Abstract

We present an accurate numerical description of frequency-conversion processes that are based on the nonlinear interaction between three pulsed electromagnetic fields. Several interesting phenomena arising in these processes are described in detail. The numerical method used is a new modified Runge–Kutta method employing characteristics. This method can be used to solve numerically sets of coupled hyperbolic differential equations in two variables, if the coefficients of the derivatives are constant. In this paper three examples of frequency-conversion processes are presented that are strongly influenced by the combined effects of phase mismatch, group-velocity differences, and depletion of the electromagnetic fields.

© 1989 Optical Society of America

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References

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  1. S. L. Shapiro, Appl. Phys. Lett. 13, 19 (1968).
    [Crossref]
  2. W. Kranitzky, K. Ding, A. Seilmeier, and W. Kaiser, Opt. Commun. 34, 483 (1980).
    [Crossref]
  3. H. Hermann and W. Sohler, J. Opt. Soc. Am. B 5, 278 (1988).
    [Crossref]
  4. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).
  5. J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, Phys. Rev. 127, 1918 (1962).
    [Crossref]
  6. J. T. Manassah, J. Opt. Soc. Am. B 4, 1235 (1987).
    [Crossref]
  7. R. C. Eckardt and J. Reintjes, IEEE J. Quantum Electron. QE-20, 1178 (1984).
    [Crossref]
  8. J. Comly and E. Garmire, Appl. Phys. Lett. 12, 7 (1968).
    [Crossref]
  9. S. A. Akhmanov, A. S. Chirkin, A. I. Kovrigin, R. V. Khokhlov, and A. P. Sukhorukov, IEEE J. Quantum Electron. QE-4, 598 (1968).
    [Crossref]
  10. A. M. Weiner, IEEE J. Quantum Electron. QE-19, 1276 (1983).
    [Crossref]
  11. J. T. Manassah and O. R. Cockings, Opt. Lett. 12, 1005 (1987);J. T. Manassah, Appl. Opt. 27, 4365 (1988).
    [Crossref] [PubMed]
  12. G. A. Bukauskas, V. I. Kabelka, A. S. Piskarskas, and Yu. A. Stabinis, Sov. J. Quantum Electron. 4, 290 (1974).
    [Crossref]
  13. R. Danelyus, G. Dikchyus, V. Kabelka, A. Piskarskas, A. Stabinis, and Ya. Yasevichyute, Sov. J. Quantum Electron. 7, 1360 (1977).
    [Crossref]
  14. N. C. Kothari and X. Carlotti, J. Opt. Soc. Am. B 5, 756 (1988).
    [Crossref]
  15. A. Yariv, Optical Electronics (Holt Saunders, New York, 1985).
  16. R. L. Street, Analysis and Solution of Partial Differential Equations (Wadsworth, Belmont, Calif., 1973).
  17. F. T. Arecchi and E. O. Schulz-Dubois, Laser Handbook (North-Holland, Amsterdam, 1972), Vol. 1.

1988 (2)

1987 (2)

1984 (1)

R. C. Eckardt and J. Reintjes, IEEE J. Quantum Electron. QE-20, 1178 (1984).
[Crossref]

1983 (1)

A. M. Weiner, IEEE J. Quantum Electron. QE-19, 1276 (1983).
[Crossref]

1980 (1)

W. Kranitzky, K. Ding, A. Seilmeier, and W. Kaiser, Opt. Commun. 34, 483 (1980).
[Crossref]

1977 (1)

R. Danelyus, G. Dikchyus, V. Kabelka, A. Piskarskas, A. Stabinis, and Ya. Yasevichyute, Sov. J. Quantum Electron. 7, 1360 (1977).
[Crossref]

1974 (1)

G. A. Bukauskas, V. I. Kabelka, A. S. Piskarskas, and Yu. A. Stabinis, Sov. J. Quantum Electron. 4, 290 (1974).
[Crossref]

1968 (3)

S. L. Shapiro, Appl. Phys. Lett. 13, 19 (1968).
[Crossref]

J. Comly and E. Garmire, Appl. Phys. Lett. 12, 7 (1968).
[Crossref]

S. A. Akhmanov, A. S. Chirkin, A. I. Kovrigin, R. V. Khokhlov, and A. P. Sukhorukov, IEEE J. Quantum Electron. QE-4, 598 (1968).
[Crossref]

1962 (1)

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

Akhmanov, S. A.

