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Abstract
In an ultra-intense femtosecond chirped-pulse amplification laser, the imperfect diffraction wave-fronts of the second and the third gratings of the compressor, where spatio-spectral coupling exists, could introduce a complex spatiotemporal coupling distortion (STCD) and degrade the pulsed beam in both near- and far-fields. Here, we propose a method of double-compressors for pre-compensation. By inserting a scaled down compressor (small compressor) with a deformable retro-reflection mirror into the beam-line, the frequency-dependent wave-front distortion, i.e., the complex STCD, could be removed. We simulate the results in two different ultra-intense femtosecond lasers with 80 and 400 nm bandwidths for comparison, and near ideal focused peak intensities could be obtained in both cases. Meanwhile, the influences of several miss-matching effects, which might appear in engineering, are also analyzed and discussed for applications.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The ultra-intense femtosecond laser right now becomes a powerful tool to explore unknowns in the fields of high-field physics, astrophysics, material science etc [1,2], especially after the invention of the chirped pulse amplification (CPA) [3] and the optical parametric CPA (OPCPA) [4]. Since the first Petawatt-class (PW, 1015 W) femtosecond laser of 0.85 PW and 33 fs obtained in Japan in 2003 [5], two 10 PW-class femtosecond lasers are achieved in Europe and China in 2019, respectively [6,7]. In near future, several 100 PW-class femtosecond lasers are going to be constructed worldwide [8,9]. However, in a PW-class femtosecond laser, because of the large-aperture beam (e.g., 200 to 500 mm) and the ultra-short pulse (e.g., 10 to 30 fs), the space-dependent complex amplitude distortion (i.e., STCD) cannot be avoided completely [10–15]. In engineering, because large amounts of transmission optics (e.g., lens) and dispersion optics (e.g., grating or prism) are used, STCD could be easily introduced during the processes of energy/power amplification, beam propagation, pulse shaping, etc [14–17]. The STCD in an ultra-intense femtosecond laser has already been experimentally observed in both near- and far-fields (i.e., the temporally compressed pulsed beam after the compressor and the spatially focused pulsed beam at the focal spot) [13,18–27]. The most general forms of STCD are pulse-front tilt and curvature [28–32], which can be introduced by a prism and a lens respectively due to the difference of group and phase velocities in transmission mediums [11,16]. For the pulse-front tilt and curvature, the pulse-front is distorted in space, while the space-dependent pulse duration almost remains unchanged. In this case, such STCD could be easily removed by some conjugate designs or adaptive optics elements [33,34].
Recently, a complex STCD induced by the imperfect diffraction wave-front of the second and/or third gratings in a four-grating compressor, where spatio-spectral coupling exists, was found in theory [14,15] and characterized in experiment [18]. Comparing with the pulse-front distortion, this complex STCD is more complex and then difficult to remove. For the intensity distribution, apart from a space-dependent pulse-front distortion, a space-dependent pulse-profile distortion could also be found. For the complex amplitude distribution, a space-dependent phase distortion, for example space-dependent group-delay, group-delay dispersion and high-order dispersion in the space-spectrum domain, would be produced. Because the influence of the amplitude spatial non-uniformity, comparing with that of the phase spatial non-uniformity, is weak, we neglect it in this paper. Consequently, according to the description, it can be found that this complex STCD cannot be simply removed by recently available optics/methods only in space or time or spectrum domain.
In this paper, we proposed a simple method of double-compressors (i.e., main compressor and small compressor), which could pre-compensate this complex STCD. A deformable mirror is positioned at the spatio-spectral coupling plane of the small compressor to produce a conjugate wave-front to match the diffraction wave-front of the second and the third gratings in the main compressor. While comparing with a traditional grating-based 4-f pulse shaper [35,36], the modulation in this method is at the spatio–spectral coupling plane (i.e., not at the Fourier plane), where spatial and spectral properties are coupled. The simulation shows that, no matter for a recently popular ~80 nm bandwidth or a future ~400 nm bandwidth ultra-intense femtosecond laser, this method works very well and the majority amount of the complex STCD (and it induced degradation) could be pre-compensated. Besides that, for engineering application, several potential miss-matching problems are analyzed, as well.
