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Abstract
J-KAREN-P is a high-power laser facility aiming at the highest beam quality and irradiance for performing state-of-the art experiments at the frontier of modern science. Here we approached the physical limits of the beam quality: diffraction limit of the focal spot and bandwidth limit of the pulse shape, removing the chromatic aberration, angular chirp, wavefront and spectral phase distortions. We performed accurate measurements of the spot and peak fluence after an f/1.3 off-axis parabolic mirror under the full amplification at the power of 0.3 PW attenuated with ten high-quality wedges, resulting in the irradiance of ~1022 W/cm2 and the Strehl ratio of ~0.5.
© 2017 Optical Society of America
1. Introduction
The performance of ultrafast petawatt-class lasers [1–3] gradually and steadily improved in recent years [4–15]. Such lasers are based on the Chirped Pulse Amplification (CPA) concept [16], where low-energy pulses are stretched from femto- or picoseconds to nanoseconds, amplified, up-collimated, and finally compressed with a vacuum compressor. One of the most important characteristics of a laser is its peak irradiance at focus, I0, which determines the laser-matter interaction regime. For high-power lasers, the irradiances are so high that electrons driven by the laser field are ultrarelativistic and the dimensionless amplitude is a0 >> 1. The laser irradiance is I0 ≈a02 × 1.37 × 1018 W/cm2 × (μm/λ)2 for linear polarization; here λ is the wavelength. Most high-power laser experiments have been performed with I0 ~1018 to 1021 W/cm2 (a0 ~1 to 30) and a few with I0 > 1021 W/cm2. To advance the field, irradiances of 1022 W/cm2 and higher are required. The awaited effects include efficient radiation pressure dominant ion acceleration, I0 > 1022 W/cm2 [17], radiation damping regime, I0 > 3 × 1023 (μm/λ)4/3 W/cm2 [18], and quantum electrodynamics effects, I0 > 6 × 1024 W/cm2 [19], up to vacuum breakdown at I0 > 1026 W/cm2 [20]. Other important parameters are the pulse duration and focal spot width. Importantly, the minimum duration is limited by the spectrum bandwidth, and the tightest spot is limited by diffraction. The highest possible irradiance is attained when these limits are reached. However, even approaching these limits is challenging, especially for high-power lasers (see below). In addition, some applications are extremely sensitive to the pulse shape and/or spot quality; for example, laser-solid target interactions suffer from prepulses and pedestals at different time scales [21], while laser-gas interactions are adversely affected by spot imperfections [22].
The final beam diameter of high-power lasers is typically ~20-30 cm to prevent compressor grating damage. The large beam diameter makes achieving a tightly focused beam with clean pulse compression challenging. The major difficulties include wavefront and spectral phase distortions degrading the focal spot and temporal pulse shape, and various kinds of spatiotemporal couplings [23, 24] affecting the laser field distribution in complicated ways. The most typical and detrimental couplings are chromatic aberration and angular chirp. All these effects, if not dealt with carefully, result in poor focal spot and pulse shape, i.e. low irradiance. The irradiance can easily drop by orders of magnitude, destroying the main advantage of high-power lasers.
In this work we optimized the beam quality of the J-KAREN-P laser [15] and approached the bandwidth limit of the pulse shape (peak power of ~0.9 of the limit) and diffraction limit of the focal spot (Strehl ratio of ~0.5 determined from the accurate focal spot measurements under conditions identical to real experiments apart from ten wedges attenuating the 0.3 PW beam). This allowed us to achieve at-focus peak irradiance of ~1022 W/cm2. To our knowledge, this is the highest-quality accurately-characterized petawatt-class laser beam, and the highest accurately estimated at-focus irradiance demonstrated to date.
We also report on challenges and methods to overcome them. In particular, we found that wavefront correction is challenging at the beam edges and requires a stable beamline with alignment reproducible down to microradian level; that irradiance calculated from wavefront is always significantly overestimated; that the final compressor alignment must be performed in vacuum; and that the real irradiance is typically drastically lower than the estimate of E0/(τFWHM XFWHM YFWHM), where E0 is the pulse energy, τFWHM is the Full Width at Half Maximum pulse duration, and XFWHM and YFWHM are the spot widths.
2. Requirements
Usually used characteristics of the spot and pulse, such as FWHM, are not adequate when the goal is to approach the physical limits and achieve maximum irradiance. This is because a major part of the pulse energy can be (and often is) outside of the main spot or main pulse.
