1 Introduction and motivation

Using the technique of chirped-pulse amplification (CPA) [1], the generation of laser pulses with high pulse energy and ultra-short pulse duration from table-top systems has become possible. Focussing these pulses to focal spots having diameters of a few μm only, peak intensities in excess of 1021W/cm2 can be reached in the laboratory. Irradiating targets with such intensities has lead to spectacular results in the field of laser-driven particle acceleration over the last couple of years [2–5]. However, the requirements e.g. on the control of the temporal and spatial parameters of the laser pulses have increased accordingly to achieve reproducible and comparable experimental conditions for such high-intensity experiments.

One of the important parameters influencing the focussability of the compressed high-power laser pulses is the residual angular chirp (AC), which is defined as the derivative dθ/dλ of the amplitude-weighted wavefront tilt θ in the plane wave approximation with respect to the wavelength λ. In the literature, AC is usually given in units of μrad/nm. AC arises mainly when the pulses pass through wedged components throughout the laser system or when pulse stretcher and compressor are not perfectly aligned with respect to each other. AC increases both the pulse duration in the focal plane due to pulse front tilt and the focal spot area due to wavelength-dependent propagation directions [6–8].

In [7], Pretzler et al. introduce the ratio of intensity reduction R of peak intensities I AC and I 0 with and without angular chirp in the in x-direction of the focus of a spatial and temporal Gaussian pulse:

$${} R = \frac{I_{\rm AC}}{I_0}\Big|_{x=0} = \xi^2 = \Big(1+\frac{C_{\rm A} f \delta\lambda}{w_0}\Big)^2,$$
(1)

where C A is the angular chirp in μrad/nm, f is the focal length, δλ is the FWHM-bandwidth of the pulse and w 0 is the Gaussian beam radius of the focus. The collimated beam radius w 1 before focusing is given by

$$w_1 \approx f\frac{\lambda}{\pi w_0},$$
(2)

which is derived from the relation tanθ = w 1/f = λ/(π w 0) and holds true for f ≫ w 1. An analogous approach is the numerical calculation of the Fourier transform of a multispectral spatial Gaussian pulse with a wavelength dependend phase tilt, which is derived from a given AC value.

Several methods such as field correlation [7], imaging spectrography [8–10], spectral interferometry [11, 12], and polychromatic far-field methods [13, 14] have been used in the past to measure the amount of residual AC. With field correlation and spectral interferometry alignment methods, results have been achieved in the range of 0.1 μrad/nm with Ti:Sa systems and a spectral width of 80…180 nm.

However, interferometric methods may suffer from systematic errors due to higher-order lateral chromatic aberrations. Such errors can, e.g., arise, if the two replica of the laser pulse are overlapped in the near field in order to achieve interference fringes. Such aberration may originate from gratings not having a perfectly plane surface. Then, the second grating will be hit at different lateral positions by different wavelengths. An uneven grating surface then results in higher Zernike-order aberrations differing with wavelength, which may distort the results of interferometric measurements.

At the POLARIS laser facility [14], we have developed and implemented a novel far-field approach, since methods measuring the center of gravity of the far field are insensitive to higher-order lateral chromatic aberrations for top-hat shaped near fields. The centers of gravity of the Fourier transforms of all Zernike polynomials except tip and tilt (the quantities we are interested in) are then centered at the origin of the coordinate system. In that case, looking for the centers of gravity of foci at different wavelengths bypasses the influence of wavelength-dependent higher-order Zernike polynomials, even if the focus is split into several hot spots as suggested in [12].

In the extreme case of laser beam profiles exhibiting several hot spots paired with higher-order chromatic wavefront aberration, the center of gravity of the monochromatic foci can change with wavelength. In a pure polychromatic wavefront analysis without consideration of the beam profile, this chromatic aberration part of the angular chirp cannot be detected.

Divoký and Straka [10] report a typical spatial accuracy of image positions of less than 10 μm for multi-shot near-field measurements using the center of gravity method, which is a convincing result considering the angular beam profiles presented in [10]. In [13], a residual angular chirp of 0.013 μrad/nm was reported using beams with two alignment wavelengths 6 nm apart during an iterative AC compensation process where all rotational degrees of freedom of one compressor grating were corrected for in an alternating multi-shot sequence. However, the excellent resolution of the method could only be achieved by rotating the compressor gratings forth for the measurement and back for compression.