S. A. Akhmanov, A. S. Chirkin, A. I. Kovrigin, R. V. Khokhlov, and A. P. Sukhorukov, IEEE J. Quantum Electron. QE-4, 598 (1968).
[Crossref]

Arecchi, F. T.

F. T. Arecchi and E. O. Schulz-Dubois, Laser Handbook (North-Holland, Amsterdam, 1972), Vol. 1.

Armstrong, J. A.

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

Bloembergen, N.

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

Bukauskas, G. A.

G. A. Bukauskas, V. I. Kabelka, A. S. Piskarskas, and Yu. A. Stabinis, Sov. J. Quantum Electron. 4, 290 (1974).
[Crossref]

Carlotti, X.

Chirkin, A. S.

S. A. Akhmanov, A. S. Chirkin, A. I. Kovrigin, R. V. Khokhlov, and A. P. Sukhorukov, IEEE J. Quantum Electron. QE-4, 598 (1968).
[Crossref]

Cockings, O. R.

Comly, J.

J. Comly and E. Garmire, Appl. Phys. Lett. 12, 7 (1968).
[Crossref]

Danelyus, R.

R. Danelyus, G. Dikchyus, V. Kabelka, A. Piskarskas, A. Stabinis, and Ya. Yasevichyute, Sov. J. Quantum Electron. 7, 1360 (1977).
[Crossref]

Dikchyus, G.

R. Danelyus, G. Dikchyus, V. Kabelka, A. Piskarskas, A. Stabinis, and Ya. Yasevichyute, Sov. J. Quantum Electron. 7, 1360 (1977).
[Crossref]

Ding, K.

W. Kranitzky, K. Ding, A. Seilmeier, and W. Kaiser, Opt. Commun. 34, 483 (1980).
[Crossref]

Ducuing, J.

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

Eckardt, R. C.

R. C. Eckardt and J. Reintjes, IEEE J. Quantum Electron. QE-20, 1178 (1984).
[Crossref]

Garmire, E.

J. Comly and E. Garmire, Appl. Phys. Lett. 12, 7 (1968).
[Crossref]

Hermann, H.

Kabelka, V.

R. Danelyus, G. Dikchyus, V. Kabelka, A. Piskarskas, A. Stabinis, and Ya. Yasevichyute, Sov. J. Quantum Electron. 7, 1360 (1977).
[Crossref]

Kabelka, V. I.

G. A. Bukauskas, V. I. Kabelka, A. S. Piskarskas, and Yu. A. Stabinis, Sov. J. Quantum Electron. 4, 290 (1974).
[Crossref]

Kaiser, W.

W. Kranitzky, K. Ding, A. Seilmeier, and W. Kaiser, Opt. Commun. 34, 483 (1980).
[Crossref]

Khokhlov, R. V.

S. A. Akhmanov, A. S. Chirkin, A. I. Kovrigin, R. V. Khokhlov, and A. P. Sukhorukov, IEEE J. Quantum Electron. QE-4, 598 (1968).
[Crossref]

Kothari, N. C.

Kovrigin, A. I.

S. A. Akhmanov, A. S. Chirkin, A. I. Kovrigin, R. V. Khokhlov, and A. P. Sukhorukov, IEEE J. Quantum Electron. QE-4, 598 (1968).
[Crossref]

Kranitzky, W.

W. Kranitzky, K. Ding, A. Seilmeier, and W. Kaiser, Opt. Commun. 34, 483 (1980).
[Crossref]

Manassah, J. T.

Pershan, P. S.

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

Piskarskas, A.

R. Danelyus, G. Dikchyus, V. Kabelka, A. Piskarskas, A. Stabinis, and Ya. Yasevichyute, Sov. J. Quantum Electron. 7, 1360 (1977).
[Crossref]

Piskarskas, A. S.

G. A. Bukauskas, V. I. Kabelka, A. S. Piskarskas, and Yu. A. Stabinis, Sov. J. Quantum Electron. 4, 290 (1974).
[Crossref]

Reintjes, J.

