Abstract

A theoretical treatment of the excitonic enhancement of two-photon absorption is presented for quasi-two-dimensional quantum-well structures. The two-photon absorption coefficient and the nonlinear index of refraction are calculated for TM and TE polarizations. Significant enhancement is observed in the TM case as a result of excitonic effects. This treatment differs from previous ones in that simple analytical expressions are derived. Moreover, we consider the asymptotic behavior as the size of the quantum well increases toward a three-dimensional system.

© 1995 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. Sheik-Bahae, D. C. Hutchings, D. Hagan, and E. W. Van Stryland, IEEE J. Quantum Electron. 27, 1296 (1991).
    [CrossRef]
  2. J. Khurgin, "Nonlinear response of the semiconductor quantum confined structures near and below the middle of the bandgap," J. Opt. Soc. Am. B 11, 624 (1994).
    [CrossRef]
  3. B. S. Wherret, Proc. R. Soc. London Ser. A 390, 373 (1983).
    [CrossRef]
  4. J. S. Aitchison, A. H. Keen, C. N. Ironside, A. Villeneuve, and G. I. Stegeman, Electron. Lett. 27, 1709 (1991).
    [CrossRef]
  5. A. Villeneuve, C. C. Yang, P. G. Wigley, G. I. Stegeman, J. S. Aitchison, and C. N. Ironside, Appl. Phys. Lett. 62, 147 (1992).
    [CrossRef]
  6. S. Hunter, F. Kiamilev, S. Esener, D. Parthenopoulos, and P. Rentzepis, Appl. Opt. 29, 2058 (1990).
    [CrossRef] [PubMed]
  7. K. W. Delong and G. I. Stegeman, Appl. Phys. Lett. 57, 2063 (1990).
    [CrossRef]
  8. H. N. Spector, Phys. Rev. B 35, 5876 (1987).
    [CrossRef]
  9. A. Pasquarello and A. Quattropani, Phys. Rev. B 38, 6206 (1988).
    [CrossRef]
  10. A. Shimizu, Phys. Rev. B 40, 1403 (1989).
    [CrossRef]
  11. A. Shimizu, T. Ogawa, and H. Sakaki, Phys. Rev. B 45, 11338 (1992).
    [CrossRef]
  12. K. Tai, Phys. Rev. Lett. 62, 1784 (1989).
    [CrossRef] [PubMed]
  13. H. K. Tsang, R. V. Penty, I. H. White, R. S. Grant, and W. Sibbett, Appl. Phys. Lett. 59, 3440 (1991).
    [CrossRef]
  14. C. C. Yang, A. Villeneuve, G. I. Stegeman, C.-H. Lin, and H.-H. Lin, IEEE J. Quantum Electron. 29, 2934 (1993).
    [CrossRef]
  15. J. M. Luttinger and W. Kohn, Phys. Rev. 7, 869 (1955).
    [CrossRef]
  16. E. O. Kane, J. Phys. Chem. Solids 1, 249 (1957).
    [CrossRef]
  17. G. Bastard, E. E. Mendez, L. L. Chang, and L. Esaki, Phys. Rev. B 26, 1807 (1982).
    [CrossRef]
  18. D. S. Chemla and D. A. B. Miller, J. Opt. Soc. Am. B 2, 1155 (1985).
    [CrossRef]
  19. R. J. Elliott, Phys. Rev. 108, 1384 (1957).
    [CrossRef]
  20. P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge U. Press, Cambridge, 1990).
    [CrossRef]
  21. M. Shinada and S. Sugano, J. Phys. Soc. Jpn. 21, 1936 (1966).
    [CrossRef]

1994 (1)

Aitchison, J. S.

J. S. Aitchison, A. H. Keen, C. N. Ironside, A. Villeneuve, and G. I. Stegeman, Electron. Lett. 27, 1709 (1991).
[CrossRef]

A. Villeneuve, C. C. Yang, P. G. Wigley, G. I. Stegeman, J. S. Aitchison, and C. N. Ironside, Appl. Phys. Lett. 62, 147 (1992).
[CrossRef]

Bastard, G.

G. Bastard, E. E. Mendez, L. L. Chang, and L. Esaki, Phys. Rev. B 26, 1807 (1982).
[CrossRef]

Butcher, P. N.

P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge U. Press, Cambridge, 1990).
[CrossRef]

Chang, L. L.

G. Bastard, E. E. Mendez, L. L. Chang, and L. Esaki, Phys. Rev. B 26, 1807 (1982).
[CrossRef]

Chemla, D. S.

D. S. Chemla and D. A. B. Miller, J. Opt. Soc. Am. B 2, 1155 (1985).
[CrossRef]

Cotter, D.