2. Complex STCD and pre-compensation
For a CPA laser, the dispersion/spectral-phase management determines if the shortest/Fourier-transform-limit pulse could be reconstructed at the output or not. In the ideal case, the compressor should completely cancel the dispersion introduced by the stretcher and dispersion materials.
However, in an ultra-intense femtosecond laser, as shown in Fig. 1, the beam aperture at the compressor is expanded to sub-meter for power/energy scaling, and the spatial non-uniformity here cannot be neglected anymore. At the second and the third gratings (G2 and G3) of the compressor, the large-aperture beam spectrally separates in space, and then spatio-spectral coupling appears. The diffraction wave-fronts of the second and the third gratings (G2 and G3) would introduce a spectrum-dependent wave-front distortion. The problem is that the recent available spectral phase control device usually has no spatial resolution [37–39], and then it is impossible to obtain dispersion-free at every spatial position across the beam aperture.
Fig. 1 Schematic of an ultra-intense CPA laser with double-compressors for complex STCD pre-compensation. The 2-dimensional case for a selected y-axis position is shown.
Our method is inserting a scaled down compressor (a small compressor compared with the main compressor) with a deformable retro-reflection mirror into the beam-line. In the small compressor, the deformable mirror is positioned after the second grating (i.e., the middle of two-passes or at the spatio-spectral coupling plane). If the wave-front generated by the deformable mirror in the small compressor could compensate the wave-front introduced by the second and the third gratings (G2 & G3) in the main compressor, in theory the complex STCD could be perfectly removed. To describe this method in detail, here we define a parameter ε, the ratio of the angular dispersion induced spectral separation LΔλ in the spatio-spectral coupling plane (i.e., between the second and the third gratings G2 & G3) and the input beam diameter D (see Fig. 1), which quantitatively determines the spatio-spectral coupling.
where, βr, βb and β0 are the diffraction angles of the lowest frequency (red light), the highest frequency (blue light) and the center frequency ω0 at the first grating G1, G is the perpendicular distance of each parallel grating pair, α is the incident angle at the first grating G1, d is the grating constant, and ΔT/Δλ is the total chirp rate of the compressor (i.e., the ratio of the duration change ΔT and the bandwidth Δλ of the corresponding chirped pulse). When the small compressor and the main compressor have a same parameter ε, they would possess a same spatio-spectral coupling. If the generated wave-front by the deformable mirror in the small compressor and the overlap diffraction wave-front of the second and the third gratings (G2 & G3) in the main compressor are conjugate, the complex STCD would be perfectly pre-compensated. Figure 1 shows that we can measure the overlap diffraction wave-front of the second and the third gratings (G2 & G3) in same geometry as used in the main compressor in advance, which is then used as the input of the deformable mirror control.To satisfy the matching of the small and the main compressors, in the following sections, two compressors are designed with same grating constant d, same incident angle α, and same parameter ε (i.e., same G/D with same d and α). The chirp ratio ΔT/Δλ of the small compressor is chosen as 1% of that of the main compressor, and consequently, comparing with the main compressor, the small compressor is scaled down 100 times in size. And a perfect image relay system is assumed to image the pre-compensation in the small compressor into the main compressor to reduce the influence of propagation diffraction.
3. Simulation results
3.1 In an 80 nm bandwidth laser
In order to verify the performance of the method, we firstly simulate it in an 80 nm bandwidth ultra-intense femtosecond laser, which actually is based on the recently popular PW-class femtosecond lasers with Ti:sapphire and/or OPCPA amplification mechanism. The center wavelength is 800 nm, the frequency-spectrum is 6-order super-Gaussian with an 80 nm bandwidth, the chirp ratios of the main and the small compressors are 2 ns/80 nm and 20 ps/80 nm (i.e., that of the stretcher is 2.02 ns/80 nm), the input beams at the main and the small compressors are 6-order super-Gaussian with a 400 mm and a 4 mm diameters, the grating density is 1480 g/mm, the incident angle is 46.3°, the grating pair perpendicular distance of the main and the small compressors are 1500 mm and 15 mm, and the parameter ε is 0.44. The detailed parameters in the simulation are given in Table 1.
