Abstract

We present a model for second-harmonic generation in a two-dimensional array of quantum dots. We show that the combined effect of the electromagnetic local field of the array and the intrinsic electronic resonances that arise from spatial confinement in the quantum dot produces great enhancements of the second-order nonlinear susceptibilities, hence giving a high efficiency of second-harmonic generation.

© 1995 Optical Society of America

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  1. We list several conference proceedings in which both experimental and theoretical examples and many further references can be found: C. R. Leavens and R. Taylor, eds., Interfaces, Quantum Wells and Superlattices (Plenum, New York, 1988);M. A. Reed and W. P. Kirk, eds., Nanostruc-ture Physics and Fabrication (Academic, New York, 1989);Surf. Sci. 228 and 229 (1990);and special issue on optics of nanostructures, Phys. Today,  46(6) (1993).
    [CrossRef]
  2. Y. Miyamoto, M. Cao, Y. Shingai, K. Furuya, Y. Suematsu, K. G. Ravikumar, and S. Arai, Jpn. J. Appl. Phys. 26, 225 (1987).
    [CrossRef]
  3. Ch. Sikorski and U. Merkt, Phys. Rev. Lett. 62, 2164 (1989).
    [CrossRef] [PubMed]
  4. W. L. Schaich and B. S. Mendoza, Phys. Rev. B 45, 14279 (1992).
    [CrossRef]
  5. R. Kubo, J. Phys. Soc. Jpn. 12, 570 (1957).
    [CrossRef]
  6. Although the effects of temperature should warrant a full study, we expect that they should be of the order of kBT/ΔE(where ΔE is the interlevel excitation-energy difference); then, as long as T ≪ ΔE/kB, the effects of temperature should be negligible.See, for example, B. S. Mendoza and Y. C. Lee, Phys. Rev. B 40, 12063 (1989).
    [CrossRef]
  7. G. Senatore and K. R. Subbaswamy, Phys. Rev. A 35, 2440 (1987).
    [CrossRef] [PubMed]
  8. D. J. Moss, E. Ghahramani, J. E. Sipe, and H. M. van Driel, Phys. Rev. B 41, 1542 (1990).
    [CrossRef]
  9. V. Mizrahi and J. E. Sipe, J. Opt. Soc. Am. B 5, 660 (1988).
    [CrossRef]
  10. A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1975), Chap. 16.
  11. A. Guerrero, “Second-harmonic generation by a system of quantum dots,” master’s thesis, Centro de Investigaciones en Optica, León, México, 1994).
  12. B. F. Levine and C. G. Louie, Appl. Phys. Lett. 20, 272 (1972).
    [CrossRef]
  13. E. Rosencher and Ph. Bois, Phys. Rev. B 44, 11315 (1991).
    [CrossRef]

1992 (1)

W. L. Schaich and B. S. Mendoza, Phys. Rev. B 45, 14279 (1992).
[CrossRef]

1991 (1)

E. Rosencher and Ph. Bois, Phys. Rev. B 44, 11315 (1991).
[CrossRef]

1990 (1)

D. J. Moss, E. Ghahramani, J. E. Sipe, and H. M. van Driel, Phys. Rev. B 41, 1542 (1990).
[CrossRef]

1989 (2)

Although the effects of temperature should warrant a full study, we expect that they should be of the order of kBT/ΔE(where ΔE is the interlevel excitation-energy difference); then, as long as T ≪ ΔE/kB, the effects of temperature should be negligible.See, for example, B. S. Mendoza and Y. C. Lee, Phys. Rev. B 40, 12063 (1989).
[CrossRef]

Ch. Sikorski and U. Merkt, Phys. Rev. Lett. 62, 2164 (1989).
[CrossRef] [PubMed]

1988 (1)

1987 (2)

G. Senatore and K. R. Subbaswamy, Phys. Rev. A 35, 2440 (1987).
[CrossRef] [PubMed]

Y. Miyamoto, M. Cao, Y. Shingai, K. Furuya, Y. Suematsu, K. G. Ravikumar, and S. Arai, Jpn. J. Appl. Phys. 26, 225 (1987).
[CrossRef]

1972 (1)

B. F. Levine and C. G. Louie, Appl. Phys. Lett. 20, 272 (1972).
[CrossRef]

1957 (1)

R. Kubo, J. Phys. Soc. Jpn. 12, 570 (1957).
[CrossRef]

Arai, S.