It is worth noting that if a wavelength-tuneable test laser beam is used for AC determination, it is merely the test beam which is chirp-minimized after the alignment, not necessarily the actual laser pulse. On its way from the oscillator to the target, the main beam passes stretcher, compressor, and several wedge-shaped optics to avoid unwanted back-reflections possibly leading to prepulses [15]. These wedges are prone to imprint AC on the laser pulses. Furthermore, the chromatic aberration part of the angular chirp mentioned above may introduce additional angular chirp. Thus, even if the compressor gratings have been perfectly parallelized with a polychromatic probe laser, the high-power laser pulses are not necessarily free of AC when leaving the compressor.

As a consequence, it is of great interest to improve the accuracy of the measurement of AC while using the laser pulses itself instead of a test beam, preferably with a single-shot method.

2 Setup for direct angular chirp compensation

AC manifests itself in the focal plane as a lateral displacement of the focal spot while the wavelength of the laser is varied. Depending on the laser parameters, this chromatic displacement can be very small but nevertheless have an impact on the peak intensity achieved in the focus.

The measurements described in this paper were carried out with attenuated laser pulses generated with the POLARIS laser [14], whose laser parameters were used for the calculation in Fig. 1 (central wavelength of 1.03 μm, a FWHM bandwidth of 12 nm, and a Gaussian transverse profile with a 1/e-diameter of 140 mm). The line density of the gratings used in the stretcher and compressor is 1,480 lines/mm, the angles of incidence and the outgoing angles are 60.68° and 40.73° for the central wavenlength of 1,030 nm, respectively. For the far-field diagnostics presented in this paper, we used an f = 5 m focusing lens, imaging the beam onto a CCD chip with pixels of 6.45 μm width and height. For high-power experiments, the beam is focused onto the target by off-axis parabolas with either 30 or 90 cm focal length.

Fig. 1
figure 1

Calculated peak intensity in the focus of a non-aberrated laser pulse as a function of the angular chirp (AC) for a bandwidth-limited laser pulse with 1.03 μm central wavelength, a bandwidth of 12 nm (FWHM), and a Gaussian transverse profile with a 1/e-diameter of 140 mm. A peak intensity of more than 90 % of the AC-free pulse is reached with an AC of less than 0.1 μrad/nm

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Using these values, the focus peak intensity is reduced to 90 % of the theoretical limit for a value of AC of 0.10 μrad/nm. Such small amounts of angular chirp are hardly detectable by direct analysis of the focal image. Two partial beams with different wavelengths become separated in the far field only if the product of AC and wavelength difference is greater than the diffraction limited resolution, which is 6.5 μrad for the Gaussian beam having the above given parameters. Hence, for the parameters given in in Fig. 1 and with a residual AC of 0.1 μrad/nm, a wavelength difference of at least 65 nm would be necessary to resolve such bi-chromatic foci. Nevertheless, the area of the focal spot is already significantly increased for considerably smaller products of AC and FWHM-bandwidth, thus reducing the intensity in the focal spot.

The sensitivity of the focus displacement measurement can be improved if beams at different central wavelengths (cf. Fig. 2) can be intentionally tilted with respect to each other by additional optics (cf. Figs. 3, 4) such that their foci are separated and their centers of gravity of the intensity distribution can be determined independently in the far field. In order to achieve this, two reversible modifications are applied to the laser beamline:

  1. 1.

    The spectrum of the laser is modified in order to generate quasi-monochromatic partial beams with different central wavelengths.

  2. 2.

    Within the compressor, certain spectral components of the dispersed laser beam are intentionally deflected by a small angle.

Fig. 2
figure 2

Measured spectral intensity distribution obtained with different apertures inside the stretcher where the beam is laterally dispersed. These narrow spectrum beams are used successively in order to calibrate and measure spectrally dependent focus positions and shapes. For modification III the gain of the amplification chain was increased

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Fig. 3
figure 3

Top view of the compressor setup for the AC measurement—alignment of the auxiliary mirror with spectral modification II. The D-shaped auxiliary mirror is driven into the beam, until half of the beam is reflected. Then, it is aligned with respect to the end mirror as described in Sect. 2

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Beams exhibiting different spectral bandwidths can be generated from the initial frequency spectrum e.g. by placing obstructing objects inside the stretcher in a plane where the beam is laterally dispersed. In Fig. 2, three different modifications of our pulses’ spectrum are shown:

  1. I

    The original spectrum

  2. II

    The central part of the spectrum (1,028 nm…1,032 nm) cut out of the initial beam by using a slit.

