Abstract

The evolution of the phase of a weak signal wave (ω2) copropagating with a strong gate wave (ω1) in a centrosymmetric medium is studied analytically and numerically for the case when the two waves are involved in a nearly phase-matched third-order process that generates a third wave (ω3=2ω1+ω2). It is shown that the signal wave collects, in addition to the phase shift that is due to the ω2=ω1+ω2-ω1 process, an extra nonlinear phase shift caused by cascading of two third-order nonlinear processes, ω3=2ω1+ω2 and ω2=ω3-2ω1. The attractive feature of this new type of cross-phase modulation is the fact that one can control its magnitude and sign by changing the wave-vector mismatch of the sum-frequency mixing process and the gate intensity.

© 1999 Optical Society of America

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  1. G. P. Agrawal, Nonlinear Fiber Optics (Academic, Boston, Mass., 1989).
  2. S. Trillo, S. Wabnitz, E. Wright, and G. Stegeman, Opt. Lett. 13, 871 (1988).
    [CrossRef] [PubMed]
  3. N. Ansari, R. Sammut, and H.-T. Tran, J. Opt. Soc. Am. B 13, 1419 (1996).
    [CrossRef]
  4. S. Dinev, A. Dreischuh, and I. Ivanova, J. Mod. Opt. 39, 667 (1992).
    [CrossRef]
  5. H. Ma, A. S. L. Gomes, and C. B. de Araujo, Appl. Phys. Lett. 59, 2666 (1991).
    [CrossRef]
  6. M. Sheik-Bahae, J. Wang, R. De Salvo, D. Hagan, and E. Van Stryland, Opt. Lett. 17, 258 (1992).
    [CrossRef] [PubMed]
  7. J. Wang, M. Sheik-Bahae, A. Said, D. Hagan, J. David, and E. Van Stryland, J. Opt. Soc. Am. B 11, 1009 (1994).
    [CrossRef]
  8. A. Dreischuh, I. Buchvarov, E. Eugenieva, S. Dinev, and V. Marinov, Opt. Quantum Electron. 28, 1187 (1996).
    [CrossRef]
  9. D. C. Hutchings, J. S. Aitchison, and C. N. Ironside, Opt. Lett. 18, 793 (1993).
    [CrossRef] [PubMed]
  10. A. L. Belostotsky, A. S. Leonov, and A. V. Meleshko, Opt. Lett. 19, 856 (1994).
    [CrossRef] [PubMed]
  11. G. Assanto, Z. Wang, D. Hagan, and E. Van Stryland, Appl. Phys. Lett. 67, 2120 (1995).
    [CrossRef]
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    [CrossRef] [PubMed]
  14. S. Saltiel, K. Koynov, P. Tzankov, A. Boardman, and S. Tanev, Phys. Rev. A 57, 3028 (1998).
    [CrossRef]
  15. P. Tzankov, “Precise analytical description of three and four-photon mixing—stationary approach,” M.S. thesis (University of Sofia, Sofia, Bulgaria, 1997).
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    [CrossRef]
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    [CrossRef]
  19. B. L. Lawrence, M. Cha, J. U. Kang, W. Toruellas, G. Stegeman, G. Baker, J. Meth, and S. Etemad, Electron. Lett. 30, 447 (1994).
    [CrossRef]
  20. S. M. Saltiel, S. Y. Goldberg, and D. Huppert, Opt. Commun. 84, 189 (1991).
    [CrossRef]
  21. K. Koynov, P. Mitev, I. Buchvarov, and S. Saltiel, Pure Appl. Opt. 4, 89 (1995).
    [CrossRef]
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    [CrossRef]
  24. D. L. Williams, D. P. West, and T. A. King, Opt. Commun. 148, 208 (1998).
    [CrossRef]
  25. X. G. Huang and M. R. Wang, Opt. Commun. 150, 235 (1998).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

1998 (4)

S. Saltiel, K. Koynov, P. Tzankov, A. Boardman, and S. Tanev, Phys. Rev. A 57, 3028 (1998).
[CrossRef]

D. L. Williams, D. P. West, and T. A. King, Opt. Commun. 148, 208 (1998).
[CrossRef]

X. G. Huang and M. R. Wang, Opt. Commun. 150, 235 (1998).
[CrossRef]

R. A. Sammut, A. V. Buryak, and Y. S. Kivshar, J. Opt. Soc. Am. B 15, 1488 (1998).
[CrossRef]

1997 (2)