Y. Miyamoto, M. Cao, Y. Shingai, K. Furuya, Y. Suematsu, K. G. Ravikumar, and S. Arai, Jpn. J. Appl. Phys. 26, 225 (1987).
[CrossRef]

Bois, Ph.

E. Rosencher and Ph. Bois, Phys. Rev. B 44, 11315 (1991).
[CrossRef]

Cao, M.

Y. Miyamoto, M. Cao, Y. Shingai, K. Furuya, Y. Suematsu, K. G. Ravikumar, and S. Arai, Jpn. J. Appl. Phys. 26, 225 (1987).
[CrossRef]

Furuya, K.

Y. Miyamoto, M. Cao, Y. Shingai, K. Furuya, Y. Suematsu, K. G. Ravikumar, and S. Arai, Jpn. J. Appl. Phys. 26, 225 (1987).
[CrossRef]

Ghahramani, E.

D. J. Moss, E. Ghahramani, J. E. Sipe, and H. M. van Driel, Phys. Rev. B 41, 1542 (1990).
[CrossRef]

Guerrero, A.

A. Guerrero, “Second-harmonic generation by a system of quantum dots,” master’s thesis, Centro de Investigaciones en Optica, León, México, 1994).

Kubo, R.

R. Kubo, J. Phys. Soc. Jpn. 12, 570 (1957).
[CrossRef]

Lee, Y. C.

Although the effects of temperature should warrant a full study, we expect that they should be of the order of kBT/ΔE(where ΔE is the interlevel excitation-energy difference); then, as long as T ≪ ΔE/kB, the effects of temperature should be negligible.See, for example, B. S. Mendoza and Y. C. Lee, Phys. Rev. B 40, 12063 (1989).
[CrossRef]

Levine, B. F.

B. F. Levine and C. G. Louie, Appl. Phys. Lett. 20, 272 (1972).
[CrossRef]

Louie, C. G.

B. F. Levine and C. G. Louie, Appl. Phys. Lett. 20, 272 (1972).
[CrossRef]

Mendoza, B. S.

W. L. Schaich and B. S. Mendoza, Phys. Rev. B 45, 14279 (1992).
[CrossRef]

Although the effects of temperature should warrant a full study, we expect that they should be of the order of kBT/ΔE(where ΔE is the interlevel excitation-energy difference); then, as long as T ≪ ΔE/kB, the effects of temperature should be negligible.See, for example, B. S. Mendoza and Y. C. Lee, Phys. Rev. B 40, 12063 (1989).
[CrossRef]

Merkt, U.

Ch. Sikorski and U. Merkt, Phys. Rev. Lett. 62, 2164 (1989).
[CrossRef] [PubMed]

Miyamoto, Y.

Y. Miyamoto, M. Cao, Y. Shingai, K. Furuya, Y. Suematsu, K. G. Ravikumar, and S. Arai, Jpn. J. Appl. Phys. 26, 225 (1987).
[CrossRef]

Mizrahi, V.

Moss, D. J.

D. J. Moss, E. Ghahramani, J. E. Sipe, and H. M. van Driel, Phys. Rev. B 41, 1542 (1990).
[CrossRef]

Ravikumar, K. G.

Y. Miyamoto, M. Cao, Y. Shingai, K. Furuya, Y. Suematsu, K. G. Ravikumar, and S. Arai, Jpn. J. Appl. Phys. 26, 225 (1987).
[CrossRef]

Rosencher, E.

