The influence of the active gain medium on the spectral amplitude and phase of amplified, femtosecond pulses in a laser system is studied. Results from a case study of a 15-petawatt laser based on Nd-doped mixed glasses show that gain-induced atomic phase shifts will distort the pulses, reducing their peak power. It is also shown that a phase compensation solution is possible and the corresponding coefficients are calculated.
© 2011 OSA
Petawatt (PW) laser pulses are vital for a variety of present and future scientific studies such as high energy and attosecond science, particle acceleration, laser plasma interaction, relativistic physics, x-ray generation, and others . These pulses are produced today using the well-established technique of chirped-pulse amplification (CPA), which has been tremendously successful over the past 25 years .
To further increase the peak power, more energy and shorter pulses are needed. While the energy output of the laser system is linked to its overall size, the pulse duration is related to the bandwidth that the system can amplify. Gain narrowing  is the dominant effect limiting the bandwidth of PW laser amplifiers today. For example, although the Nd-doped glass (APG-1 phosphate, 27.8 nm Lorentzian linewidth) used in the first petawatt laser system  had enough bandwidth to amplify 20 fs pulses, the final pulse duration was 440 fs due to gain narrowing. Even lasers based on Ti:Sapphire crystals, known for their extremely wide bandwidth, suffer from gain narrowing . To circumvent this shortcoming, three different approaches can be used. The first takes advantage of optical parametric CPA amplifiers (OPCPAs) in hybrid laser systems where part of the gain is shared with the parametric amplifier that does not suffer from gain narrowing. This technique is mostly used in the front end of a petawatt laser for which moderate-energy and high-quality pump lasers have been developed . Second is to use multiple gain media to amplify sequential parts of the spectrum . The third approach involves the use of spectral filters to “push down” on the amplification near the peak wavelength and amplify more of the side wavelengths [5, 8].
The latter approach comes at the expense of the output energy. Nd:glass lasers are, however, very energetic and can easily produce more energy than what is needed for a 100-fs, multi-PW system. Hence, an affordable loss in the output energy carries the benefit of increased bandwidth and, therefore, shorter pulses. All three above-mentioned approaches to fight gain narrowing have been successfully employed in a mixed-glass CPA system that produced 1.1 PW laser pulses . Higher peak power lasers, 10 to 15 PW in a single beam, could be built in the next 5 to 10 years by following this path and by exploiting the energetic nature of glass amplifiers.
The process of light amplification in such PW lasers is also influenced by the gain-induced dispersion, also called the atomic phase shift (APS) . This phase shift is better known for causing frequency pulling in high gain lasers  but it can also change the phase of a pulse in a laser amplifier . Therefore, it can be potentially disruptive in femtosecond CPA systems  that are very sensitive to small phase disturbances.
The APS develops in an inverted amplifying medium where the gain is intimately connected to the index of refraction, as it is dictated by the Kramers-Kronig relations. Its influence in a CPA system is illustrated in Fig. 1 that shows a representation of the refractive index of a laser medium versus wavelength for the cases where the medium is passive (no population inversion) and active (inverted, providing gain). The dispersion of passive optical components, illustrated by the dotted black line in Fig. 1, is taken into consideration in the stretcher-amplifier-compressor calculations needed for the design of a CPA system . The dispersion introduced by an active medium, shown in Fig. 1 by the black continuous line, is generally not considered in the laser design. Figure 1 also shows the overlap of this index of refraction with the spectrum of a long pulse such as a nanosecond or a picoseconds pulse (red line), and the spectrum of a short pulse, such as a sub-100 fs pulse (blue dashed line). An important thing to notice is that the red line spectrum overlaps only with the linear part of the index of refraction, and that is the case for the majority of the CPA systems today. However, the blue line spectrum, being wider, it also overlaps with the nonlinear parts of the refractive index, as it is the case for CPA lasers where the gain narrowing is mitigated and large bandwidths are amplified. The stretched pulse that propagates through this nonlinear part of the index of refraction will acquire a phase shift (APS) that cannot be easily removed in the compressor. Both the amplitude and shape of this APS are important for the recompression of the pulse. Moreover, since almost all PW laser systems operate in or near saturation, it is difficult to provide a quick estimate of the shape of this phase shift. For single-shot and low repetition systems, the phase distortion is hard to quantify and measure since it develops only when the power amplifiers are fired. It therefore becomes indubitable that, for high-gain, large bandwidth CPA lasers, a careful investigation is needed to establish the magnitude of the distortion introduced by the APS on the compressed pulse and find possible compensation solutions. In this paper, such a study is performed on a feasible, 15 PW mixed glass laser.
2. Laser design
The case study laser uses two types of amplifiers, one based on phosphate glass and the other on silicate glass. The laser architecture is hybrid, with a front-end OPCPA followed by Nd:glass power amplifiers. This system uses all three of the above mentioned approaches to minimize gain narrowing. The components of the laser system are shown in Fig. 2 .