1996 (3)

G. I. Stegeman, D. J. Hagan, and L. Torner, Opt. Quantum Electron. 28, 1691 (1996).
[CrossRef]

N. Ansari, R. Sammut, and H.-T. Tran, J. Opt. Soc. Am. B 13, 1419 (1996).
[CrossRef]

A. Dreischuh, I. Buchvarov, E. Eugenieva, S. Dinev, and V. Marinov, Opt. Quantum Electron. 28, 1187 (1996).
[CrossRef]

1995 (2)

G. Assanto, Z. Wang, D. Hagan, and E. Van Stryland, Appl. Phys. Lett. 67, 2120 (1995).
[CrossRef]

K. Koynov, P. Mitev, I. Buchvarov, and S. Saltiel, Pure Appl. Opt. 4, 89 (1995).
[CrossRef]

1994 (4)

B. L. Lawrence, M. Cha, J. U. Kang, W. Toruellas, G. Stegeman, G. Baker, J. Meth, and S. Etemad, Electron. Lett. 30, 447 (1994).
[CrossRef]

R. Sheik, Opt. Quantum Electron. 26, 415 (1994).
[CrossRef]

J. Wang, M. Sheik-Bahae, A. Said, D. Hagan, J. David, and E. Van Stryland, J. Opt. Soc. Am. B 11, 1009 (1994).
[CrossRef]

A. L. Belostotsky, A. S. Leonov, and A. V. Meleshko, Opt. Lett. 19, 856 (1994).
[CrossRef] [PubMed]

1993 (2)

D. C. Hutchings, J. S. Aitchison, and C. N. Ironside, Opt. Lett. 18, 793 (1993).
[CrossRef] [PubMed]

C. Schwan, A. Penzkofer, N. J. Marx, and K. N. Drexhage, Appl. Phys. B: Photophys. Laser Chem. 57, 203 (1993).
[CrossRef]

1992 (3)

S. Dinev, A. Dreischuh, and I. Ivanova, J. Mod. Opt. 39, 667 (1992).
[CrossRef]

M. Sheik-Bahae, J. Wang, R. De Salvo, D. Hagan, and E. Van Stryland, Opt. Lett. 17, 258 (1992).
[CrossRef] [PubMed]

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, IEEE J. Quantum Electron. 28, 2631 (1992).
[CrossRef]

1991 (2)

S. M. Saltiel, S. Y. Goldberg, and D. Huppert, Opt. Commun. 84, 189 (1991).
[CrossRef]

H. Ma, A. S. L. Gomes, and C. B. de Araujo, Appl. Phys. Lett. 59, 2666 (1991).
[CrossRef]

1988 (1)

1987 (1)

W. Lenpacher and A. Penzkofer, Appl. Phys. B: Photophys. Laser Chem. 44, 133 (1987).
[CrossRef]

1976 (1)

C. Sauteret, J.-P. Herrmann, R. Frey, F. Pradere, J. Ducuing, R. H. Baughman, and R. R. Chance, Phys. Rev. Lett. 36, 956 (1976).
[CrossRef]

Aitchison, J. S.

Ansari, N.

Assanto, G.

G. Assanto, Z. Wang, D. Hagan, and E. Van Stryland, Appl. Phys. Lett. 67, 2120 (1995).
[CrossRef]

Baker, G.

B. L. Lawrence, M. Cha, J. U. Kang, W. Toruellas, G. Stegeman, G. Baker, J. Meth, and S. Etemad, Electron. Lett. 30, 447 (1994).
[CrossRef]

Baughman, R. H.

C. Sauteret, J.-P. Herrmann, R. Frey, F. Pradere, J. Ducuing, R. H. Baughman, and R. R. Chance, Phys. Rev. Lett. 36, 956 (1976).
[CrossRef]

Belostotsky, A. L.

Boardman, A.

S. Saltiel, K. Koynov, P. Tzankov, A. Boardman, and S. Tanev, Phys. Rev. A 57, 3028 (1998).
[CrossRef]

S. Saltiel, S. Tanev, and A. Boardman, Opt. Lett. 22, 148 (1997).
[CrossRef] [PubMed]

Buchvarov, I.