R. C. Eckardt and J. Reintjes, IEEE J. Quantum Electron. QE-20, 1178 (1984).
[Crossref]

Schulz-Dubois, E. O.

F. T. Arecchi and E. O. Schulz-Dubois, Laser Handbook (North-Holland, Amsterdam, 1972), Vol. 1.

Seilmeier, A.

W. Kranitzky, K. Ding, A. Seilmeier, and W. Kaiser, Opt. Commun. 34, 483 (1980).
[Crossref]

Shapiro, S. L.

S. L. Shapiro, Appl. Phys. Lett. 13, 19 (1968).
[Crossref]

Shen, Y. R.

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).

Sohler, W.

Stabinis, A.

R. Danelyus, G. Dikchyus, V. Kabelka, A. Piskarskas, A. Stabinis, and Ya. Yasevichyute, Sov. J. Quantum Electron. 7, 1360 (1977).
[Crossref]

Stabinis, Yu. A.

G. A. Bukauskas, V. I. Kabelka, A. S. Piskarskas, and Yu. A. Stabinis, Sov. J. Quantum Electron. 4, 290 (1974).
[Crossref]

Street, R. L.

R. L. Street, Analysis and Solution of Partial Differential Equations (Wadsworth, Belmont, Calif., 1973).

Sukhorukov, A. P.

S. A. Akhmanov, A. S. Chirkin, A. I. Kovrigin, R. V. Khokhlov, and A. P. Sukhorukov, IEEE J. Quantum Electron. QE-4, 598 (1968).
[Crossref]

Weiner, A. M.

A. M. Weiner, IEEE J. Quantum Electron. QE-19, 1276 (1983).
[Crossref]

Yariv, A.

A. Yariv, Optical Electronics (Holt Saunders, New York, 1985).

Yasevichyute, Ya.

R. Danelyus, G. Dikchyus, V. Kabelka, A. Piskarskas, A. Stabinis, and Ya. Yasevichyute, Sov. J. Quantum Electron. 7, 1360 (1977).
[Crossref]

Appl. Phys. Lett. (2)

S. L. Shapiro, Appl. Phys. Lett. 13, 19 (1968).
[Crossref]

J. Comly and E. Garmire, Appl. Phys. Lett. 12, 7 (1968).
[Crossref]

IEEE J. Quantum Electron. (3)

S. A. Akhmanov, A. S. Chirkin, A. I. Kovrigin, R. V. Khokhlov, and A. P. Sukhorukov, IEEE J. Quantum Electron. QE-4, 598 (1968).
[Crossref]

A. M. Weiner, IEEE J. Quantum Electron. QE-19, 1276 (1983).
[Crossref]

R. C. Eckardt and J. Reintjes, IEEE J. Quantum Electron. QE-20, 1178 (1984).
[Crossref]

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

Opt. Commun. (1)

W. Kranitzky, K. Ding, A. Seilmeier, and W. Kaiser, Opt. Commun. 34, 483 (1980).
[Crossref]

Opt. Lett. (1)

Phys. Rev. (1)

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

Sov. J. Quantum Electron. (2)

G. A. Bukauskas, V. I. Kabelka, A. S. Piskarskas, and Yu. A. Stabinis, Sov. J. Quantum Electron. 4, 290 (1974).
[Crossref]

R. Danelyus, G. Dikchyus, V. Kabelka, A. Piskarskas, A. Stabinis, and Ya. Yasevichyute, Sov. J. Quantum Electron. 7, 1360 (1977).
[Crossref]

Other (4)

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).

A. Yariv, Optical Electronics (Holt Saunders, New York, 1985).

R. L. Street, Analysis and Solution of Partial Differential Equations (Wadsworth, Belmont, Calif., 1973).

F. T. Arecchi and E. O. Schulz-Dubois, Laser Handbook (North-Holland, Amsterdam, 1972), Vol. 1.

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Figures (7)

Fig. 1
Fig. 1

Characteristics of the three fields in the z, η plane.

Fig. 2
Fig. 2

Analytical and numerical calculated shapes of a pulse generated by second-harmonic generation in LiNbO3. The fundamental pulse has a pulse duration of 1 psec and a wavelength of 1064 nm (Nd:YAG).