E. Rosencher and Ph. Bois, Phys. Rev. B 44, 11315 (1991).
[CrossRef]

Schaich, W. L.

W. L. Schaich and B. S. Mendoza, Phys. Rev. B 45, 14279 (1992).
[CrossRef]

Senatore, G.

G. Senatore and K. R. Subbaswamy, Phys. Rev. A 35, 2440 (1987).
[CrossRef] [PubMed]

Shingai, Y.

Y. Miyamoto, M. Cao, Y. Shingai, K. Furuya, Y. Suematsu, K. G. Ravikumar, and S. Arai, Jpn. J. Appl. Phys. 26, 225 (1987).
[CrossRef]

Sikorski, Ch.

Ch. Sikorski and U. Merkt, Phys. Rev. Lett. 62, 2164 (1989).
[CrossRef] [PubMed]

Sipe, J. E.

D. J. Moss, E. Ghahramani, J. E. Sipe, and H. M. van Driel, Phys. Rev. B 41, 1542 (1990).
[CrossRef]

V. Mizrahi and J. E. Sipe, J. Opt. Soc. Am. B 5, 660 (1988).
[CrossRef]

Subbaswamy, K. R.

G. Senatore and K. R. Subbaswamy, Phys. Rev. A 35, 2440 (1987).
[CrossRef] [PubMed]

Suematsu, Y.

Y. Miyamoto, M. Cao, Y. Shingai, K. Furuya, Y. Suematsu, K. G. Ravikumar, and S. Arai, Jpn. J. Appl. Phys. 26, 225 (1987).
[CrossRef]

van Driel, H. M.

D. J. Moss, E. Ghahramani, J. E. Sipe, and H. M. van Driel, Phys. Rev. B 41, 1542 (1990).
[CrossRef]

Yariv, A.

A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1975), Chap. 16.

Appl. Phys. Lett. (1)

B. F. Levine and C. G. Louie, Appl. Phys. Lett. 20, 272 (1972).
[CrossRef]

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

J. Phys. Soc. Jpn. (1)

R. Kubo, J. Phys. Soc. Jpn. 12, 570 (1957).
[CrossRef]

Jpn. J. Appl. Phys. (1)

Y. Miyamoto, M. Cao, Y. Shingai, K. Furuya, Y. Suematsu, K. G. Ravikumar, and S. Arai, Jpn. J. Appl. Phys. 26, 225 (1987).
[CrossRef]

Phys. Rev. A (1)

G. Senatore and K. R. Subbaswamy, Phys. Rev. A 35, 2440 (1987).
[CrossRef] [PubMed]

Phys. Rev. B (4)

D. J. Moss, E. Ghahramani, J. E. Sipe, and H. M. van Driel, Phys. Rev. B 41, 1542 (1990).
[CrossRef]

Although the effects of temperature should warrant a full study, we expect that they should be of the order of kBT/ΔE(where ΔE is the interlevel excitation-energy difference); then, as long as T ≪ ΔE/kB, the effects of temperature should be negligible.See, for example, B. S. Mendoza and Y. C. Lee, Phys. Rev. B 40, 12063 (1989).
[CrossRef]

W. L. Schaich and B. S. Mendoza, Phys. Rev. B 45, 14279 (1992).
[CrossRef]

E. Rosencher and Ph. Bois, Phys. Rev. B 44, 11315 (1991).
[CrossRef]

Phys. Rev. Lett. (1)

Ch. Sikorski and U. Merkt, Phys. Rev. Lett. 62, 2164 (1989).
[CrossRef] [PubMed]

Other (3)

We list several conference proceedings in which both experimental and theoretical examples and many further references can be found: C. R. Leavens and R. Taylor, eds., Interfaces, Quantum Wells and Superlattices (Plenum, New York, 1988);M. A. Reed and W. P. Kirk, eds., Nanostruc-ture Physics and Fabrication (Academic, New York, 1989);Surf. Sci. 228 and 229 (1990);and special issue on optics of nanostructures, Phys. Today,  46(6) (1993).
[CrossRef]

A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1975), Chap. 16.

