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J. Eur. Opt. Society-Rapid Publ. 22, 4( 2026)
Calculations for our experimental wavelengths with L c = 50mmyields s; i = 0.67 ps. This walk-off time is directly related to the spectral acceptance in the low parametric gain limit( DFG) assuming a narrowband pump wave, m ¼ 0:886 = s s; i which gives about 44 cm �1, equivalent to 16 nm for the signal and 25 nm for the idler spectral bandwidths in the case of the 50-mm long PPLN. The parametric gain C broadens these values by a factor of
~ 0. 6( CL c) 1 / 2 [ 1 ]. For the maximum pump level applied
in the experiments with the 50-mm long PPLN crystal corresponding to a spatially averaged pump intensity of
~ 10. 5 MW / cm2, the increased parametric gain bandwidth
is ~ 1.33 44 cm �1 = 59cm �1 for a single pass parametric amplifier, i. e., only 33 % larger compared to DFG. Although often used, these simple estimates ignore pump depletion and present an oversimplification being derived from analytical parametric gain expressions ignoring GVM and GVD, with the final results appearing ironically exactly in terms of GVM or, when the GVM is vanishing, in terms of the next term, GVD [ 1 ]. Strictly speaking all such analytical estimates are applicable for tunable monochromatic waves but not for ultrashort pulses or for broadband radiation in general. Usually, broadband nanosecond parametric oscillators will have narrower output bandwidths compared to the parametric amplification bandwidth due to multiple passes, however, saturation effects at higher conversion efficiency lead to the opposite trend. Thus, realistic estimations for the OPO and NRO output bandwidths, in particular taking into account depletion and back conversion, can be derived only from numerical simulations.
In the simulations we used the actual cavity mirror parameters from the experiments with the 50-mm long PPLN crystal. For normal incidence, we calculated the wavelength dependence of the diffraction efficiency( reflectivity) of the home-made VBG following the analytical formula presented in [ 24 ]. Alternatively, we modelled the VBG as a super-Gaussian( n = 4) shapednotchfilter in the frequency domain. We established that the shape of this filter was not critical and the simulation results were mainly determined by its FWHM. Following [ 24 ], a FWHM of 0.5 nm and a peak reflectivity of 99 % were assumed for the home-made VBG used.
The maximum pump power applied in the NRO experiments with the 50-mm long PPLN crystal corresponds to a pump pulse energy of 0.95 mJ. Thus, in the plane-wave simulations we used a spatially averaged peak pump intensity of ~ 10.5 MW / cm 2, equal to one half of the pump on-axis peak intensity. For the temporal shape of the pump pulse we assumed a Gaussian dependence with a FWHM of 7.5 ns in accordance with the experiment.
Alternatively, to emulate a Gaussian spatial pump beam distribution within the plane wave model while keeping the same temporal pump intensity dependence as in the true plane wave consideration we also used an approach we called quasi-spatial Gaussian( QSG). The beam computational area is divided into 10 annular segments of constant pump intensity corresponding to the Gaussian radial dependence. In the QSG simulation, the peak on-axis pump intensity is used as a parameter. For the experiment with the 50-mm long PPLN crystal, the maximum on-axis pump intensity amounts to 21 MW / cm 2, i. e., twice the average
Figure 11. NRO output spectra for the 3-mm thick PPLN calculated for a single pulse using the QSG approach for broadband operation.
value used in the true plane wave approach. The 1 / e 2 pump beam radius is ~ 0.6 mm, as in the experiment, and the radial interval is chosen as 60.19 lm. The output energies are obtained by adding the contributions of all rings and the output spectra are obtained by a similar summation but using the ring areas( proportional to the radial distance from the beam center) as a weight function. The radial interval is chosen in such a way that for the chosen number of segments the pump intensity in the outermost ring is below the NRO threshold. We confirmed in preliminary tests that the final results converge at yet smaller intervals with larger number of segments.
We established that the QSG approach leads to a closer agreement with the experimental results with respect to the input output energetic performance. Concerning the simulation of the NRO output spectra, essential deviations from the true plane wave approach were observed only in the broadband case with the spectra computed by the QSG approach being broader. No significant differences were observed in the narrowband case, which can be explained by the VBG acting as a very narrow frequency filter dominating all other spectral shaping effects. While in the true planewaveapproachweaveragedthefinalspectralresults over multiple pulse simulations, in the QSG approach some spectral averaging takes place even when only one pump pulse is simulated.
Figure 11 presents the spectra obtained by such a single QSG simulation in the broadband case, and Figure 12a presents the spectra obtained by QSG simulation averaged over 10 pulses in the narrowband case for the 50-mm long PPLN crystal. The simulation provides output spectral data with high resolution corresponding to a wavelength step of ~ 0.01 nm for the signal and ~ 0.015 nm for the idler. The narrowband spectra were additionally smoothed to account for the finite experimental spectral resolution( ~ 0.5 nm). This was realized by convolving the computed spectra with a Gaussian apparatus function with the corresponding FWHM in the frequency domain, followed by conversion to wavelength units to plot Figure 12b. In fact this smoothing procedure made the averaging over multiple pulses starting from quantum noise( i. e., multiple pulse simulations) redundant.
The simulation results provided strong support for the experimental observations in terms of spectral behavior demonstrating that the VBG acting on the signal wave