P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge U. Press, Cambridge, 1990).
[CrossRef]

Delong, K. W.

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

Elliott, R. J.

R. J. Elliott, Phys. Rev. 108, 1384 (1957).
[CrossRef]

Esaki, L.

G. Bastard, E. E. Mendez, L. L. Chang, and L. Esaki, Phys. Rev. B 26, 1807 (1982).
[CrossRef]

Esener, S.

S. Hunter, F. Kiamilev, S. Esener, D. Parthenopoulos, and P. Rentzepis, Appl. Opt. 29, 2058 (1990).
[CrossRef] [PubMed]

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.

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

Hunter, S.

S. Hunter, F. Kiamilev, S. Esener, D. Parthenopoulos, and P. Rentzepis, Appl. Opt. 29, 2058 (1990).
[CrossRef] [PubMed]

Hutchings, D. C.

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

Ironside, C. N.

A. Villeneuve, C. C. Yang, P. G. Wigley, G. I. Stegeman, J. S. Aitchison, and C. N. Ironside, Appl. Phys. Lett. 62, 147 (1992).
[CrossRef]

J. S. Aitchison, A. H. Keen, C. N. Ironside, A. Villeneuve, and G. I. Stegeman, Electron. Lett. 27, 1709 (1991).
[CrossRef]

Kane, E. O.

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

Keen, A. H.

J. S. Aitchison, A. H. Keen, C. N. Ironside, A. Villeneuve, and G. I. Stegeman, Electron. Lett. 27, 1709 (1991).
[CrossRef]

Khurgin, J.

Kiamilev, F.

S. Hunter, F. Kiamilev, S. Esener, D. Parthenopoulos, and P. Rentzepis, Appl. Opt. 29, 2058 (1990).
[CrossRef] [PubMed]

Kohn, W.

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

Lin, C.-H.

C. C. Yang, A. Villeneuve, G. I. Stegeman, C.-H. Lin, and H.-H. Lin, IEEE J. Quantum Electron. 29, 2934 (1993).
[CrossRef]

Lin, H.-H.

C. C. Yang, A. Villeneuve, G. I. Stegeman, C.-H. Lin, and H.-H. Lin, IEEE J. Quantum Electron. 29, 2934 (1993).
[CrossRef]

Luttinger, J. M.

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

Mendez, E. E.

G. Bastard, E. E. Mendez, L. L. Chang, and L. Esaki, Phys. Rev. B 26, 1807 (1982).
[CrossRef]

Miller, D. A. B.

D. S. Chemla and D. A. B. Miller, J. Opt. Soc. Am. B 2, 1155 (1985).
[CrossRef]

Ogawa, T.

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

Parthenopoulos, D.

S. Hunter, F. Kiamilev, S. Esener, D. Parthenopoulos, and P. Rentzepis, Appl. Opt. 29, 2058 (1990).
[CrossRef] [PubMed]

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]

Quattropani, A.

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

Rentzepis, P.

S. Hunter, F. Kiamilev, S. Esener, D. Parthenopoulos, and P. Rentzepis, Appl. Opt. 29, 2058 (1990).
[CrossRef] [PubMed]

Sakaki, H.

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

Sheik-Bahae, M.

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

Shimizu, A.

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

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

Shinada, M.

M. Shinada and S. Sugano, J. Phys. Soc. Jpn. 21, 1936 (1966).
[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]

Spector, H. N.

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

Stegeman, G. I.

C. C. Yang, A. Villeneuve, G. I. Stegeman, C.-H. Lin, and H.-H. Lin, IEEE J. Quantum Electron. 29, 2934 (1993).
[CrossRef]

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

J. S. Aitchison, A. H. Keen, C. N. Ironside, A. Villeneuve, and G. I. Stegeman, Electron. Lett. 27, 1709 (1991).
[CrossRef]

A. Villeneuve, C. C. Yang, P. G. Wigley, G. I. Stegeman, J. S. Aitchison, and C. N. Ironside, Appl. Phys. Lett. 62, 147 (1992).
[CrossRef]

Stryland, E. W. Van

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

Sugano, S.

M. Shinada and S. Sugano, J. Phys. Soc. Jpn. 21, 1936 (1966).
[CrossRef]

Tai, K.

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

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]

Villeneuve, A.

A. Villeneuve, C. C. Yang, P. G. Wigley, G. I. Stegeman, J. S. Aitchison, and C. N. Ironside, Appl. Phys. Lett. 62, 147 (1992).
[CrossRef]

J. S. Aitchison, A. H. Keen, C. N. Ironside, A. Villeneuve, and G. I. Stegeman, Electron. Lett. 27, 1709 (1991).
[CrossRef]

C. C. Yang, A. Villeneuve, G. I. Stegeman, C.-H. Lin, and H.-H. Lin, IEEE J. Quantum Electron. 29, 2934 (1993).
[CrossRef]

Wherret, B. S.