A. Dreischuh, I. Buchvarov, E. Eugenieva, S. Dinev, and V. Marinov, Opt. Quantum Electron. 28, 1187 (1996).
[CrossRef]

K. Koynov, P. Mitev, I. Buchvarov, and S. Saltiel, Pure Appl. Opt. 4, 89 (1995).
[CrossRef]

Buryak, A. V.

Byer, R. L.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, IEEE J. Quantum Electron. 28, 2631 (1992).
[CrossRef]

Cha, M.

B. L. Lawrence, M. Cha, J. U. Kang, W. Toruellas, G. Stegeman, G. Baker, J. Meth, and S. Etemad, Electron. Lett. 30, 447 (1994).
[CrossRef]

Chance, R. R.

C. Sauteret, J.-P. Herrmann, R. Frey, F. Pradere, J. Ducuing, R. H. Baughman, and R. R. Chance, Phys. Rev. Lett. 36, 956 (1976).
[CrossRef]

David, J.

de Araujo, C. B.

H. Ma, A. S. L. Gomes, and C. B. de Araujo, Appl. Phys. Lett. 59, 2666 (1991).
[CrossRef]

De Salvo, R.

Dinev, S.

A. Dreischuh, I. Buchvarov, E. Eugenieva, S. Dinev, and V. Marinov, Opt. Quantum Electron. 28, 1187 (1996).
[CrossRef]

S. Dinev, A. Dreischuh, and I. Ivanova, J. Mod. Opt. 39, 667 (1992).
[CrossRef]

Dreischuh, A.

A. Dreischuh, I. Buchvarov, E. Eugenieva, S. Dinev, and V. Marinov, Opt. Quantum Electron. 28, 1187 (1996).
[CrossRef]

S. Dinev, A. Dreischuh, and I. Ivanova, J. Mod. Opt. 39, 667 (1992).
[CrossRef]

Drexhage, K. N.

C. Schwan, A. Penzkofer, N. J. Marx, and K. N. Drexhage, Appl. Phys. B: Photophys. Laser Chem. 57, 203 (1993).
[CrossRef]

Ducuing, J.

C. Sauteret, J.-P. Herrmann, R. Frey, F. Pradere, J. Ducuing, R. H. Baughman, and R. R. Chance, Phys. Rev. Lett. 36, 956 (1976).
[CrossRef]

Etemad, S.

B. L. Lawrence, M. Cha, J. U. Kang, W. Toruellas, G. Stegeman, G. Baker, J. Meth, and S. Etemad, Electron. Lett. 30, 447 (1994).
[CrossRef]

Eugenieva, E.

A. Dreischuh, I. Buchvarov, E. Eugenieva, S. Dinev, and V. Marinov, Opt. Quantum Electron. 28, 1187 (1996).
[CrossRef]

Fejer, M. M.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, IEEE J. Quantum Electron. 28, 2631 (1992).
[CrossRef]

Frey, R.

C. Sauteret, J.-P. Herrmann, R. Frey, F. Pradere, J. Ducuing, R. H. Baughman, and R. R. Chance, Phys. Rev. Lett. 36, 956 (1976).
[CrossRef]

Goldberg, S. Y.

S. M. Saltiel, S. Y. Goldberg, and D. Huppert, Opt. Commun. 84, 189 (1991).
[CrossRef]

Gomes, A. S. L.

H. Ma, A. S. L. Gomes, and C. B. de Araujo, Appl. Phys. Lett. 59, 2666 (1991).
[CrossRef]

Hagan, D.

Hagan, D. J.

G. I. Stegeman, D. J. Hagan, and L. Torner, Opt. Quantum Electron. 28, 1691 (1996).
[CrossRef]

Herrmann, J.-P.

C. Sauteret, J.-P. Herrmann, R. Frey, F. Pradere, J. Ducuing, R. H. Baughman, and R. R. Chance, Phys. Rev. Lett. 36, 956 (1976).
[CrossRef]

Huang, X. G.

X. G. Huang and M. R. Wang, Opt. Commun. 150, 235 (1998).
[CrossRef]

Huppert, D.

S. M. Saltiel, S. Y. Goldberg, and D. Huppert, Opt. Commun. 84, 189 (1991).
[CrossRef]

Hutchings, D. C.

Ironside, C. N.

Ivanova, I.

