Abstract

The two-photon absorption coefficient and the intensity-dependent refractive index associated with it are calculated for semiconductor quantum wells and quantum wires. It is shown analytically that in the limit of large dimensions the nonlinear response of quantum-confined structures becomes isotropic and approaches the nonlinear response of the bulk semiconductor despite the different gauges used in the calculations. The theoretical results are compared with the experiment, and the practical implications are discussed.

© 1994 Optical Society of America

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  1. D. A. B. Miller, J. S. Weiner, and D. S. Chemla, IEEE J. Quantum Electron. QE-22, 1816 (1986).
    [CrossRef]
  2. D. S. Chemla, T. C. Damen, D. A. B. Miller, A. C. Gossard, and W. Wiegmann, Appl. Phys. Lett. 44, 16 (1984).
    [CrossRef]
  3. A. Shimizu and K. Fujii, Nonlinear Opt. 1, 149 (1991).
  4. B. S. Wherrett, Proc. R. Soc. London A 390, 373 (1983).
    [CrossRef]
  5. K. W. Delong and G. I. Stegeman, Appl. Phys. Lett. 57, 2063 (1990).
    [CrossRef]
  6. M. Sheik-Bahae, D. C. Hutchings, D. J. Hagan, and E. W. Van Stryland, IEEE J. Quantum Electron. 27, 1296 (1991).
    [CrossRef]
  7. B. S. Wherrett, J. Opt. Soc. Am. B 1, 67 (1984).
    [CrossRef]
  8. M. Sheik-Bahae, D. J. Hagan, and E. W. Van Stryland, Phys. Rev. Lett. 65, 96 (1990).
    [CrossRef] [PubMed]
  9. H. N. Spector, Phys. Rev. B 35, 5876 (1987).
    [CrossRef]
  10. A. Pasquarello and A. Quattropani, Phys. Rev. B 38, 6206 (1988).
    [CrossRef]
  11. A. Shimizu, Phys. Rev. B 40, 1403 (1989).
    [CrossRef]
  12. A. Shimizu, T. Ogawa, and H. Sakaki, Phys. Rev. B 45, 11338 (1992).
    [CrossRef]
  13. K. Tai, Phys. Rev. Lett. 62, 1784 (1989).
    [CrossRef] [PubMed]
  14. H. K. Tsang, R. V. Penty, I. H. White, R. S. Grant, and W. Sibbett, Appl. Phys. Lett. 59, 3440 (1991).
    [CrossRef]
  15. R. Cingolani, M. Lepore, R. Tommasi, I. M. Catalano, H. Lage, D. Heitmann, K. Ploog, A. Shimizu, H. Sakaki, and T. Ogawa, Phys. Rev. Lett. 69, 1277 (1992).
    [CrossRef]
  16. J. M. Luttinger and W. Kohn, Phys. Rev. 7, 869 (1955).
    [CrossRef]
  17. E. O. Kane, J. Phys. Chem. Solids 1, 249 (1957).
    [CrossRef]
  18. M. L. Cohen and J. R. Chelikowsky, Electronic Structure and Optical Properties of Semiconductors (Springer-Verlag, Berlin, 1988), p. 49.
  19. S. T. Ho, G. E. Soccolich, M. N. Islam, W. S. Hobson, A. F. J. Levi, and R. E. Slisher, Appl. Phys. Lett. 59, 2558 (1991).
    [CrossRef]

1992 (2)

A. Shimizu, T. Ogawa, and H. Sakaki, Phys. Rev. B 45, 11338 (1992).
[CrossRef]

R. Cingolani, M. Lepore, R. Tommasi, I. M. Catalano, H. Lage, D. Heitmann, K. Ploog, A. Shimizu, H. Sakaki, and T. Ogawa, Phys. Rev. Lett. 69, 1277 (1992).
[CrossRef]

1991 (4)

H. K. Tsang, R. V. Penty, I. H. White, R. S. Grant, and W. Sibbett, Appl. Phys. Lett. 59, 3440 (1991).
[CrossRef]

S. T. Ho, G. E. Soccolich, M. N. Islam, W. S. Hobson, A. F. J. Levi, and R. E. Slisher, Appl. Phys. Lett. 59, 2558 (1991).
[CrossRef]