B. S. Wherret, Proc. R. Soc. London Ser. 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]

Wigley, P. G.

A. Villeneuve, C. C. Yang, P. G. Wigley, G. I. Stegeman, J. S. Aitchison, and C. N. Ironside, Appl. Phys. Lett. 62, 147 (1992).
[CrossRef]

Yang, C. C.

A. Villeneuve, C. C. Yang, P. G. Wigley, G. I. Stegeman, J. S. Aitchison, and C. N. Ironside, Appl. Phys. Lett. 62, 147 (1992).
[CrossRef]

C. C. Yang, A. Villeneuve, G. I. Stegeman, C.-H. Lin, and H.-H. Lin, IEEE J. Quantum Electron. 29, 2934 (1993).
[CrossRef]

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

Other (20)

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

J. S. Aitchison, A. H. Keen, C. N. Ironside, A. Villeneuve, and G. I. Stegeman, Electron. Lett. 27, 1709 (1991).
[CrossRef]

A. Villeneuve, C. C. Yang, P. G. Wigley, G. I. Stegeman, J. S. Aitchison, and C. N. Ironside, Appl. Phys. Lett. 62, 147 (1992).
[CrossRef]

S. Hunter, F. Kiamilev, S. Esener, D. Parthenopoulos, and P. Rentzepis, Appl. Opt. 29, 2058 (1990).
[CrossRef] [PubMed]

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

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]

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

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

C. C. Yang, A. Villeneuve, G. I. Stegeman, C.-H. Lin, and H.-H. Lin, IEEE J. Quantum Electron. 29, 2934 (1993).
[CrossRef]

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

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

G. Bastard, E. E. Mendez, L. L. Chang, and L. Esaki, Phys. Rev. B 26, 1807 (1982).
[CrossRef]

D. S. Chemla and D. A. B. Miller, J. Opt. Soc. Am. B 2, 1155 (1985).
[CrossRef]

R. J. Elliott, Phys. Rev. 108, 1384 (1957).
[CrossRef]

P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge U. Press, Cambridge, 1990).
[CrossRef]

M. Shinada and S. Sugano, J. Phys. Soc. Jpn. 21, 1936 (1966).
[CrossRef]

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

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1
Fig. 1

The four types of transition involved in the TPA in the TM case. (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. (d) Electron–hole pair creation, two hole intersubband transitions, recombination.

Fig. 2
Fig. 2

TPA in a GaAs QW of 5-nm width for the TM case. The excitonic contribution is shown separately from the electron–hole pair contribution.

Fig. 3
Fig. 3

Nonlinear index of refraction (n2) for a GaAs QW 5-nm width for the TM case. The excitonic contribution is shown separately from the electron–hole pair contribution.

Fig. 4
Fig. 4

The four types of transition involved in TPA in the TE case. (a) Electron–hole pair creation, two electron intraband transitions, recombination. (b) Electron–hole pair creation, electron intraband transition, hole intraband transition, recombination. (c) Electron–hole pair creation, hole intraband transition, electron intraband transition, recombination. (d) Electron–hole pair creation, two hole intraband transitions, recombination.

Fig. 5
Fig. 5

TPA in a GaAs QW of 5-nm width for the TE case. The excitonic contribution is shown separately from the electron–hole pair contribution. Note that the excitonic curve is magnified by 104.

Equations (35)

