Abstract

We present a detailed theoretical description of a broadband omnidirectional light concentrator and absorber with cylinder geometry. The proposed optical “trap” captures nearly all the incident light within its geometric cross-section, leading to a broad range of possible applications – from solar energy harvesting to thermal light emitters and optoelectronic components. We have demonstrated that an approximate lamellar black-hole with a moderate number of homogeneous layers, while giving the desired ray-optical performance, can provide absorption efficiencies comparable to those of ideal devices with a smooth gradient in index.

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  1. V. M. Shalaev, W. S. Cai, U. K. Chettiar, H. K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metamaterials,” Opt. Lett. 30(24), 3356–3358 (2005).
    [CrossRef]
  2. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
    [CrossRef] [PubMed]
  3. Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical Hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express 14(18), 8247–8256 (2006).
    [CrossRef] [PubMed]
  4. A. Salandrino and N. Engheta, “Far-Field Subdiffraction Optical Microscopy Using Metamaterial Crystals: Theory and Simulations,” Phys. Rev. B 74(7), 075103 (2006).
    [CrossRef]
  5. N. M. Litchinitser and V. M. Shalaev, “Metamaterials: transforming theory into reality,” J. Opt. Soc. Am. B 26(12), B161–B169 (2009).
    [CrossRef]
  6. N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100(20), 207402 (2008).
    [CrossRef] [PubMed]
  7. T. V. Teperik, F. J. García de Abajo, A. G. Borisov, M. Abdelsalam, P. N. Bartlett, Y. Sugawara, and J. J. Baumberg, “Omnidirectional absorption in nanostructured metal surfaces,” Nat. Photonics 2(5), 299–301 (2008).
    [CrossRef]
  8. E. E. Narimanov and A. V. Kildishev, “Optical black hole: broadband omnidirectional light absorber,” Appl. Phys. Lett. 95(4), 041106 (2009).
    [CrossRef]
  9. L. D. Landau, and E. M. Lifshitz, Mechanics, Pergamon Press, Oxford, (1976).
  10. R. K. Luneburg, Mathematical Theory of Optics, University of California Press, Berkeley, 1964, p. 12.
  11. E. J. Post, Formal Structure of Electromagnetics: General Covariance and Electromagnetics, 1962, p. 152.
  12. S. Gradshteyn, and I. M. Ryzhik, Tables of integrals, series and products, Academic Press, New York, CD-ROM Edition, 1994, Eq. (8).511.4.
  13. M. Born, and E. Wolf, Principles of Optics, Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University Press, New York, 1999).
  14. G. P. Agrawal and D. N. Pattanayak, “Gaussian beam propagation beyond the paraxial approximation,” J. Opt. Soc. Am. 69(4), 575 (1979).
    [CrossRef]
  15. M. Abramowitz, and I. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York, Dover, (1972).
  16. V. Kildishev, U. K. Chettiar, Z. Jacob, V. M. Shalaev, and E. E. Narimanov, “Materializing a binary hyperlens design,” Appl. Phys. Lett. 94(7), 071102 (2009).
    [CrossRef]
  17. S. D. Conte, and C. W. Boor, “Elementary Numerical Analysis: An Algorithmic Approach, 3rd edition,” 1980.
  18. H. Hulst, Light scattering by small particles, Dover, 1981, p. 309.

2009 (3)

E. E. Narimanov and A. V. Kildishev, “Optical black hole: broadband omnidirectional light absorber,” Appl. Phys. Lett. 95(4), 041106 (2009).
[CrossRef]

V. Kildishev, U. K. Chettiar, Z. Jacob, V. M. Shalaev, and E. E. Narimanov, “Materializing a binary hyperlens design,” Appl. Phys. Lett. 94(7), 071102 (2009).
[CrossRef]

N. M. Litchinitser and V. M. Shalaev, “Metamaterials: transforming theory into reality,” J. Opt. Soc. Am. B 26(12), B161–B169 (2009).
[CrossRef]