A. Guerrero, “Second-harmonic generation by a system of quantum dots,” master’s thesis, Centro de Investigaciones en Optica, León, México, 1994).

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Figures (10)

Fig. 1
Fig. 1

2D array of QD’s, where the triangles, dots and crosses represent the three different sublattices, 1, 2, and 3, respectively. The lattice period along x is 2b and along y is b, and the distance between 1 and 2 is a. The inset shows the geometry of the QD.

Fig. 2
Fig. 2

Geometry of the incident beam, where P and S polarization vectors, θ is the angle of incidence, and ϕ is the azimuthal angle (see text for details).

Fig. 3
Fig. 3

Plot of α/α0 [Eq. (15)] versus ω/ω0 for the QD parameters given in the text.

Fig. 4
Fig. 4

Plot of χ 1 , 2 , 3 ( d , Q ) [see Eqs. (27) and (34), and Appendix A] versus ω/ω0.The prefactor (1/32π2) (1/e)α02[(1/16 π2) (1/e)α02] for the dipolar (quadrupolar) susceptibilities has been taken out (see text for details).

Fig. 5
Fig. 5

SHG efficiency versus ã = a/b and W = ω / ω 0 for pP polarization. We plot on a linear scale normalized to its maximum value of 1.8 × 10−16 cm2/W, which occurs at ã = 0.318 and W = 0.97.

Fig. 6
Fig. 6

SHG efficiency versus ã = a/b and W = ω / ω 0 for pS polarization. We plot on a linear scale normalized to its maximum value of 0.41 × 10−16 cm2/W, which occurs at ã = 0.212 and ω = 2.13.

Fig. 7
Fig. 7

Three upper panels, plots of the imaginary part of γ (solid curves), 1/det(T2ω) (dashed curves), and their product (dotted curves) versus ω/ω0 [see relation (76)], for the xxx, xyy, and yxy Cartesian components. Note that the left-hand scale is for the solid curves, the first scale on the right is for the dashed curves, and the rightmost scale is for the dotted curves. The three scales are in arbitrary units. Bottom panel, SHG efficiency versus ω/ω0 for pP polarization. We plot on a logarithmic scale. The maximum value corresponds to the highest peak in Fig. 5 (see text for details).

Fig. 8
Fig. 8

Same as Fig. 7, except the maximum value corresponds to the second highest peak in Fig. 5 (see text for details).

Fig. 9
Fig. 9

Same as Fig. 7, except the maximum value corresponds to the highest peak in Fig. 6 (see text for details).

Fig. 10
Fig. 10

SHG efficiency versus θ and ϕ for pP polarization. We have chosen ã = 0.318 and, W = 0.97 which correspond to the highest peak in Fig. 5. We plot on a linear scale normalized to its maximum value of 1.8 × 10−16 cm2/W (see text for details).

Equations (84)