  3. III

    The spectrum including the outer parts of the initial spectrum only (below 1,020 nm and above 1,036 nm) by obstructing the central parts of the spectrum with a suitable beam block in the pulse stretcher.

At the compressor end mirror, for the lateral separation of the partial beams from the opposite ends of the laser pulse spectrum (spectral modification III) after the second grating pass (cf. Fig. 4) the following equation must be satisfied

Fig. 4
figure 4

Top view of the compressor setup for the AC measurement—alignment of the second compressor grating with spectral modification III

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$$S [\tan \alpha_o(\lambda_b)-\tan\alpha_o(\lambda_r)] > D \sin\alpha_i,$$
(3)

where D is the beam diameter, S is the perpendicular grating distance (the z-component of the compressor grating separation, cf. Fig. 5), and α i and α o are the angles of the incident and emitted beams, respectively. λ b and λ r denote the upper and lower limit value for the transmitted wavelengths at the blue and red end of the spectral modification III, respectively. The larger the difference λ r  − λ b and the compressor grating separation S, the easier condition (1) is fulfilled.

3 Method of measurement and alignment

In this section, the spectral modifcations II and III are used successively in order to calibrate the setup and to measure wave length-dependent focal spot positions. Rotations of optics around the axis perpendicular to the plane of diffraction in the compressor are designated as tilt (see Fig. 5 for illustration). Rotations around the axis, which is defined by the intersection of the plane of diffraction and the surface of the rotated optics, are designated as tip. Rotations around the orthogonal third axis are referred to as twist.

Fig. 5
figure 5

Scheme of the tiled grating compressor, which was used during the measurements presented in Sect. 4. Reference grating and mosaic grating together form the second grating of a Treacy setup. The mosaic grating is freely adjustable with respect to the fixed reference in all relevant degrees of freedom. Both gratings together are mounted onto a superordinate translation and rotation stage in order to adjust the tiled grating block with respect to the in-/outcoupling grating. The relevant degrees of freedom and orientation of the coordinate system used in this paper are depicted at the bottom right of the figure

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In a first step of the alignment procedure, a D-shaped auxiliary tilt/tip mirror (cf. Fig. 3) needs to be inserted into the compressor setup and be parallelized with respect to the compressor end mirror which is also a prerequisite for being able to measure the initial AC of the laser pulse. While using the quasi-monochromatic spectral modification II, this auxiliary mirror is positioned in front of the end mirror in a way that one half of the beam hits the auxiliary mirror while the other half propagates to the end mirror and is reflected there (cf. Fig. 3).

For the case of conical mounting (i.e., the incident pulse is tilted with respect to the plane of diffraction of the compressor gratings in order to separate the incident from the outgoing pulse), the auxiliary mirror should be positioned as close to the end mirror as possible in order to minimize vertical beam offset. For compressor systems incorporating a roof-top end mirror, the auxiliary mirror should also be an adjustable roof-top mirror with a similar vertical offset.

In our experimental setup, a tiled grating (cf. Fig. 5) was used for the second and third grating pass. Instead of an auxiliary mirror, the adjustable mosaic grating was utilized for the separation of the partial beams introduced in Sect. 2, producing similar results as the configuration with the auxiliary mirror.

For the determination of a misalignment of the auxiliary mirror with respect to the end mirror, the focus of the modified compressor output is observed. Tip alignment starts by intentionally tilting the auxiliary mirror, until two different focal spots can be distinguished. Tip mismatch is eliminated by matching the centers of gravity in tip direction by tipping the auxiliary mirror. Figure 6a, b shows the initial state of tiled gratings adjustment with +20 μrad tilt of the auxiliary mirror added and after tip correction.

Fig. 6
figure 6

Far-field alignment of the parallelism of reference and adjustable grating (end mirror and auxiliary mirror in the case of monolithic compressor gratings) using spectrum modification II. Thin lines next to the foci indicate their centers of gravity. The foci are elliptical, since the corresponding near fields are D-shaped. a Initial state of auxiliary mirror adjustment, +20 μrad tilt to adjustable grating added, b after tip correction, c initial state after step b with +20 μrad tip of adjustable grating added, d after tilt correction

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Now, after the auxiliary mirror has been aligned in tip direction, it is intentionally tipped and then matched in tilt direction by tilting the mirror. Figure 6c, d show the initial tilt alignment with +20 μrad tip of the adjustable grating added, and after a tilt correction, respectively. After setting the tilt and tip to their new reference values, auxiliary and end mirror planes are parallel. The achievable accuracy of parallelization is in the sub-μrad domain, when the focal length of the imaging optics is in the meter scale and the centers of gravity can be determined with sub-pixel accuracy from the images.