Table 1. Simulation parameters of double-compressors in an 80 nm bandwidth laser.
Refer to the measured diffraction wave-front of a meter-sized grating in our lab (see Fig. 1), we construct an overlap diffraction wave-front of the second and the third gratings (G2 & G3) as shown by the blue curve in Fig. 2(a), which contains both slowly varying deformation and high-frequency random noises. This wave-front will be used as the input in the following simulation.
Fig. 2 (a) In the spatio-spectral coupling plane, second and third gratings induced wave-front distortion (blue curve) in main compressor, deformable mirror generated pre-compensation (red curve) in small compressor, and angular dispersion induced spectrum-dependent beam separation. (b) (i-iii) Without and (iv-vi) with pre-compensation, (i) (iv) electric-field and (ii) (v) intensity distributions in the near-field, and (iii) (vi) spectral phases at x = 100, 0, and −100 mm.
In engineering, the measurement of the overlap diffraction wave-front of the second and the third gratings (G2 & G3) is like this (see Fig. 1): gratings are positioned in the same geometry as used in the main compressor; a monochromatic light with the center wavelength of the signal experiences diffractions by two gratings; and then the wave-front distortion is measured and imaged onto the spatio-spectral coupling plane (multiplied by cosα/cosβ0).
According to the pre-measurement, the deformable mirror in the small compressor generates a conjugate wave-front for pre-compensation, as shown by the red curve in Fig. 2(a). Because the inter-actuator space of the deformable mirror here is 200 μm, the generated wave-front only contains the slowly varying deformation, and the high-frequency random noises are not included. The small compressor here is 100 times smaller than the main compressor, and then the aperture of the pre-compensation wave-front is also 100 times smaller than that of the distortion wave-front in the main compressor. Figure 2(a) gives the spectrum-dependent beam separation in the spatio-spectral coupling plane, which is almost same in both small and main compressors due to a same parameter ε and perfect image relay. Here, perfect image relay is reasonable, because the low-frequency deformation produced in the small compressor could pass spatial filters and arrive at the main compressor.
If we neglect the spectrum-independent wave-front (i.e., space-dependent only), which is not the topic studied in this paper and easy to be removed by deformable mirrors, Figs. 2(b)(i) and 2(b)(ii) show the simulated electric-field and intensity distributions after the main compressor. The detailed simulation method can be found in our previous works [14,15]. For observation, the frequency of the carrier wave has been multiplied by 0.3 to avoid fast oscillations of the electric-field in time. It can be found that both the electric-field and the intensity distributions distort in space and time, which is due to the space-dependent spectral-phase distortion (i.e., the complex STCD). Figure 2(b)(iii) shows, even if a traditional spectral phase control is used before/after the stretcher (see Fig. 1), only the on-axis position of x = 0 possesses dispersion-free, and the other positions, e.g., x = 100 and −100 mm, still possess residual spectral phase distortions. However, when the pre-compensation in the small compressor is added and imaged into the main compressor, Figs. 2(b)(iv)-2(b)(vi) show that near distortion-free electric-field distribution, intensity distribution and spectral-phase could be achieved at every spatial position. The tiny spectral phase distortion in Fig. 2(b)(vi) is due to the un-compensated high-frequency random noises of the overlap diffraction wave-front of the second and the third gratings, whose influence is negligible.
Figure 3 shows the result when the near-field pulsed beam given in Fig. 2(b) is focused onto the focal spot by an ideal focusing optics with a 2 m focal length. Figures 3(a) and 3(b) show the electric-field and the intensity distributions corresponding to the results in Figs. 2(b)(i) and 2(b)(ii), and electric-field distortion and peak intensity decreasing can be found. When the pre-compensation in the small compressor is added, comparing with Figs. 3(a) and 3(b), Figs. 3(d) and 3(e) show that both the electric-field and the intensity distributions in the far-field are improved a lot. And, Figs. 3(c) and 3(f) show the pulses at ξ = 0 and the focal spots at t = 0 without and with the pre-compensation, and the peak intensity in the far-field could be improved from around 0.69 to around 0.98.
Fig. 3 (a) (b) Without and (d) (e) with pre-compensation, (a) (d) electric-field and (b) (e) intensity distributions in the far-field, (c) pulses at ξ = 0, and (f) spots at t = 0 without and with pre-compensation.