A. Shimizu and K. Fujii, Nonlinear Opt. 1, 149 (1991).

M. Sheik-Bahae, D. C. Hutchings, D. J. Hagan, and E. W. Van Stryland, IEEE J. Quantum Electron. 27, 1296 (1991).
[CrossRef]

1990 (2)

M. Sheik-Bahae, D. J. Hagan, and E. W. Van Stryland, Phys. Rev. Lett. 65, 96 (1990).
[CrossRef] [PubMed]

K. W. Delong and G. I. Stegeman, Appl. Phys. Lett. 57, 2063 (1990).
[CrossRef]

1989 (2)

A. Shimizu, Phys. Rev. B 40, 1403 (1989).
[CrossRef]

K. Tai, Phys. Rev. Lett. 62, 1784 (1989).
[CrossRef] [PubMed]

1988 (1)

A. Pasquarello and A. Quattropani, Phys. Rev. B 38, 6206 (1988).
[CrossRef]

1987 (1)

H. N. Spector, Phys. Rev. B 35, 5876 (1987).
[CrossRef]

1986 (1)

D. A. B. Miller, J. S. Weiner, and D. S. Chemla, IEEE J. Quantum Electron. QE-22, 1816 (1986).
[CrossRef]

1984 (2)

D. S. Chemla, T. C. Damen, D. A. B. Miller, A. C. Gossard, and W. Wiegmann, Appl. Phys. Lett. 44, 16 (1984).
[CrossRef]

B. S. Wherrett, J. Opt. Soc. Am. B 1, 67 (1984).
[CrossRef]

1983 (1)

B. S. Wherrett, Proc. R. Soc. London A 390, 373 (1983).
[CrossRef]

1957 (1)

E. O. Kane, J. Phys. Chem. Solids 1, 249 (1957).
[CrossRef]

1955 (1)

J. M. Luttinger and W. Kohn, Phys. Rev. 7, 869 (1955).
[CrossRef]

Catalano, I. M.

R. Cingolani, M. Lepore, R. Tommasi, I. M. Catalano, H. Lage, D. Heitmann, K. Ploog, A. Shimizu, H. Sakaki, and T. Ogawa, Phys. Rev. Lett. 69, 1277 (1992).
[CrossRef]

Chelikowsky, J. R.

M. L. Cohen and J. R. Chelikowsky, Electronic Structure and Optical Properties of Semiconductors (Springer-Verlag, Berlin, 1988), p. 49.

Chemla, D. S.

D. A. B. Miller, J. S. Weiner, and D. S. Chemla, IEEE J. Quantum Electron. QE-22, 1816 (1986).
[CrossRef]

D. S. Chemla, T. C. Damen, D. A. B. Miller, A. C. Gossard, and W. Wiegmann, Appl. Phys. Lett. 44, 16 (1984).
[CrossRef]

Cingolani, R.

R. Cingolani, M. Lepore, R. Tommasi, I. M. Catalano, H. Lage, D. Heitmann, K. Ploog, A. Shimizu, H. Sakaki, and T. Ogawa, Phys. Rev. Lett. 69, 1277 (1992).
[CrossRef]

Cohen, M. L.

M. L. Cohen and J. R. Chelikowsky, Electronic Structure and Optical Properties of Semiconductors (Springer-Verlag, Berlin, 1988), p. 49.

Damen, T. C.

D. S. Chemla, T. C. Damen, D. A. B. Miller, A. C. Gossard, and W. Wiegmann, Appl. Phys. Lett. 44, 16 (1984).
[CrossRef]

Delong, K. W.

K. W. Delong and G. I. Stegeman, Appl. Phys. Lett. 57, 2063 (1990).
[CrossRef]

Fujii, K.

A. Shimizu and K. Fujii, Nonlinear Opt. 1, 149 (1991).

Gossard, A. C.

D. S. Chemla, T. C. Damen, D. A. B. Miller, A. C. Gossard, and W. Wiegmann, Appl. Phys. Lett. 44, 16 (1984).
[CrossRef]

Grant, R. S.

H. K. Tsang, R. V. Penty, I. H. White, R. S. Grant, and W. Sibbett, Appl. Phys. Lett. 59, 3440 (1991).
[CrossRef]

Hagan, D. J.

