We report on a novel approach of ultra-broad bandwidth parametric amplification around degeneracy. A bandwidth of up to 400-nm centered around 800 nm is amplified in a BBO crystal by using chirped pump pulses with a bandwitdth as broad as 10 nm. A supercontinuum signal is generated in a microstructured fiber, having to first order a quadratic chirp, which is necessary to ensure temporal overlap of the interacting waves over this broad bandwidth. Furthermore, we discuss the potential of this approach for an octave-spanning parametric amplification.
©2005 Optical Society of America
Intense few-cycle optical pulses have attracted considerable attention due to their numerous applications ranging from high-order-harmonic generation and the generation of soft X-rays to probing of ultrafast processes in physics, photo-chemistry and photo-biology. Two different approaches have been considered to realize intense sub-5-fs pulses. The first is based on self-phase modulation induced spectral broadening of already amplified short laser pulses in glass or hollow core waveguides followed by pulse compression techniques [1–3]. The second approach relies on the direct amplification of the sufficient bandwidth required for a few-cycle optical pulse, which is offered by non-collinearly phase-matched optical parametric amplification (NOPA) .
In general, optical parametric amplification (OPA) utilizes instantaneous nonlinear interaction of spatially and temporally overlapped pump and signal pulses. OPA is receiving much attention due to its favorable characteristics. Depending on the crystal, the interaction type, interacting wavelengths and pump intensity an enormous gain of up to 108 can be achieved. Parametric amplification can offer large gain bandwidth and/or large tunability. Furthermore, OPAs have the potential for very high average powers as a result of negligible thermal load which is inherent to parametric processes. High peak powers are feasible by the use of large aperture crystals, consequently, Petawatt laser systems based on parametric amplifiers have been discussed .
Abandoning collinear interaction geometry of the three waves offers an access to ultra-broad amplification bandwidths, discovered by Gale and co-workers in a synchronously pumped optical parametric oscillator configuration . It has been recognized that in a non-collinear beam geometry the group-velocity mismatch between pump, signal and idler waves can be efficiently compensated. In such a configuration the group velocity of the faster traveling idler wave is projected at an angle onto the direction of the slower traveling signal wave, therefore, the temporal overlap is improved (to the detriment of spatial overlap, except in special cases where the non-collinear angle corresponds to the walk-off angle) and a broad phase-matching bandwidth is achieved . Figure 1 shows the calculated type 1 phase-matching curve for non-collinear interaction in a BBO crystal, calculated using the SNLO software . Assuming a monochromatic pump wavelength of 400 nm, i.e. the second harmonic of the Ti:sapphire laser, and a pump tilt or non-collinearity angle α of 3.7°, one can observe a flat phase-matching curve at a constant internal signal angle of ~ 27.6°. Thus, in this context, phase-matching is achieved for signal wavelengths ranging from ~540 nm to ~ 740 nm, corresponding to 160 THz bandwidth. The generation of μJ-level sub-5-fs pulses has been demonstrated by using this amplification bandwidth followed by refined pulse compression techniques .
Even larger amplification bandwidths of up to 250 THz have been discussed employing more sophisticated experimental strategies such as the implication of a pump with a tilted wavefront , angular dispersion of the pump beam  or multiple pump beams . The use of large bandwidth chirped pump pulses has also been discussed theoretically in the context of a chirp compensation scheme in the non-degenerate case .
In this contribution, we report on the ultra-broad bandwidth parametric amplification at degeneracy. Bandwidth enhancement is simply achieved by using a broadband pump with a chirp set to optimize the temporal overlap with the different frequency components of the signal. In terms of complexity, the herein discussed scheme requires only spectral phase management of the pump while a scheme such as pulse front matching  needs both spectral phase (temporal dispersion) and angular dispersion management. Up to 400 nm bandwidth is amplified around 800 nm wavelength in a BBO crystal.
2. Broadband parametric amplification around degeneracy
In general, the above discussed condition of group-velocity matching between signal and idler is inherently satisfied at degeneracy due to the equal frequencies of signal and idler waves. The black curve in Fig. 2(a) shows the calculated phase-matching conditions in a type 1 BBO crystal pumped at 410 nm around degeneracy in a collinear configuration (α=0°). As shown by the vertical black dashed lines, at an internal signal angle of 28.3° phase-matching can be achieved from 770 to 870 nm (about 40 THz). The discussed phase-matching curves are based on a monochromatic pump wave. An enormous bandwidth enhancement can be obtained by applying a broadband pump. This is the basic idea of ultra-broadband parametric amplification around degeneracy as shown by the red curves in Fig. 2a. The solid line represents phase-matching curve for monochromatic pump wavelength at 410 nm, and a non-collinearity angle of 0°. It is revealed that at an internal signal angle of ~28.3° phase-matching is given in a wavelength range from 640 to 1060 nm (red vertical dashed lines), corresponding to a bandwidth of 200 THz. Therefore, our scheme is similar to pump tuning of OPA’s, but uses a broadband pump instead of a tunable one.