Finally, the spectral modification III is applied. Starting from an initial, but most likely not perfect compressor alignment we displace one rotational degree of freedom of the auxiliary mirror in order to generate two separated focal spots having two different central wavelengths. Angular chirp in tilt is measured by misaligning the auxiliary mirror with respect to its tip, but it is now corrected by turning the first grating around the tilt-axis, until the centers of gravity of the two foci match in tilt direction (see Fig. 7a,b). The outgoing angle for the central wavelength is simultaneaously rearranged by tilting the end mirror by the proper amount.

Fig. 7
figure 7

Minimization of the angular chirp of the whole laser system including the compressor using far-field observation with spectrum modification III. a Initial state of adjustment, +10 μrad tip to adjustable grating added, b AC in the xz-plane was eliminated by tilting the first compressor grating by a certain amount

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Residual AC in the vertical direction can be corrected in a similar way. A misalignment of the gratings in tip or twist directions introduces the conical mounting case, i.e., there now exists a non-vanishing grating constant also in the y-direction, which increases with the sine of the conical mounting angle, whereas the change of the grating constant in x-direction decreases with the cosine. The non-vanishing grating constant in y-direction introduces a wavelength dependend change of the k y -vector after the grating pass, whereas the k x -vector remains almost unchanged due to the cosine-dependence, which has been demonstrated in [8]. Again, the general change of the outgoing vector (relative to the direction of the central wave length) in y-direction is compensated by tipping the end mirror, while the non-zero wavelength dependend rise of the grating equation in y-direction accounts for the AC generated.

In the vertical AC compensation step, the D-mirror is used to displace the partial beams horizontally in order to measure the AC generated or compensated in the vertical direction. AC in tip direction is corrected by displacing tip and twist similarly [7], until the centers of gravity match in y-direction.

Regarding systematic errors, it is crucial to assure that the observation planes for the target focus and the AC-measurement focus are both precisely situated in the waist of the focused beam. This is especially important during the monochromatic alignment phase (cf. Fig. 3) when the focussed partial beams have different propagation directions. In this case, a focal plane mismatch leads to non-parallelism of auxiliary and end mirror after an apparently correct alignment. To reach a perfect focal position, a modified Bahtinov mask (cf. Fig. 8) can be used. Again, the centers of gravity are used to determine, whether the first-order foci are arranged in a perfect rectangle. Thus, the accuracy of this method is comparable to that of the AC measurement. However, due to the half split character of the mask, higher-order Zernike polynomials are of significance. To make sure that the focusing is not deteriorated by aberrations, the mask should be turned by 90° after the focal plane is found with the initial orientation, cross-checking the results.

Fig. 8
figure 8

a Modified Bahtinov mask used for finding the focal plane precisely. b Focus pattern slightly out of focal plane. c Focus pattern in the focal plane. Pictures b and c are results of a simulation

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Furthermore, the orientations of the rotational axes of the D-shaped mirror/adjustable grating need to be checked against those of the grating which is moved to compensate for residual AC. If there is a measurable mismatch of these axes for the applied angular separation of the polychromatic foci during the measurement, it should be taken account of by rotating one of the coordinate systems during the analysis of the measurement in order to match them. The same procedure is recommended for a rotation of the far-field camera frame against the rotation axes of the D-mirror/gratings. Alternatively, the CCD chip can be rotated. Also, non-orthogonality of the tilt/tip/twist axes can be treated by a rotation of the coordinate axes in a first-order approximation. However, high-precision optical mounts usually show a very good orthogonality of their rotational degrees of freedom, so the effects were not measureable in our setup.

It should also be noted that for the pulse duration at the focal position to be as short as possible, the overall wave front needs to be as flat as possible. In order to achieve this, a closed loop adaptive optics systems by NightN coupled with a high-sensitivity wave front sensor by Phasics is applied at the POLARIS system. Also lateral chromatic aberration of the imaging optics should be taken into account and—if possible—be avoided by using appropriately corrected optics.

4 Results

Figure 9 shows data of an exemplary measurement set of the AC behind the compressor, as it depends on the rotational displacement of the first grating around the tilt and tip axes. Each data point in the graph is accumulated of 30 single measurements, which were aquired back-to-back after separating the partial foci one time as described in Sect. 3 The error bars are showing the 95 % confidence interval and indicate that the AC can be measured with an accuracy of ±0.019 μrad/nm with a 30-shot statistics (equivalent to 0.19 pixel width with a focal diameter of around 8 pixels) and of ±0.104 μrad/nm for a single shot, respectively. The root mean square deviations of the data set values against the linear fits are 0.013 μrad/nm for tilt and 0.022 μrad/nm for tip, respectively.