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U 0 ( r ) = { r R 0 z ± h 0 / 2 0 otherwise ,
Ĥ 0 = P 2 2 m * + U 0 .
Ĥ 0 ψ l m n = E l m n ψ l m n ,
ψ l m n ( r , θ , z ) = 2 R 0 J m [ ν m l ( r / R 0 ) ] J m + 1 ( ν m l ) 1 2 π exp ( i m θ ) ( 2 h 0 ) 1 / 2 × cos [ n π h 0 ( z + h 0 2 ) ] ,
E l m n = 2 2 m * ( π 2 n 2 h 0 2 + ν m l 2 R 0 2 ) ,
δ O ( t ) = 1 i t d s [ O ( t ) , h ( s ) ] ,
h = H [ exp ( i ω t ) + exp ( i ω t ) ] ,
δ O ( t ) = δ O ( ω ) [ exp ( i ω t ) + exp ( i ω t ) ] ,
δ O ( 1 ) ( ω ) = 1 i 0 d τ exp ( i ω t ) [ O ( τ ) , H ] ,
δ O ( 1 ) ( ω ) = 1 β [ O 0 β H β 0 ω ω β 0 H β 0 O β 0 ω + ω β 0 ] ,
H = e x ̂ j E j ( L ) ½ e x ̂ j x ̂ k ( j E k ) L m ̂ j B j ( L ) ,
O e x ̂ i p i .
δ p i ( 1 ) ( ω ) = α i j ( ω ) E j ( L ) + β i j k ( j E k ) L .
β i j k 0 ( by inversion symmetry ) ,
α ( ω ) = α 0 l 2 ω l 1 ω l 1 2 ω 2 Δ 1 2 ( l , 1 ; 1 , 0 ) ,
ω l m = ν l m 2 ν 10 2 1 ,
E l m = ω 0 ω l m ,
α 0 = 4 π R 0 3 ( e 2 / R 0 ω 0 )
Δ N ( l , m ; l , m ) = 0 1 d x J m ( ν l m x ) J m + 1 ( ν l m ) x N + 1 J m ( ν l m x ) J m + 1 ( ν l m )
δ Q i j ( 1 ) ( ω ) = α i j k ( Q ) ( ω ) E k ( L ) + β i j k l ( Q ) ( k E l ) L .
p ( 1 ) Q ( 1 ) a R 0 2 = ( a / R 0 ) 2 a ,
p i ( 1 ) ( ω ) = α i j ( ω ) E j ( L )
δ O ( 2 ) ( 2 ω ) = ( 1 i ) 2 0 d τ 1 exp ( i ω τ 1 ) τ 1 d τ 2 exp ( i ω τ 2 ) × [ [ O , H ( τ 1 ) ] , H ( τ 2 ) ] = 1 2 α , β [ O 0 α H α β H β 0 ( 2 ω ω α 0 ) ( ω ω β 0 ) + H 0 α H α β O β 0 ( 2 ω + ω α 0 ) ( ω + ω β 0 ) H 0 α O α β H β 0 ( ω + ω α 0 ) ( ω ω β 0 ) ] .
δ p i ( 2 ) ( 2 ω ) = χ i j k ( d ) ( ω ) E j ( L ) E k ( L ) + χ i j k l ( d ) ( ω ) E j ( L ) ( k E l ) L + χ i j k l m ( d ) ( ω ) ( j E k ) L ( l E m ) L .
χ i j k ( d ) ( ω ) = 0 ( by inversion symmetry ) ,
χ i j k l m ( d ) ( ω ) = 0 ( by inversion symmetry ) .
χ i j k l ( d ) = χ 1 ( d ) δ i j δ j k δ k l + [ χ 2 ( d ) ( δ i k δ j l + δ i l δ j k ) + χ 3 ( d ) δ i j δ k l ] × ( 1 δ i j δ j k δ k l )