As a measure of the pulse quality, we use the effective duration:
where P(t) is the power and P0 is the peak power; thus, τeff equals the area under the normalized power curve p(t). Ideally, τeff equals the effective duration of the bandwidth-limited pulse. Since E0 = ∫P(t)dt = P0∫p(t)dt, the peak power is:We also define the effective spot area, Aeff, and radius, reff:
Here F(x,y) is the focal spot fluence distribution and F0 is the peak fluence. Thus, Aeff equals the volume under the normalized fluence surface f(x,y). Because E0 = ∫F(x,y)dxdy = F0∫f(x,y)dxdy, the peak fluence and irradiance areThe Strehl ratio S is the ratio of peak fluences of the real and diffraction-limited spots; here we do not include time-domain effects. In an ideal case, S = 1. For a real spot,In the absence of spatiotemporal couplings, Eqs. (2) and (4) give correct expressions for peak power, fluence, and irradiance which follow from the definitions of effective pulse width (1) and spot radius (3); calculations of these values are possible from the measured dependences p(t) and f(x,y). In contrast, the “equation” I0 = P0/(XFWHM YFWHM), sometimes used in the literature, gives incorrect (overestimated) results. For example, even without the time-domain effects, for the spot shown in Fig. 1(a) (Table 1, left column), it gives 1.5 × 1022 W/cm2, an order of magnitude overestimate compared to the correct estimate of 1.6 × 1021 W/cm2.
Fig. 1 J-KAREN-P focal spot after the f/1.3 OAP, the color scales are linear with the maxima equal the corresponding Strehl ratios. (a) Lens-based beam transport and expansion, severe chromatic aberration. (b) Mirror-based beam transport and expansion, wavefront corrected, angular chirp not compensated. (c) Angular chirp compensated. (a) and (c): booster amplifier operating at the power of 0.3 PW attenuated with 8 and 10 wedges, respectively; (b): 30 TW power amplifier.
Table 1. Focal spot parameters at the peak power of 0.3 PW.
Wavefront distortion directly affects the focal spot quality; distortion before the compressor also leads to spatiotemporal coupling. For high-frequency noise-like wavefront distortions with rms error σ, the Strehl ratio can be estimated as exp[-(ηπσ/λ)2], where η ~2 [25]; for our measurements, η ≈1.85 gave a good fit to data. This requires σ << 100 nm; as sigma grows, the low-intensity halo and satellite peaks around the main spot increase, decreasing the Strehl ratio. Low-order wavefront distortions must be minimized as well. Although in principle astigmatism and not perfect beam collimation (i.e. divergence) before the compressor can be compensated to a large extent after the compressor with the final off-axis parabolic (OAP) mirror, this would result in non-perfect pulse compression (spatiotemporal coupling) as different beamlets would have different incidence angles on the compressor gratings.
Similarly to the wavefront, spectral phase distortions should be minimized to achieve clean pulses with shortest duration.
Spatiotemporal couplings result in spot and pulse degradation and must also be avoided, especially with wide-bandwidth femtosecond lasers. The use of lenses must be minimized due to chromatic aberration and radial group delay (propagation time difference [26]). The angular dispersion should be smaller than the diffraction limit: CΔλ << λ/Ø, where C is the angular chirp [μrad/nm], Δλ is bandwidth and Ø is beam diameter. The angular chirp is especially harmful for femtosecond petawatt lasers due to their wide bandwidth and large beam diameter, which requires C << 0.1 μrad/nm, i.e. ~10 μrad alignment accuracy of all grating angles. With gratings in vacuum, this alignment is challenging.
We started from the spot shown in Fig. 1(a); the spot parameters are listed in Table 1 (left). This was not as good as in our previous experiments [27–30] with the J-KAREN [31] laser due to higher power and larger beam diameter [the J-KAREN-P laser, Fig. 1(a) case: lens-based expander up to Ø135 mm and final mirror-based expander up to Ø280 mm; the previous J-KAREN system: Ø40-50 mm and Ø150 mm, respectively].
3. J-KAREN-P beamline and alignment beams
To meet the requirements formulated in Section 2, we constructed the beamline serving as (i) the radiation-safe beam transport from the laser hall to the experimental hall through several rooms with a kink preventing radiation propagation from the experimental hall back to the laser hall; (ii) beam expander from 90 up to 280 mm; (iii) wavefront corrector employing a deformable mirror (DM), wavefront sensors, and adaptive loop; (iv) chirped pulse compressor reducing the pulse duration from ~1 ns down to ~30 fs, approaching the bandwidth limit; (v) beam delivery system to two independent target areas; (vi) monitor system for reproducible and time-efficient everyday beam alignment; (vii) several permanently available laser diode (LD) alignment beams for efficient work in parallel on the laser, beamline, and experimental areas; (viii) insertable attenuator for accurate measurement of full-power J-KAREN-P pulses with energy up to 60 J; pulses with energy of up to 20 J were used in this work. The beamline schematic is shown in Fig. 2.