2008 (2)

N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100(20), 207402 (2008).
[CrossRef] [PubMed]

T. V. Teperik, F. J. García de Abajo, A. G. Borisov, M. Abdelsalam, P. N. Bartlett, Y. Sugawara, and J. J. Baumberg, “Omnidirectional absorption in nanostructured metal surfaces,” Nat. Photonics 2(5), 299–301 (2008).
[CrossRef]

2006 (2)

A. Salandrino and N. Engheta, “Far-Field Subdiffraction Optical Microscopy Using Metamaterial Crystals: Theory and Simulations,” Phys. Rev. B 74(7), 075103 (2006).
[CrossRef]

Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical Hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express 14(18), 8247–8256 (2006).
[CrossRef] [PubMed]

2005 (1)

2000 (1)

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
[CrossRef] [PubMed]

1979 (1)

Abdelsalam, M.

T. V. Teperik, F. J. García de Abajo, A. G. Borisov, M. Abdelsalam, P. N. Bartlett, Y. Sugawara, and J. J. Baumberg, “Omnidirectional absorption in nanostructured metal surfaces,” Nat. Photonics 2(5), 299–301 (2008).
[CrossRef]

Agrawal, G. P.

Alekseyev, L. V.

Bartlett, P. N.

T. V. Teperik, F. J. García de Abajo, A. G. Borisov, M. Abdelsalam, P. N. Bartlett, Y. Sugawara, and J. J. Baumberg, “Omnidirectional absorption in nanostructured metal surfaces,” Nat. Photonics 2(5), 299–301 (2008).
[CrossRef]

Baumberg, J. J.

T. V. Teperik, F. J. García de Abajo, A. G. Borisov, M. Abdelsalam, P. N. Bartlett, Y. Sugawara, and J. J. Baumberg, “Omnidirectional absorption in nanostructured metal surfaces,” Nat. Photonics 2(5), 299–301 (2008).
[CrossRef]

Borisov, A. G.

T. V. Teperik, F. J. García de Abajo, A. G. Borisov, M. Abdelsalam, P. N. Bartlett, Y. Sugawara, and J. J. Baumberg, “Omnidirectional absorption in nanostructured metal surfaces,” Nat. Photonics 2(5), 299–301 (2008).
[CrossRef]

Cai, W. S.

Chettiar, U. K.

V. Kildishev, U. K. Chettiar, Z. Jacob, V. M. Shalaev, and E. E. Narimanov, “Materializing a binary hyperlens design,” Appl. Phys. Lett. 94(7), 071102 (2009).
[CrossRef]

V. M. Shalaev, W. S. Cai, U. K. Chettiar, H. K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metamaterials,” Opt. Lett. 30(24), 3356–3358 (2005).
[CrossRef]

Drachev, V. P.

Engheta, N.

A. Salandrino and N. Engheta, “Far-Field Subdiffraction Optical Microscopy Using Metamaterial Crystals: Theory and Simulations,” Phys. Rev. B 74(7), 075103 (2006).
[CrossRef]

García de Abajo, F. J.

T. V. Teperik, F. J. García de Abajo, A. G. Borisov, M. Abdelsalam, P. N. Bartlett, Y. Sugawara, and J. J. Baumberg, “Omnidirectional absorption in nanostructured metal surfaces,” Nat. Photonics 2(5), 299–301 (2008).
[CrossRef]

Jacob, Z.

V. Kildishev, U. K. Chettiar, Z. Jacob, V. M. Shalaev, and E. E. Narimanov, “Materializing a binary hyperlens design,” Appl. Phys. Lett. 94(7), 071102 (2009).
[CrossRef]

Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical Hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express 14(18), 8247–8256 (2006).
[CrossRef] [PubMed]

Kildishev, A. V.

Kildishev, V.