O i j = e x ̂ i x ̂ j
δ Q i j ( 2 ) ( 2 ω ) = χ i j k l ( Q ) ( ω ) E k ( L ) E l ( L ) + γ i j k l m ( Q ) ( ω ) E k ( L ) ( l E m ) L + Γ i j k l m n ( Q ) ( ω ) ( k E l ) L ( m E n ) L .
γ i j k l m ( Q ) ( ω ) = 0 ( by inversion symmetry ) ,
χ ( Q ) E E / Γ ( Q ) E E ( a / R 0 ) 2 ,
p ( 2 ) / Q ( 2 ) 1 / a .
δ Q i j ( 2 ) ( 2 ω ) = χ i j k l ( Q ) ( ω ) E k ( L ) E l ( L ) ,
χ i j k l ( Q ) = χ 1 ( Q ) δ i j δ j k δ k l + [ χ 2 ( Q ) ( δ i k δ j l + δ i l δ j k ) + χ 3 ( Q ) δ i j δ k l ] × ( 1 δ i j δ j k δ k l ) .
p i = α i j ( ω ) E j ( L ) + χ i j k l ( d ) ( 2 ω ) E j ( L ) ( k E l ) L ,
h 0 < R 0 < a c / ω .
M i = i ( 1 / R ) ,
M i j = i j ( 1 / R ) = i M j ,
M i j k = i j k ( 1 / R ) = i M j k .
Φ ( x , x ) = p i M i + ½ Q i j M i j + ,
E i ( x , x ) = i Φ ( x , x ) = M i j p j + ½ M i j k Q j k + ,
i E j ( x , x ) = M i j k p k + .
δ p i ( 1 ) ( ω ) p i ( 1 ) ( ρ ) = α ( ω ) E i ( ρ , L ) ,
ρ = n 1 a 1 x ̂ + n 2 a 2 ŷ
E i ( ρ , L ) = E i ( A ) + ρ M i j ( ρ ρ ) p j ( 1 ) ( ρ ) ,
p i ( 1 ) ( ρ ) = α ( ω ) [ E i ( A ) + ρ M i j ( ρ ρ ) p j ( 1 ) ( ρ ) ] .
ρ M i j ( ρ ρ ) ρ M i j ( ρ ρ ) i j ( , ) ,
p i ( 1 ) ( ) = α ( ω ) [ E i ( A ) + i j ( , ) p j ( 1 ) ( ) ] ,
δ p i ( 2 ) ( 2 ω ) p i ( nl ) ( l ) ,
p i ( tot ) ( ρ ) = p i ( nl ) ( ρ ) + α ( 2 ω ) [ E i ( Q ) ( ρ ) + ρ M i j ( ρ ρ ) p j ( tot ) ( ρ ) ] ,
[ δ i j δ α ( 2 ω ) i j ( , ) ] p j ( tot ) ( ) = χ i j k l ( d ) E j ( loc ) ( ω , ) k l m ( , ) p m ( 1 ) ( ω , ) + ½ α ( 2 ω ) i j k ( , ) χ j k l m ( Q ) E l ( loc ) ( ω , ) E m ( loc ) ( ω , ) ,
ρ M i j k ( ρ ρ ) = i j k ( , ) ,
E i ( loc ) ( ω , ) = α 1 ( ω ) p i ( 1 ) ( ω , )
( T n ω ) i j = δ i j δ α ( n ω ) i j ( , ) ,
( T n ω 1 ) i k ( T n ω ) k j = ( T n ω ) i k ( T n ω 1 ) k j = δ i j δ .
p i ( 1 ) ( ) = α ( ω ) ζ i j E j ( A ) ,
ζ i j = ( T ω 1 ) i j .