Fig. 2 J-KAREN-P beamline after the final amplifier; the beamline is covered to reduce air fluctuations. M1 to M7: monitors recording the far and near fields of the beam transmitted through dielectric mirrors. DM: deformable mirror (ILAO) with 52 actuators. W1, W5, W6: wavefront sensors (HASO32). P: periscope changing the polarization from vertical to horizontal. × 1.5 and × 2.1: beam expanders consisting of convex and concave spherical mirrors, with the corresponding expansion ratios. Boxed numbers to : alignment CCDs observing (through dielectric mirrors) diffraction patterns from a reproducibly insertable small aperture, directly or with a small-diameter camera lens. OAP5 to OAP7: high-quality f = 2000 mm off-axis parabolic mirrors (f/7, 8° off-axis angle, Tydex) focusing the 280 mm beam for alignment and characterization. OAP8: in-vacuum f = 350 mm (f/1.3, 45° off-axis angle, Okamoto Optics) OAP focusing the beam on target. T1 and T2: two target areas with short-focus and long-focus optics, respectively. S5 to S8: focal spot monitors with high-quality Mitutoyo apochromatic objective lenses and low-noise 12 bit CCDs (Spiricon SP620U). LD1: Ø90 mm dual-wavelength (779 and 830 nm) laser diode retro-alignment beam injected into the main beamline with a reproducibly insertable mirror on a translation stage. LD7: Ø280 mm triple-wavelength (405, 640, and 785 nm) LD retro-alignment beam injected into the beamline through a dielectric mirror. The line thickness corresponds to the beam diameters of 90, 135, and 280 mm. The line colors correspond to the chirped ns pulse (light red), compressed femtosecond pulse (dark red), alignment LD beams (blue), and attenuation paths (orange) consisting of several reproducibly-insertable wedges with high surface quality confirmed with interferometry. The dashed lines denote beams transmitted through dielectric mirrors.
For accurate wavefront correction, which will be discussed in Section 5, the day-to-day beamline reproducibility must be at the 10 µrad level. This was challenging as the beam passed through several rooms and kinks, and the drift of many mirror holders was not negligible. Furthermore, the temperature sensitivity was ~50 µrad/K, and the existing ± 1 K stabilization was therefore insufficient. To achieve the required beamline reproducibility, we set up the monitors M1 to M7 (Fig. 2), consisting of near and far field images with the sensitivity from ~10 µrad down to a few µrad for the 90 and 280 mm beams, respectively. In addition, four alignment CCDs denoted by boxed numbers to served for precise alignment of beam through the final expander. We note that standard beam monitors consisting of focusing optics and near and far field cameras could not be used in these places due to the lack of space. For alignment, we used an insertable reproducible beam block with a 12 mm diameter hole. The diffraction patterns recorded by the cameras were very sensitive to any deviations of the beam from the reference alignment, so the deviations could be minimized. These alignment steps were typically performed several times a month; the alignment to monitors M1 to M7 was performed daily.
The performance of the alignment system allowed us to conduct experiments with minimum day to day spot and pulse variations.
For efficient beamline optimization and experimental setup preparation we constructed a retro-alignment system [32] consisting of dual-wavelength and triple-wavelength Laser Diode (LD) beams at the 90 mm and 280 mm diameter stages (Fig. 2). The majority of alignment was performed with these LD beams.
4. Chromatic aberration removal
We identified the chromatic aberration caused by six lens pairs used for image relay and beam up-collimation from 90 up to 135 mm as the main reason of the poor original focal spot quality [Fig. 1(a)]. Physical optics propagation calculation (Zemax software) showed that these lenses led to a Strehl ratio of 0.1. The radial group delay (propagation time difference [26]) in the lenses decreased the intensity even further by stretching the pulse. We re-built the beamline (Fig. 2) using mirror-based expanders with astigmatism compensation [33] to drastically reduce the chromatic effects. The incidence angles of the mirrors were calculated using the analytical expressions and optimized with Zemax. The remaining lenses at the initial stage of the beamline did not cause significant chromatic effects. Indeed, the largest radial group delay, ~10 fs edge to center, was caused by the FemtoPower expander. This resulted in negligible pulse shape elongation at focus, even for a flat-top spatial distribution. Assuming a 30 fs FWHM pulse and 15 mm flat-top beam diameter, the integrated pulse at focus would have ~30.7 fs FWHM duration and peak power of 0.98 of the original. The same expander also caused the largest residual transverse chromatic aberration. The physical optics propagation calculation showed that this resulted in a Strehl ratio of 0.9, i.e. much higher than our final value.
