J. Eur. Opt. Society-Rapid Publ. 21, 33( 2025) 37
do not arise in Figure 5c. The only bright spot is the delta function located on axis. This result shows that the ideal autocorrelation peaks( delta function) can only be obtained when two patterns are identical in all aspects( phase noise and amplitude), and here the added phase noise dramatically increases the mismatch between the two patterns.
Finally, Figure 5d shows the result when the hologram used in Figure 5c is multiplied by the phase-only function e �i / 1ðx
Þ, which is a single complex-conjugate version of the
original pattern used to calculate the horizontal triplicator hologram. In this case, the bright spot corresponding to the delta function is shifted to the horizontal positive first diffraction order, where the perfect phase cancellation in the hologram plane leads to a phase-only correlation peak. The circle is reproduced in the central zero-order, and centered at locations ðx 0 0; y0 0Þ. However, the zero-order circle contains a different added phase noise r R with respect to the latter two circles. The remaining terms correspond to higher-order convolutions, which appear as broader, radially symmetric noisy light distributions. The phase-only autocorrelation peak shown in Figure 5d represents thus an interesting efficient method for sensing and analyzing a given input beam. Note that this distribution represents a generalization to arbitrary functions of the 2D vortex triplicator array reported in [ 12 ], where spiral phases were used as the functions / 1 ðxÞ and / 2 ðxÞ, and the input vortex beam could be detected.
6 Conclusions
In summary, we have generalized the optimum phase-only triplicator diffraction grating profile to an arbitrary phase-only hologram. We have shown that this triplicator hologram generates three equally-energetic terms comprising ideally 92.6 % of the input energy. The ± 1st diffraction orders yield the hologram reconstruction and its inverted complex-conjugate version, while the zero-order gives a bright spot. We have then demonstrated how this can be used to simultaneously obtain the correlation and the convolution of the pattern encoded in the triplicator hologram with another pattern, which is reproduced on the zeroorder. The result has been further generalized by combining two triplicator gratings along the horizontal and vertical directions, respectively. The resulting phase hologram generates an array of 3 3 equi-energetic terms, with two replicas of the encoded patterns aligned in the horizontal and vertical diffraction orders, and two correlation terms and two convolution terms in the diagonal and antidiagonal diffraction orders. All these properties have been experimentally verified using a high-resolution phase-only LCOS-SLM. Such triplicator holograms could find applications in the detection and quantitative evaluation of structured light beams.
Funding
We acknowledge financial support from Ministerio de Ciencia e Innovación, Spain( ref. PID2021-126509OB-C22 and PDC2022- 133332-C22).
Conflicts of interest This work has no financial or non-financial competing interests.
Data availability statement Data will be made available on request.
Author contribution statement
All coauthors contributed to the paper. SG contributed with the computer calculations and with the realization of the experiments. MMSL, PGM and IM contributed to the conceptualization and the design of the experiments. All authors contributed to the analysis of the results. IM contributed to writing the manuscript, and all authors contributed to the revision of the manuscript.
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