E i loc ( ) = ζ i j E j ( A ) .
P i tot ( ) = χ i j k ( ) E j ( A ) E k ( A ) ,
χ i j k ( ) = ( T 2 ω 1 ) i m Γ m j k ,
Γ i j k = χ i r t l ( d ) ζ r j t l s α ( ω ) ζ s k + ½ α ( 2 ω ) i m t χ m t r s ( Q ) ζ r j ζ s k .
P i tot = p i tot ( ) = χ i j k eff E j ( A ) E k ( A ) ,
χ i j k eff = χ i j k ( ) .
= P ( 2 ω ) / P 2 ( ω ) ,
P 2 ω ( z , t ) = P 2 ω δ ( z 0 + ) exp ( i Ω t ) + c.c. ,
P i 2 ω = χ i j k eff E j ( z = 0 ) E k ( z = 0 ) ,
= 32 π 2 ω 2 c 3 A sec 2 ( θ ) | e 2 ω · χ eff : e ω e ω | 2 ,
e ω = ( ŝ t 0 m s ŝ + p ̂ m t 0 m p p ̂ 0 ) · ê in ,
e 2 ω = ê out · [ ŝ ( 1 + R 0 m s ) ŝ + P ̂ 0 + ( P ̂ 0 + + P ̂ 0 R 0 m p ) ] ,
m ( ω ) = ( ω 2 ω L 2 ω 2 ω T 2 ) ,
sin 2 ( Δ k L / 2 ) / ( Δ k L / 2 ) 2 ,
x x x , x y y = y x y = y y x ;
i j k ( 2 , 3 ) = i j k ( , ) = 0 ,
i j k ( 1 , 2 ) = i j k ( 1 , 3 ) = 0 for a = b / 2 ,
χ x x x eff , χ x y y eff , χ y x y eff = χ y y x eff ,
χ eff 1 det ( T 2 ω ) γ ,
χ 0 = 1 32 π 2 1 e n 0 α 0 3 0 2 b 4 ,
η = χ 0 χ bulk | χ ( ω ) | ,
χ 1 ( d ) = 1 32 π 2 1 e α 0 2 l , l { 2 Δ 1 ( l , 1 ; 1 , 0 ) Δ 1 ( l , 1 ; l , 0 ) Δ 2 ( l , 0 ; 1 , 0 ) × [ 1 ( 2 ω ω l 1 ) ( ω ω l 0 ) 1 ( ω + ω l 1 ) ( ω ω l 0 ) + 1 ( 2 ω + ω l 1 ) ( ω + ω l 0 ) 1 ( ω + ω l 0 ) ( ω ω l 1 ) ] + Δ 1 ( l , 1 ; 1 , 0 ) Δ 1 ( l , 1 ; l , 2 ) Δ 2 ( l , 2 ; 1 , 0 ) × [ 1 ( 2 ω ω l 1 ) ( ω ω l 2 ) 1 ( ω + ω l 1 ) ( ω ω l 2 ) + 1 ( 2 ω + ω l 1 ) ( ω + ω l 2 ) 1 ( ω + ω l 2 ) ( ω ω l 1 ) ] + 3 Δ 1 ( l , 1 ; 1 , 0 ) Δ 1 ( l , 1 ; l , 0 ) Δ 2 ( l , 1 ; l , 1 ) × [ 1 ( ω ω l 1 ) ( 2 ω ω l 1 ) + 1 ( 2 ω + ω l 1 ) ( ω + ω l 1 ) ] } ,
χ 2 ( d ) = 1 32 π 2 1 e α 0 2 l , l { Δ 1 ( l , 1 ; 1 , 0 ) Δ 1 ( l , 1 ; l , 2 ) Δ 2 ( l , 2 ; 1 , 0 ) [ 1 ( 2 ω + ω l 1 ) ( ω + ω l 2 ) + 1 ( 2 ω + ω l 1 ) ( ω + ω l 2 ) ] + Δ 1 ( l , 1 ; 1 , 0 ) Δ 1 ( l , 1 ; l , 0 ) Δ 2 ( l , 1 ; l , 1 ) [ 1 ( 2 ω ω l 1 ) ( ω ω l 1 ) + 1 ( 2 ω + ω l 1 ) ( ω ω l 1 ) ] Δ 1 ( l , 1 ; 1 , 0 ) Δ 1 ( l , 1 ; l , 2 ) Δ 2 ( l , 2 ; l , 0 ) [ 1 ( ω + ω l 1 ) ( ω ω l 2 ) + 1 ( ω + ω l 2 ) ( ω ω l 1 ) ] } ,