5. Wavefront correction
Our target was to obtain a high-quality 280 mm diameter beam before the compressor and use the subsequent low-distortion beamline to achieve simultaneously clean pulse compression and tight focusing.
Firstly, we measured with the Zygo interferometer every flat large mirror and compressor grating, all fixed in their holders. Mirrors with low surface quality were replaced. Distortions caused by holders were removed by optimizing the holding method and strength.
Secondly, we tuned the final beam expander (135 to 280 mm) to achieve a collimated beam without astigmatism. The beam collimation was checked with a shearing interferometer, while the astigmatism was measured with an f = 3000 mm lens and HASO32 wavefront sensor. The astigmatism-free beam was essential because at the next step we used an OAP for beam focusing, which could compensate for the astigmatism, if it existed, so it would pass unnoticed.
Next, we used a state-of-the-art adaptive loop consisting of a deformable mirror situated after the final amplifier (90 mm beam), two HASO32 wavefront sensors, and WaveTune software (ImagineOptic), to correct the wavefront distortions. The 1st wavefront sensor, W1 in Fig. 2, measured the wavefront of the beam transmitted through a dielectric mirror soon after the deformable mirror; an f = 2500 mm and f = 120 mm lenses down-collimated the beam and imaged the deformable mirror surface to the wavefront sensor.
The 2nd sensor, W5 in Fig. 2, was placed before the compressor. The 280 mm beam was focused by a high-quality f = 2000 mm alignment OAP (Tydex) and an f = −500 mm lens imaged the deformable mirror surface to the sensor; imaging provided better wavefront correction results compared to a non-imaging setup. The negative lens was selected because, in contrast to a positive lens, it caused negligible additional aberrations. The path to the 2nd sensor was opened by removing the mirror before the compressor. Thus it could not be used during the beam delivery to the target, but when open it measured the true wavefront. In contrast, the 1st sensor was usable any time. We note that the W6 sensor was also used for the wavefront measurement at an early stage of the work; however, at the final stage, when the compressor and the beamline after the compressor were in vacuum, it was impractical to remove the turning mirror after the compressor. We therefore made the final wavefront measurements with the W5 sensor situated in air.
The adaptive loop included a calibration, closed loop to obtain aberration-free spherical wavefront at the 2nd sensor (W5), and measurement of a reference wavefront at the 1st sensor. This reference wavefront was subsequently used for the local adaptive loop, including only the deformable mirror and the 1st wavefront sensor for which the calibration was performed separately. The local loop was used when the beam was delivered to the compressor or target chamber and the 2nd wavefront sensor was therefore not available.
After several attempts, we identified the following essential points. Firstly, the wavefront measurement noise must be minimized; we enclosed the beamline into boxes reducing air turbulence, and used up to 1000 frame averaging. Secondly, the beamline must be stable down to tens µrad level, which we achieved using the monitors M1 to M7 and CCDs to , as discussed in Section 3. This allowed the use of adaptive loop calibration and reference wavefront for many months. Without accurate alignment, the calibration of the adaptive loop quickly became unusable. We note that even the full alignment procedure including all monitors and cameras to was significantly faster than the adaptive loop re-calibration; the latter would also require the removal of the final mirror before the compressor to open the undistorted beamline to wavefront sensor W5. Thirdly, even state-of-the-art wavefront sensors gave errors at the low-irradiance beam edges; this gave pointing and divergence errors after the closed loop. We limited the wavefront correction area down to Ø90 mm at the W1 stage, neglecting a few mm periphery (note that the beam characterization was performed using the full aperture without this limitation). This reduced pointing errors down to tens µrad, which were corrected manually.
As a result, the rms wavefront error of the LD beam before the compressor reduced to 0.04 µm. The Strehl ratio calculated from the measured wavefront increased to 0.94. We checked with the shearing interferometer that the beam before the compressor was collimated.
After wavefront correction, we measured the focal spot before the compressor using the alignment OAP (without the f = −500 mm lens) and a high-quality Mitutoyo microscope objective lens. This gave the Strehl ratio of 0.78. In Appendix B we discuss the difference between the Strehl ratios obtained from the measured wavefront and focal spot.
All wavefront correction steps above were performed with the alignment LD beam, as discussed in Section 3.
Finally, we corrected the Ti:Sapphire beam aberrations using the 30 TW mode, without pumping the booster amplifier. We operated the local closed loop (deformable mirror – 1st sensor with the reference wavefront obtained with the alignment LD beam) and measured the Ti:Sapphire beam wavefront before the compressor (with the 2nd sensor). As the beam passed through focus before the sensor, the pulses were sufficiently attenuated after amplifiers to avoid nonlinear effects. The rms wavefront error decreased from 0.2 to 0.3 μm before the correction [Fig. 3(a)], down to 0.07 μm [Fig. 3(b)]. The Strehl ratio calculated from wavefront increased up to ~0.87.
