3 Phase plate characterization

The topography of the central section with an area of $10\;{\mathrm{mm}}\times 10\;{\mathrm{mm}}$ of the phase plate was measured with the optical profiler. This was necessary for assessing whether the step heights were equal, as required in the ideal SPP case. The topography, shown in Figure 2, was obtained by stitching more than 2300 smaller, overlapping scans of the surface.

Figure 2 A $10\;{\mathrm{mm}}\times 10\;{\mathrm{mm}}$ optical profiler image with a height range of 3 $\unicode{x3bc}$ m. The height was exaggerated to observe the curved background and the stitching artifacts.

The curved background of the image hinders the calculation of the step height distribution. For background removal, the gradient of the surface was first calculated. The discontinuities in the gradient image generated at the step borders were then replaced by interpolating between the neighboring areas. Finally, the background was obtained by integrating the filtered gradient image.

The height distribution was obtained using a Gaussian kernel density estimator (KDE) on the image data without the background. A KDE was preferred over a histogram because the maximum of the latter depends on the choice of bin size and starting position. For each data point, the KDE method adds up the values of a kernel function (in this case a Gaussian) centered at that point. The resulting sum is an estimate of the distribution function.

For extracting the average heights of the steps, the result of the KDE was fitted with the sum of 15 Gaussian functions (not related to the kernel of the KDE), corresponding to the intermediate step heights, where the centers of the Gaussians are allowed to vary during the fitting process.

The maximum height difference across the phase plate is 1.754 $\unicode{x3bc}$ m. For FS (refractive index $n=1.4534$ at $\lambda =795$ nm), this yields a maximum phase shift of $2\pi$ (one wavelength optical path difference), for a beam with an initially flat wavefront. The $2\pi$ phase shift is considered also in the design of helical phase plates in reflection^[ Reference Kim, Bae, Kim, Kim, Chang, Jeon, Choi and Nam^64]. It should be noted that each of the 16 terraces of the SPP corresponds to a constant phase of the wavefront.

Assuming 15 equal step heights, the maximum height of 1.754 $\unicode{x3bc}$ m should yield a single step height of 116.9 nm. This value, as well as the overall step height, is close to that proposed by Sueda et al. ^[ Reference Sueda, Miyaji, Miyanaga and Nakatsuka^47], also using an FS plate in transmission. The measured step heights have the same average value but with a standard deviation of 14.7 nm. They range between 101 and 167 nm, with the largest gap occurring between the eighth and the ninth steps. The distribution of the height error to the mean is shown in Figure 3.

Figure 3 Relative error of the step heights across the phase plate, compared with the average value.

Figure 4 Wavefront distortion as measured with the Shack–Hartmann setup: (a) initial wavefront; (b) wavefront after masking out the discontinuities.

The complementary analysis of the wavefront was performed over a 70 mm central aperture. This was required as it is known that thin optical components often introduce residual wavefront distortions (RWDs) as a consequence of imperfect polishing of the surfaces. Figure 4(a) shows the wavefront measurement result. The RWD was estimated in the form of a polynomial obtained from the wavefront measured with the S-H setup by first masking out the discontinuities generated at the step boundaries using the wavefront sensor’s software (Figure 4(b)). The resulting surface is approximated with a coefficient of determination ${R}^2=0.81$ by the following function:

(3) $$\begin{align}f\left(x,y\right)={10}^{-2}\sum \limits_{i,j=0}^3{a}_{i,j}{x}^i{y}^j, \end{align}$$

with ${a}_{i,j}$ being the elements of the matrix:

(4) $$\begin{align}\left(\begin{array}{cccc}{a}_{0,0}& 3.24& -1.50& -0.164\\ {}2.59& -0.250& -0.111& 0.014\\ {}-2.11& -0.029& -0.103& 0.046\\ {}-0.244& 0.036& 0.093& -0.039\end{array}\right),\end{align}$$

where $x$ , $y$ are in centimeters, $f\left(x,y\right)$ is in micrometers and the center of the phase plate is in the origin. The value of ${a}_{0,0}$ is irrelevant since it only adds a piston term to the phase.