Considering the phase-matching curves in Fig. 2(a) one can see that signal wavelengths around 800 nm are phase-matched to a pump wavelength of 410 nm. As the pump wavelength is decreased, phase-matching occurs for two signal wavelengths which drift away from the central wavelength. In that way, signal wavelengths of 640 and 1060 nm are matched to pump field components at 400 nm. The phase-matching map that is inherent to this broadband pump configuration is summarized in Fig. 2(b). It is now clear that efficient ultra-broadband amplification will set unique relation between the temporal distribution of the pump frequencies i.e. its chirp and the chirp of the signal. Consequently, a linear chirp in the pump pulse (quadratic spectral phase) requires a signal with a cubic phase to provide temporal overlap of phase-matched spectral components, or vice versa.
Indeed, in the herein presented scheme on broadband parametric amplification the signal is a supercontinuum generated in a photonic crystal fiber . Such a supercontinuum has in first order a cubic phase, because the wavelengths are spread over both sides around the zero-dispersion point of the photonic crystal fiber. This is illustrated in figure 6 showing a typical spectrogram (wavelength vs. time) of supercontinuum generated in a photonic crystal fiber possessing a zero-dispersion wavelength of 810 nm and a length of 20 cm. The pulse propagation in the photonic crystal fiber is simulated using split-step Fourier method to numerically solve an extended nonlinear Schrödinger equation . The pulse input parameters of the simulation are adapted to the experiment described below, i.e., 40 fs pulse duration, center wavelength 810 nm and 1.5 nJ pulse energy. As shown, wavelengths around the zero-dispersion point travel at the leading edge of the ultra-broadband continuum with a quadratic chirp (cubic phase). The red vertical lines in figure 6 illustrate which pump (around 400 nm) and signal wavelength (continuum) are phase-matched in a parametric amplifier applying the above discussed configuration.
Figure 6 is indeed very similar to Fig. 2(b), except that the X-axis is now a time delay instead of the pump wavelength. Actually, an optimal parametric amplification would require the 2 plots to overlap perfectly. This is then achieved by choosing the proper time delay of each pump wavelength so that their corresponding phase-matched signal components overlap.
3. Experiment and results
The experimental setup of ultra-broadband parametric amplification around degeneracy, as shown in Fig. 3, consists of a Ti:sapphire amplifier system, a photonic crystal fiber to generate the broad-bandwidth signal, a frequency doubling BBO crystal, a prism pair to chirp the pump pulses and a second BBO crystal for parametric amplification.
The Ti:sapphire amplifier system provides 500 μJ of 40 fs pulses with a repetition rate of 1 kHz at 810 nm. A small fraction of the power is launched into a photonic crystal fiber with a zero-dispersion wavelength of 810 nm and a core diameter of 2 μm (Crystal Fibre NL-2.0-810). The beam is attenuated using two 4% beam samplers and a half-wave plate in combination with a polarization beam splitter. Finally, just few nJ are launched into the PCF generating a continuum which extends from <450 nm to >1600 nm. Figure 4 shows the continuum produced in 20 cm fiber length in the wavelength range of interest.
Most of the initial pulses (FWHM = 28 nm) is frequency-doubled in type I BBO (cut: θ=29°, ϕ = 0°) crystal. Because the second harmonic phase-matching bandwidth is crystal length dependent  the right length has to be chosen. Figure 5 shows the spectrum of the frequency doubled Ti:sapphire laser system output for several BBO crystal thicknesses. Using a 200 μm thick doubling crystal a bandwidth of 10 nm is achieved centered at 405 nm. The conversion efficiency is about 20 % obtained even in this thin crystal and with a collimated beam having a diameter of about 2 mm. These UV pulses constitute the pump field in the parametric amplifier described herein.
To achieve temporal overlap between pump and signal pulses over the entire phase-matched bandwidth the UV pump radiation is positively chirped in a sequence of fused silica prisms. The prism distance is just in the order of few cm, therefore, material dispersion is dominating, resulting in a positive chirp. The prism arrangement has a second important intention, it gives a simple possibility to remove the non-converted infrared light in the pump beam. Because we are working at degeneracy any residual 800 nm radiation would deplete the parametric gain.