Fig. 3 J-KAREN-P laser wavefront (a-c) and corresponding calculated spots (point-spread functions, PSFs) (d-f) at several positions; the values in the frames (a-c) show the rms wavefront errors, the PSF maxima in (d-f) equal the Strehl ratios calculated from the corresponding wavefronts. (a) Wavefront before the deformable mirror, beam Ø90 mm. (b) After the deformable mirror, beam Ø90 mm; the reference wavefront is subtracted, i.e. the data shown is the prediction of the wavefront which should be obtained before the compressor. (c) Before the compressor, beam diameter Ø280 mm, after the f/7 alignment OAP, rms = 0.07 μm, PV = 0.65 μm; ideally, this wavefront should be same as in (b); the small deviations from this shape, especially at the beam edges, are due to imperfect wavefront correction.
6. Wavefront and Strehl ratio evolution
We measured the wavefront of the alignment LD1 laser and the Ti:Sapphire J-KAREN-P beam at several beamline positions. Some of the data for the Ti:Sapphire beam are shown in Fig. 3; the data for the LD1 demonstrated similar trend with somewhat lower wavefront errors and higher Strehl ratios.
Figure 3(b) shows the wavefront after the DM with the reference wavefront subtracted, i.e. the data shown correspond to the predicted wavefront before the compressor. Figure 3(c) shows the measured wavefront before the compressor. It looked similar to the prediction, except some additional distortions at the beam edges, which led to an rms error increase from 0.04 μm to 0.07 μm and a calculated Strehl ratio decrease from 0.93 down to 0.87.
The Strehl ratios of the alignment LD1 and Ti:Sapphire beams, measured just after the deformable mirror, before the compressor, and on target, are shown in Fig. 4. The following points can be noted: (i) The Strehl ratio determined from the measured wavefront was always overestimated. (ii) The LD1 spot was always better, probably due to lower wavefront fluctuations and absence of chromatic aberration and angular chirp. (iii) Before the angular chirp compensation, the spot measured with narrow band-pass filter was notably better than the full-spectrum spot; this difference disappeared after the angular chirp compensation, see Section 7. (iv) There was a moderate degradation of the spot before the compressor compared to the original spot just after the DM, although the measured wavefront remained nearly identical; this was probably due to imperfect match between the adaptive loop DM – W5 and local loop DM – W1, i.e. imperfect reference wavefront, and gradual increase of the wavefront fluctuations as the beam propagated through the beamline. (v) The LD spot on target was slightly degraded compared to the one before the compressor, probably due to mirror and grating imperfections which could not be completely avoided.
Fig. 4 Strehl ratios measured at different beamline positions. LD is the CW alignment beam LD1, 30 TW is the 10 Hz Ti:Sapphire alignment beam (power amplifier), 0.3 PW is the 0.1 Hz main Ti:Sapphire beam (booster amplifier), BP 800 is Ti:Sapphire beam measured with a 10 nm band-pass filter in front of CCD (all other Ti:Sapphire data points are obtained with the full bandwidth). Data labeled 'Wavefront' (open symbols) correspond to calculation of the Strehl ratio from wavefront measurements; Data labeled 'Spot' (filled symbols) correspond to the Strehl ratio obtained from the measured focal spot distributions (see Appendix B “Note on the Strehl Ratio Calculation”). For the Strehl ratio obtained from spot, the filled circles with error bars denote average and standard deviation, while the triangles denote maximum Strehl ratio achieved in the corresponding data series. The horizontal black dashed line corresponds to the Strehl ratio level before chromatic aberration removal.
7. Angular chirp removal
We measured and corrected the angular chirp at both stages of our double CPA system. The angular chirp after the 1st CPA (FemtoPower) was measured with a device [34] consisting of an etalon transmitting three isolated wavelengths within the laser spectrum, diffraction grating, and objective lens. The angular chirp of ~1.6 µrad/nm resulted in an estimated Strehl ratio of 0.86 (Ref [23].). After the FemtoPower compressor realignment, the angular chirp was reduced down to 0.9 µrad/nm, which gave the estimated Strehl ratio of 0.94.
Secondly, we performed a two-step compressor alignment with the final accuracy of ~10 μrad. During the first step, all three angles of each of the four gratings were aligned in air with the accuracy of several tens of μrad. Several methods to do this exist [3]. We developed a method based on autocollimation with an absolute vertical reference, which will be described elsewhere. Pumping down the compressor caused additional misalignment, resulting in spot elongation [Fig. 1(b)]. At first glance, the angular chirp effect on the spot looked similar to astigmatism, however, this elongation was not observed with a narrow band-pass filter, and it could not be removed by an OAP adjustment (we note that it is possible to make spot to appear more round, but that would lead to an even larger spot and lower irradiance).
