Abstract

The solution of the coupled-wave equations for second-harmonic generation in a near-resonant three-level system is extended to include absorption. It is shown, within second-order perturbation theory, that double resonance is the optimal conversion condition, despite absorption enhancement. We extend the solution numerically, using nonperturbative susceptibilities derived within the rotating-wave approximation, to saturating intensities and discuss the modifications to the perturbative conclusions as well as the regimes of validity for the various approximations.

© 1994 Optical Society of America

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  1. J. A. Armstrong, N. Bloembergen, J. Ducuing, P. S. Pershan, Phys. Rev. 127, 1918 (1962).
    [CrossRef]
  2. M. M. Fejer, S. J. B. Yoo, R. L. Byer, A. Harwit, J. S. Harris, Phys. Rev. Lett. 62, 1041 (1989).
    [CrossRef] [PubMed]
  3. E. Rosencher, P. Bois, J. Nagle, S. Delaitre, Electron. Lett. 25, 1063 (1989).
    [CrossRef]
  4. Z. Chen, D. Cui, M. Li, C. Jiang, J. Zhou, G. Yang, Appl. Phys. Lett. 61, 2401 (1992).
    [CrossRef]
  5. P. Boucaud, F. H. Julien, J. Phys. III 1, 13 (1991).
  6. Z. Ikonic, V. Milanovic, D. Tjapkin, IEEE J. Quantum Electron. 25, 54 (1989).
    [CrossRef]
  7. T. A. DeTemple, L. A. Bahler, J. Osmundsen, Phys. Rev. A 24, 1950 (1981).
    [CrossRef]
  8. G. Almogy, M. Segev, A. Yariv, Opt. Lett. 19, 1192 (1994).
    [CrossRef] [PubMed]
  9. G. Almogy, A. Shakouri, A. Yariv, Appl. Phys. Lett. 63, 2720 (1993).
    [CrossRef]
  10. P. Boucaud, F. H. Julien, D. D. Yang, J. M. Lourtoiz, E. Rosencher, P. Bois, Opt. Lett. 16, 199 (1991).
    [CrossRef] [PubMed]
  11. G. Almogy, M. Segev, A. Yariv, Phys. Rev. B 48, 10950 (1993).
    [CrossRef]
  12. E. Rosencher, J. Appl. Phys. 73, 1909 (1993).
    [CrossRef]

1994 (1)

1993 (3)

G. Almogy, A. Shakouri, A. Yariv, Appl. Phys. Lett. 63, 2720 (1993).
[CrossRef]

G. Almogy, M. Segev, A. Yariv, Phys. Rev. B 48, 10950 (1993).
[CrossRef]

E. Rosencher, J. Appl. Phys. 73, 1909 (1993).
[CrossRef]

1992 (1)

Z. Chen, D. Cui, M. Li, C. Jiang, J. Zhou, G. Yang, Appl. Phys. Lett. 61, 2401 (1992).
[CrossRef]

1991 (2)

1989 (3)

Z. Ikonic, V. Milanovic, D. Tjapkin, IEEE J. Quantum Electron. 25, 54 (1989).
[CrossRef]

M. M. Fejer, S. J. B. Yoo, R. L. Byer, A. Harwit, J. S. Harris, Phys. Rev. Lett. 62, 1041 (1989).
[CrossRef] [PubMed]

E. Rosencher, P. Bois, J. Nagle, S. Delaitre, Electron. Lett. 25, 1063 (1989).
[CrossRef]

1981 (1)

T. A. DeTemple, L. A. Bahler, J. Osmundsen, Phys. Rev. A 24, 1950 (1981).
[CrossRef]

1962 (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing, P. S. Pershan, Phys. Rev. 127, 1918 (1962).
[CrossRef]

Almogy, G.

G. Almogy, M. Segev, A. Yariv, Opt. Lett. 19, 1192 (1994).
[CrossRef] [PubMed]

G. Almogy, M. Segev, A. Yariv, Phys. Rev. B 48, 10950 (1993).
[CrossRef]

G. Almogy, A. Shakouri, A. Yariv, Appl. Phys. Lett. 63, 2720 (1993).
[CrossRef]

Armstrong, J. A.

J. A. Armstrong, N. Bloembergen, J. Ducuing, P. S. Pershan, Phys. Rev. 127, 1918 (1962).
[CrossRef]

Bahler, L. A.

T. A. DeTemple, L. A. Bahler, J. Osmundsen, Phys. Rev. A 24, 1950 (1981).
[CrossRef]

Bloembergen, N.

J. A. Armstrong, N. Bloembergen, J. Ducuing, P. S. Pershan, Phys. Rev. 127, 1918 (1962).
[CrossRef]

Bois, P.

Boucaud, P.

Byer, R. L.

M. M. Fejer, S. J. B. Yoo, R. L. Byer, A. Harwit, J. S. Harris, Phys. Rev. Lett. 62, 1041 (1989).
[CrossRef] [PubMed]

Chen, Z.

Z. Chen, D. Cui, M. Li, C. Jiang, J. Zhou, G. Yang, Appl. Phys. Lett. 61, 2401 (1992).
[CrossRef]

Cui, D.