The chirped broadband pump and the supercontinuum are overlapped in a second 3 mm long type 1 BBO crystal (cut: θ=29°, ϕ = 0°). An off-axis parabolic mirror is used to image the continuum into the second BBO crystal to avoid chromatic aberrations of the broadband signal. A small angle of less than 1° between pump and signal is used to separate signal and idler after amplification. The basic consideration of phase-matching made above are still valid, the phase-matching curves are just slightly red-shifted, resulting in a possible amplification bandwidth ranging from 630 nm to 1030 nm. Furthermore, the idler and signal waves having opposite phases, it is important for both stability and recompression to separate the idler produced by direct amplification of the above 800 nm input components from light generated during the amplification of sub-800nm input.
Figure 7 shows the obtained amplified spectrum for two different delay positions and a photonic crystal fiber length of 20 cm. As shown, at the two delay positions either the wings (630/960 nm) or the central part (800 nm) of the phase-matched bandwidth is amplified. The difference in the delay positions is 600 μm corresponding to a time delay of 2 ps. These spectra can be interpreted by considering numerical simulations of the continuum dynamics in the photonic crystal fiber, as plotted in Fig. 6. At the output of the fiber, the delay between the 800 nm components and the 630/1000 nm is about 2.0 ps. On the other hand, the UV pulses are stretched in the prism sequence to about 550 fs (5 cm fused silica, assuming -1100 ps/(nm∙km) at 400 nm ). Therefore, the pump pulse is too short to amplify the entire stretched spectrum at once. Instead, the present configuration is analog to a cross-correlation FROG measurement  where the pump operates as a temporal gate in the wave-mixing process.
There are two approaches to achieve parametric amplification over the entire bandwidth. First is to stretch the UV pump pulses to 2.0 ps or, second, to shorten the photonic crystal fiber in order to reduce the time delay between the center and the wings components of the continuum. The main reason why we have chosen the second approach is the final goal of the work, i.e., the generation of ultra-short intense laser pulses. According to  the supercontinuum radiation is recompressible if the continuum is generated with very short pulses (sub-50-fs) in very short fibers (<5 cm) due to reduced spectral phase instabilities. Therefore, we have shortened the photonic crystal fiber down to 5 cm. The corresponding spectrogram is shown in Fig. 8.
The resulting parametrically amplified spectrum is shown in Fig. 9. At a single delay position a spectrum ranging from 630 to 1030 nm is amplified, corresponding to about 400 nm or 200 THz phase-matched bandwidth. The non-optimized parametric gain is ~104 and the conversion efficiency is less than 10%, resulting in ~5 μJ broad-bandwidth pulses.
4. Conclusions and Outlook
We have demonstrated a new technique of ultra-broad bandwidth parametric amplification around degeneracy. The approach is based on amplification with a broadband pump involving a control of the chirp of either the pump, the signal or both in order to achieve temporal overlap of the phase-matched components over the entire bandwidth. A spectrum ranging from 630 nm to 1030 nm is amplified in a type 1 BBO crystal, corresponding to a bandwidth of 400 nm (200 THz). To our knowledge, this is the largest bandwidth ever reported for parametric amplification at degeneracy. The recompression and the improvement of efficiency will be subject of future activities.
It is needed to be pointed out that the chirp control, herein achieved by a linearly chirped pump and a quadratically chirped signal out of a photonic crystal fiber, can also be obtained by phase-control devices such as a Dazzler or a spatial light modulator (SLM) giving more flexibility and avoiding the difficulties arising from the needed compression of the supercontinuum.
Furthermore, the approach allows for the parametric amplification of an octave spanning spectral width. This can be achieved by adding an angular dispersion to the broad-band pump, as illustrated in Fig. 10. The black curves showing once again the phase-matching conditions as discussed in the herein presented experiment, assuming a 10 nm broad pump and a constant non-collinearity angle of 1°. By adding angular dispersion over the same pump bandwidth a significant phase-matching bandwidth enhancement can be realized. Assuming for the 400 nm pump components a non-collinearity angle of 0.5° and for 410 nm an angle of 2° the phase-matching curves are shifted away resulting at a constant internal signal angle of 27.1° a possible phase-matching ranging from 580 nm to 1240 nm, i.e. more than one octave in the optical spectrum.
This work was supported by the Conseil Régional d’Aquitaine and the European Community (FEDER). Furthermore, the authors thank Thomas Schreiber for preparing the supercontinuum spectrograms.
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