Fig. 3
Fig. 3

Simulation of the shape and intensity of a 286.3-nm pulse, generated by sum-frequency mixing of a 4-psec 532-nm pulse of 200 μJ/cm2 and a 620-nm pulse in a 12-mm KDP crystal. The delay between the pulses is equal to zero. Pulse duration and energy per square centimeter of the 620-nm pulse: (a) 400 fsec and 50 nJ/cm2, (b) 800 fsec and 100 nJ/cm2.

Fig. 4
Fig. 4

Simulation of the idler and signal pulse generated by parametric amplification of an idler pulse of 10 pJ/cm2 with a 20-psec 1064-nm pulse of 11 mJ/cm2 in a 4-cm LiNbO3 crystal. There is no delay between the pulses. Wavelength of idler and signal: (a) 2200 and 2061 nm, (b) 4000 and 1450 nm.

Fig. 5
Fig. 5

As in Fig. 4 but with a 6-cm LiNbO3 crystal.

Fig. 6
Fig. 6

As in Figs. 4 and 5 but with an 8-cm LiNbO3 crystal.

Fig. 7
Fig. 7

Calculated cross-correlation signal of a 0.7-psec 266-nm pulse of 10 μJ/cm2 and a 1.0-psec 1064-nm pulse of 100 μJ/cm2 for five different lengths of a KDP crystal.

Equations (9)

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i ( z , t ) = E i ( z , t ) exp [ i ( k i z ω i t ) ] ,
( z + 1 υ 1 g t ) E 1 = i ω 1 χ eff ( 2 ) 2 n 1 c E 2 * A 3 , ( z + 1 υ 2 g t ) E 2 = i ω 2 χ eff ( 2 ) 2 n 2 c E 1 * A 3 , ( z + 1 υ 3 g t ) A 3 = i ω 3 χ eff ( 2 ) 2 n 3 c E 1 E 2 + i Δ k A 3 ,
( z ) B 1 = i ω 1 χ eff ( 2 ) 2 n 1 c B 2 * B 3 , [ z + ( 1 υ 2 g 1 υ 1 g ) η ] B 2 = i ω 2 χ eff ( 2 ) 2 n 2 c B 1 * B 3 , [ z + ( 1 υ 3 g 1 υ 1 g ) η ] B 3 = i ω 3 χ eff ( 2 ) 2 n 3 c B 1 B 2 + i Δ k B 3 .
z f n ( z , η ) = C n η f n ( z , η ) + F n { f m ( z , η ) } ,
f n ( z + h , η ) = f n ( z , η ) + h z f n ( z , η ) + h 2 2 2 z 2 f n ( z , η ) + h 3 6 3 z 3 f n ( z , η ) + h 4 24 4 z 4 f n ( z , η ) + .
2 z 2 f n ( z , η ) = C n 2 2 η 2 f n ( z , η ) + ( C s + C m ) f s F n { f m ( z , η ) } × η 2 f s ( z , η ) + f s F n { f m ( z , η ) } F s { f p ( z , η ) } .
K I n ( z , η ) = F n { f m ( z , η ) } , K II n ( z + h 2 , η ) = F n { f m ( z , η + C m h 2 ) + h 2 K I m ( z , η + C m h 2 ) } , K III n ( z + h 2 , η ) = F n { f m ( z , η + C m h 2 ) + h 2 K II m ( z + h 2 , η ) ) } , K IV n ( z + h , η ) = F n { f m ( z , η + C m h ) + h K III m ( z + h 2 , η + C m h 2 ) } , f n ( z + h , η ) = f n ( z , η + C n h ) + ( h 6 ) [ K I n ( z , η + C n h ) + 2 K II n ( z + h 2 , η + C n h 2 ) + 2 K III n ( z + h 2 , η + C n h 2 ) + K IV n ( z + h , η ) ] .
E 3 ( z , η ) = i ω 3 χ eff ( 2 ) | E 1 max | 2 2 n 3 c δ exp ( i Δ k η / δ ) × η η + δ z exp ( 4 ln 2 y 2 ) exp ( i Δ k y / δ ) d y ,
τ x 2 = τ cross 2 τ a 2 .

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