S. Dinev, A. Dreischuh, and I. Ivanova, J. Mod. Opt. 39, 667 (1992).
[CrossRef]

Jundt, D. H.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, IEEE J. Quantum Electron. 28, 2631 (1992).
[CrossRef]

Kang, J. U.

B. L. Lawrence, M. Cha, J. U. Kang, W. Toruellas, G. Stegeman, G. Baker, J. Meth, and S. Etemad, Electron. Lett. 30, 447 (1994).
[CrossRef]

King, T. A.

D. L. Williams, D. P. West, and T. A. King, Opt. Commun. 148, 208 (1998).
[CrossRef]

Kivshar, Y. S.

Koynov, K.

S. Saltiel, K. Koynov, P. Tzankov, A. Boardman, and S. Tanev, Phys. Rev. A 57, 3028 (1998).
[CrossRef]

K. Koynov, P. Mitev, I. Buchvarov, and S. Saltiel, Pure Appl. Opt. 4, 89 (1995).
[CrossRef]

Lawrence, B. L.

B. L. Lawrence, M. Cha, J. U. Kang, W. Toruellas, G. Stegeman, G. Baker, J. Meth, and S. Etemad, Electron. Lett. 30, 447 (1994).
[CrossRef]

Lenpacher, W.

W. Lenpacher and A. Penzkofer, Appl. Phys. B: Photophys. Laser Chem. 44, 133 (1987).
[CrossRef]

Leonov, A. S.

Ma, H.

H. Ma, A. S. L. Gomes, and C. B. de Araujo, Appl. Phys. Lett. 59, 2666 (1991).
[CrossRef]

Magel, G. A.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, IEEE J. Quantum Electron. 28, 2631 (1992).
[CrossRef]

Marinov, V.

A. Dreischuh, I. Buchvarov, E. Eugenieva, S. Dinev, and V. Marinov, Opt. Quantum Electron. 28, 1187 (1996).
[CrossRef]

Marx, N. J.

C. Schwan, A. Penzkofer, N. J. Marx, and K. N. Drexhage, Appl. Phys. B: Photophys. Laser Chem. 57, 203 (1993).
[CrossRef]

Meleshko, A. V.

Meth, J.

B. L. Lawrence, M. Cha, J. U. Kang, W. Toruellas, G. Stegeman, G. Baker, J. Meth, and S. Etemad, Electron. Lett. 30, 447 (1994).
[CrossRef]

Mitev, P.

K. Koynov, P. Mitev, I. Buchvarov, and S. Saltiel, Pure Appl. Opt. 4, 89 (1995).
[CrossRef]

Penzkofer, A.

C. Schwan, A. Penzkofer, N. J. Marx, and K. N. Drexhage, Appl. Phys. B: Photophys. Laser Chem. 57, 203 (1993).
[CrossRef]

W. Lenpacher and A. Penzkofer, Appl. Phys. B: Photophys. Laser Chem. 44, 133 (1987).
[CrossRef]

Pradere, F.

C. Sauteret, J.-P. Herrmann, R. Frey, F. Pradere, J. Ducuing, R. H. Baughman, and R. R. Chance, Phys. Rev. Lett. 36, 956 (1976).
[CrossRef]

Said, A.

Saltiel, S.

S. Saltiel, K. Koynov, P. Tzankov, A. Boardman, and S. Tanev, Phys. Rev. A 57, 3028 (1998).
[CrossRef]

S. Saltiel, S. Tanev, and A. Boardman, Opt. Lett. 22, 148 (1997).
[CrossRef] [PubMed]

K. Koynov, P. Mitev, I. Buchvarov, and S. Saltiel, Pure Appl. Opt. 4, 89 (1995).
[CrossRef]

Saltiel, S. M.

S. M. Saltiel, S. Y. Goldberg, and D. Huppert, Opt. Commun. 84, 189 (1991).
[CrossRef]

Sammut, R.

Sammut, R. A.

Sauteret, C.

C. Sauteret, J.-P. Herrmann, R. Frey, F. Pradere, J. Ducuing, R. H. Baughman, and R. R. Chance, Phys. Rev. Lett. 36, 956 (1976).
[CrossRef]

Schwan, C.

C. Schwan, A. Penzkofer, N. J. Marx, and K. N. Drexhage, Appl. Phys. B: Photophys. Laser Chem. 57, 203 (1993).
[CrossRef]

Sheik, R.

