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J. Eur. Opt. Society-Rapid Publ. 21, 17( 2025)
Figure 1. Flow chart of the iterative Fourier transform algorithm( IFTA) and the iterative wave-optical propagation algorithm( IWPA). The IFTA uses a forward / inverse Fourier transform to calculate the forward / inverse propagation of light through the optical system while the IWPA calculates a full( inverse) wave-optical propagation of the light.
controlled via a camera-based control loop [ 36, 37 ]. As OPT allows to simulate the spatially resolved non-linear amplification within the optical amplifier, the IWPA enables the compensation of the amplifier influence for arbitrary target distributions provided the amplifier is sufficiently characterized and the target distribution can be realized in conjunction with the optical amplifier.
To demonstrate this effect, we present simulation results for a simplified 2f-setup with a focal length f = 200 mm( cf. Fig. 2). An ideal amplifier is placed halfway between the phase mask and the focusing lens. The target plane is positioned in the focal plane of the focusing lens to allow a comparison with a phase mask calculated via IFTA. The target intensity distribution is a homogeneous distribution developed to increase the achievable surface rates for laser polishing( cf. Fig. 3)[ 38 ]. The amplification of the input beam I in( x, y) is modeled as
I out ðx; yÞ ¼ I S W G
0 exp ðI in ðx; yÞ = I S ÞI in ðx; yÞ
I S ð1Þ
with the lateral coordinates( x, y), the output beam I out, the saturation intensity I S and the small signal gain G 0 [ 39, 40 ]. W( x) denotes the Lambert W function. The amplifier is assumed to be a Nd: YAG-crystal with saturation intensity I S = 20 W / mm 2 and a small signal gain G 0 = 5 [ 41, 42 ]. The original input beam in front of the phase mask is a Gaussian beam with a diameter( 1 / e 2 intensity) of 6 mm and a power of 100 W. Note that the input beam I in in equation( 1) is the beam as it is diffracted by the phase mask onto the amplifier at a distance
of f / 2 behind the phase mask and is not simply assumed to be the original Gaussian input beam. The phase mask, amplifier and the focusing lens are simulated with a field size of 12.8 mm 12.8 mm with a resolution of 1023 1023 pixels. The pixel number stems from implementation details of the IWPA via OPT( cf. Sect. 2) that are optimized for pixel counts along each dimension of 2 n � 1; n 2 N. The target plane is simulated with a field size of 1.5 mm 1.5 mm with the same pixel count. The phase levels of the phase mask are continuous, i. e. no discretization to the phase levels was applied.
Figure 4 compares the simulated beam shaping results for phase masks calculated via IFTA and IWPA respectively for the optical setup in Figure 2. The corresponding phase masks are shown in Figure 5. For both phase masks, the resulting intensity distribution is simulated with OPT for the optical setup with optical amplifier. As the IFTA cannot consider the amplifier, the resulting phase mask leads to a superelevation at the edges of the intensity distribution. This is caused by the non-linear amplification( Eq.( 1)) that leads to a higher relative amplification for the lower intensities at the edge of the input beam compared to the center of the beam. The IWPA can take the amplifier into account and calculates a phase mask that pre-compensates for the subsequent non-linear amplification. The resulting intensity distributions are evaluated based on their normalized root mean square deviation( RMSD) from the target intensity distribution I target:
RMSD ¼
1 maxðI target Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 X � 2
I n � I target; n: ð2Þ N n