Influence of optical inhomogeneity on the parametric

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Jul 4, 2015 - (Submitted August 7,1981). Kvantovaya Elektron. (Moscow) 9,1414-1420 (July (982). A theoretical and experimental investigation is reported of ...
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Influence of optical inhomogeneity on the parametric amplification of picosecond light pulses in lithium niobate crystals

This content has been downloaded from IOPscience. Please scroll down to see the full text. 1982 Sov. J. Quantum Electron. 12 900 (http://iopscience.iop.org/0049-1748/12/7/A19) View the table of contents for this issue, or go to the journal homepage for more

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Influence of optical inhomogeneity on the parametric amplification of picosecond light pulses in lithium niobate crystals S. A. MagnitskiT, V. I. Pryalkin, V. G. Tunkin, and A. I. Kholodnykh M. V. Lomonosov State University, Moscow (Submitted August 7,1981) Kvantovaya Elektron. (Moscow) 9,1414-1420 (July (982) A theoretical and experimental investigation is reported of the influence of a longitudinal optical inhomogeneity of a lithium niobate crystal on the parametric amplification of picosecond light pulses. It is shown that an increase in the gain reduces the influence of the inhomogeneity on the effective length of the crystal. This makes it possible to use longer crystals in picosecond optical parametric oscillators, in spite of the fact that the inhomogeneity of such crystals makes them unsuitable for nanosecond parametric oscillators. PACS numbers: 42.65.Cq, 42.20. - y

1. Characteristics of optical parametric oscillators for the nanosecond range are largely determined by the inhomogeneity of nonlinear crystals, particularly of lithium niobate which suffers from a considerable longitudinal inhomogeneity of the birefringence resulting from fluctuations during growth. This inhomogeneity affects directly the parametric gain G: in the case of an ideal nonlinear crystal pumped by a plane monochromatic wave when the amplification is sufficiently strong, we have1

where y 1 2 are the nonlinear coupling coefficients; //>, / 1 2 , and ln are the power densities of the pump wave, of the parametrically generated waves, and of the parametric noise, respectively; A' is the phase mismatch between the interacting waves; lCT is the length of the crystal. The amplification is maximal for A' = 0. The inhomogeneity makes it impossible to satisfy this condition simultaneously throughout a crystal. This r e duces the gain and we have to calculate it then using expressions more complex than Eq. (1). The influence of the inhomogeneity of a crystal on the parametric gain can be allowed for phenomenologically, as has been done in the case of second harmonic generation1 by introducing an effective length lett of a crystal, which can then be used to rewrite Eq. (1) in the form It is shown theoretically in Refs. 2 and 3 that the parametric gain of linearly inhomogeneous crystals (which include lithium niobate because of growth inhomogeneities) reaches saturation in the weak amplification range (G 2 1) when the length of a crystal attains a certain value and then the gain falls. The stronger the inhomogeneity, the shorter the length of a crystal in which the gain maximum is reached and the smaller this maximum. This can be explained qualitatively using the relationship (1) and allowing for the fact that the phase mismatch is A'^ const and that it increases on increase in the length of the crystal. A general solution of a differential equation for the parametric gain of linearly inhomogeneous media is obtained in Ref. 4 for the case of weak amplification, 900

Sov. J. Quantum Electron. 12(7), July 1982

which is usually encountered in nanosecond optical parametric oscillators, and in the case of strong amplification, typical of picosecond parametric oscillators. The results of experimental investigations of the influence of the inhomogeneity of lithium niobate crystals on the parametric gain and generation of nanosecond pulses in optical parametric oscillators were reported in Refs. 5 and 6. The results of these investigations were found to be in qualitative agreement with the theoretical calculations of Refs. 2 and 3. In the case of picosecond parametric oscillators the pump power density is between two and three orders of magnitude higher than in nanosecond oscillators and this is accompanied by a reduction in the relative importance of the term with A' in the argument of the exponential function in Eq. (1). We can therefore expect the influence of the inhomogeneity on the parametric gain to decrease in the case of picosecond oscillators. This is the topic considered below. We shall analyze the solutions obtained in Ref. 4 for the case of strong amplification and calculate the specific dependences of the parametric gain and, consequently, of lttt because of Eq. (2) on the length and inhomogeneity of crystals, and we shall show that the results differ considerably from those obtained in the case of weak amplification. We shall also report an experimental verification of these relationships in the case of lithium niobate crystals and show that the experimental results are in good agreement with calculations. This will give us quantitative information on the degree of influence of the inhomogeneity of nonlinear crystals on the parametric gain in picosecond oscillators. 2. It is assumed in these calculations that the rate of change of the birefringence B is constant along a crystal, i.e., we shall postulate that dB/dz = const = *. As in Ref. 3, we shall call * the inhomogeneity parameter. The solution for the amplitude At of a parametrically generated wave of wavelength A obtained in Ref. 4 on the assumption that the pump wave is plane and monochromatic is /I 1 =exp(—U-)

\IX, ix-)],

w h e r e F i s t h e d e g e n e r a t e h y p e r g e o m e t r i c function w i t h

0049-1748/82/070900-04$04.10

© 1982 American Institute of Physics

,900

a complex argument; A= £+ \; Xx= - t ^ / 2 * ; #* = y^yjAn); M = I ; Hi = k * 2 = **72; j | = / - e / « i traveling coordinate;

0.2

0.5

r

i

Ip. GW/cm2

A is the initial mismatch of the wave vector of the interacting waves. If A= 0, the form of the above solution simplifies greatly, but the case A= 0 does not correspond to the gain maximum in inhomogeneous media. We determined the maximum parametric gain Gmix by varying the phase mismatch A within the necessary limits. We found the dependence of Gmax on the length of LiNbO3 crystals, which was within the range 1-5 cm, and we did this for inhomogeneity parameters * = 10"6 - 10"4 cm"1 and different pump power densities I,= 0.2-2 GW/cm2. The value of Alo was deduced from the power density of the parametric noise1: In= hi>3c~*dQdu (hv is the photon energy of the amplified wave; dQ, is the solid angle of divergence of the radiation from an optical parametric oscillator; dv is the spectral width of the gain profile). We assumed that dQ.dv^l/12^. (Ref. 7). In our experiments, we found that IJ.lcr= 1 cm) = 40 mW/cm2. The calculations were carried out on an EC 10-10 computer. Figure 1 shows the dependences of the effective length on the geometric length of a crystal calculated for different values of the inhomogeneity parameter in the degenerate interaction case Ax = X2= 2.12 /n; the properties of lithium niobate are taken from the review in Ref. 8. The continuous curves in Fig. 1 are calculated for Ip= 2.1 GW/cm2 and the dashed curves are taken from Ref. 3 and they apply to the weak amplification case. The difference between the dependences for the two amplification regimes is demonstrated by a comparison of curves 4 and 5 plotted for approximately the same value of = 5-xlO"5 cm"1. In the case of short

'cr.

c m

FIG. 1. Theoretical dependences of Zeff of lithium niobate on the real crystal length la for Ip= 2.1 GW/cm2 (continuous curves) and in the weak amplification regime 3 (dashed curves). *(cm"'): 1) 0; 2) 5xl0' 6 ; 3) 10"5; 4) 5xlVO,

pump power density for each pulse. This was done by determining simultaneously the pump energy Wp and the energy of a second harmonic W2u generated in a KDP crystal 4 in the phase-matched regime under conditions of low conversion efficiency. Since J2ucc./J, we could use the relationship Wia/Wp