V. Kildishev, U. K. Chettiar, Z. Jacob, V. M. Shalaev, and E. E. Narimanov, “Materializing a binary hyperlens design,” Appl. Phys. Lett. 94(7), 071102 (2009).
[CrossRef]

Landy, N. I.

N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100(20), 207402 (2008).
[CrossRef] [PubMed]

Litchinitser, N. M.

Mock, J. J.

N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100(20), 207402 (2008).
[CrossRef] [PubMed]

Narimanov, E.

Narimanov, E. E.

V. Kildishev, U. K. Chettiar, Z. Jacob, V. M. Shalaev, and E. E. Narimanov, “Materializing a binary hyperlens design,” Appl. Phys. Lett. 94(7), 071102 (2009).
[CrossRef]

E. E. Narimanov and A. V. Kildishev, “Optical black hole: broadband omnidirectional light absorber,” Appl. Phys. Lett. 95(4), 041106 (2009).
[CrossRef]

Padilla, W. J.

N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100(20), 207402 (2008).
[CrossRef] [PubMed]

Pattanayak, D. N.

Pendry, J. B.

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
[CrossRef] [PubMed]

Sajuyigbe, S.

N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100(20), 207402 (2008).
[CrossRef] [PubMed]

Salandrino, A.

A. Salandrino and N. Engheta, “Far-Field Subdiffraction Optical Microscopy Using Metamaterial Crystals: Theory and Simulations,” Phys. Rev. B 74(7), 075103 (2006).
[CrossRef]

Sarychev, A. K.

Shalaev, V. M.

Smith, D. R.

N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100(20), 207402 (2008).
[CrossRef] [PubMed]

Sugawara, Y.

T. V. Teperik, F. J. García de Abajo, A. G. Borisov, M. Abdelsalam, P. N. Bartlett, Y. Sugawara, and J. J. Baumberg, “Omnidirectional absorption in nanostructured metal surfaces,” Nat. Photonics 2(5), 299–301 (2008).
[CrossRef]

Teperik, T. V.

T. V. Teperik, F. J. García de Abajo, A. G. Borisov, M. Abdelsalam, P. N. Bartlett, Y. Sugawara, and J. J. Baumberg, “Omnidirectional absorption in nanostructured metal surfaces,” Nat. Photonics 2(5), 299–301 (2008).
[CrossRef]

Yuan, H. K.

Appl. Phys. Lett. (2)

E. E. Narimanov and A. V. Kildishev, “Optical black hole: broadband omnidirectional light absorber,” Appl. Phys. Lett. 95(4), 041106 (2009).
[CrossRef]

V. Kildishev, U. K. Chettiar, Z. Jacob, V. M. Shalaev, and E. E. Narimanov, “Materializing a binary hyperlens design,” Appl. Phys. Lett. 94(7), 071102 (2009).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Nat. Photonics (1)

T. V. Teperik, F. J. García de Abajo, A. G. Borisov, M. Abdelsalam, P. N. Bartlett, Y. Sugawara, and J. J. Baumberg, “Omnidirectional absorption in nanostructured metal surfaces,” Nat. Photonics 2(5), 299–301 (2008).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. B (1)

A. Salandrino and N. Engheta, “Far-Field Subdiffraction Optical Microscopy Using Metamaterial Crystals: Theory and Simulations,” Phys. Rev. B 74(7), 075103 (2006).
[CrossRef]

Phys. Rev. Lett. (2)

N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100(20), 207402 (2008).
[CrossRef] [PubMed]

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
[CrossRef] [PubMed]

Other (8)

S. D. Conte, and C. W. Boor, “Elementary Numerical Analysis: An Algorithmic Approach, 3rd edition,” 1980.

H. Hulst, Light scattering by small particles, Dover, 1981, p. 309.

M. Abramowitz, and I. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York, Dover, (1972).

L. D. Landau, and E. M. Lifshitz, Mechanics, Pergamon Press, Oxford, (1976).

R. K. Luneburg, Mathematical Theory of Optics, University of California Press, Berkeley, 1964, p. 12.

E. J. Post, Formal Structure of Electromagnetics: General Covariance and Electromagnetics, 1962, p. 152.

S. Gradshteyn, and I. M. Ryzhik, Tables of integrals, series and products, Academic Press, New York, CD-ROM Edition, 1994, Eq. (8).511.4.

