Teaching Fourier optics through ray matrices

In this work we examine the use of ray-transfer matrices for teaching and for deriving some topics in a Fourier optics course, exploiting the mathematical simplicity of ray matrices compared to diffraction integrals. A simple analysis of the physical meaning of the elements of the ray matrix provides a fast derivation of the conditions to obtain the optical Fourier transform. We extend this derivation to fractional Fourier transform optical systems, and derive the order of the transform from the ray matrix. Some examples are provided to stress this point of view, both with classical and with graded index lenses. This formulation cannot replace the complete explanation of Fourier optics provided by the wave theory, but it is a complementary tool useful to simplify many aspects of Fourier optics and to relate them to geometrical optics.

[1]  S. A. Collins.  Lens-System Diffraction Integral Written in Terms of Matrix Optics , 1970 .

[2]  Karlton Crabtree,et al.  Fractional Fourier transform optical system with programmable diffractive lenses. , 2003, Applied optics.

[3]  L. Bernardo.  ABCD MATRIX FORMALISM OF FRACTIONAL FOURIER OPTICS , 1996 .

[4]  A. Thetford.  Introduction to Matrix Methods in Optics , 1976 .

[5]  R A Lilly,et al.  Ray-matrix approach for diffractive optics. , 1993, Applied optics.

[6]  H. Ozaktas,et al.  Fractional Fourier transforms and their optical implementation. II , 1993 .

[7]  Rainer G. Dorsch,et al.  Fractional Fourier transform used for a lens-design problem. , 1995, Applied optics.

[8]  Henri H. Arsenault,et al.  Factorization of the transfer matrix for symmetrical optical systems , 1983 .

[9]  M. Teich,et al.  Fundamentals of Photonics , 1991 .

[10]  J. Shamir,et al.  Root and power transformations in optics , 1995 .

[11]  A. Lohmann.  Image rotation, Wigner rotation, and the fractional Fourier transform , 1993 .

[12]  H. Ozaktas,et al.  Fractional Fourier transforms and their optical implementation. II , 1993 .

[13]  Joseph Shamir,et al.  First-order optics—a canonical operator representation: lossless systems , 1982 .

[14]  Ronald N. Bracewell,et al.  The Fourier Transform and Its Applications , 1966 .

[15]  K. W. Cattermole.  The Fourier Transform and its Applications , 1965 .

[16]  Toshimitsu Asakura,et al.  Optical Fourier-transform theory based on geometrical optics , 2002 .

[17]  Maria Josefa Yzuel.  Basic education in optics for physicists , 1992, Other Conferences.

[18]  J. Goodman.  Introduction to Fourier optics , 1969 .

[19]  A. Gerrard,et al.  Introduction to Matrix Methods in Optics , 1975 .