M. Sheik-Bahae, D. C. Hutchings, D. J. Hagan, and E. W. Van Stryland, IEEE J. Quantum Electron. 27, 1296 (1991).
[CrossRef]

M. Sheik-Bahae, D. J. Hagan, and E. W. Van Stryland, Phys. Rev. Lett. 65, 96 (1990).
[CrossRef] [PubMed]

Heitmann, D.

R. Cingolani, M. Lepore, R. Tommasi, I. M. Catalano, H. Lage, D. Heitmann, K. Ploog, A. Shimizu, H. Sakaki, and T. Ogawa, Phys. Rev. Lett. 69, 1277 (1992).
[CrossRef]

Ho, S. T.

S. T. Ho, G. E. Soccolich, M. N. Islam, W. S. Hobson, A. F. J. Levi, and R. E. Slisher, Appl. Phys. Lett. 59, 2558 (1991).
[CrossRef]

Hobson, W. S.

S. T. Ho, G. E. Soccolich, M. N. Islam, W. S. Hobson, A. F. J. Levi, and R. E. Slisher, Appl. Phys. Lett. 59, 2558 (1991).
[CrossRef]

Hutchings, D. C.

M. Sheik-Bahae, D. C. Hutchings, D. J. Hagan, and E. W. Van Stryland, IEEE J. Quantum Electron. 27, 1296 (1991).
[CrossRef]

Islam, M. N.

S. T. Ho, G. E. Soccolich, M. N. Islam, W. S. Hobson, A. F. J. Levi, and R. E. Slisher, Appl. Phys. Lett. 59, 2558 (1991).
[CrossRef]

Kane, E. O.

E. O. Kane, J. Phys. Chem. Solids 1, 249 (1957).
[CrossRef]

Kohn, W.

J. M. Luttinger and W. Kohn, Phys. Rev. 7, 869 (1955).
[CrossRef]

Lage, H.

R. Cingolani, M. Lepore, R. Tommasi, I. M. Catalano, H. Lage, D. Heitmann, K. Ploog, A. Shimizu, H. Sakaki, and T. Ogawa, Phys. Rev. Lett. 69, 1277 (1992).
[CrossRef]

Lepore, M.

R. Cingolani, M. Lepore, R. Tommasi, I. M. Catalano, H. Lage, D. Heitmann, K. Ploog, A. Shimizu, H. Sakaki, and T. Ogawa, Phys. Rev. Lett. 69, 1277 (1992).
[CrossRef]

Levi, A. F. J.

S. T. Ho, G. E. Soccolich, M. N. Islam, W. S. Hobson, A. F. J. Levi, and R. E. Slisher, Appl. Phys. Lett. 59, 2558 (1991).
[CrossRef]

Luttinger, J. M.

J. M. Luttinger and W. Kohn, Phys. Rev. 7, 869 (1955).
[CrossRef]

Miller, D. A. B.

D. A. B. Miller, J. S. Weiner, and D. S. Chemla, IEEE J. Quantum Electron. QE-22, 1816 (1986).
[CrossRef]

D. S. Chemla, T. C. Damen, D. A. B. Miller, A. C. Gossard, and W. Wiegmann, Appl. Phys. Lett. 44, 16 (1984).
[CrossRef]

Ogawa, T.

R. Cingolani, M. Lepore, R. Tommasi, I. M. Catalano, H. Lage, D. Heitmann, K. Ploog, A. Shimizu, H. Sakaki, and T. Ogawa, Phys. Rev. Lett. 69, 1277 (1992).
[CrossRef]

A. Shimizu, T. Ogawa, and H. Sakaki, Phys. Rev. B 45, 11338 (1992).
[CrossRef]

Pasquarello, A.

A. Pasquarello and A. Quattropani, Phys. Rev. B 38, 6206 (1988).
[CrossRef]

Penty, R. V.

H. K. Tsang, R. V. Penty, I. H. White, R. S. Grant, and W. Sibbett, Appl. Phys. Lett. 59, 3440 (1991).
[CrossRef]

Ploog, K.

R. Cingolani, M. Lepore, R. Tommasi, I. M. Catalano, H. Lage, D. Heitmann, K. Ploog, A. Shimizu, H. Sakaki, and T. Ogawa, Phys. Rev. Lett. 69, 1277 (1992).
[CrossRef]

Quattropani, A.

A. Pasquarello and A. Quattropani, Phys. Rev. B 38, 6206 (1988).
[CrossRef]

Sakaki, H.