Z. Chen, D. Cui, M. Li, C. Jiang, J. Zhou, G. Yang, Appl. Phys. Lett. 61, 2401 (1992).
[CrossRef]

Delaitre, S.

E. Rosencher, P. Bois, J. Nagle, S. Delaitre, Electron. Lett. 25, 1063 (1989).
[CrossRef]

DeTemple, T. A.

T. A. DeTemple, L. A. Bahler, J. Osmundsen, Phys. Rev. A 24, 1950 (1981).
[CrossRef]

Ducuing, J.

J. A. Armstrong, N. Bloembergen, J. Ducuing, P. S. Pershan, Phys. Rev. 127, 1918 (1962).
[CrossRef]

Fejer, M. M.

M. M. Fejer, S. J. B. Yoo, R. L. Byer, A. Harwit, J. S. Harris, Phys. Rev. Lett. 62, 1041 (1989).
[CrossRef] [PubMed]

Harris, J. S.

M. M. Fejer, S. J. B. Yoo, R. L. Byer, A. Harwit, J. S. Harris, Phys. Rev. Lett. 62, 1041 (1989).
[CrossRef] [PubMed]

Harwit, A.

M. M. Fejer, S. J. B. Yoo, R. L. Byer, A. Harwit, J. S. Harris, Phys. Rev. Lett. 62, 1041 (1989).
[CrossRef] [PubMed]

Ikonic, Z.

Z. Ikonic, V. Milanovic, D. Tjapkin, IEEE J. Quantum Electron. 25, 54 (1989).
[CrossRef]

Jiang, C.

Z. Chen, D. Cui, M. Li, C. Jiang, J. Zhou, G. Yang, Appl. Phys. Lett. 61, 2401 (1992).
[CrossRef]

Julien, F. H.

Li, M.

Z. Chen, D. Cui, M. Li, C. Jiang, J. Zhou, G. Yang, Appl. Phys. Lett. 61, 2401 (1992).
[CrossRef]

Lourtoiz, J. M.

Milanovic, V.

Z. Ikonic, V. Milanovic, D. Tjapkin, IEEE J. Quantum Electron. 25, 54 (1989).
[CrossRef]

Nagle, J.

E. Rosencher, P. Bois, J. Nagle, S. Delaitre, Electron. Lett. 25, 1063 (1989).
[CrossRef]

Osmundsen, J.

T. A. DeTemple, L. A. Bahler, J. Osmundsen, Phys. Rev. A 24, 1950 (1981).
[CrossRef]

Pershan, P. S.

J. A. Armstrong, N. Bloembergen, J. Ducuing, P. S. Pershan, Phys. Rev. 127, 1918 (1962).
[CrossRef]

Rosencher, E.

E. Rosencher, J. Appl. Phys. 73, 1909 (1993).
[CrossRef]

P. Boucaud, F. H. Julien, D. D. Yang, J. M. Lourtoiz, E. Rosencher, P. Bois, Opt. Lett. 16, 199 (1991).
[CrossRef] [PubMed]

E. Rosencher, P. Bois, J. Nagle, S. Delaitre, Electron. Lett. 25, 1063 (1989).
[CrossRef]

Segev, M.

G. Almogy, M. Segev, A. Yariv, Opt. Lett. 19, 1192 (1994).
[CrossRef] [PubMed]

G. Almogy, M. Segev, A. Yariv, Phys. Rev. B 48, 10950 (1993).
[CrossRef]

Shakouri, A.

G. Almogy, A. Shakouri, A. Yariv, Appl. Phys. Lett. 63, 2720 (1993).
[CrossRef]

Tjapkin, D.

Z. Ikonic, V. Milanovic, D. Tjapkin, IEEE J. Quantum Electron. 25, 54 (1989).
[CrossRef]

Yang, D. D.

Yang, G.

Z. Chen, D. Cui, M. Li, C. Jiang, J. Zhou, G. Yang, Appl. Phys. Lett. 61, 2401 (1992).
[CrossRef]

Yariv, A.

G. Almogy, M. Segev, A. Yariv, Opt. Lett. 19, 1192 (1994).
[CrossRef] [PubMed]

G. Almogy, M. Segev, A. Yariv, Phys. Rev. B 48, 10950 (1993).
[CrossRef]

G. Almogy, A. Shakouri, A. Yariv, Appl. Phys. Lett. 63, 2720 (1993).
[CrossRef]

Yoo, S. J. B.

M. M. Fejer, S. J. B. Yoo, R. L. Byer, A. Harwit, J. S. Harris, Phys. Rev. Lett. 62, 1041 (1989).
[CrossRef] [PubMed]

Zhou, J.

