Nov 1, 1997 - ever, through second-order cascading an OPA can be a highly efficient medium for cascaded four-wave mixing. Recently an OPA was shown to ...
November 1, 1997 / Vol. 22, No. 21 / OPTICS LETTERS
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Near-degenerate cascaded four-wave mixing in an optical parametric amplifier A. Varanaviˇcius, A. Dubietis, A. Berˇzanskis, R. Danielius, and A. Piskarskas Laser Research Center, Building 3, Sauletekio ˙ Avenue 9, 2040 Vilnius, Lithuania Received June 19, 1997 Efficient generation of new spectral components owing to second-order cascading in a seeded broadband b-barium borate type I phase-matching parametric amplifier is demonstrated. One can vary the number and magnitude of these components by changing amplification bandwidth (wavelength) and phase-matching conditions. The phenomenon is treated theoretically by use of a formalism developed previously for the case of cascaded self-diffraction. 1997 Optical Society of America
Optical parametric amplif iers (OPA’s) have been considered mostly in light of their connection to the generation of broadly tunable light pulses. However, through second-order cascading an OPA can be a highly efficient medium for cascaded four-wave mixing. Recently an OPA was shown to support the self-diffraction of chirped pulses that leads to the generation of new pulses with enhanced chirp.1 An attractive feature of the OPA is that it is capable of boosting the energy of the signal in which the cascaded interaction results above the level of the incident signals. The interaction of closely spaced frequency components (near-degenerate four-wave mixing) is of particular interest because of its practical application to processing of optical signals in frequency-multiplexing- and shifting schemes (see Ref. 2 for a comprehensive review). An experimental realization of the sum-frequency generation scheme was demonstrated several years ago.3 Recently a cascaded frequency-mixing effect under special seed conditions in an optical parametric oscillator was reported.4 In this Letter we present numerical and experimental results of a study of the cascaded frequency-mixing effect in a type I b-barium borate (BBO) OPA. We observed as many as three new spectral components in both signal and idler branches with amplitudes comparable with those of the originally amplified signal and idler waves. The experimental configuration is depicted in Fig. 1. The 0.9-ps seed signal with energy of as much as 5 mJ, tunable within 635 –1055 nm, was generated in a triple-pass traveling-wave BBO (type II phasematching) parametric generator (TOPAS Model 530, Light Conversion Ltd., Vilnius, Lithuania) pumped by a 1-ps-long second-harmonic pulse of the Nd:glass laser. The OPA (8-mm-long BBO crystal, type I phase matching) was set to operate in a large-signal amplification mode with a gain factor of ,20 in saturation. Equal amounts of pump energy were sent to the parametric generator and the OPA by the beam splitter. The pump intensity for the OPA was set by a beamsize-reducing telescope up to 15 GWycm2 . In simple terms, the cascaded frequency-mixing effect occurs as a consequence of interaction between the signal and the idler pulses: The signal pulse gen0146-9592/97/211603-03$10.00/0
erates its second harmonic, which interacts with the idler pulse, producing radiation at vs1 2vs 2 vi . The new frequency component is then amplif ied by the intense pump pulse, producing a conjugate wave at vi1 vp 2 vs1 . Thereafter the process repeats as vs2 2vs1 2 vi , etc. This interaction character is preserved as long as the new frequencies remain within the amplification bandwidth. It should be noted that cascaded interactions of this type are allowed only in the type I phase-matching conf iguration. Evidently, the frequency-mixing process described above can be considered degenerate four-wave mixing if one omits the intermediate fields. The only difference from pure four-wave mixing is that here one of the required waves is generated through the three-wave parametric amplification process. The overall effect resembles the self-diffraction5 that can be seen as a generation of new frequencies in the space domain. Thus the theoretical approach developed to describe self-diffraction6 can be directly applied to describe time-domain frequency mixing in the OPA; i.e., we describe the interacting waves of each frequency as a single pulse with modulation in time. Thus we are to conf ine the theoretical investigation to the framework of a common system of equations for three-wave interaction (interaction type oo – e) in the slowly varying amplitude approximation: h1 ≠3 A1 ≠A1 ig1 ≠2 A1 ≠A1 1 1u 1 ≠z ≠t 2 ≠t2 6 ≠t3 2isA A p exps2iDkzd , 2
2
≠A2 ig2 ≠2 A2 h2 ≠3 A2 1 1 ≠z 2 ≠t2 6 ≠t3 2isA1 2 expsiDkzd ,
(1)
Fig. 1. Experimental setup: BS, 50% beam splitter; HR’s, highly ref lective mirrors for pump wavelength; DM, dichroic mirror, transparent for the seed signal wavelength and ref lective for that of the pump. 1997 Optical Society of America
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OPTICS LETTERS / Vol. 22, No. 21 / November 1, 1997
where A1 and A2 denote the slowly varying amplitudes of the signal and the pump pulses, respectively, s is the coupling constant that is due to second-order susceptibility, z is the propagation coordinate, and Dk is the phase mismatch. The parameters u, g1, 2 , and h1, 2 are related to group-velocity mismatch and dispersion; thus Eqs. (1) are written in the third-order approximation of dispersion theory. The spectral structure of complex envelopes A1 and A2 may consist of several well-distinguished spectral peaks that can be identified as new frequency components. The complex envelope of the initial signal pulse can be described for z 0 as µ 2 ∂ 2t 0 1 iDvt , (2) A1 exp t1 2
in the time domain. Each of these parametric processes has its own phase-matching conditions, and the most efficient generation of a certain component is determined by its frequency, the set of parameters u, g1, 2 , and h1, 2 , and the phase-mismatch Dk. In that way we can change the intensities of the frequency components by tuning the phase mismatch. Figure 2(b) illustrates the situation when the crystal
