Ultrashort-pulse evolution inside a optical parametric oscillator is described by using a nonlinear-envelope-equation approach, eliminating the assumptions of fixed frequencies and a single χ(2) process associated with conventional solutions based on the three coupled-amplitude equations. By treating the interacting waves as a single propagating field, the experimentally-observed behaviors of singly and doubly-resonant OPOs are predicted across near-octave-spanning bandwidths, including situations where the nonlinear crystal provides simultaneous phasematching for multiple nonlinear processes.
©2011 Optical Society of America
Optical parametric oscillators (OPOs) are conventionally analyzed by solving the three coupled-amplitude equations , which describe the exchange of energy between the pump, signal and idler waves due to the nonlinearity arising from the χ(2) susceptibility in non-centrosymmetric media. Despite the assumptions of monochromatic waves implicit in the coupled-amplitude equations, they can be used successfully to model interactions between ultrashort pulses [2–5], with the proviso that the pulse bandwidth remains a small fraction of the centre frequency of each pulse (the slowly-varying amplitude approximation). For parametrically-coupled interactions that involve more than three waves (e.g. cascaded parametric generation) or which involve ultra-broadband pulses, the coupled-amplitude equations no longer provide a useful description. Fortunately, the very recently discovered χ(2) nonlinear envelope equation (NEE)  provides, for the first time, a rigorous means of analyzing ultra-broadband pulse evolution in a χ(2) medium, similar to the octave-spanning supercontinuum models which have been implemented for photonic-crystal fibers  by using the now established χ(3) NEE . The emergence of the χ(2) NEE as an analytical technique for studying ultra-broadband χ(2) interactions is particularly timely because of several parallel experimental observations of ultra-broadband conversion in quasi-phasematched (QPM) interactions, notably in periodically-poled lithium niobate (PPLN) waveguides  and MgO:PPLN OPOs [10, 11]. The χ(2) NEE has been shown to predict the structure of an octave-spanning supercontinuum generated in a PPLN waveguide , and has also been found to agree with single-pass experiments examining multi-step parametric processes . A major attraction of envelope equations is that they can be solved with much less computational effort than a full-field analysis based on Maxwell’s equations, and indeed the χ(3) NEE has been validated by direct comparison with Maxwell’s equations for single-cycle optical pulses . For this reason, the χ(2) NEE therefore represents a powerful new method for studying ultra-broadband pulse evolution in systems incorporating optical feedback, such as an OPO. To our knowledge, no prior work has been reported in which the χ(2) NEE is applied in this context, however the rapid progress in degenerate modelocked OPOs – which exploit ultra-broadband χ(2) interactions for generating mid-infrared frequency-combs [13, 14] – motivates the development of a rigorous theoretical description of these devices.
Here we report a numerical model, based on the χ(2) NEE, which describes ultra-broadband pulse evolution in an OPO synchronously-pumped by a femtosecond laser. The gain materials used in the simulations are exclusively QPM media, in which the polarity of χ(2) is modulated along the length of the medium with a period equal to twice the coherence length of the intended χ(2) process. The longitudinal grating formed in this way can also quasi-phasematch any higher-order process whose coherence length is an odd sub-harmonic of the grating period and, when the magnitude of either the fields or of χ(2) is sufficiently high, this effect leads to multiple simultaneous nonlinear processes. Effects like this are readily observed when the intensities of the interacting fields are enhanced inside the high-finesse cavity of an OPO. We demonstrate the existence of steady-state and periodic solutions for broadband pulses propagating in an OPO, supported by comparisons with experimental data from a singly-resonant non-degenerate tandem OPO and a doubly-resonant degenerate OPO.
To allow direct comparison with experiment we concentrated on synchronously-pumped OPOs operated with only the nonlinear crystal present in the cavity, which is a typical experimental configuration. In this case, the propagation of a resonant pulse can be represented by two operations: dispersive propagation and gain inside the nonlinear medium (modeled by the NEE) followed by filtering with a complex spectral band-pass filter which is used to represent output coupling, parasitic losses, mirror reflectivity and cavity delay. The simulation uses a definition of the complex electric field envelope of the form , where is the inverse Fourier transform of the positive frequency components of the physical electric field, is an arbitrary reference frequency and is the wavevector measured at . Propagation in the transparency region of the nonlinear medium was modeled by ,Eq. (1) was implemented by a split-step Fourier method , in which the linear part of the equation (the left-hand side) was solved in the frequency domain and the nonlinear part (the right-hand side) was integrated by using a second-order Runge-Kutta scheme. Optical feedback was incorporated by the complex spectral amplitude filter,Eq. (1), completing one complete cavity roundtrip. Since a real OPO is seeded by quantum noise, the numerical model is bootstrapped by a broadband, low-intensity signal pulse centred at a wavelength known to be phasematched by the grating. When the net gain of the cavity is sufficiently high, this seed pulse can develop quickly into a steady state after 20 – 50 cavity roundtrips. The pump field is chosen to be transform-limited Gaussian pulse with a FWHM duration matching experiment, and a field amplitude of where I is the pulse intensity averaged across its beam radius and n is the refractive index at the pump wavelength.