χ 3 ( d ) = 1 32 π 2 1 e α 0 2 l , l { 2 Δ 1 ( l , 1 ; 1 , 0 ) Δ 1 ( l , 1 ; l , 0 ) Δ 2 ( l , 0 ; 1 , 0 ) × [ 1 ( 2 ω ω l 1 ) ( ω ω l 0 ) 1 ( ω + ω l 1 ) ( ω ω l 0 ) + 1 ( 2 ω + ω l 1 ) ( ω + ω l 0 ) 1 ( ω + ω l 0 ) ( ω ω l 1 ) ] Δ 1 ( l , 1 ; 1 , 0 ) Δ 1 ( l , 1 ; l , 2 ) Δ 2 ( l , 2 ; 1 , 0 ) × [ 1 ( 2 ω ω l 1 ) ( ω ω l 2 ) 1 ( ω + ω l 1 ) ( ω ω l 2 ) + 1 ( 2 ω + ω l 1 ) ( ω + ω l 2 ) 1 ( ω + ω l 2 ) ( ω ω l 1 ) ] + Δ 1 ( l , 1 ; 1 , 0 ) Δ 1 ( l , 1 ; l , 0 ) Δ 2 ( l , 1 ; l , 1 ) × [ 1 ( 2 ω ω l 1 ) ( ω ω l 1 ) + 1 ( 2 ω + ω l 1 ) ( ω + ω l 1 ) ] .
χ 1 ( Q ) = 1 16 π 2 1 e α 0 2 × l , l { 2 Δ 1 ( l , 1 ; 1 , 0 ) Δ 1 ( l , 1 ; l , 0 ) Δ 2 ( l , 0 ; 1 , 0 ) × [ 1 ( 2 ω + ω l 0 ) ( ω + ω l 1 ) + 1 ( 2 ω ω l 0 ) ( ω ω l 1 ) ] + Δ 1 ( l , 1 ; 1 , 0 ) Δ 1 ( l , 1 ; l , 2 ) Δ 2 ( l , 2 ; 1 , 0 ) × [ 1 ( 2 ω + ω l 2 ) ( ω + ω l 1 ) 1 ( 2 ω ω l 2 ) ( ω ω l 1 ) ] 3 Δ 1 ( l , 1 ; 1 , 0 ) Δ 1 ( l , 1 ; 1 , 0 ) Δ 2 ( l , 1 ; l , 1 ) × [ 1 ( ω + ω l 1 ) ( ω ω l 1 ) ] } ,
χ 2 ( Q ) = 1 16 π 2 1 e α 0 2 × l , l { Δ 1 ( l , 2 ; l , 1 ) Δ 1 ( l , 1 ; 1 , 0 ) Δ 2 ( l , 2 ; 1 , 0 ) × [ 1 ( 2 ω ω l 2 ) ( ω ω l 1 ) + 1 ( 2 ω + ω l 2 ) ( ω + ω l 1 ) ] Δ 1 ( l , 1 ; 1 , 0 ) Δ 1 ( l , 1 ; 1 , 0 ) Δ 2 ( l , 1 ; l , 1 ) × [ 1 ( ω + ω l 1 ) ( ω ω l 1 ) ] } ,
χ 3 ( Q ) = 1 16 π 2 1 e α 0 2 × l , l { 2 Δ 2 ( l , 0 ; 1 , 0 ) Δ 1 ( l , 0 ; l , 1 ) Δ 1 ( l , 1 ; 1 , 0 ) × [ 1 ( 2 ω ω l 0 ) ( ω ω l 1 ) + 1 ( 2 ω + ω l 0 ) ( ω + ω l 1 ) ] Δ 2 ( l , 2 ; 1 , 0 ) Δ 1 ( l , 2 ; l , 1 ) Δ 1 ( l , 1 ; 1 , 0 ) × [ 1 ( 2 ω ω l 2 ) ( ω ω l 1 ) + 1 ( 2 ω + ω l 2 ) ( ω + ω l 1 ) ] Δ 1 ( l , 1 ; 1 , 0 ) Δ 2 ( l , 1 ; l , 1 ) Δ 1 ( l , 1 ; 1 , 0 ) × [ 1 ( ω + ω l 1 ) ( ω ω l 1 ) ] .

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