In the second in-vacuum step we adjusted the last grating rotation around its vertical axis and groove tilt to optimize the focal spot [35] (Fig. 5). The angular chirp due to some grating misalignment was compensated by the last grating, which caused some imperfections in the time domain; however, due to the small adjustment values, ~50 and 60 µrad, the estimated worst-case elongation was ~1 fs. The spot measurement with a band-pass filter confirmed absence of the chromatic aberration and angular dispersion after this procedure. We note that a high initial spot quality (before the in-vacuum grating adjustment) was essential. Indeed, if the wavefront was not corrected well, the angular chirp effect would not be clearly noticeable; on the other hand, if the initial angular chirp was significant, the final grating correction would be large and therefore cause significant time-domain effects.
Fig. 5 Angular chirp compensation by fine tuning of the last compressor grating. (a) Grating rotation about its vertical axis. (b) Grooves tilt. The squares with error bars show 10-shot average with std. deviations, the circles 10-shot maxima, the lines model [23], see Appendix C. The insets show measured spots with misaligned compressor.
8. Pulse compression
We performed pulse compression in three steps. Firstly, the grating separation was optimized to maximize the white light generation in a glass block, after which the pulse shape was measured with a home-made Transient Grating FROG [36], resulting in a 72 fs FWHM pulse with a 5 ps tail. The measured spectral phase was used for incidence angle and grating separation correction, resulting in τFWHM = 28 fs, τeff = 40 fs. Secondly, we used commercial self-referenced spectral interferometry (Wizzler) for measurement and feedback to an acousto-optic programmable dispersive filter (Dazzler). Note that before the feedback loop of the Wizzler was engaged it was essential to remove the ps tail which was outside of the Wizzler measurement range. The feedback gave a clean pulse with τFWHM = 29.1 ± 0.7 fs and effective duration τeff = 34.4 ± 1.1 fs (Fig. 6). These were close to the bandwidth limit, 28 and 31 fs, respectively.
Fig. 6 J-KAREN-P pulse (Wizzler, 156 single shots). (a) Normalized laser power, gray area shows shot-to-shot standard deviation, dashed line shows bandwidth-limited pulse. (b) Effective and FWHM pulse width history; horizontal lines show the bandwidth limits.
During the first two pulse compression steps we measured the pulse duration in the experimental chamber using a few mm diameter sample of the beam operating at 30 TW and attenuated after amplification with wedges. Due to the negligible spatiotemporal coupling which we confirmed by spot measurements at both CPA stages of the laser system, the sample gave adequate representation of the whole beam. Although during this measurement the booster amplifier was not pumped, the difference between the 30 TW and 0.3 PW modes was not significant because we made the beam diameter at all amplification and propagation stages large enough to suppress the nonlinear effects. We proved that these effects were indeed small by comparison of the focal spots obtained at the 30 TW and 0.3 PW modes, see the next section. The effect of the nonlinearity on the spectral phase was even smaller. The spectral bandwidth was slightly narrower for the 0.3 PW mode due to the red-shift during amplification; using the measured spectral phase and the spectra of the 30 TW and 0.3 PW modes, we calculated the pulse shapes, which were almost identical, with an effective width difference of ~1 fs or less.
Thirdly, to confirm the negligible influence of the spatiotemporal effects on the pulse duration and shape, we down-collimated the full beam in the long-focus target area (T2 in Fig. 2) using an f = 2600 mm and f = 25.4 mm OAPs and measured ~80% of the beam with the Wizzler, which showed nearly identical effective pulse duration and structure of satellite pulses.
During all pulse duration measurements the dispersion of 2 mm thick window and air (~4.3 m for TG FROG, ~1.3 m for Wizzler) was subtracted. In the case of TG FROG it was done manually by post-processing; in the case of Wizzler it was done automatically using the software option.
The measurements presented here were performed before the experiments. To ensure that the pulse duration was the same during the experimental period, we checked that the pulse shape was reproducible day to day over a few months period.
