 0=1. The particles are arranged into simple cubic lattice centered at the point r=0.

Equations (29)

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ρ ( r , ω ) = 2 ω ε 0 π c 2 Im [ Tr G ̂ ( r , r ) ] ,
Δ G ̂ ( r , r , k ) G ̂ ( r , r , k ) + ω 2 c 2 ε ( r ) G ̂ ( r , r , k ) = I ̂ n = 1 N δ ( r r r n ) e i k · r n .
G ̂ ( r + r n , r + r n , k ) = G ̂ ( r , r , k ) e i k · ( r n r n ) ,
G ̂ ( r , r ) = 1 V BZ BZ G ̂ ( r , r , k ) d k .
G ̂ ( r , r , k ) = G ̂ 0 ( r , r , k )
ω 2 c 2 ( I ̂ + k 0 2 ) G ̂ ( 0 , r , k ) n = 1 N ( ε n ε 0 ) e i k · r n f n ( q ) e i q · ( r r n ) ( k 0 2 q 2 ) d q ,
G ̂ 0 ( r , r , k ) = 1 4 π ( I ̂ + k 0 2 ) n = 1 N e i k 0 r r r n r r r n e i k · r n ,
G ̂ ( r , r ) = G ̂ 0 ( r , r ) + α k 0 2 V BZ ( I ̂ + k 0 2 ) n = 1 N e i k 0 R n R n BZ G ̂ ( 0 , r , k ) e i k · r n d k ,
G ̂ ( r , r ) = G ̂ 0 ( r , r ) + α k 0 2 ( I ̂ + k 0 2 ) n = 1 N e i k 0 R n R n G ̂ ( r n , r ) ,
G ̂ ( 0 , r , k ) = β G ̂ 0 ( 0 , r , k ) + α β n = 2 N 3 I 1 n I 1 n 1 R 1 n 3 e i k · r n G ̂ ( 0 , r , k )
G ̂ ( r j , r ) = β G ̂ 0 ( r j , r ) + α β n = 1 , n j N 3 I jn I jn 1 R jn 3 G ̂ ( r n , r ) ,
β = ( 2 ε 0 + ε 1 3 ε 0 ) 1 , I jn = r j r n r j r n ,
G ̂ ( 0 , r ) = β [ G ̂ 0 ( 0 , r ) + α β n = 2 N 3 I 1 n I 1 n 1 r 1 n 3 G ̂ 0 ( r n , r ) + ,
α 2 β n = 2 N 3 I 1 n I 1 n 1 r 1 n 3 k = 1 , k n N 3 I nk I nk 1 r nk 3 G ̂ ( r k , r ) ] ,
ε ( r ) = ε 0 + n = 1 N ( ε n ε 0 ) f n ( r r n ) ,
f n ( r r n ) = { 1 , inside particle 0 , outside particle .
Δ G ̂ ( r , r , k ) G ̂ ( r , r , k ) + k 0 2 G ̂ ( r , r , k ) +
ω 2 c 2 n = 1 N ( ε n ε 0 ) f n ( r r n ) G ̂ ( r , r , k ) = I ̂ n = 1 N δ ( r r r n ) e i k · r n ,
f n ( r r n ) G ̂ ( r , r , k ) f n ( r r n ) G ̂ ( r n , r , k ) .
Δ G ̂ ( r , r , k ) G ̂ ( r , r , k ) + k 0 2 G ̂ ( r , r , k ) +
ω 2 c 2 n = 1 N ( ε n ε 0 ) f n ( r r n ) G ̂ ( 0 , r , k ) e i k · r n = I ̂ n = 1 N δ ( r r r n ) e i k · r n ,
G ̂ ( q , r , k ) = ( I ̂ q q k 0 2 ) n = 1 N { e i k · r n i q · ( r + r n ) 8 π 3 ( k 0 2 q 2 )
ω 2 c 2 n = 1 N ( ε n ε 0 ) G ̂ ( 0 , r , k ) f n ( q ) e i ( k q ) · r n ( k 0 2 q 2 ) } ,
G ̂ ( r , r , k ) = G ̂ 0 ( r , r , k )
ω 2 c 2 ( I ̂ + k 0 2 ) G ̂ ( 0 , r , k ) n = 1 N ( ε n ε 0 ) e i k · r n f n ( q ) e i q · ( r r n ) ( k 0 2 q 2 ) d q ,
G ̂ 0 ( r , r , k ) = 1 4 π ( I ̂ + k 0 2 ) n = 1 N e i k 0 r r r n r r r n e i k · r n ,
G ̂ ( r , r ) = G ̂ 0 ( r , r )
ω 2 c 2 V BZ ( I ̂ + k 0 2 ) n = 1 N ( ε n ε 0 ) BZ G ̂ ( 0 , r , k ) e i k · r n d k f n ( q ) e i q . ( r r n ) ( k 0 2 q 2 ) d q ,
G ̂ 0 ( r , r ) = 1 4 π ( I ̂ + k 0 2 ) e i k 0 r r r r ,

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