M. Born, and E. Wolf, Principles of Optics, Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University Press, New York, 1999).

Supplementary Material (1)

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Figures (6)

Fig. 1
Fig. 1

(a) the schematics of the system; panels (b), (c), (d), and (e) show light ray trajectories for p equal to −1, 1, 2, and 3 respectively, where core radius r c much smaller than shell radius r s (r c = r s).

Fig. 2
Fig. 2

Schematic geometry of the layered structure.

Fig. 3
Fig. 3

(Media 1) Ideal black hole with εs = 2.1, εc = 12, γc = 0.7, r s = 20 µm, and r c = r ssc)½ = 8.367 µm. The Gaussian beam with free-space wavelength λ = 1.5µm and full width w = 2λ is focused at x0 = 0, and (a) y0 = 1.5r s; (b) y0 = r s; (c) y0 = 0.75r s, and (d) y0 = 0. Range of modes used in series from (a) to (d): [-50;50], [-140;-40], [-170;-70], [-230;-130]

Fig. 4
Fig. 4

Simulated field maps for a lamellar “black hole” optical concentrator and absorber (εs = 2.1, r s = 20 µm, and r c = 8.367 µm) vs. the total number of layers, l. The device is illuminated by a plane wave with free-space wavelength λ = 1.5µm. (a) l = 3, and 72% absorption efficiency, (b) l = 5, and 84% absorption efficiency, (c) l = 9, and 90% absorption efficiency, (d) l = 17, and 94% absorption efficiency. Panel (e) depicts the reference case of the ideal black hole with smooth gradient, and 99% absorption efficiency.

Fig. 5
Fig. 5

Field patterns of the black hole device (ε s = 2.1, r s = 20 µm, and r c = 8.367 µm) illuminated by a TE-polarized plane wave with a free-space wavelength of (a) 1.5 µm, (b) 3.0 µm, (c) 4.5 µm, and (d) 6.0 µm.

Fig. 6
Fig. 6

(a) Absorption efficiency Q a vs. the ratio λ/r s obtained for the TE- and TM-polarized plane wave; (b) a more detailed plot for the ratios from 0.075 to 0.15.

Equations (69)