3. Experimental validation
We tested the model against two previously reported femtosecond OPOs, both operating over a substantial bandwidth unable to be accurately described by the conventional coupled-amplitude equations. The first system was an idler-resonant tandem OPO, pumped by a 150-fs-duration Ti:sapphire laser at 845 nm and configured in a cavity with high-reflectivity at a wavelength of 2.3 µm . The gain medium was a 2.6-mm-long dual-grating PPLN crystal, with the first 1 mm composed of multiple uniform gratings with periods from 22.60 – 23.09 µm, which provided quasi-phasematching for optical parametric generation from 845 nm to a (non-resonant) signal at ~1.25 µm and an idler at ~2.6 µm. The second 1.6-mm-long section contained multiple uniform gratings with periods from 25.23 – 34.68 µm, which were quasi-phasematched for difference-frequency-generation (DFG) between the signal and idler pulses produced in the first section of crystal. As this system was a standing-wave resonator we included in the cavity filter the material dispersion corresponding to a second pass through the nonlinear crystal every roundtrip. Starting from a weak 50-fs seed pulse centred at 2.3µm, the simulation reached steady-state in around 30 cavity roundtrips, which can be seen from the logarithmic plot of the spectral evolution of the OPO shown in Fig. 1(a) .
The output of the tandem OPO covers a spectral bandwidth of 1800 nm, comprising strong components at the signal, idler and DFG wavelengths, together with weaker but detectable outputs due to other χ(2) processes. An insight into the origin of these outputs is given by Fig. 1(b) which presents a logarithmic plot of the spectral evolution of the field in the OPO crystal once steady-state has been reached. In the first section of the crystal the pump pulse (a) can be seen converting into a signal pulse at 1.32 µm (b) and an idler pulse at 2.35 µm (c). In the second section of the crystal, difference frequency mixing between the signal and idler pulses leads to a mid-infrared pulse at 3.0 µm (d), which interacts with the pump to create a near-infrared pulse at 1.17 µm (e). The grating period of the second section is nearly phasematched for second-harmonic generation of 3.0 µm, and consequently a second-harmonic pulse can be seen at 1.5 µm (f), which mixes with the intense pump pulse to produce a weak 2-µm output (g). A direct comparison of the output-coupled mid-infrared spectrum recorded from the OPO and that predicted by the simulation is shown in Fig. 2 , and reveals a number of similarities. The sharp features predicted by the simulation are not resolved by the low-resolution mid-infrared spectrometer used in the experiment, however the bandwidths, positions and shapes of the spectra are very comparable, giving a high degree of confidence in the accuracy of the simulation. The solid line in the simulation result in Fig. 2 shows the exact spectrum obtained from the model, while the dashed line shows this spectrum after filtering (by convolution) with a 35-nm FWHM Gaussian band-pass filter. The measured spectrum in Fig. 3 was recorded with a spectrometer whose best resolution was 17 nm, however to improve the signal level we used it without entrance or exit slits, which is realistically expected to increase this value to a figure similar to that applied numerically.
In a second comparison with experiment we simulated the doubly-resonant configuration matching the degenerate OPO described in . In this system, 1.56-µm sub-85-fs pulses were used to pump a doubly-resonant ring-cavity based on a 0.5-mm-long MgO:PPLN gain crystal and configured with metal mirrors for ultra-broadband operation across the 2 – 4 µm wavelength band. Simulating a doubly-resonant OPO is simply matter of setting the bandwidth of to be wide enough to cover the signal and idler wavelengths generated by the crystal. We used a cavity filter with high-reflectivity from 2.0 – 4.0 µm, sufficient to cover the parametric gain bandwidth of the 34.8-µm period crystal used in . Steady-state was reached in 20 cavity roundtrips, starting from a 50-fs seed pulse centered at twice the pump wavelength. The spectral evolution of the OPO is shown in Fig. 3, with the steady-state spectrum (Fig. 3(a)) comprising three strong peaks from 2.8 – 3.5 µm, in agreement with the data recorded from the actual OPO operated without intracavity dispersion compensation.
The simulation results presented so far used a cavity delay chosen so that the wavelength component of the resonant pulse which corresponded to exact quasi-phasematching was always returned with a fixed group delay, relative to the pump pulse injected at the start of each roundtrip. This approach led to steady-state behaviour after a small number of roundtrips, representing a single-point attractor in terms of nonlinear dynamics. For cavity-length adjustments far from this position we observed an alternative solution which took the form of a limit-cycle attractor, never reaching steady-state, but instead converging on a periodic behaviour. An example of this solution is shown in Fig. 4 for the degenerate OPO adjusted with a cavity delay 8 fs greater (2.4 µm longer) than in the previous result. The spectrum is similar to the steady-state result, however energy is continuously exchanged between different frequencies in the resonant pulse, with the spectrum reproducing itself with a period of around 26 cavity roundtrips. A qualitative explanation of this behaviour can be made by analogy with simple-harmonic motion, in which a displacement of the pulse centre wavelength feels a “restoring force” due to the phasematching and dispersion properties of the crystal. This oscillatory behaviour can also be observed in simulations of a singly-resonant OPO and so is not uniquely associated with the doubly-resonant oscillator. Previously reported oscillatory behaviour in a singly-resonant femtosecond OPO operated in the region of zero group-delay dispersion  may be explained by the model, but a detailed examination of this result lies outside the scope of this paper.
We have applied a nonlinear envelope equation to describe the dynamics of ultrashort pulse evolution inside an OPO. Unlike previous approaches based on the three coupled-amplitude equations, our model makes no distinction between the interacting waves – treating them as a single propagating field – enabling it to predict the behavior of an OPO across exceptionally broad bandwidths, and in scenarios where the nonlinear crystal provides simultaneous phasematching for multiple processes. The close agreement between the predictions of the simulation and experimental results from singly and doubly-resonant OPOs shows that the model accurately describes the complex nonlinear behavior of these systems.
This work was supported by the UK EPSRC under grant number EP/H000011/1.
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