3.2 In a 400 nm bandwidth laser
Recently, some few-optical-cycle or single-optical-cycle intense lasers and methods are developed, which might be an efficient way for the peak power/intensity further scaling [40–43]. In this case, we secondly simulate the above case in a 400 nm ultra-broadband laser. The center wavelength is 1000 nm, the frequency-spectrum is 6-order super-Gaussian with an around 400 nm bandwidth, the chirp ratios of the main and the small compressors are 2 ns/400 nm and 20 ps/400 nm (i.e., that of the stretcher is 2.02 ns/400 nm), the input beams at the main and the small compressors are 6-order super-Gaussian with a 400 mm and a 4 mm diameters, the grating density is 800 g/mm, the incident angle is 40°, the grating pair perpendicular distance of the main and the small compressors are 1130 mm and 11.3 mm, and the parameter ε is 0.75. The detailed parameters are given in Table 2.
Table 2. Simulation parameters of double-compressors in a 400 nm bandwidth laser.
Figure 4(a) shows the spectrum-dependent beam separation in the spatio-spectral coupling plane is increased due to an ultra-broadband spectrum. We still neglect the spectrum-independent wave-front, and Figs. 4(b)(i) and 2(b)(ii) show the simulated electric-field and intensity distributions after the main compressor. Both the electric-field and the intensity distributions possess distortions in space and time. Figure 4(b)(iii) shows that the spectral phases at the off-axis positions of x = 100 and −100 mm possess fast oscillations due to the increasing of the parameter ε. When the pre-compensation is added in the small compressor, Figs. 4(b)(iv)-4(b)(vi) show that near distortion-free electric-field distribution, intensity distribution and spectral-phase could also be achieved at each spatial position. Figure 4(c) gives the results when the near-field results given in Fig. 4(b) are focused onto the focal spot by an ideal focusing optics with a 2 m focal length. Figures 4(c)(i) and 4(c)(ii) show the far-field electric-field and intensity distributions corresponding to the near-field results in Figs. 4(b)(i) and 2(b)(ii), and electric-field distortion and peak intensity decreasing can be found. When the pre-compensation in the small compressor is added, comparing with Figs. 4(c)(i) and 4(c)(ii), Figs. 4(c)(iv) and 4(c)(v) show that near distortion-free electric-field and intensity distributions appear. Figures 4(c)(iii) and 4(c)(vi) give the pulses at ξ = 0 and the focal spots at t = 0 without and with the pre-compensation, and, similarly with the result in the section 3.1, the peak intensity is also improved from around 0.69 to around 0.98. And fortunately, no sensitivity increasing is found in an ultra-broadband laser.
Fig. 4 (a) In the spatio-spectral coupling plane, second and third gratings induced wave-front distortion (blue curve) in main compressor, deformable mirror generated pre-compensation (red curve) in small compressor, and angular dispersion induced spectrum-dependent beam separation. (b) In the near-field, (i-iii) without and (iv-vi) with pre-compensation, (i) (iv) electric-field and (ii) (v) intensity distributions, and (iii) (vi) spectral phases at x = 100, 0, and −100 mm. (c) In the far-field, (i) (ii) without and (iv) (v) with pre-compensation, (i) (iv) electric-field and (ii) (v) intensity distributions, (iii) pulses at ξ = 0, and (vi) spots at t = 0 without and with pre-compensation
4. Sensitivities to compensation miss-matching
To promote the application of this method, in this section, we will analyze several possible miss-matching problems, which may occur in engineering.
4.1 Miss-matching of the parameter ε
From the above, the parameter ε is a key factor which determines the spatio-spectral coupling and accordingly the pre-compensation of the complex STCD. Equation (1) shows that the measurement error of the input beam diameter D at the small compressor would influence the parameter ε. Unfortunately, at the small compressor, because the beam diameter is small, the accurate measurement of the beam diameter is usually challenged. For a fixed angular dispersion induced spectral separation LΔλ (i.e., a fixed perpendicular distance G) in the small compressor, when the beam diameter D is zooming, the aperture of the pre-compensated wave-front by the deformable mirror in the small and that in the main compressors can be considered to be zooming [see Fig. 5(a)].