Z. Chen, D. Cui, M. Li, C. Jiang, J. Zhou, G. Yang, Appl. Phys. Lett. 61, 2401 (1992).
[CrossRef]

Appl. Phys. Lett. (2)

Z. Chen, D. Cui, M. Li, C. Jiang, J. Zhou, G. Yang, Appl. Phys. Lett. 61, 2401 (1992).
[CrossRef]

G. Almogy, A. Shakouri, A. Yariv, Appl. Phys. Lett. 63, 2720 (1993).
[CrossRef]

Electron. Lett. (1)

E. Rosencher, P. Bois, J. Nagle, S. Delaitre, Electron. Lett. 25, 1063 (1989).
[CrossRef]

IEEE J. Quantum Electron. (1)

Z. Ikonic, V. Milanovic, D. Tjapkin, IEEE J. Quantum Electron. 25, 54 (1989).
[CrossRef]

J. Appl. Phys. (1)

E. Rosencher, J. Appl. Phys. 73, 1909 (1993).
[CrossRef]

J. Phys. III (1)

P. Boucaud, F. H. Julien, J. Phys. III 1, 13 (1991).

Opt. Lett. (2)

Phys. Rev. (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing, P. S. Pershan, Phys. Rev. 127, 1918 (1962).
[CrossRef]

Phys. Rev. A (1)

T. A. DeTemple, L. A. Bahler, J. Osmundsen, Phys. Rev. A 24, 1950 (1981).
[CrossRef]

Phys. Rev. B (1)

G. Almogy, M. Segev, A. Yariv, Phys. Rev. B 48, 10950 (1993).
[CrossRef]

Phys. Rev. Lett. (1)

M. M. Fejer, S. J. B. Yoo, R. L. Byer, A. Harwit, J. S. Harris, Phys. Rev. Lett. 62, 1041 (1989).
[CrossRef] [PubMed]

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Figures (2)

Fig. 1
Fig. 1

Second-harmonic amplitude conversion efficiency: E2ω(zmax)/Eω(0) and conversion length (zmax) are shown versus normalized amplitudes for (a) moderate and (b) intense fundamental harmonics. The solid curves are the results of the perturbative numerical solutions (Pth), the short-dashed curves are the transparent approximation (NA), and the long-dashed curves (only in the low-intensity case) are the nondepleted approximation (ND). The filled and open circles in (b) are the results of the nonperturbative treatment (Full) for the double-resonant (Δω = 0) and the highly detuned (Δω = 100) cases, respectively.

Fig. 2
Fig. 2

Numerically derived second-harmonic amplitudes: E2ω (z)/Eω (0) versus normalized propagation lengths for several detunings at (a) a perturbative (Ωω = 0.01) and (b) a saturating (Ωω = 10) normalized intensity. PM, Artificially imposed phase matching.

Equations (15)

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χ ( ω , ω ) ( 1 ) = N μ 10 2 T 10 ћ 0 1 ( Δ ω 10 i ) ,
χ ( 2 ω , 2 ω ) ( 1 ) = N μ 20 2 T 20 ћ 0 1 ( Δ ω 20 i ) ,
χ ( 2 ω , ω , ω ) ( 2 ) = N μ 10 μ 21 μ 20 T 10 T 20 2 ћ 2 0 1 ( Δ ω 10 i ) ( Δ ω 20 i ) ,
α ω ω 2 nc Im [ χ ( ω , ω ) ( 1 ) ] ,
β 2 ω ( ω / nc ) { n 2 ω 2 n ω 2 + Re [ χ ( 2 ω , 2 ω ) ( 1 ) ] Re [ χ ( ω , ω ) ( 1 ) ] } ,
d E 2 ω d z = ( α 2 ω + i β 2 ω ) E 2 ω i ω nc χ ( 2 ω , ω , ω ) ( 2 ) E ω 2 ,
d E ω d z = α ω E ω i ω 2 nc χ ( ω , 2 ω , ω ) ( 2 ) E 2 ω E ω * ,
k ( ω / c ) { n ω 2 + Re [ χ ( ω , ω ) ( 1 ) ] } 1 / 2 ω n / c
E 2 ω ( z ) = i ω nc χ ( 2 ω , ω , ω ) ( 2 ) ( α ω 2 α ω ) [ exp ( 2 α ω z ) exp ( α 2 ω z ) ] E ω ( 0 ) 2 .
z max = 1 α 2 ω 2 α ω ln ( α 2 ω 2 α ω ) ,
| E 2 ω ( z max ) E ω ( 0 ) | = ω nc E ω ( 0 ) | χ ( 2 ω , ω , ω ) ( 2 ) | [ 1 α 2 ω ( 2 α ω α 2 ω ) 2 α ω α 2 ω - 2 α ω ] .
| E 2 ω ( z max ) E ω ( 0 ) | = 1 e | χ ( 2 ω , ω , ω ) ( 2 ) | E ω ( 0 ) Re [ χ ( 2 ω , 2 ω ) ( 1 ) ] ,
| E 2 ω ( z max ) E ω ( 0 ) | = 1 e μ 21 E ω ( 0 ) 2 ћ T 2 1 e Ω 21 ( 0 ) T 2 Ω ω / e ,
l c 2 π β 2 ω = 1 α 2 ω ( π Δ ω ) .
Ω ω = Ω ij T ij < ½ T ij / T j j [ ( 1 + Δ ω i j 2 ) ] 1 / 2 ,

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