4 Impact of staircase-like phase plate imperfections on focus

For beam propagation simulations, a square region with a size of 150 mm and a lateral resolution of 2501 pixels was used. Figure 5(a) shows the eighth-order super-Gaussian irradiance distribution, with a full-width at half-maximum of 48.5 mm and wavelength 795 nm, and the 100 TW HPLS output irradiance pattern (Figure 5(b)). The staircase-like phase distribution is shown in Figure 5(c). The polynomial background from Equation (3), in units of wavelength, is shown in Figure 5(d).

Figure 5 Starting field distribution of irradiance (a), (b) and equi-distributed phase (c). Image (d) shows the phase distortion corresponding to the fitted polynomial. The squares have a size of 80 mm. The irradiance distributions in (a) and (b) are normalized to the same power and share the same intensity scale. The unit of phase distributions is one wavelength.

For the first set of simulations, a spiral phase filter, with the distribution of step heights as measured with the profiler (Figure 3) and having the polynomial background given by Equation (3), was applied to the super-Gaussian and HPLS fields. The resulting field was propagated for a distance of 40 m to a perfect lens with a focal distance of 150 mm. The irradiance distribution was then calculated in the back focal plane of the lens, where the computed square region size reached a value of 22 $\unicode{x3bc}$ m. Values of 15.1% RSD and 17.6% RSD of the circular section were obtained for a super-Gaussian beam and the HPLS beam, respectively (see Figures 6(a) and 6(e)). If the phase filter has a flat background instead, the values drop by about 10% RSD and the focal spot becomes more annular, as shown in Figures 6(b) and 6(f).

Figure 6 Simulated irradiance profiles in focus and the relative standard deviation (in percent RSD) of the circular sections through the points with the highest value. The $2\pi$ phase jump in the initial phase distribution was, for all images calculated, in the upper left quadrant, at an angle of 36.6° with the vertical. The rows index the initial spatial distribution of the field: super-Gaussian (a)–(d) or measured in the HPLS (e)–(h). The columns index the step height distribution across the spiral: nonuniform as measured, with the wavefront background being polynomial (a), (e) or flat (b), (f); uniform between the total height difference measured (c), (g); uniform, with corrected total height (d), (h). For the last two columns, no polynomial distortion of the wavefront was applied. The images share the same range of values for the color map.

For further refining the focus quality, the step height distribution was considered to be uniform. Neither the super-Gaussian nor the HPLS beam achieved a better focal pattern (Figures 6(c) and 6(g)), and a modulation in the ring shape remains, even for the perfect super-Gaussian field.

Trying to pinpoint the source of the remaining modulation, it was realized that it lies in the distribution of step heights across the staircase. There are two possibilities of approximating a continuous phase plate with a uniformly stepped structure. In the first case, shown in Figure 7(a), which corresponds to the SPP as measured with the optical profiler, the largest phase jump corresponds indeed to one wavelength as for a continuous plate. To simplify the drawing, a four-step SPP was considered. The phase, however, is not uniformly distributed, because the phases on the first and last steps are equivalent, being one wavelength apart, thus effectively doubling the area of the zero phase-shift step. The correct step height distribution (Figure 7(b)), as presented also in the work of Longman et al. ^[ Reference Longman, Salgado, Zeraouli, Apiñaniz, Pérez-Hernández, Eltahlawy, Volpe and Fedosejevs^50], results in a phase plate whose overall height is lower than for a continuous plate. The difference comes from the last step of height $l\lambda /N$ which, as seen in Figure 7(b), is ‘virtual’. It follows that the previously determined height of 1.754 $\unicode{x3bc}$ m should be divided between 16 equal steps, instead of 15; hence the heights of the steps for the next simulations were decreased by 7.3 nm, from 116.9 to 109.6 nm.

Figure 7 Two ways of approximating a continuous phase plate of $l=1$ (dashed green line) with a discrete, $N=4$ -step plate (orange in (a) and blue in (b)) and the resulting optical path length difference (OPLD) as a function of the azimuth $\phi$ . The largest phase jump is $\lambda$ in (a), but in (b) it is $\lambda \left(1-1/N\right)$ .

Figure 8 Coefficient of variation of the circular section in the focus as a function of the relative lateral shift between the phase plate and the HPLS beam.