Fig. 5 (a) When the aperture of the pre-compensated wave-front produced by the deformable mirror is zooming, the focused peak intensities If as functions of the zooming coefficient p1 for (b) the 80 nm and (c) the 400 nm bandwidth lasers without and with the pre-compensation.
In the 80 nm bandwidth laser, Fig. 5(b) shows, even with the pre-compensation, the focused peak intensity in the far-field decreases with increasing or reducing the zooming coefficient p1. When p1 is larger than 1.25 or smaller than 0.75 (i.e., > 25% change), the pre-compensation effect disappears. If further increasing or reducing p1, the pre-compensation would become superimposed distortion, and an even lower focused peak intensity than that without the pre-compensation would appear. In the 400 nm bandwidth laser, Fig. 5(c) shows that the situation is similar, but the sensitivity to the zooming coefficient p1 is slightly increased.
Similarly, for a fixed input beam diameter D (i.e., a fixed aperture of the pre-compensated wave-front by the deformable mirror) in the small compressor, because of some alignment errors, when the angular dispersion induced spectral separation LΔλ (e.g., the parallel grating pair perpendicular distance G) is zooming, the parameter ε of the small compressor will also deviate from that of the main compressor [see Fig. 6(a)]. In the 80 nm bandwidth laser, Fig. 6(b) shows that the focused peak intensity in the far-field with the pre-compensation decreases with increasing or reducing the zooming coefficient p2. And, like the result in Fig. 5, in the 400 nm bandwidth laser, Fig. 6(c) shows, the sensitivity to the zooming coefficient p2 is also slightly increased. However, comparing with the case in Figs. 5(b) and 5(c), the sensitivity in the Figs. 6(b) and 6(c) is lower, which means the sensitivity to the zooming of the angular dispersion induced spectral separation LΔλ is slightly lower than that of the input beam diameter D.
Fig. 6 (a) When the angular dispersion induced spectral separation LΔλ is zooming, the focused peak intensities If as functions of the zooming coefficient p2 for (b) the 80 nm and (c) the 400 nm bandwidth lasers without and with the pre-compensation.
Consequently, we can found that: firstly, in the 80 nm bandwidth laser, the sensitivity of the pre-compensation to the miss-matching of the input beam diameter D is slightly higher than that to the miss-matching of the angular dispersion induced spectral separation LΔλ; and secondly, the sensitivity to the miss-matching of both the input beam diameter D and the angular dispersion induced spectral separation LΔλ is slightly higher in the 400 nm bandwidth laser than in the 80 nm bandwidth laser.
4.2 Miss-matching of the deformable mirror’s transverse location
Another miss-matching problem is the offsets of the deformable mirror’s transverse location, which might be frequently found in engineering. Figure 7(a) shows when the deformable mirror (or it produced pre-compensation wave-front) has offsets, the perfect compensation condition cannot be obtained. In the 80 nm bandwidth laser, Fig. 7(b) shows, the focused peak intensity in the far-field with the pre-compensation decreases with increasing the offsets. When the amount of offsets is larger than ± 5% of the input beam diameter D, the pre-compensation effect disappears and the superimposed distortion begins to appear. Comparing with Fig. 7(b), Fig. 7(c) shows, in the 400 nm bandwidth laser, the sensitivity to the offsets is slightly reduced. It is because that the parameter ε (i.e., the angular dispersion induced spectral separation LΔλ) in the 400 nm bandwidth laser of 0.75 is larger than that in the 80 nm bandwidth laser of 0.44.
Fig. 7 (a) When the deformable mirror has transverse offsets, the focused peak intensities If as functions of the offsets for (b) the 80 nm and (c) the 400 nm bandwidth lasers without and with the pre-compensation. D is the beam diameter.
While comparing with the results in the section 4.1, the sensitivity to the miss-matching of the deformable mirror’s transverse location (offsets) is much higher than that of the parameter ε. Then, the offsets should be controlled less than ± 5% of the input beam diameter D. In this case, to increase the control accuracy, the beam diameter, and accordingly the overall size, of the small compressor could be slightly increased for detailed engineering consideration.