The front-end produces 60-fs, Gaussian, transform-limited pulses centered at 1069 nm having 1.9 Joules of energy. These numbers are comparable to those already demonstrated by the Texas Petawatt Laser . Since high-fidelity and high-energy pump lasers for OPCPA systems are still under development, the pulses are here further amplified by Nd-glass amplifiers. The size, small signal gains and all other parameters used in this study are typical for disk amplifiers that are currently operating at the Jupiter Laser Facility (JLF) at Lawrence Livermore National Laboratory . Each amplifier is made of two modules each containing two double-passed sets of disk amplifiers. The active areas of the phosphate and silicate amplifiers are 63.6 cm2 (9-cm diameter) and 339.8 cm2 (20.8-cm diameter), respectively. Preceding each power amplifier, there are two spectral filters (see Fig. 2) used to reduce gain narrowing. Finally, the chirped pulses are passed through the 4-pass compressor in order to become femtosecond pulses. The transmission of the compressor is assumed 86% at all wavelengths. Other linear system losses (optics, diagnostics, etc.) are considered to be 5%.
The operation of the laser has been studied with a one-dimensional computer code that models the amplification, gain narrowing, saturation and APS. The optical properties of the laser glasses  were taken to be identical to those of Nd:Phosphate (APG-1, 27.8 nm linewidth), and Nd:Silicate (K-824, 38.2 nm linewidth). The stimulated emission cross sections were considered Lorentzian functions with the above-mentioned linewidths. The amplification of the phosphate and the silicate glasses peaks at 1053.9 nm and 1064.5 nm, respectively. To expedite the computer modeling, these glasses, operating at ambient temperature, were assumed to be 100% homogeneous .
3. Spectrum and phase evolution
The spectrum of the pulse is shaped by every module of amplification and filtering. Figure 3 shows the evolution of the spectrum starting with the initial 60-fs pulse and ending with the spectrum after the final (silicate) amplifier. A close look at these spectra reveals strong gain pulling towards the peak amplification wavelengths, 1053.9 nm and 1064.5 nm. The width and position of the spectral filters were purposely set to maximize the width of the silicate spectrum (Fig. 3, line s), which is also considered the final, compressed spectrum having a 24 nm full-width-at-half-maximum (FWHM) bandwidth. A transform-limited pulse with this spectrum would be 86 fs long. However, due to the APS, the amplified pulse does not have a flat phase and will not compress to 86 fs. The sum Δϕ of the phase shifts introduced by each amplifier is shown below in Eq. (1):
Here ωp, Δωp and ωs, Δωs are the peak angular frequencies and linewidths of the phosphate and the silicate glasses, respectively. The power gain coefficients Gp and Gs, respectively, are functions of the frequency because of the Lorentzian lineshapes and the gain saturation. The phase shifts introduced by each amplifier as well as their sum are shown in Fig. 4 . The peak-to-peak phase excursion is: ΔϕPP = 6.8 rad. The total phase shift Δϕ is a residual phase that is imprinted on the stretched pulse. Other residual phase terms sometimes generated by imperfect stretcher-amplifier-compressor designs or low-quality laser optics are neglected to reveal the influence of the APS effect alone.
4. Pulse Compression
The influence of this residual phase on the compressibility of the pulse is presented in Fig. 5 . The line (i) shows the shape of the initial, 60-fs pulse. Due to the APS accumulated during amplification (see total phase curve, Fig. 4) and assuming the same spectrum, the initial pulse shape would change to that shown by line (ip). The difference between lines (i) and (ip) in Fig. 5 is only meant to point out the effect (on the 60-fs pulse) of a pure phase distortion from flat to the APS.
However, the spectrum of the amplified pulse changes as well, as previously shown in Fig. 3, line (s). An FFT reconstruction from this spectrum and with the APS is presented in Fig. 5, line (a). This is the temporal profile of the amplified pulse. This pulse is distorted and broken in two parts, with a significant amount of its energy in the early part. To recover the pulse shape and increase the peak power, a phase modulator can be used. Using a 4th order polynomial fit to the total phase shown in Fig. 4 as a starting point, the best 2nd, 3rd and 4th order compensation coefficients were found by minimizing the pulse duration retrieved in the computer code. Applying the phase compensation according to these coefficients (2.7×103 fs2, 2.4×105 fs3, and 1×107 fs4, respectively), the pulse shape improves significantly with 96.6% of the energy concentrated in a single pulse, as shown by line (ac) in Fig. 5. The FWHM of this pulse is 88 fs, which is only 2 fs longer than the transform-limited pulse width. The peak power of this pulse is 15 PW and exceeds the peak power of the uncompensated pulse by 61%. If plotted on a logarithmic scale (not shown), a contrast of 10−12 is reached at −1 ps and +2.5 ps by the leading edge and by the trailing edge, respectively. The width of the pulse at the 10−3 level is 532 fs and at the 10−6 level is 1198 fs. Overall, if the OPCPA provides a high-quality, strong seeding pulse , the contrast should be good enough for the majority of high energy density experiments. It is important to note that this 4th order polynomial phase compensation is being discussed here because it can be easily and quickly applied with commercial phase modulators. Higher accuracy devices able to compensate for arbitrary phase shapes could yield better results once the APS has been calculated.