A. Shimizu, T. Ogawa, and H. Sakaki, Phys. Rev. B 45, 11338 (1992).
[CrossRef]

R. Cingolani, M. Lepore, R. Tommasi, I. M. Catalano, H. Lage, D. Heitmann, K. Ploog, A. Shimizu, H. Sakaki, and T. Ogawa, Phys. Rev. Lett. 69, 1277 (1992).
[CrossRef]

Sheik-Bahae, M.

M. Sheik-Bahae, D. C. Hutchings, D. J. Hagan, and E. W. Van Stryland, IEEE J. Quantum Electron. 27, 1296 (1991).
[CrossRef]

M. Sheik-Bahae, D. J. Hagan, and E. W. Van Stryland, Phys. Rev. Lett. 65, 96 (1990).
[CrossRef] [PubMed]

Shimizu, A.

A. Shimizu, T. Ogawa, and H. Sakaki, Phys. Rev. B 45, 11338 (1992).
[CrossRef]

R. Cingolani, M. Lepore, R. Tommasi, I. M. Catalano, H. Lage, D. Heitmann, K. Ploog, A. Shimizu, H. Sakaki, and T. Ogawa, Phys. Rev. Lett. 69, 1277 (1992).
[CrossRef]

A. Shimizu and K. Fujii, Nonlinear Opt. 1, 149 (1991).

A. Shimizu, Phys. Rev. B 40, 1403 (1989).
[CrossRef]

Sibbett, W.

H. K. Tsang, R. V. Penty, I. H. White, R. S. Grant, and W. Sibbett, Appl. Phys. Lett. 59, 3440 (1991).
[CrossRef]

Slisher, R. E.

S. T. Ho, G. E. Soccolich, M. N. Islam, W. S. Hobson, A. F. J. Levi, and R. E. Slisher, Appl. Phys. Lett. 59, 2558 (1991).
[CrossRef]

Soccolich, G. E.

S. T. Ho, G. E. Soccolich, M. N. Islam, W. S. Hobson, A. F. J. Levi, and R. E. Slisher, Appl. Phys. Lett. 59, 2558 (1991).
[CrossRef]

Spector, H. N.

H. N. Spector, Phys. Rev. B 35, 5876 (1987).
[CrossRef]

Stegeman, G. I.

K. W. Delong and G. I. Stegeman, Appl. Phys. Lett. 57, 2063 (1990).
[CrossRef]

Tai, K.

K. Tai, Phys. Rev. Lett. 62, 1784 (1989).
[CrossRef] [PubMed]

Tommasi, R.

R. Cingolani, M. Lepore, R. Tommasi, I. M. Catalano, H. Lage, D. Heitmann, K. Ploog, A. Shimizu, H. Sakaki, and T. Ogawa, Phys. Rev. Lett. 69, 1277 (1992).
[CrossRef]

Tsang, H. K.

H. K. Tsang, R. V. Penty, I. H. White, R. S. Grant, and W. Sibbett, Appl. Phys. Lett. 59, 3440 (1991).
[CrossRef]

Van Stryland, E. W.

M. Sheik-Bahae, D. C. Hutchings, D. J. Hagan, and E. W. Van Stryland, IEEE J. Quantum Electron. 27, 1296 (1991).
[CrossRef]

M. Sheik-Bahae, D. J. Hagan, and E. W. Van Stryland, Phys. Rev. Lett. 65, 96 (1990).
[CrossRef] [PubMed]

Weiner, J. S.

D. A. B. Miller, J. S. Weiner, and D. S. Chemla, IEEE J. Quantum Electron. QE-22, 1816 (1986).
[CrossRef]

Wherrett, B. S.

B. S. Wherrett, J. Opt. Soc. Am. B 1, 67 (1984).
[CrossRef]

B. S. Wherrett, Proc. R. Soc. London A 390, 373 (1983).
[CrossRef]

White, I. H.

H. K. Tsang, R. V. Penty, I. H. White, R. S. Grant, and W. Sibbett, Appl. Phys. Lett. 59, 3440 (1991).
[CrossRef]

Wiegmann, W.

