Elektron. (Mosc.) 2, 434-436 (February 1975). Conditions are found for the simultaneous phase and group matching in three-photon parametric interaction ...
Group matching of interacting light pulses in nonlinear crystals V. I. Kabelka, A. S. Piskarskas, A. Yu. Stabinis, and R. L. Sher V. Kapsukas State University, Vil'nyus (Submitted July 17, 1974; resubmitted October 16, 1974) Kvant. Elektron. (Mosc.) 2, 434-436 (February 1975)
Conditions are found for the simultaneous phase and group matching in three-photon parametric interaction between ultrashort light packets in KDP, LiNbO 3 , and LiIO3 crystals. The exchange of energy in the parametric interaction between ultrashort light pulses (and in the generation of second harmonics) is largely governed by the mismatch of the group velocities of the packets or pulses involved (see, for example, ref. 1). This exchange is most effective when the packets are group-matched. In the parametric interaction of light pulses under group-matching conditions we should observe not only an efficient energy exchange but also a considerable narrowing of the signal pulses, which is limited only by the dispersion broadening of the wave packets and the reaction of the signal pulses on the pump wave. In this case the interaction between the packets has several special features similar, because of the space—time analogy, to those found in the parametric interaction between diffracted light beams.2 It would be interesting to consider the possibility of the coexistence of the phase and group matching in nonlinear crystals. We shall consider the possibility of the group matching in KDP, LiNbO3, and LiIO3 crystals. We have to use the dispersion relationships governing the refractive index n = /(X). The necessary relationship for KDP is taken from ref. 3. The dispersions of the refractive indices of LiNbO3 and LiIO3 are tabulated in refs. 4 and 5. We used these relationships in numerical calculations on a computer, which gave us smooth dependences n = /(X) in the transparency regions of the LiNbO3 and LiIO3 crystals in the form of the Sellmeier equation: 2
— XQ—Cl2,
(1)
where the constants A, B, C, and XQ are listed in Table 1. The conditions for the group matching in uniaxial crystals such as KDP, LiNbO3, and LiIO3 can be reduced
OA
0.6
0.8
1.0
TABLE 1
Crystal
Polarization of wave
LiNbO 3
0
Li 10,
0
e e
A
B
2.449866 2.415597 2.083648 1.673463
2.476150 2.162165 1.332068 1,245229
XO.M
C,M-2 0.027493 0.022073 0.008525 0.003641
0.2174 0.2112 0.1879 0.1680
to the conditions for the compatibility of the equations xrt1cos81+(l — x)«2cos 62=n3cos03, xfljSin 61+(1 — x)n 2 sin 6 2 =« 3 sin 83,
\
('
where the subscripts 1 and 2 represent the signal pulses and the subscript 3 represents the pump pulse; x = X 3 /\j; 0t 2 3 are the angles between the optic axis of the crystal and the corresponding wave vectors; u = c/n[l+ (X/n) • (dn/dX)] is the group velocity; c is the velocity of light. The system (2) is valid for the vector and scalar (0t = 02 = 63) three-photon parametric interactions. The first two equations in this system represent the phase matching and the last equation represents the group matching. We used the computer to find the conditions for the compatibility of the equations in the system (2) . We found that in the case of the e— oo interaction the group matching was obtained only for a degenerate parametric interaction (K = 0.5), which was due to the nature of the dispersion of the group velocities in the crystals under consideration. Figure 1 gives the conditions for the coexistence of the phase and group matching in KDP, LiNbO3, and LiIO3 crystals under degenerate pa-
1.2 I,,
Fig. 1. Dependence of the angle between the direction of the pump wave vector and the optic axis of a crystal 63 (a) and of the angle between the directions of the pump and signal wave vectors |6 3 — 6 t | (b) on the pump wavelength at which group phase matching is obtained in KDP (1), LiNbO3 (2). and LiIO3 (3) crystals. 255
Sov. J. Quant. Electron., Vol. 5, No. 2, August 1975
Copyright © 1975 American Institute of Physics
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rametric interaction conditions. For a given value of the pump wavelength X3 there are definite directions of the wave vectors of the pump and signal waves for which the group matching can be obtained. The range of the pump wavelengths in which the group matching is possible is 0.39-0.52 n for KDP, 0.77-1.02 n for LiNbO3, and 0.431.08 ii for LiIO3. The limit on the side of short pump wavelengths corresponds to the direction of the wave vector of the pump wave which is perpendicular to the optic axis of the crystal and the limit on the side of long wavelengths corresponds to the scalar (collinear) interaction. Thus, in all crystals the group matching in the scalar interaction of waves occurs for just one pump wavelength. •
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We can naturally expect the conditions for the exis-
Sov. J. Quant. Electron., Vol. 5, No. 2. August 1975
tence of the group matching to be more stringent in the case of the e—oe and e—eo interactions. A numerical analysis shows that it is impossible to ensure the coexistence of the phase and group matching of light pulses interacting in a KDP crystal for any value of x.
'A. P. Sukhorukov, A. K. Shchednova, Zh. Eksp. Teor. Fiz., 60, 1251(1971) [Sov. Phys.-JETP, 33, 677 (1971)]. 2 Yu. N. Karamzin and A. P. Sukhorukov, Preprint No. 43 [in Russian], Institute of Applied Mathematics, Academy of Sciences of the USSR, Moscow (1974). ¥. Zernike, Jr., J. Opt. Soc. Am., 54, 1215 (1964). 4 G. D. Boyd, W. L. Bond, and H. L. Carter, J. Appl. Phys., 38, 1941 (1967). 5 S. Umegaki, S. I. Tanaka, T. Uchiyama, and S. Yabumoto, Opt. Commun., 3, 244 (1971).
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