Another consequence of the APS is the delay of the peak of the amplified pulse with respect to that of the un-amplified (when the glass amplifiers are not being fired) pulse . That can be inferred from Fig. 5 where the phase compensated pulse “arrives” later by 160 fs than any flat-phase pulse such as the initial 60-fs pulse. This fact could be detrimental in single shot experiments where a low-energy probe pulse derived before the amplification stages is commonly employed for pump-probe measurements. A phase modulator may be able to “remove” the temporal delay between the probe and the main pulses but it would have to be placed after the probe pickoff optics and before the glass amplifiers.
The phase of the pulse can also be affected by the self-phase modulation process. The peak value of this phase, also called the “B-integral,” was calculated after each pass of amplification. The nonlinear indexes of refraction were n2 = 1.13×10−13 esu and n2 = 3.44×10−13 esu for phosphate and silicate glasses, respectively . A stretching factor of 385 ps/nm, typically provided by a double-pass, Offner-type stretcher , was also used in this calculation to determine the intensity of the pulse as a function of wavelength.
Figure 6(a) shows the accumulated (a sum over all passes in each amplifier) B-integral values in the phosphate and silicate glasses. Also shown is the overall value which is the sum of the two and that has a peak value of 0.8 near 1052 nm. Such a low B-integral value in a 15 PW laser is a positive consequence of fighting gain narrowing and amplifying large bandwidths. The 24-nm stretched pulse is, therefore, quite long (9.24 ns), and that helps reduce its intensity. Thus, it can be concluded that the self-phase modulation of the stretched pulse will not affect its compressibility. It should be also noted that the maximum fluences in the phosphate and silicate amplifiers are 3.8 J/cm2 and 4.7 J/cm2, respectively. For this 9.24 ns stretched pulse, both fluence values are below the damage threshold for high-quality, inclusion-free glass .
Small fluctuations in the laser gain, typical of pulsed systems, could change the spectrum and the APS. This can adversely affect the amplified spectrum and the phase of the pulse and make the phase compensation impossible. A statistical study was done to quantify the stability of the laser system under the influence of the APS. Assuming a typical 1% variation (standard deviation/mean value) in the initial small signal gain of every glass amplifier in the laser, a statistical study of 300 shots was performed. The output energy and pulse duration were calculated for all shots with only one set of phase compensation factors, precisely, those used to optimize the pulse compression shown in Fig. 5.
The results of the study are shown in Fig. 6 (b). The variations of the output energy, power, bandwidth and pulse duration are: 3.6%, 3.1%, 0.67%, and 0.38%, respectively. Therefore, fluctuations in the small signal gain will only produce minimal changes in the laser output. Overall, Fig. 6 argues for the feasibility of the test laser that is significantly affected by the APS.
Another study was undertaken in an attempt to “gauge” the magnitude of the APS effect on the pulse compression at smaller levels of laser power. This is a difficult task because the APS effect depends on the laser design, and, in principle, the operation of each laser system needs to be modeled in order to exactly determine the distortions of the compressed pulse. This study targeted two different laser designs, both producing 1 PW pulses, the first (system A) with a small bandwidth, and the second (system B), with a large bandwidth. Both systems have the same front-end as shown in Fig. 2 but some subsequent amplifiers and filters were removed or modified. In both cases, the compressed pulses are much less distorted compared to the 15 PW laser with no pulse splitting being observed. Mainly, the distortion shows up as an increase in the pulse FWHM duration. The results are shown in Table 1 where Δλ is the pulse bandwidth, λC is the centroid wavelength, Δt is the pulse width, ΔtTL is the transform-limited pulse width, ΔϕPP is the peak-to-peak value of the APS, and RTL is the Δt/ΔtTL ratio. As expected, the APS affects more the system with larger bandwidth (system B) because of the higher gain spent to generate this bandwidth and because of the overlapping of the spectrum with the nonlinear parts of the phase shift. Both systems can be phase compensated and pulse durations near ΔtTL can be obtained.
In conclusion, the atomic phase shift was calculated for a feasible, high-gain, mixed-glass laser system. It is shown that this phase shift is large enough to significantly distort the compressed pulse. The pulse shape can be improved and the peak power increased by 61% with a phase modulator. The pulse duration would then be 88 fs and the peak power approximately 15 PW. Modern cooling of the glass disks coupled with efficient diode pumping could make these laser systems the preferred tool for high energy density science.
These results have, in fact, important implications for any multi-PW laser. Namely, they show that gain-related issues such as gain narrowing and gain induced dispersion can be overcome and intense, sub-100 fs pulses can be generated in a single beam by a moderate size laser. It appears that the ultimate limiting factors for the peak power of such systems remain the B-integral and the damage threshold of optics.
This work was performed under the auspices of the US Department of Energy by Lawrence Livermore National Laboratory under contract DE-AC52-07NA27344.
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