J. Eur. Opt. Society-Rapid Publ. 2025, 21, 33 Ó The Author( s), published by EDP Sciences, 2025 https:// doi. org / 10.1051 / jeos / 2025027 Available online at: https:// jeos. edpsciences. org
Journal of the European Optical Society-Rapid Publications
RESEARCH ARTICLE
Triplicator phase-only hologram and its use as a snapshot optical convolver and correlator
Shang Gao 1, María del Mar Sánchez-López 1, 2,*
, Pascuala García-Martínez 3, and Ignacio Moreno 4, 1
1 |
Instituto de Bioingeniería, Universidad Miguel Hernández, 03202 Elche, Spain |
2 |
Departamento de Física Aplicada, Universidad Miguel Hernández, 03202 Elche, Spain |
3 |
Departament d’ Optica i Optometria i Ciencies de la Visió, Universitat de València, 46100 Burjassot, Spain |
4 |
Departamento de Ciencia de Materiales Óptica y Tecnología Electrónica, Universidad Miguel Hernández, 03202 Elche, Spain |
Received 24 April 2025 / Accepted 28 May 2025
Abstract. In this work we extend the optimum phase triplicator diffraction grating profile to a phase-only hologram. This modified hologram generates three equally-energetic orders with the highest diffraction efficiency, yielding a direct version and an inverted complex conjugate version of the target pattern in the ± 1st orders, respectively, and a delta function in the DC zero order. When combined with another phase-only function, the resulting hologram yields convolution and correlation terms in the ± 1st orders, respectively, while the target function encoded in the added phase appears in the zero order. Experimental results obtained with a high-resolution liquid-crystal phase-only spatial light modulator( SLM) demonstrate the proposed design.
Keywords: Diffraction Gratings, Computer-Generated Holograms, Convolution, Correlation.
1 Introduction
A triplicator is a diffraction grating that generates three equally intense 0th and ± 1st diffraction orders. The analytical derivation of the optimum phase-only triplicator presented by Gori et al. in [ 1 ] provided the theoretical phase profile leading to an achievable maximum total efficiency of 92.6 % by adding up the three target diffraction orders. Another significative contribution was recently introduced by Gori and coworkers, where affine transformations of the original profile were applied to yield nonaligned equi-intense orders with the same theoretical efficiency [ 2 ].
Being a continuous phase profile, the experimental realization of the optimum triplicator was not an easy task. Borghi et al. [ 3 ] derived a four-level triplicator design with unequal period spacing that shows a maximum high efficiency( 91.0 %) very close to the optimal value and could be fabricated using lithography. Modern high-resolution phase-only liquid crystal spatial light modulators( SLM) [ 4, 5 ] are devices that enable continuous phase modulation, accurately controlled by a computer through the addressed gray level image. Hence, the continuous profile can be accurately reproduced in such pixelated displays, and the optimum triplicator profile has been experimentally
* Corresponding author: mar. sanchez @ umh. es demonstrated when displayed with large enough periods [ 6, 7 ]. When the gratings are displayed with shorter periods, quantization reduces the efficiency. The generation of binary triplicator phase gratings operating at the SLM spatial resolution limit has also been recently analyzed [ 8 ].
The optimum triplicator function derived by Gori et al. [ 1 ] is finding applications in combination with other functions. For example, it is employed in parallel signal processing [ 9 ] and for developing trifocal diffractive intraocular lenses [ 10, 11 ]. In combination with spiral phases embedded in orthogonal directions, it has been applied for realizing a 3 3 array of optical vortices with different topological charges [ 12 ]. The triplicator profile has also been applied to design a metasurface that generates interleaved vortexbeamsatdifferentwavelengths [ 13 ]. Furthermore, using triplicators of varying periods in a phase-shifting interferometer has allowed us to retrieve the phase values of tailored phase gratings encoded on an LCOS-SLM [ 14 ].
In this work, we go a step beyond the triplicator grating and apply the optimum triplicator profile to a phase-only Fourier transform( FT) computer-generated hologram( CGH). It is shown that the application of the triplicator profile to the hologram’ s phase raises three equallyenergetic terms: the hologram’ s direct object reconstruction, an inverted complex-conjugate version, and a delta function. Then, it is shown that this new triplicator hologram can be used as an optical convolver and optical
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