D. S. Chemla, T. C. Damen, D. A. B. Miller, A. C. Gossard, and W. Wiegmann, Appl. Phys. Lett. 44, 16 (1984).
[CrossRef]

Appl. Phys. Lett. (4)

K. W. Delong and G. I. Stegeman, Appl. Phys. Lett. 57, 2063 (1990).
[CrossRef]

D. S. Chemla, T. C. Damen, D. A. B. Miller, A. C. Gossard, and W. Wiegmann, Appl. Phys. Lett. 44, 16 (1984).
[CrossRef]

H. K. Tsang, R. V. Penty, I. H. White, R. S. Grant, and W. Sibbett, Appl. Phys. Lett. 59, 3440 (1991).
[CrossRef]

S. T. Ho, G. E. Soccolich, M. N. Islam, W. S. Hobson, A. F. J. Levi, and R. E. Slisher, Appl. Phys. Lett. 59, 2558 (1991).
[CrossRef]

IEEE J. Quantum Electron. (2)

D. A. B. Miller, J. S. Weiner, and D. S. Chemla, IEEE J. Quantum Electron. QE-22, 1816 (1986).
[CrossRef]

M. Sheik-Bahae, D. C. Hutchings, D. J. Hagan, and E. W. Van Stryland, IEEE J. Quantum Electron. 27, 1296 (1991).
[CrossRef]

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

J. Phys. Chem. Solids (1)

E. O. Kane, J. Phys. Chem. Solids 1, 249 (1957).
[CrossRef]

Nonlinear Opt. (1)

A. Shimizu and K. Fujii, Nonlinear Opt. 1, 149 (1991).

Phys. Rev. (1)

J. M. Luttinger and W. Kohn, Phys. Rev. 7, 869 (1955).
[CrossRef]

Phys. Rev. B (4)

H. N. Spector, Phys. Rev. B 35, 5876 (1987).
[CrossRef]

A. Pasquarello and A. Quattropani, Phys. Rev. B 38, 6206 (1988).
[CrossRef]

A. Shimizu, Phys. Rev. B 40, 1403 (1989).
[CrossRef]

A. Shimizu, T. Ogawa, and H. Sakaki, Phys. Rev. B 45, 11338 (1992).
[CrossRef]

Phys. Rev. Lett. (3)

K. Tai, Phys. Rev. Lett. 62, 1784 (1989).
[CrossRef] [PubMed]

M. Sheik-Bahae, D. J. Hagan, and E. W. Van Stryland, Phys. Rev. Lett. 65, 96 (1990).
[CrossRef] [PubMed]

R. Cingolani, M. Lepore, R. Tommasi, I. M. Catalano, H. Lage, D. Heitmann, K. Ploog, A. Shimizu, H. Sakaki, and T. Ogawa, Phys. Rev. Lett. 69, 1277 (1992).
[CrossRef]

Proc. R. Soc. London A (1)

B. S. Wherrett, Proc. R. Soc. London A 390, 373 (1983).
[CrossRef]

Other (1)

M. L. Cohen and J. R. Chelikowsky, Electronic Structure and Optical Properties of Semiconductors (Springer-Verlag, Berlin, 1988), p. 49.

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Figures (7)

Fig. 1
Fig. 1

Geometry of (a) the QW’s and (b) the QWR’s.

Fig. 2
Fig. 2

Four types of transition involved in TPA of the TM-polarized light in QW’s: (a) electron–hole pair creation, two electron intersubband transitions, recombination; (b) electron–hole pair creation, electron intersubband transition, hole intersubband transition, recombination; (c) electron–hole pair creation, hole intersubband transition, electron intersubband transition, recombination; and (d) electron–hole pair creation, two hole intersubband transitions, recombination.

Fig. 3
Fig. 3

Four types of transition involved in TPA of the TE-polarized light in QW’s: (a) electron–hole pair creation, two electron intrasubband transitions, recombination; (b) electron–hole pair creation, electron intrasubband transition, hole intrasubband transition, recombination; (c) electron–hole pair creation, hole intrasubband transition, electron intrasubband transition, recombination; and (d) electron–hole pair creation, two hole intrasubband transitions, recombination.

Fig. 4
Fig. 4

TPA in GaAs QW’s of different width: (a) 5 nm, (b) 10 nm, and (c) 40 nm. The solid curve shows TM polarization; the dashed curve, TE polarization; and the dotted curve, bulk TPA (for comparison).

Fig. 5
Fig. 5

TPA in GaAs QWR’s of different width: (a) 5 nm, (b) 10 nm, and (c) 40 nm. The solid curve shows TM polarization; the dashed curve, TE polarization; and the dotted curve, bulk TPA (for comparison).

