Uniform and efficient beam shaping for high-energy lasers: supplement

Phase-only beam shaping with liquid crystal on silicon spatial light modulators (SLM) allows modulating the wavefront dynamically and generating arbitrary intensity patterns with high efficiency. Since this method cannot take control of all degrees of freedom, a speckle pattern appears and drastically impairs the outcome. There are several methods to overcome this issue including algorithms which directly control phase and amplitude, but they suffer from low efficiency. Methods using two SLMs yield excellent results but they are usually limited in the applicable energy due to damage to the SLM's backplane. We present a method which makes use of two SLMs and simultaneously gives way for high-energy laser applications. The algorithm and setup are designed to keep the fluence on the SLMs low by distributing the light over a large area. This provides stability against misalignment and facilitates experimental feasibility while keeping high efficiency.

The spherical lens term f defocus is applied to the target distribution. f defocus , the distance a and the corresponding lens term f 2 change with the target size s target and the chosen defocus d 2 . The two presented configurations of the main publication are indicated in red (d 2 = 50 mm) and blue (d 2 = −50 mm). In both cases the target size s target stays the same. In our presented implementation the value for the distance d 1 is chosen close to the focal length f 1 (d 1 = 180 mm). ( distances and angles are not to scale) The relation between the size of the target object 2s target , the size on the second SLM 2s SLM 2 , and the applied focal length f 2 can be determined with the matrix method: Equation S1 can be solved for f 2 , where θ SLM 2 is substituted with θ SLM 2 = s SLM 2 d 1 , corresponding to the largest angle of incidence (compare Figure 1 in the main document). Finally, f 2 reads: The effective defocus of the target distribution changes with the applied focal length f 2 on the SLM. Since the field on the second SLM is derived by back-propagation starting from the target distribution, we need to calculate the corresponding defocus f defocus . In that way, the focal length f 2 is automatically encoded in the resulting wave front. The defocus is defined by: Thereby, a is the distance from the second lens to the position of the focus (compare Figure S1). At this point, the calculated height s focus needs to vanish (s focus = 0) and we can take the definition of Equation S1 to calculate a. Therefore, s target is substituted with s focus and we define a to be the last propagation distance instead of d 2 + f 3 . In that way we can determine an equation for a and solve Equation S3: The calculated distance f defocus is the focal length of the target wave front:

ALIGNMENT
There has to be a perfect mapping between the shaped amplitude and the applied correction wave front. We perform the alignment whilst we image the plane of the second SLM on a camera. This shows the shaped intensity pattern. If a phase mask is applied on the second SLM, the contours of this pattern slightly emerge. We choose the phase mask to be identical to the target distribution. This gives the opportunity to compare the shaped outcome with the theoretical expectations. If those structures match perfectly, the setup is well-aligned. A good choice for a target pattern is a grid with several crossings as illustrated in Figure S2. If the gird appears blurred, the calculated propagation distance might not agree with the experimental setup such that the target intensity pattern appears in a slightly shifted plane. Either the setup or the distances in the algorithm can be adjusted but it is usually easier to adapt the propagation path in the algorithm and see if the outcome improves. Besides the axial alignment, the two patterns need to be adjusted laterally. This can be done with the two mirrors in the setup. It makes sense to use two mirrors since this not only allows for shifting the shaped intensity pattern laterally but also for correcting slight rotations. Ideally, the two patterns match over the whole area. We observed a slight mismatch in the aspect ratio between the shaped and expected structure. Therefore it is useful to display a grid since it is sensitive to a distortion of the pattern. It is likely that those effects appear since the beam impinges on the SLM under a certain angle and it is practically impossible to work completely distortion-free. However, a simple compression/stretching can easily be compensated beforehand by giving the GS algorithm a target distribution with the opposite stretching/compression.

EVALUATION OF DIFFERENT RECTANGLES
The recorded rectangular structures in Figure S3 are the basis for the evaluation of the efficiency η, beam uniformity U, and flatness factor FF. The inner outline marks the area for the calculation of beam uniformity U and flatness factor FF. It is slightly smaller than the shaped structure to exclude the abrupt transition at the edges. The outer rectangle includes the full signal area to calculate the efficiency η. The results can be found in Table 1 in the main publication.

A. Simulation parameters
The simulation area is preliminarily defined by the dimensions and pixel size of the SLM (960x960 px with a pitch of 9.2 µm). To take diffracted light under large angles along the propagation path into account, the simulation area is increased with zeropadding by 1000 px in each direction. The FT is energy conserving and light cannot leave the boundaries of the modelled area. The padded range is chosen to contain the light field over the whole propagation path. The simulation area could be chosen even larger but it has proven experimentally that this is sufficient for the chosen path lengths, major diffraction angles and optics in the setup. Otherwise the calculation time increases rapidly. After the phase masks are calculated, we probe them on the incoming light field and simulate the expected outcome. To counter potential numerical instability we use subsampling. Therefore the initial light field is meshed with finer steps ( 1 4 of the SLM's pixel size). The resolution of the iteratively calculated phase masks is limited by the resolution of the SLM and is thus only subsampled by repeating the same pixel value within the initial pixel area. This step is done to test the phase masks for potentially under-sampled structures which would significantly impair the quality in the experimental result. The calculation of the two phase masks takes about 1 min on a 4-core Intel i5-7500 CPU at 3.4 GHz with 8 Gb RAM (the individual matrices are 2960x2960 complex double). Calculating the simulated results with subsampling (11840x11840 complex double) takes around 5 min.

B. Evaluation of misalignment on rectangular structures
We addressed different kinds of misalignment on simulated rectangular structures. Even though we already showed misalignment on snowflakes in the main publication, we would like to add a simulation for rectangular structures in Figure S4. The beam uniformity U and flatness factor FF can be calculated easily on a plain structure. It is worth noticing that a misalignment does not cause major drops in efficiency for our proposed method in contrast to a direct Fourier relation. The development of the beam quality parameters depends on the applied misalignment. They remain relatively constant for a lateral misalignment in case of our proposed method in contrast to a setup in a direct Fourier relation. The beam uniformity shows a similar behaviour in case of axial misalignment. The flatness factor drops more rapidly for our proposed method due to pronounced overshoots at the edges. A rotational misalignment around the z-axis affects our proposed method more strongly than a setup in direct Fourier relation. In general, a misalignment seems to cause more blurry structures for a direct Fourier relation whereas our proposed method starts to suffer from emerging speckle.