Wolf, Kurt BernardoAlieva Krasheninnikova, Tatiana2023-06-202023-06-202008-02-011084-752910.1364/JOSAA.25.000365https://hdl.handle.net/20.500.14352/51265© 2008 Optical Society of America. T. Alieva acknowledges the Spanish Ministry of Education and Science for financial support (project TEC 2005- 02180/MIC). K. B. Wolf acknowledges the support of the SEP-CONACYT (México) project IN102603 “Óptica Matemática.” The authors are grateful to the UCM/ UNAM Collaboration Agreement for making this joint work possible. We appreciate Guillermo Krötzsch for assistance with the graphics, and Luis Edgar Vicent for Figs. 2 and 5.Hermite-Gauss and Laguerre-Gauss modes of a continuous optical field in two dimensions can be obtained from each other through paraxial optical setups that produce rotations in (four-dimensional) phase space. These transformations build the SU(2) Fourier group that is represented by rigid rotations of the Poincare sphere. In finite systems, where the emitters and the sensors are in N x N square pixellated arrays, one defines corresponding finite orthonormal and complete sets of two-dimensional Kravchuk modes. Through the importation of symmetry from the continuous case, the transformations of the Fourier group are applied on the finite modes.engjournal articlehttp://dx.doi.org/10.1364/JOSAA.25.000365http://www.opticsinfobase.org/open access535Fractional fourier-transformsOrbital angular-momentumSystemsOscillatorGeometryDynamicsÓptica (Física)2209.19 Óptica Física