4.3 Miss-matching of the image relay
The above simulation is based on perfect image relay, i.e., the pre-compensated wave-front at the spatio-spectral coupling plane in the small compressor is perfectly imaged at that in the main compressor, and the propagation diffraction is neglected. In engineering, it is impossible to obtain such perfect case without any distortion, and then we now analyze the influence of the deviation of the image relay.
Figure 8(a) shows, to reduce the propagation diffraction effect, four 4-f telescopes are inserted between the small and the main compressors for image relay and beam expansion. The ABCD propagation matrix for the ideal case is given by
where, fi (i = 1 to 8) is the focal length of the lenses from the small compressor to the main compressor. Based on the parameters (2 × , 2 × , 5 × , and 5 × ) given in Fig. 8(a), the result of Eq. (3) is equal to [100, 0; 0, 1/100], and it is perfect image relay without any diffraction-induced wave-front distortion, which means only the beam aperture is increased by 100 times and the beam divergent angle is reduced to 1/100. Apart from these, no other diffraction effects/errors would be introduced. However, when the input (object) plane of a telescope deviates from the output (image) plane of its previous telescope, or the small/main compressor deviates from the perfect object/image plane, the propagation diffraction induced distortion would appear. Figure 8(b) shows that, in the 80 nm bandwidth laser, the focused peak intensity in the far-field with the pre-compensation gradually decreases with increasing the deviation distance Δz. However, the degradation is very small, for example of a deviation distance of ± 2000 times beam diameter (i.e., ± 8, ± 16, ± 32, ± 160, and ± 800 m at the positions before the first, the second, the third, the fourth telescopes, and after the fourth telescope, respectively), the degradation of the focused peak intensity is very small, and the result is still much better than that without the pre-compensation. In the case of the 400 nm ultra-broadband laser, Fig. 8(c) shows the result is similar with the one in the 80 nm bandwidth laser as shown in Fig. 8(b), i.e., the influence is still negligible.Fig. 8 (a) Schematic of the image relay system consisting four 4-f telescopes between the mall and the main compressors, and in a broadband laser it should use reflection telescopes for achromatic design. The focused peak intensities If as functions of the deviation Δz away from the perfect object/image planes for (b) the 80 nm and (c) the 400 nm bandwidth lasers with the pre-compensation. D is the beam diameter where the deviation occurs. The case without the pre-compensation is also shown for comparison.
Next, let’s consider the influence of transmission mediums in the beam-line, e.g., Ti:sapphire crystals in a Ti:sapphire laser or nonlinear crystals in an OPCPA laser or window glasses of vacuum pipes/chambers. Figures 9(a)-9(c) illustrate the beam propagation in free-space, through a medium with a refractive index of n2, and through a tilted medium with a tilt angle of θ1. The ABCD propagation matrices of Figs. 9(b) and 9(c) are respectively given by [44]
andwhere, θ2 is the refractive angle in the medium, and p3 and p4 are two coefficients here. Equations (4) and (5) show that the transmission medium in the beam-line would change the optical path from the input plane to the output plane, and the optical path change is also spectrum-dependent (i.e., chromatic error). Figure 9(d) shows the spectrum-dependent coefficients p3 and p4 in the Ti:sapphire crystal as functions of the signal wavelength from 600 to 1200 nm. The spectrum-dependent difference of the optical path change (600 nm spectral range) is only around 0.9%, and the coefficient p4 with a tilt angle θ1 of 5° has no obvious extra-distortion. Figure 9(e) shows the result in the BBO crystal, when the signal is the ordinary-light (o-light). The spectrum-dependent difference of the optical path change (600 nm spectral range) is slightly increased to around 1%, and no obvious extra-distortion could be found in the case of the coefficient p4 (θ1 is 5°). In this case, for different spectral components, the optical path changes from the input plane to the output plane are almost the same, and then the propagation diffraction induced wave-front distortions would be almost the same, too. Consequently, the influence of transmission mediums in the beam-line can be classified into the case discussed above, i.e., the real object/image plane deviates away from the ideal position. Figures 8(b) and 8(c) show that in both the 80 nm and the 400 nm bandwidth lasers the tolerance to a short deviation distance Δz is very high.Fig. 9 Schematic of propagation in (a) free-space, (b) through a medium and (c) through a θ1 angle tilted medium. The refractive index in free-space and medium is n1 and n2, respectively. Spectrum-dependent coefficients p3 and p4 (θ1 = 5°) in (d) Ti:sapphire and (e) BBO crystal (o-light), respectively, and the result is normalized by the value at 800 nm.