where t1 denotes the pulse duration, A1 0 is the pulse amplitude, and Dv vs 2 vp y2 is the pulse’s frequency detuning from the carrier frequency. The initial condition for the pump wave is µ 2∂ 2t . 0 (3) A2 exp t2 2 We analyzed frequency mixing through second-order cascaded processes in an OPA by means of Eqs. (1) with the initial conditions in expressions (2) and (3). Such an analysis takes into account all possible frequency components within the gain bandwidth of an OPA, which is determined by the dispersion parameters u, g1, 2 , and h1, 2 . The newly generated frequencies are seen as equidistantly placed peaks in the time-domain spectrum of the appropriate pulse: Z ` An st, zdexps2ivt 0 ddt, n 1, 2 . (4) Sn sv, zd 2`
The set of Eqs. (1) was solved by means of a splitstep method incorporating the fourth-order Runge– Kuta integration of nonlinear equations in the time domain and dispersive evaluation of pulses and their propagation relative to each other in the spectral domain. In our experimental situation we were dealing with pulses whose durations were suff iciently long (of the order of 1 ps). Thus it may seem sufficient to take into account group-velocity mismatch only, which would enable one to neglect terms proportional to g1, 2 and h1, 2 in Eqs. (1). In support of the argument for retaining these terms, we can mention that in our theoretical approach we must properly describe a wide phase-matching band, whereas the new frequency components are generated within a broad frequency interval. The OPA output spectrum is plotted in Fig. 2(a). We have observed as many as three new spectral components with wavelengths shorter than that of the signal (1029 nm), with the phase matching optimized for the signal amplification. The same picture should be observable in the idler branch (not shown here, as a silica photodiode array of the spectrum analyzer OMA III has no absorption for wavelengths that are .1100 nm). Following our model, we can consider frequency mixing as a variety of parametric processes taking place
Fig. 2. (a) OPA output spectrum at the signal wavelength. The signal at 1029 nm is accompanied by newly generated spectral components s1, s2, and s3 at 977, 932, and 890 nm, respectively. DF 0 is the phase-matching condition for the seed signal amplification. (b) The same situation with an addition of phase mismatch sDF 0.1±d.
Fig. 3. Spectrum of the OPA seeded by a 970-nm signal at (a) DF 0 and (b) DF 0.1±.
November 1, 1997 / Vol. 22, No. 21 / OPTICS LETTERS
Fig. 4. Semilogarithmic plot of the OPA output spectrum at pump wavelength. 2vs and 2vi are the frequencydoubled signal and idler, respectively; 2vs1 and 2vi1 are the frequency-doubled new spectral components.
is rotated from the optimal amplification point for the signal by DF 0.1±. Figure 3 shows the OPA output spectrum obtained when we tuned the seed signal wavelength to 970 nm. Only one newly generated spectral component, at 870 nm, was observed. Besides having the phase-matching constraints, the amplitudes of the newly generated spectral components depend strongly on the seed signal intensity. Our experimental set of parameters ensures a large-signal amplification mode, i.e., the OPA gain saturation for the original signal. Low seed signal energies substantially degrade the quantum eff iciency of the amplification process, mostly at the expense of newly generated spectral components. To prove that a real cascaded process was taking place we measured the spectrum in the proximity of the OPA pump line (Fig. 4). Despite rather strong wings, which occur because of light scattering in the spectrometer, one can distinguish peaks at frequencies of the second harmonics of the respective signals, which is clear indication of the cascading effect. The calculated spectra in Figs. 2 –4 are shown by dashed curves. The theoretical and experimental re-
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sults coincide qualitatively, which enables us to state that the nature of the process is well understood. We have chosen the parameters for the numerically simulated curves to give the best coincidence of experimental and theoretical results, and these parameters were set to be equal to the experimental ones within the error bars. To demonstrate this phenomenon simultaneously in both the space and the time domains, we injected the seed signal into an OPA noncollinearly in the plane noncritical to phase matching. Then all the spectral components emerged from the OPA crystal at different angles, def ined by the wave-vector geometry in the selfdiffraction process. In conclusion, we have demonstrated efficient generation of new spectral components that are due to cascaded second-order processes in a seeded broadband parametric amplif ier. The phenomenon can be seen to some extent as the analog of self-diffraction, with the difference that it takes place in the time domain. The effect leads to a great enrichment of a pulse spectrum, so it has a high potential for practical applications. The number and intensity of the frequency components generated by the cascading can be varied by modification of the phase-matching conditions and the seed signal wavelength. We are indebted to G. Valiulis for providing us the software TLPI v6, which enabled us to calculate all necessary parameters of the BBO crystal. References 1. R. Danielius, A. Dubietis, and A. Piskarskas, Opt. Commun. 133, 277 (1997). 2. G. I. Stegeman, D. J. Hagan, and L. Torner, Opt. Quantum Electron. 28, 1691 (1996). 3. H. Tan, G. P. Banf i, and A. Tomaselli, Appl. Phys. Lett. 63, 2472 (1993). 4. K. Schneider and S. Schiller, Opt. Lett. 22, 363 (1997). 5. R. Danielius, P. Di Trapani, A. Dubietis, A. Piskarskas, D. Podenas, and G. P. Banf i, Opt. Lett. 18, 574 (1993). 6. A. Berˇzanskis, R. Danielius, A. Dubietis, A. Piskarskas, and A. Stabinis, Appl. Phys. B 60, 421 (1995).