The CV for the super-Gaussian beam was thus reduced to 1.5% RSD, a value limited by aliasing and diffraction at the simulated field boundaries. With 8001-pixel resolution and the initial space size of 600 mm, the RSD of the circular section in focus is only 0.3% (Figure 6(d)).

In practice, however, the doughnut focus quality for the HPLS beam is limited by the nonuniform cross-sectional irradiance distribution. Even with the correct distribution of the step heights, the focus quality remains at 7.8% RSD (Figure 6(h)). It should be noted that an incorrectly distributed step height like that in Figure 7(a) is, in effect, a phase plate that acts correctly at a different (longer) wavelength. Equivalently, it can be considered as having a non-integer charge for the working wavelength. For the investigated SPP, the correct wavelength would be 847 nm.

When performing the propagation simulations, the angular position of the largest phase jump in the spiral phase distribution was chosen arbitrarily, avoiding multiples of $\pi /2$ . It was set at an azimuth of 126.6°, like in Figures 4(a) and 5(c). Any asymmetry resulting from a nonuniform irradiance distribution of the starting field, like in Figure 5(b), or from a non-integer charge^[ Reference Longman and Fedosejevs^48], can be partially compensated with a lateral displacement of the phase mask. The position of the phase mask was therefore chosen to be the one that gives the minimum value of the RSD in focus. As an example, Figure 8 shows the dependency of the RSD value on the displacement for a flat background phase mask with step height distribution as measured with the profiler (Figure 6(f)). It can be approximated by a quadratic two-dimensional polynomial up to a relative root mean square of residuals of 0.13%. The optimization of the phase mask position was done for all the cases considered. The figure also shows the sensitivity, about 0.4 mm, of the doughnut quality with respect to the position of the phase plate.

Figure 9 Circular sections (a) through the irradiance profiles in focus for several cases, and azimuthal integration (b) of the respective profiles. The legend refers to the images from Figure 4 and their respective RSDs.

The circular profiles obtained at the radius of maximum value for the azimuthally integrated irradiance profiles are shown in Figure 9(a). The best scenario (super-Gaussian, flat wavefront, optimized step heights) is depicted by the curve denoted Figure 6(d). The low-amplitude, fast modulations present in the profile are coming from numerical noise, due to the limited size and resolution of the computed input field, and set the limit of calculations. For the other three cases (Figures 6(b), 6(e) and 6(h)) analyzed in Figure 9, slower modulations are observed. For the Figure 6(b) case (super-Gaussian, measured nonuniform steps), the associated modulation is 3%, while for the Figure 6(h) case (measured beam profile, optimized step heights) the modulation is higher at 7.8%. When adding the polynomial wavefront to nonuniform steps and a measured beam profile, the RSD is 17.6%. It should be noted that a DM can compensate the wavefront background, leading to an achievable RSD of 8.4% (Figure 6(f)).

The azimuthally integrated irradiance profiles are shown in Figure 9(b). One can observe the overall decrease in the irradiance across the ring, more than 30%, in the worst case analyzed (Figure 6(e)). This can significantly impact the estimated particle acceleration processes^[ Reference Brabetz, Busold, Cowan, Deppert, Jahn, Kester, Roth, Schumacher and Bagnoud^37]. Analyzing the polynomial background of the wavefront and removing it using a DM would lead to a higher quality doughnut focus. Both can be achieved, for example, using the ZEHBRO method^[ Reference Ohland, Posor, Eisenbarth, Iancu, Ungureanu, Ursescu and Bagnoud^40]. Comparing the cases in Figures 6(b) and 6(d) in Figure 9(b) shows that the step height distribution does not impact significantly the integrated peak irradiance around the ring. A nonuniform irradiance profile of the laser, however, with or without the wavefront background, illustrated respectively in Figures 6(e) and 6(h), leads to a significant decrease in the irradiance. This decrease can be used to quantify the doughnut quality, like the Strehl ratio in the case of vortex-free pulses.

A final remark is related to the depth of the doughnut hole, which was indicated as a signature of distortions in the case of helical pulses with spatial-temporal couplings^[ Reference Talposi, Iancu and Ursescu^63]. The relative depth of the hole in the worst-case scenario (see Figure 9(b)) is only 4%, making it difficult to distinguish in experiments.