Fig. 6
Fig. 6

Nonlinear refractive index in GaAs QW’s of different width: (a) 5 nm, (b) 10 nm, and (c) 40 nm. The solid curve shows TM polarization; the dashed curve, TE polarization; and the dotted curve, bulk TPA (for comparison).

Fig. 7
Fig. 7

Nonlinear refractive index in GaAs QWR’s of different width: (a) 5 nm, (b) 10 nm, and (c) 40 nm. The solid curve shows TM polarization; the dashed curve, TE polarization; and the dotted curve, bulk TPA (for comparison).

Tables (1)

Tables Icon

Table 1 Comparison of TPA Strength for Different Polarizations in the Infinitely Wide QW’sa

Equations (38)

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

E ν , n = π 2 2 n 2 2 m ν , d 2 ,
E ν , = 2 k 2 2 m ν , ,
ν , n = u ν Ψ n ( z ) exp ( i k r ) ,
u e = i S ;             u h = 1 2 X + i Y ; u l = 1 6 X + i Y - ( 2 3 ) 1 / 2 Z .
s = n e , k , e ; n ν , k , ν ,
χ ( 3 ) ( ω ; - ω , ω , ω ) = 1 0 ( e m 0 ω ) 4 g , s 1 , s 2 , s 3 p g , s 1 p s 1 , s 2 p s 2 , s 3 p s 3 , g ( E s 1 - E g - ω ) ( E s 3 - E g - ω ) ( E s 2 - E g - 2 ω - i Γ ) ,
s 1 = n e , n ν , k ,
s 3 = l e , l ν , k ,
p g , s 1 = P 2 ( x ^ + y ^ ) δ n e , n h δ ν , h + P 2 ( x ^ + y ^ ) δ n e , n l δ ν , l - ( 2 3 ) 1 / 2 P z ^ δ n e , n l δ ν , l ,
s 2 = m e , m ν , k .
p s 1 , s 2 = i m 0 m c 4 m e n e m e 2 - n e 2 δ m e , n e + 2 k - 1 δ m ν , n ν z ^ - i m 0 m n u , 4 m ν n ν m ν 2 - n ν 2 δ m ν , n ν + 2 k - 1 δ m e , n e z ^ + i m 0 m c k δ m e , n e δ m ν , n ν ( x ^ + y ^ ) + i k m 0 m ν , δ m ν , n ν δ m e , n e ( x ^ + y ^ ) ,
χ 1 ( 3 ) ( ω ; - ω , ω , ω ) = 2 3 e 4 P 2 0 m 0 2 ω 2 k n z n , n ± 1 2 E g + E e , n ± 1 + E l , n - 2 ω - i Γ × ( 1 E g + E e , n + E l , n - ω - 1 E g + E e , n ± 1 + E l , n ± 1 - ω ) 2 .
Im [ χ ( 3 ) ( ω ; - ω , ω , ω ) ] = 2 3 e 4 P 2 0 m 0 2 ω 4 μ l , μ l , 2 d 3 ( ω ) 2 × n [ ( 2 n ± 1 ) - ( 2 n ± 1 ) - 1 ] 2 × Θ ( 2 ω - E g - E l , n - E c , n ± 1 ) ,
β ( ω ) = 2 ω η n ω 2 c Im [ χ ( 3 ) ( ω ; - ω , ω , ω ) ] ,
β ( ω ) = ( 32 π α ) 2 3 n ω 2 [ E 11 ( l ) ] 2 d E g 3 ω μ l , μ l , [ F ( N 1 ) + F ( N 2 ) ] ( E g 2 ω ) 4 ,
E 11 ( l ) = E l , 1 + E e , 1 = 2 ω 2 2 μ l , d 2
F ( x ) = 4 / 3 x 3 - 7 / 3 x - 1 / 4 d 2 d x 2 ln [ Γ ( x + 1 / 2 ) ] ,