9. Resulting focal spot and irradiance
We measured the on-target focal spot attained after the f = 350 mm (f/1.3) off-axis parabola (Okamoto Optics) using a high-quality carefully-aligned apochromatic Mitutoyo objective lens which accepted the whole beam [Fig. 1(c) and Table 1]. The 0.1 Hz, 0.3 PW beam after amplification in the booster amplifier was supplied to the target area under the usual experimental conditions (compressor and femtosecond beamline in vacuum). The beam was attenuated with 10 high surface quality wedges [10-20 nm rms surface error measured (in holders) with the Zygo interferometer]. The resulting effective spot radius was 1.0 μm, the Strehl ratio and peak irradiance were 0.46 ± 0.06 and (0.93 ± 0.12) × 1022 W/cm2, exceeding 0.5 and 1022 W/cm2 in some shots. Due to insignificant spatiotemporal coupling, we calculated the irradiance by separating the temporal and spatial domains. The FWHM and FW1/e2 spot sizes were close to the diffraction-limited values; the main reason for the Strehl ratio degradation down to ~0.5 was low-intensity halo containing ~40% energy, as shown in Fig. 1(c). We note that our Strehl ratio calculation method using (4) takes this halo into account. We also note that without pumping the final amplifier, at the power level of 30 TW, the measured Strehl ratio was slightly higher, 0.54 ± 0.05, which confirmed that the nonlinear effects were relatively small.
10. Discussion and outlook
We plan to further increase the irradiance with an additional 0.1 Hz, 80 mm diameter amplifier with a 120 mm diameter Ti:Sapphire crystal, which, according to the calculations, will produce 60 J pulses [15]. The large-diameter beamline presented here was designed to accommodate this energy. In addition, there is room for higher quality wavefront correction. Finally, an OAP with smaller f-number can be used, which can be as small as 0.6 [4] or even 0.4 (plasma-mirror) [37]. Thus more than an order of magnitude irradiance enhancement is within reach.
The methods described here are useful for present and especially next-generation petawatt lasers with even higher powers [38, 39] and for focusing and characterization of high-quality x-ray pulses [40].
11. Conclusion
We reported on the recent reconstruction of the J-KAREN-P beamline, removal of the chromatic aberration and angular chirp, wavefront correction, and pulse compression. We approached the diffraction limit with a Strehl ratio of ~0.5, and effective duration just 10% larger than the bandwidth limit. We were also not far from a λ2 spot, with a FWHM spot width of 1.3 μm and effective radius of 1.0 μm, i.e. ~1.3λ. We achieved an irradiance of ~1022 W/cm2 during 0.3 PW operation, which is the highest accurately estimated irradiance demonstrated to date.
Appendix A List of mistakes leading to wrong irradiance estimates
Here we list up typical mistakes leading to irradiance overestimates; note that often the resulting overestimate can reach an order of magnitude or more.
Issue A1): Too small beam diameter resulting in large nonlinearities preventing correct measurements and even preventing full power operation; thus, the wavefront, spot, and pulse can be measured at low power but they are totally different at high power, or high power shots are impossible at all. It is surprising how common this mistake is. Our beam diameters at all the amplifiers and the final beam diameter (280 mm) were chosen to avoid this.
Issue A2): The beamline is often not reproducible day to day. Thus, measurements of the spot and pulse a day before have no relation to experiments. We avoided this by using sufficiently accurate alignment system; that allowed us to get reproducible results during a several-month-long experimental campaign which will be described elsewhere.
Issue A3): The spatiotemporal couplings are rarely avoided. We removed them by separately tuning the CPA1 and CPA2 parts of our laser so that the residual errors were negligible: ~ 1 fs estimated pulse elongation and negligible spot degradation (we additionally measured the spot with narrow band-pass filter and found no difference).
Issue B1): The spot is typically measured at low power mode, which always gives overestimated quality, with the difference increasing quickly with degree of system nonlinearity, i.e. especially severe with small diameter beams, see A1). The equivalent plane imaging through mirrors is also subject to errors. In fact, we are not aware of other groups which are using ten wedges with high (measured) surface quality to measure the real spot at the full power operation.
Issue B2): Measurement of the spot with insufficient resolution, using low dynamic range CCD, and choosing camera settings so that the low-intensity signal disappears. These measurements give plain zero instead of the low-intensity halo which causes significant irradiance overestimate. For example, it would give two times higher irradiance for our final spot, Fig. 1(c). We avoided this mistake by correct selection of low-noise CCD and correct background subtraction procedure (Spiricon Ultracal).
Issue B3): Inadequate optical system for the spot measurement. This includes an objective lens with insufficient numerical aperture, or with significant chromatic aberration, or inaccurately aligned; this also includes narrow bandpass filters in the measurement line which hide chromatic aberration and angular chirp effects, which in turn leads to spot quality overestimate. We avoided these mistakes by using carefully-aligned high-quality high-NA objective lens and by not using bandpass filters during the final spot measurement; in contrast, we temporarily inserted a bandpass filter to detect if any chromatic effects were present in the beam.
Issue C1): Using the following “equations” for the peak power and irradiance: P0 = E0/τFWHM and I0 = P0/(XFWHM×YFWHM). We demonstrated that these are wrong as they always overestimate the power and irradiance. We suggested a method free from this mistake, and used it through the manuscript.