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

E n , v = π 2 2 n 2 2 m v , d 2 ,
E v , = 2 K 2 2 m v , ,
v , n = u v ϕ n ( z ) exp ( i K · r ) ,
ϕ n ( z ) = 2 / d sin n π z d .
Ψ n , m ( r , z e , z h ) = ϕ n ( z e ) ϕ m ( z h ) f l ( r , θ ) ,
f 1 s ( r , θ ) = 2 / π λ 1 s 2 exp ( - r / λ 1 s ) ,
f 2 p ( r , θ ) = 2 3 π λ 2 p 4 r exp ( - r / λ 2 p ± i θ ) ,
χ ( 3 ) ( ω ; - ω , ω , ω ) = 1 0 ( e m 0 ω ) 4 × g , s 1 , s 2 , s 3 P g , s 1 η ^ P s 1 , s 2 η ^ P s 2 , s 3 η ^ P s 3 , g η ^ ( E s 1 - ω ) ( E s 2 - 2 ω - i Γ ) ( E s 3 - ω ) .
Z n , n ± 1 = Ψ n z Ψ n ± 1 = 2 d π 2 [ 1 - ( 2 n ± 1 ) 2 ] .
Ψ n , m P g = i P cv m 0 ω 8 π λ ,
χ ( 3 ) ( ω ; - ω , ω , ω ) = N e 4 3 0 P cv 2 π λ 2 m 0 2 ω 2 × n ± Z n , n ± 1 2 E n , n ± 1 - 2 ω - i Γ × [ 1 E n n - ω - 1 E n ± 1 , n ± 1 - ω ] 2 .
Im [ χ ( 3 ) ] = N e 4 3 ! 0 E g 2 π λ 2 d 2 μ lh , 3 ω 6 [ n = 1 N 1 ( 2 n + 1 - 1 2 n + 1 ) 2 + n = 1 N 2 ( 2 n - 1 - 1 2 n - 1 ) 2 ] n ± L ( 2 ω ) ,
L ( 2 ω ) = Γ ( 2 ω - E n , n ± 1 ) 2 + Γ 2 .
N 1 = Int [ ( ζ - μ lh , m c + μ lh , 2 m c 2 ) 1 / 2 - μ lh , m c ] ,
N 2 = Int [ ( ζ - μ lh , m c + μ lh , 2 m c 2 ) 1 / 2 - μ lh , m c ] ,
ζ = 2 ω + E ex - E g E 11 ,
E 11 = E 1 e + E 1 lh = π 2 2 2 d 2 μ lh , .
β ( ω ) = 2 ω η n ω 2 c Im [ χ ( 3 ) ] = 2 ω μ 0 n ω 2 Im [ χ ( 3 ) ] .
β ( ω ) = e 4 E g μ 0 6 π 0 λ 2 d 3 μ lh , 3 ω 5 n ω 2 [ F 1 + F 2 ] n ± L ( 2 ω ) .
F 1 = 4 / 3 ( N 1 + 1 ) 3 - 7 / 3 N 1 - 7 / 3 + π 2 / 8 - 1 / 4 d 2 / d N 1 2 ln Γ ( N 1 + 3 / 2 ) ,
F 2 = 4 / 3 ( N 2 + 1 ) 3 - 4 ( N 2 + 1 ) 2 + 5 / 3 N 2 + 8 / 3 + π 2 / 8 - 1 / 4 d 2 / d N 2 2 ln Γ ( N 2 + 1 / 2 ) .
n 2 = 3 4 0 c n 0 2 Re [ χ ( 3 ) ] .
Re [ χ ( 3 ) ] = e 4 E g 3 ! 0 2 π λ 2 d 3 μ lh , 3 ω 6 × n ± ( 2 n ± 1 - 1 2 n ± 1 ) 2 × E n , n ± 1 - 2 ω ( 2 ω - E n , n ± 1 ) 2 + Γ 2 .
χ ( 3 ) = 1 0 ( e m 0 ω ) 4 1 ( ω ) 2 2 p P g , 1 s η ^ P 1 s , 2 p η ^ P 2 p , 1 s η ^ P 1 s , g η ^ E 2 p - 2 ω - i Γ ,
χ ( 3 ) = 1 0 ( e m 0 ω ) 4 1 ( ω ) 2 × v = hh , lh l , m n q x v P c v 2 I 2 | - i x f v | r = 0 2 × 1 E 2 p ( n ) - 2 ω - i Γ .
| x f v | r = 0 2 = 2 n ( n - 1 ) π λ 4 ( n - 1 / 2 ) 5 .
Im [ χ ( 3 ) ] = 1 0 ( e m 0 ω ) 4 P cv 2 ω 2 × v q x v l , m n 2 2 n ( n - 1 ) π λ 4 ( n - 1 / 2 ) 5 × Γ [ 2 ω - E 2 p ( n ) ] 2 + Γ 2 .
Ψ n , m ( r , z e , z h ) = ψ ( z e , z h ) f l ( r , θ )
ψ ( z e , z h ) = m , n C m n ϕ n ( z e ) ϕ m ( z h ) .
2 P cv 2 π λ 2 m 0 2 ω 2 [ ( 1 m e , z e + 1 m lh , z h ) ψ ( z e , z h ) × δ ( z e - z h ) d z e d z h ] 2 .
ψ n , n + 1 ± ( z e , z h ) = 2 - 1 / 2 ϕ n ( z e ) ϕ n + 1 ( z h ) ± 2 - 1 / 2 ϕ n + 1 ( z e ) ϕ n ( z h ) .
[ ϕ n + 1 ( z e ) d d z e ϕ n ( z e ) ± ϕ n ( z e ) d d z e ϕ n + 1 ( z e ) ] d z e .
ψ 1 , 2 - ( z e , z h ) = 2 - 1 / 2 ϕ 1 ( z e ) ϕ 2 ( z h ) - 2 - 1 / 2 ϕ 2 ( z e ) ϕ 1 ( z h ) ,
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 ,

Metrics