Study On Fourier Transform Of Orthogonal Cylindrical Lens

At the beginning of the 19 th century,French mathematician J.Fourier put forward the concept of Fourier transform,which was introduced into optics by scientist P.M.Duffieux.With the progress of science and technology,higher requirements were put forward for Fourier transform by diferentresearch fields.In 1980,Namias put forward the mathematical definition and properties of fractional Fourier transform.At the end of the 20 th century,Ozaktas and Mendelov introduced fractional Fourier transform into optic.After that,a series of optical systems for fractional Fourier transform are proposed.Since then,fractional Fourier transform optics has attracted widespread attention and become a research hotspot in optical theory and application.In 1984,Szoplik et al.proposed a method of realizing deformed Fourier transform: under plane wave irradiation,a group of cylindrical lenses with different focal length and perpendicular to each other were used to realize the deformed Fourier transform.This paper chooses orthogonal cylindrical lenses as the research object.Firstly,the phase transformation characteristics of orthogonal cylindrical lenses are studied and calculated.By analyzing the physical mechanism of phase factor of orthogonal cylindrical lenses,we can see that orthogonal cylindrical lenses play different phase delay roles in x and y directions compared with traditional spherical lenses,and can change the curvature before and after orthogonal cylindrical lenses.Rate radius and medium control change the phase transition characteristics of orthogonal cylindrical lenses.Then the Fourier transform characteristics of orthogonal cylindrical lenses are derived by using the phase transformation formula and Fresnel diffraction formula of orthogonal cylindrical lenses.Morever,the complex amplitude of the optical field of the symmetrical single orthogonal cylindrical lenssystem and the biorthogonal cylindrical lens system is derived.Compared with the definition of fractional Fourier transform,it is concluded that the fractional Fourier transform and its constraints of different orders can be realized in the X and Y directions when the fractional Fourier transform is realized by using a single orthogonal cylindrical lens and a biorthogonal cylindrical lens.Finally,the reliability and feasibility of the conclusions are verified by computer simulation experiments on the basis of the above theoretical inference.