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ε ( r ) ε s Δ ε ( r )
Δ ε ( r ) = { 1 , r > r s ( r s / r ) p , r c < r r s ( ε c + ι γ ) / ε s , r r c
r c = r s ( ε s / ε c ) 1 / p .
V e f f ( r ) = 1 2 ( ω / c ) 2 [ ε s ε ( r ) ] ,
ϕ ( r ) = ϕ 0 + m r s m r d ξ C 0 ε ( m / ξ ) ξ 2 ,
r ( ϕ ) = r s × { | csc ϕ 0 cos [ p 2 2 ( ϕ ϕ p ) ] | 2 p 2 , p 2 exp [ cot ϕ 0 ( ϕ 0 ϕ ) ] , p = 2 ,
k 0 2 ε μ H = × × H ln ε × × H ,
k 0 2 ε μ E = × × E ln μ × × E .
k 2 H + 2 H = ( ln μ H ) ln ε × × H ,
k 2 E + 2 E = ( ln ε E ) ln μ × × E .
2 f + k 2 f + L f = 0.
r 2 f m + [ r + q ( r ) ] f m + [ k 2 r 2 m 2 ] f m = 0 ,
f m ( r ) = a m J m ( r ¯ ) + b m H m + ( r ¯ ) ,
f m ( r ) = τ m J m ( r ¯ ) / J m ( r ¯ s ) + ρ m H m + ( r ¯ ) / H m + ( r ¯ s ) .
f m ( r ) = c m H m ( r ¯ ) + c m + H m + ( r ¯ ) ,
r 2 f m + 3 r f m + [ ν 2 m 2 ] f m = 0 ,
r 2 f m + r f m + [ ν 2 m 2 ] f m = 0.
f m ( r ) = { a m r ξ 1 + b m r ξ 2 , if ξ 1 ξ 2 ( a m + b m ln r ) r ξ 1 , otherwise
ξ 1 , 2 = { 1 ± ι ν 2 m 2 1 , ( TE ) ± ι ν 2 m 2 , ( TM )
[ e ] r = r i = [ e ] r = r i = 0 ,
[ h ] r = r i = [ ε 1 h ] r = r i = 0 ,
f m ( r ) = a m J m ( k r ) .
f m ( r ) = b m H m + ( k r ) .
exp ( ι z cos φ ) = ι m e ι m φ J m ( z ) ,
f ( x , y ) = e ι k x = m = + a m J m ( k r ) e ι m φ .
{ 2 f + k 2 f = 0 f ( 0 , y ) = exp [ ( y / w ) 2 ]
f ( x , y ) = 1 1 a q exp [ ι k ( q y + 1 q 2 x ) ] d q + | q | > 1 a q exp [ ι k q y k q 2 1 | x | ] d q ,
σ = | | q | > 1 a q exp [ ι k q y k q 2 1 | x | ] d q | k w π 1 + e 1 4 k 2 w 2 q 2 d q = erfc ( 1 2 k w ) .
f ( x , y ) = k w 2 π 1 1 e 1 4 k 2 w 2 q 2 + ι k [ q ( y y 0 ) + 1 q 2 ( x x 0 ) ] d q ,
f ( r , ϕ ) = k w 2 π 1 1 e 1 4 k 2 w 2 q 2 ι k ( q y 0 + 1 q 2 x 0 r cos [ ϕ ψ ( q ) ] ) d q ,
f ( r , ϕ ) = m = + a m J m ( k r ) e ι m ϕ ,
f ( m ) i ( r ) = τ i T i ( r ) / T i ( r i ) + ρ i R i ( r ) / R i ( r i ) , i = 1 , l ¯ , r 1 = r 2 ,
[ 1 1 L T i , 1 L R i , 1 ] [ τ i + 1 ρ i + 1 ] = [ 1 1 L T i , 0 L R i , 0 ] diag ( F T i , F R i ) [ τ i ρ i ] ,
[ τ i + 1 ρ i + 1 ] = w m i [ τ i ρ i ] , i = 1 , l 1 ¯ ,
w m i = 1 d i + 1 [ F T i ( L R i , 1 L T i , 0 ) F R i ( L R i , 1 L R i , 0 ) F T i ( L T i , 1 L T i , 0 ) F R i ( L T i , 1 L R i , 0 ) ] ,
W [ J m , Y m ] ( x ) = 2 / π x ; W [ J m , H m ± ] ( x ) = ± ι 2 / π x ; W [ H m + , H m ] ( x ) = ι 4 / π x .
