Rodrigo Martín-Romo, José AugustoAlieva, Tatiana KrasheninnikovaCalvo Padilla, María Luisa2023-06-202023-06-202007-03-051. H. M. Ozaktas, Z. Zalevsky, and M. Alper Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (John Wiley and Sons, NY, USA (2001). 2. M. W. Beijersbergen, L. Allen, H. E. L. O. Van der Veen and J. P. Woerdman, "Astigmatic laser mode converters and transfer of orbital angular momentum”, Opt. Commun. 96, 123-132 (1993). 3. E. G. Abramochkin and V. G. Volostnikov, "Beam transformations and nontransformed beams”, Opt. Commun. 83, 123-135 (1991). 4. E. G. Abramochkin and V. G. Volostnikov, "Generalized Gaussian beams”, J. Opt. A.: Pure Appl. Opt. 6, S157- S161 (2004). 5. G. F. Calvo, "Wigner representation and geometric transformations of optical orbital angular momentum spatial modes”, Opt. Lett. 30, 1207-1209 (2005). 6. R. Simon and K. B. Wolf, "Structure of the set of paraxial optical systems”, J. Opt. Soc. Am. A 17, 342-355 (2000). 7. K. B. Wolf, Geometric Optics on Phase Space (Springer-Verlag, Berlin, 2004). 8. J. A. Rodrigo, T. Alieva, M. L. Calvo, "Optical system design for ortho-symplectic transformations in phase space”, J. Opt. Soc. Am. A 23, 2494-2500 (2006). 9. T. Alieva, V. Lopez, F. Agullo Lopez, L. B. Almeida, "The fractional Fourier transform in optical propagation problems, " J. Mod. Opt. 41, 1037-1044 (1994). 10. M. Bastiaans and T. Alieva, "First-order optical systems with unimodular eigenvalues”, J. Opt. Soc. Am. A 23, 1875-1883 (2006). 11. T. Alieva and M. Bastiaans, "Mode mapping in paraxial lossless optics”, Opt. Lett. 30, 1461-1463 (2005). 12. M. Abramowitz and I. A. Stegun, eds., Pocketbook of Mathematical Functions, Frankfurt am Main, Germany (1984). 13. I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series and products (Academic Press, NY, USA, 1996). 14. V. V. Kotlyar, S. N. Khonina, A. A. Almazov, V. A. Soifer, K. Jefimovs and J. Turunen, "Elliptic Laguerre- Gaussian beams”, J. Opt. Soc. Am. A 23, 43-56, (2006).1094-408710.1364/OE.15.002190https://hdl.handle.net/20.500.14352/51043© 2007 Optical Society of America. Spanish Ministry of Education and Science is acknowledged for financial support, project TEC2005-02180/MIC.In this work we formulate the main properties of the gyrator operation which produces a rotation in the twisting (position-spatial frequency) phase planes. This transform can be easily performed in paraxial optics that underlines its possible application for image processing, holography, beam characterization, mode conversion and quantum information. As an example, it is demonstrated the application of gyrator transform for the generation of a variety of stable modes.engjournal articlehttp://dx.doi.org/10.1364/OE.15.002190http://www.opticsinfobase.org/open access535Orbital Angular-MomentumOptical-SystemsGaussian BeamsÓptica (Física)2209.19 Óptica Física