N 1 ( 2 ) = Int { [ ζ l - 1 + μ l , 2 m c ( l ) 2 ] 1 / 2 - μ l , m c ( l ) } ,
ζ ν = 2 ω - E g E 11 ( ν ) .
β ( ) ( ω ) = 2 1 / 2 ( 64 α 3 n ω π ) 2 π ω E g 3 μ l , μ l , 3 / 2 ( 2 ω - E g ) 3 / 2 × ( E g 2 ω ) 4 .
Im [ χ ( 3 ) ( ω ; - ω , ω , ω ) ] = ν a ν 2 e 4 P 2 0 m 0 2 ω 2 ( ω ) 4 × n ( 2 ω - E g - E ν , n n ) Θ ( 2 ω - E g - E ν , n n ) ,
β ( ω ) = 2 ( 16 α n ω ) 2 E 11 ( l ) 2 d E g 3 ω [ 1 12 F ( ζ l ) + 1 4 F ( ζ h ) μ l , μ h , ] × ( E g 2 ω ) 4 ,
F ( ζ ν ) = ζ ν N ν - 1 / 3 N ν 3 + 1 / 2 N ν 2 - 1 / 6 N ν ,
N ν = Int ( ζ ν ) 1 / 2 .
β ( ) ( ω ) = 2 - 1 / 2 ( 16 α 3 n ω ) 2 π ω E g 3 1 μ l , 1 / 2 ( 2 ω - E g ) 3 / 2 × [ 1 + 3 ( μ h , μ l , ) 1 / 2 ] ( E g 2 ω ) 4 .
β bulk ( ω ) = 2 1 / 2 ( 16 α 3 n ω ) 2 π ω E g 3 1 μ l , 1 / 2 ( 2 ω - E g ) 3 / 2 × [ 3 5 ( μ h , μ l , ) 1 / 2 + 7 5 ] ( E g 2 ω ) 4 .
E ν , n , m = π 2 2 ( m 2 + n 2 ) 2 m ν , d 2 ,
E ν , x = 2 k x 2 2 m ν , .
Im [ χ ( 3 ) ( ω ; - ω , ω , ω ) ] = ν a ν e 4 P 2 0 m 0 2 ω 4 ( 2 μ ν , ) 1 / 2 μ ν , 2 d 4 ( ω ) 2 × m , n Θ ( 2 ω - E g - E ν , n , m - E ν , n ± 1 , m ) ( 2 ω - E g - E ν , n , m - E ν , n ± 1 , m ) 1 / 2 × [ ( 2 n ± 1 ) - ( 2 n ± 1 ) - 1 ] 2 ,
β ( ω ) = 1 π ( 32 α π n ω 2 ) 2 [ E 11 , 11 ( l ) ] 2 E g 3 d ω × [ 1 2 μ l , 2 μ h , 1 / 2 μ l , μ h , 3 / 2 ( F h , 1 + F h , 2 ) + 1 6 ( μ l , μ l , ) 1 / 2 ( F l , 1 + F l , 2 ) ] ( E g 2 ω ) 4 ,
F ν , 1 ( 2 ) = m , n [ ( 2 n ± 1 ) - ( 2 n ± 1 ) - 1 ] 2 × Θ [ ζ ν - m 2 - n 2 - ( 2 n ± 1 ) μ ν , m c ( ν , ) ] [ ζ ν - m 2 - n 2 - ( 2 n + 1 ) μ ν , m c ( ν , ) ] 1 / 2 ,
E 11 , 11 ν = 1 / 2 ( E ν , 1 , 1 + E c , 1 , 1 ) = 2 π 2 2 μ ν , d 2 ,
ζ ν = 2 ω - E g E 11 , 11 ν .
Im [ χ ( 3 ) ( ω ; - ω , ω , ω ) ] = a l 2 · 2 1 / 2 e 4 P 2 0 m 0 2 ω 3 ( ω ) 3 μ l , 1 / 2 n ( 2 ω - E g - E l , m , n - E c , m , n ) 1 / 2 × Θ ( 2 ω - E g - E l , m , n - E c , m , n ) .
β ( ω ) = 2 3 π ( 128 α n ω ) 2 [ E 11 , 11 ( l ) ] 2 E g 3 μ l , 3 / 2 μ l , 3 / 2 d ω F l ,
F l = m , n ( ζ l - m 2 - n 2 ) 1 / 2 Θ ( ζ l - m 2 - n 2 ) .
n 2 ( ω ) = c π 0 β ( ω ; ω ) ( ω ) 2 - ω 2 d ω ,
β ( ω ; ω ) = β ( ω ¯ ) n ( ω ¯ ) 2 n ( ω ) n ( ω ) ω ¯ 4 ( ω ω ) 2 .

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