Issue C2): Calculation of the PSF from wavefront data and using the result instead of the focal spot. It is surprising how often this is used. The PSF does not include the chromatic effects, angular chirp, and low-intensity halo around the spot which contains significant energy, especially in high power mode as significant part of the halo results from wavefront distortions due to nonlinearities.
Appendix B Note on the Strehl ratio calculation
The Strehl ratio calculated from measured wavefront was rather high. For example, for the LD before the compressor the calculation gave the Strehl ratio of 0.94. However, this was an overestimate caused by the sampling of the wavefront sensor, absence or inaccuracy of wavefront data at the beam edges, and rapid wavefront fluctuations smoothed by the averaging. The Strehl ratio calculated from the measured spot, the method see below, was 0.78±0.02 (10 frame average ± std. deviation). To give another example, during the intermediate stage, before the final wavefront correction, the calculations from the wavefront (W6 sensor) gave the Strehl ratios of 0.74 (wavefront average of 200 frames) and 0.63±0.06 (1655 single frames), while calculation from the spot gave the Strehl ratio of 0.48±0.03 (1000 single frames); note that the central part of the focal spot was calculated correctly, and it was the low-intensity halo that was not predicted from the wavefront measurements. In addition to this, for the Ti:Sapphire beam, the wavefront-based Strehl ratio did not include chromatic aberration nor angular chirp effects.
We therefore calculated the Strehl ratio using (5), i.e. as the ratio of peak fluences (4) of the measured and diffraction-limited spots for the same pulse power, which was equivalent to the reverse ratio of effective spot areas (3). The diffraction-limited focal spot was calculated as the Fourier transform of the square root of the measured near field distribution. This required high signal-to-noise ratio as the square root significantly increased the noise level.
Note that calculation of both irradiances formally included integration up to infinite limits, in (4) and during the Fourier transform. In practice we clipped the integration area, ensuring that the fluence was zero at the area edges within the measurement accuracy (except the original spot shown in Fig. 1(a) where the chromatic aberration caused low-intensity halo extending up to the CCD edges). The effect of the dark, readout, and shot noise of the CCD caused estimated ~3% Strehl ratio error. All these effects caused an estimated combined uncertainty of the Strehl ratio of below 5-10%. In addition, there were random shot-to-shot focal spot shape fluctuations; standard deviations of Strehl ratio values caused by these fluctuations are shown in this paper as the error bars.
Appendix C Modeling the effect of angular chirp
To estimate the Strehl ratio in the presence of the angular chirp, we used the approach from [23]. For a beam free from wavefront distortions (dashed lines in Fig. 5), the Strehl ratio was calculated as
where C [μrad/nm] is the angular chirp, k = (π/2 ln 2)ØΔλ/λ, Ø is the beam diameter, Δλ is the FWHM bandwidth, and λ is the wavelength. The horizontal and vertical angular chirps due to the grating misalignment were calculated as i.e. half of the corresponding expressions from [23], because we used a single-pass, four-grating compressor rather than a folded configuration. Here εx and εg are grating misalignments, i.e. horizontal rotation and groove rotation angles, N = 1480/mm is the grating groove density, α = 49.6° is the beam incidence angle, and β0 = 25.5° is the diffraction angle corresponding to the central wavelength λ = 805 nm.For the case with wavefront distortion, solid lines in Fig. 5, we extended the model and used the equation
where S0 is the Strehl ratio without the angular chirp. For a beam without wavefront distortion, S0 = 1 and (9) coincides with (6). For a significant angular chirp, kC >> 1/S0, (9) gives nearly the same results as (6), as shown in Fig. 5. For zero angular chirp, (9) gives S = S0. The model gave a reasonably good fit to both experimental data sets [Figs. 5(a) and 5(b)], with only one fitting parameter for each curve, i.e. the optimum grating angle.Funding
Japanese Ministry of Education, Culture, Sports, Science and Technology (Supplemental budget, C-PhoST); Japan Society for the Promotion of Science (JSPS) (KAKENHI JP 26707031, 26247100, 16K05506, 15F15772); Japan Science and Technology Agency (PRESTO JPMJPR16P9).
Acknowledgments
We thank the J-KAREN-P team, K. Sekiguchi, A. Kon, R. Rungsawang, I. Stefanon, M. A. Alkhimova for contribution, and H. Daido, D. Neely, C. G. R. Geddes, J. Fuchs, T. Zh. Esirkepov, J. K. Koga, A. Ya. Faenov, T. A. Pikuz, A. A. Soloviev, I. W. Choi, S. K. Lee, and X. Levecq for discussions.
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