Fig. 9
figure 9

Measured residual angular chirp as a function of tilt and tip misalignment of the first and fourth grating pass. The angular chirp values at 0 μrad displacement correspond to the initial alignment state of the compressor. Linear fits indicate, that the first/fourth grating needs to be tilted by +2 μrad and tipped by +64 μrad (depicted by the intersections of the zero-AC-line with the arrows) in order to eliminate AC. Each data point is accumulated of 30 single measurements. Error bars show the 95 % confidence interval

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Without consideration of systematic errors, the precision of the measurement is limited by the number of shots taken into account for the statistics which can be chosen almost arbitrarily for systems with a high repetition rate. With two distinct measurement series, each consisting of 400 shots within three hours a long-time drift of the angular chirp of (0.016 ± 0.006) μrad/nm could be detected.

In Fig. 9, the slope of the linear fit for tip rotation is smaller than for tilt, because the amount of angular beam displacement depends on which rotational degree of the compressor grating alignment is changed [16]. From the slope values, it can be deduced that the angular alignment precision with respect to the tilt axis has 1.5 times higher requirements than with respect to the tip axis.

In order to exploit the full potential of the method, it is crucial that the hysteresis of the positioning stages is as low as possible and well known when rotating the D-shaped mirror in one axis in order to determine the residual angular chirp with respect to the other axis. At POLARIS, the D-shaped mirror is represented by the movable grating in a tiled grating arrangement. This grating is mounted on a piezo-driven 6-axis actuator stage with non-measureable hysteresis.

In general, when choosing high-quality rotation stages, like piezo-driven actuators or hysteresis minimized mechanical stages with step-motors and high-translation high-precision gears, as well as thoroughly characterizing the hysteresis behavior of these stages, the manageable repositioning accuracy is well below the measureable limit found in this paper, as measured in our laboratory and reproduced on a daily basis.

Other than the accuracy of the AC measurement, the alignment accuracy of the gratings depends only on the relative positioning accuracy of the gratings and end mirror. Furthermore, for the alignment of the compressor gratings, there is no need for an absolute repositioning accuracy, since the gratings are aligned in-situ with this method. Here, the important parameter is the minimum angle step of the concerned rotational stages, which can be driven to very low values by using high-translation gears.

Finally, it is important that the introduction or elimination of angular chirp by grating tilting is given by the change of the first-order derivative of the grating equation and is therefore relatively insensitive to small rotations, such as when the general change of the outgoing angle for all wavelengths after tilting the second grating is compensated by the end mirror. However, from our results, we found out that the positioning stage of the tilt rotation stage of the second compressor grating had to be refined in order to take full advantage of the sensibility of the residual AC measurement presented in this paper.

For laser systems with a low repetition rate, the initial AC is measured with sufficient statistical confidence and then divided by the slope of the linear fits determined from a graph similar to Fig. 9. The corresponding quotients are then equal the amounts of rotational detuning of the first compressor grating (in tip and tilt) necessary to eliminate angular chirp. Our method also improves the accuracy of the measurement of the residual AC in the whole laser system. In the case of chromatically varying wavefront aberrations as described in section1, our far-field approach provides a first-order correction of the pointing error.

5 Conclusion

In conclusion, we have developed a high-sensitivity measurement technique to determine the residual angular chirp of a high-power laser beam after amplification and re-compression in a CPA setup. In our setup, this was achieved by using the two gratings of our tiled grating compressor as chromatic separators and observing the centers of gravity of two foci with largest possible spectral separation.

By slightly detuning the global grating parallelism in the compressor, it was possible to compensate for the initial AC of the laser. The amount of AC is monitored using the laser itself as a multi-spectral probe pulse instead of a tunable external laser source while maintaining the accuracy of polychromatic probe-beam methods.

The technique can be implemented in every CPA laser system where the lateral dispersion on the compressor end mirror is large enough to spatially separate quasi-monochromatic beams from the opposite boundary regions of the spectrum. Using this method, we were able to measure the residual AC with an accuracy better than 0.02 μrad/nm, which in theory increases the peak intensity within the POLARIS focus to more than 98 % of the theoretical limit, as compared to 90 % for an AC of 0.10 μrad/nm.