{ ρ 1 = 0 τ i + 1 = w 11 i τ i + w 12 i ρ i ρ i + 1 = w 21 i τ i + w 22 i ρ i τ l known, e .g ., a ( m ) i n c J m ( r ¯ l ) , ( i 1 , l 1 ¯ )
{ ρ ¯ 1 = 0 τ ¯ i = w 11 i + w 12 i ρ ¯ i ρ ¯ i + 1 = ( w 21 i + w 22 i ρ ¯ i ) / τ ¯ i , ( i 1 , l 1 ¯ )
{ τ i = τ i + 1 / τ ¯ i ρ i = ρ ¯ i τ i , ( i 1 , l 1 ¯ )
f = m = + f m ( r ¯ s ) exp ι m ϕ .
f m = H m + c m + H m + ,
f m = 2 J m + ( c m + 1 ) H m + ,
f i n c = 2 m J m = 2 e ι k y ,
f s c a = m = ( c m + 1 ) H m + exp ι m φ ,
S r a b s = 1 8 r ¯ s r ^ Re [ z ^ f × ( ι f × z ^ ) * ] = 1 8 r s Im ( f * f r ¯ s ) ,
Q a b s = π 4 m = Im ( f m * f m ) .
Q a b s = 1 2 r ¯ s m = ( 1 | c m + | 2 ) ;
Q s c a = 1 2 r ¯ s m = | 1 c m + | 2 .
Q a b s = 2 r ¯ s m = ( Re [ ι m b m ] + | b m | 2 ) ,
Q s c a = 2 r ¯ s m = | b m | 2 .
c m + = ρ m 3 H m + = ( H m / H m + ) [ ( w 21 2 w 11 1 + w 22 2 w 21 1 ) / ( w 11 2 w 11 1 + w 12 2 w 21 1 ) ] m .
c m + = H m + η m H m H m + + η m H m + ,
η m = β m r ¯ s [ α β m ι 1 X 1 + X ] ,
X = 1 + ι Y 1 ι Y ( r c r s ) ι 2 β m ,
Y = α β m + A r ¯ c J m ( r ¯ c ) β m J m ( r ¯ c ) .
α β m A TE 0 r ¯ s 2 m 2 1 TM 1 r ¯ s 2 m 2 1 ε c / ( ε c + ι γ c )
η m = β m r ¯ s [ α β m tan ( β m ln r c r s + tan 1 [ α β m + A r ¯ c J m ( r ¯ c ) β m J m ( r ¯ c ) ] ) ] .
H m ± ( x ) = 2 π x 2 m 2 [ 1 + O ( 1 x 2 m 2 ) ] exp [ ± ι σ m ( x ) ] ,
Q a b s 1 2 r ¯ s | m | r ¯ s ( 1 | c m + | 2 ) .
Q a b s 1 2 r ¯ s r ¯ s r ¯ s d m ( 1 | c m + | 2 ) .
c m + = exp { ι 2 [ tan 1 ( r ¯ s β m η m 1 2 r ¯ s 2 β m 3 ) σ m ( r ¯ s ) ] } .
r ¯ s β m η m = α β m tan ( β m ln r c r s + tan 1 [ α β m + A r ¯ c J m ( r ¯ c ) β m J m ( r ¯ c ) ] ) ,
tan 1 ( r ¯ s β m η m ) = β m ln r c r s tan 1 [ A r ¯ c J m ( r ¯ c ) β m J m ( r ¯ c ) ] .
tan 1 ( r ¯ s β m η m ) = β m ln r c r s tan 1 [ A ( 1 2 r ¯ c 2 β m ( r ¯ c 2 m 2 ) + r ¯ c 2 m 2 β m tan [ σ m ( r ¯ c ) ] ) ] ,
c m + = exp { ι 2 [ β m ln r c r s σ m ( r ¯ c ) + σ m ( r ¯ s ) ] } ,
| c m + | = exp { γ c ε c r ¯ s 2 m 2 } ,
Q a b s 1 2 r ¯ s r ¯ s r ¯ s ( 1 exp ( 2 γ c ε c r ¯ s 2 m 2 ) ) d m ,
Q a b s = 1 F ( 2 r ¯ s γ c ε c ) = π 2 [ I 1 ( 2 r ¯ s γ c ε c ) L 1 ( 2 r ¯ s γ c ε c ) ] ,
F ( x ) = 0 π / 2 cos θ exp ( x cos θ ) d θ = 1 π 2 [ I 1 ( x ) L 1 ( x ) ] = { 1 1 2 x + O ( x 2 ) , x 1 1 x 2 + O ( 1 x 4 ) , x 1

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