R. Sheik, Opt. Quantum Electron. 26, 415 (1994).
[CrossRef]

Sheik-Bahae, M.

Stegeman, G.

B. L. Lawrence, M. Cha, J. U. Kang, W. Toruellas, G. Stegeman, G. Baker, J. Meth, and S. Etemad, Electron. Lett. 30, 447 (1994).
[CrossRef]

S. Trillo, S. Wabnitz, E. Wright, and G. Stegeman, Opt. Lett. 13, 871 (1988).
[CrossRef] [PubMed]

Stegeman, G. I.

G. I. Stegeman, D. J. Hagan, and L. Torner, Opt. Quantum Electron. 28, 1691 (1996).
[CrossRef]

Tanev, S.

S. Saltiel, K. Koynov, P. Tzankov, A. Boardman, and S. Tanev, Phys. Rev. A 57, 3028 (1998).
[CrossRef]

S. Saltiel, S. Tanev, and A. Boardman, Opt. Lett. 22, 148 (1997).
[CrossRef] [PubMed]

Torner, L.

G. I. Stegeman, D. J. Hagan, and L. Torner, Opt. Quantum Electron. 28, 1691 (1996).
[CrossRef]

Toruellas, W.

B. L. Lawrence, M. Cha, J. U. Kang, W. Toruellas, G. Stegeman, G. Baker, J. Meth, and S. Etemad, Electron. Lett. 30, 447 (1994).
[CrossRef]

Tran, H.-T.

Trillo, S.

Tzankov, P.

S. Saltiel, K. Koynov, P. Tzankov, A. Boardman, and S. Tanev, Phys. Rev. A 57, 3028 (1998).
[CrossRef]

Van Stryland, E.

Wabnitz, S.

Wang, J.

Wang, M. R.

X. G. Huang and M. R. Wang, Opt. Commun. 150, 235 (1998).
[CrossRef]

Wang, Z.

G. Assanto, Z. Wang, D. Hagan, and E. Van Stryland, Appl. Phys. Lett. 67, 2120 (1995).
[CrossRef]

West, D. P.

D. L. Williams, D. P. West, and T. A. King, Opt. Commun. 148, 208 (1998).
[CrossRef]

Williams, D. L.

D. L. Williams, D. P. West, and T. A. King, Opt. Commun. 148, 208 (1998).
[CrossRef]

Wright, E.

Appl. Phys. B: Photophys. Laser Chem. (2)

W. Lenpacher and A. Penzkofer, Appl. Phys. B: Photophys. Laser Chem. 44, 133 (1987).
[CrossRef]

C. Schwan, A. Penzkofer, N. J. Marx, and K. N. Drexhage, Appl. Phys. B: Photophys. Laser Chem. 57, 203 (1993).
[CrossRef]

Appl. Phys. Lett. (2)

H. Ma, A. S. L. Gomes, and C. B. de Araujo, Appl. Phys. Lett. 59, 2666 (1991).
[CrossRef]

G. Assanto, Z. Wang, D. Hagan, and E. Van Stryland, Appl. Phys. Lett. 67, 2120 (1995).
[CrossRef]

Electron. Lett. (1)

B. L. Lawrence, M. Cha, J. U. Kang, W. Toruellas, G. Stegeman, G. Baker, J. Meth, and S. Etemad, Electron. Lett. 30, 447 (1994).
[CrossRef]

IEEE J. Quantum Electron. (1)

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, IEEE J. Quantum Electron. 28, 2631 (1992).
[CrossRef]

J. Mod. Opt. (1)

S. Dinev, A. Dreischuh, and I. Ivanova, J. Mod. Opt. 39, 667 (1992).
[CrossRef]

J. Opt. Soc. Am. B (3)

Opt. Commun. (3)

D. L. Williams, D. P. West, and T. A. King, Opt. Commun. 148, 208 (1998).
[CrossRef]

X. G. Huang and M. R. Wang, Opt. Commun. 150, 235 (1998).
[CrossRef]

S. M. Saltiel, S. Y. Goldberg, and D. Huppert, Opt. Commun. 84, 189 (1991).
[CrossRef]

Opt. Lett. (6)

Opt. Quantum Electron. (3)