Here, all optical aberrations of the telescopes (e.g., chromatic aberration, spherical aberration, etc.) are not considered, which actually is not the main topic studied in this paper. For example, in engineering, reflection telescopes are usually used to avoid chromatic aberration, and aspheric lenses or some compensation methods can be considered to remove spherical aberration. For the transmission telescope induced pulse-front curvature (or called radial group delay), a conjugate compensator can be used [33], which won’t obviously influence the propagation of the pre-compensated wave-front by the method [45].
Anyway, beam propagation is very complex and quite different in different high-energy or ultra-intensity laser facilities. Previously, the pre-compensation of the final output wave-front by using a deformable mirror in the frontend or after the pre-amplifiers has already been widely used in worldwide facilities [46]. The only difference in this method is that the pre-compensation is spectrum-dependent. If in case of achromatic or low-chromatic image relay, the performance/reliability should be the same.
5. Discussions
In this paper, the simulation is accomplished in the 2-dimensional domain of x-t/λ, and only the diffraction wave-front along the dispersion direction of the second and the third gratings (G2 & G3) are considered. According to our previous work [15], the wave-front along the non-diffraction direction of the second and the third gratings (G2 & G3) and that of the first and the fourth gratings (G1 & G4) will not introduce any spatio-spectral/temporal coupling distortion, and only space-dependent pulse-front and phase-front distortions (no deviation between pulse-front and phase-front) would be generated. Generally, when one (or several) deformable mirror is used to correct the phase-front (or called wave-front) for the near diffraction-limit focal spot, the pulse-front distortion would also be removed automatically. Therefore, we won’t repeat it here again.
In the simulation, the wave-front of the second grating in the small compressor is not considered. It is because that, in engineering, the diffraction wave-front of a millimeter-sized grating (e.g., 5 to 20 mm) is much better than that of a meter-sized one. The tiny wave-front error of the second grating in the small compressor is negligible. Even if it is not negligible, we can still use the deformable mirror in the small compressor to pre-compensation both the diffraction wave-front of the second grating in the small compressor and that of the second and the third gratings in the main compressor.
The section 4.1 shows the sensitivity to the miss-matching of the parameter ε is not very high, and in our opinion a 10% deviation is acceptable. Then, the small compressor can still be used to accurately adjust the pulse duration (i.e., the second order dispersion) at the output for the Fourier-transform-limit. If the small compressor is used to increase the pulse duration for experimental applications (i.e., longer than the shortest pulse duration), the influence of the complex STCD becomes negligible, and the topic studied in this paper does not need to be considered anymore.
In this paper, the size ratio of the main compressor and the small compressor is chosen as 100. As discussed in the section 4.2, to reduce the adverse influence of the miss-matching of the deformable mirror’s transverse location (offsets), this ratio can be reduced, i.e., the size of the small compressor can be increased for a perfect pre-compensation. And it would also reduce the requirement of the high spatial resolution of the deformable mirror. Meanwhile, the two compressors should be similar in geometry, and both the accurate incident angles and near-perfect parallelism would be obtained in engineering [47].
6. Conclusion
In conclusion, by inserting a scaled down compressor with a deformable retro-reflection mirror into the frontend of an ultra-intense femtosecond laser, the complex STCD induced by the imperfect wave-front of the second and the third gratings in the spatio-spectral coupling plane could be pre-compensated. The simulation shows, no matter in an 80 nm bandwidth laser or a 400 nm ultra-broadband laser, the complex STCD could be reduced to a very low level, and near ideal pulsed beams could be obtained in both near- and far-fields. Considering the application, several miss-matching effects that may appear in engineering are also analyzed and discussed. We believe the method proposed here is an efficient way to remove the adverse influence of the complex STCD for a near ideal focused peak intensity at the target.
Funding
JST-Mirai Program, Japan (JPMJMI17A1).
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