G. I. Stegeman, D. J. Hagan, and L. Torner, Opt. Quantum Electron. 28, 1691 (1996).
[CrossRef]

A. Dreischuh, I. Buchvarov, E. Eugenieva, S. Dinev, and V. Marinov, Opt. Quantum Electron. 28, 1187 (1996).
[CrossRef]

R. Sheik, Opt. Quantum Electron. 26, 415 (1994).
[CrossRef]

Phys. Rev. A (1)

S. Saltiel, K. Koynov, P. Tzankov, A. Boardman, and S. Tanev, Phys. Rev. A 57, 3028 (1998).
[CrossRef]

Phys. Rev. Lett. (1)

C. Sauteret, J.-P. Herrmann, R. Frey, F. Pradere, J. Ducuing, R. H. Baughman, and R. R. Chance, Phys. Rev. Lett. 36, 956 (1976).
[CrossRef]

Pure Appl. Opt. (1)

K. Koynov, P. Mitev, I. Buchvarov, and S. Saltiel, Pure Appl. Opt. 4, 89 (1995).
[CrossRef]

Other (5)

M. Bass, ed., Handbook of Optics (McGraw-Hill, New York, 1995), Vol. II.

J. F. Reintjes, Nonlinear Optical Parametric Processes in Liquids and Gasses (Academic, Orlando, Fla., 1984).

P. Tzankov, “Precise analytical description of three and four-photon mixing—stationary approach,” M.S. thesis (University of Sofia, Sofia, Bulgaria, 1997).

S. Saltiel, I. Buchvarov, and K. Koynov, “Nonlinear optical devices based on χ(2)(2) cascading,” in Advanced Photonics with Second-Order Nonlinear Processes, A. Boardman, ed. (Kluwer Academic, Dordrecht, The Netherlands, 1998).

G. P. Agrawal, Nonlinear Fiber Optics (Academic, Boston, Mass., 1989).

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Figures (2)

Fig. 1
Fig. 1

Cascaded NPS Δφ2,casc as a function of the normalized mismatch Δk˜ for several values of the normalized gate intensity γ0a102L. Solid curves, numerical solution of Eqs. (1); dotted curves, analytical solution obtained from Eq. (8). The ratio of the signal and the gate input intensities is a202/a102=0.01. Dashed curve, NPS that is due to CTOP when ω1=ω2 and a102=a202 (this curve is reproduced from Ref. 14).

Fig. 2
Fig. 2

Variation of (a) total NPS Δφ2 and (b) signal intensity transmittance a22/a202 with the normalized gate intensity. The parameter is the mismatch ΔkL. Dotted line, NPS that is due to pure XPM. The ratio of the signal and gate input intensities is a202 /a102=0.01.

Tables (1)

Tables Icon

Table 1 Values of the Wave-Vector Mismatches (in cm-1) for Different Processes in Lead Molybdate Crystala

Equations (18)

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

E(z, t)=12ejAj(z)exp[i(ωjt-kjz)]+c.c.,
dA1dz=iW1A1+iγ1A3A2*A1* exp(iΔkz),
dA2dz=iW2A2+iγ2A3A1*A1* exp(iΔkz),
dA3dz=iW3A3+iγ3A1A1A2 exp(-iΔkz),
aj dajdz=±γja12a2a3 sin Φ,
aj dφjdz=Wjaj+γja12a2a3ajcos Φ,
aj2+γjγ3a32=aj02,j=1,2,
γ3a12a2a3 cos Φ+Λa32+Δγ4a34=0.
Λ=(Δk+Γ1a102+Γ2a202)/2,
Δγ=(Γ3γ3-Γ1γ1-Γ2γ2)/γ3,
Γj=-2γ1j-γ2j+γ3j(j=1, 2, 3).
aj2(z)=aj02-γjγ3a32(z)(j=1, 2),
a32(z)=γ32Q2a104a202 sin2(Qz),
Δφ2γ21a102L-γ3γ2a104L2Λ[1-sinc(2ΛL)].
Δφ2=A+γ32 Ba104a2022Q2L-γ32 Ba104a202L2Q2sin(2QL)2QL-4γ2Λ+γ3Δγa2024γ2QDarctan[D tan(QL)],
A=γ21a102+γ22+γ3Δγ4γ2a202+Λ,
B=γ23-γ21γ1γ3-γ22γ2γ3+Δγ4,
D=1-γ2γ3a104Q21/2.

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