Teaching Fourier optics through ray matrices

Abstract

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.

Using ray matrices to derive analytical expressions of optical aberrations

“…However this aberration function theory can be quite annoying for students since the polynomial expressions are large, and analytical expressions are difficult to derive. Following our previous works [3,6], we believe that the application of a common tool, as the ray matrix theory, to teach different modules of an Optics course permits the students to achieve a deeper understanding of the practical aspects of its application, and provides a uniform and elegant framework. It is our purpose here to add an additional step in this sequence, and use the ray matrix formalism to explain aberrations concepts.…”

“…For instance, in Refs. [3,4] we presented a full derivation of Fourier Optics systems based on this matrix formalism. It is also commonly employed to analyze stability conditions in optical resonators [5], as well as its connection with the fractional Fourier transform [6].…”

Teaching stable two-mirror resonators through the fractional Fourier transform

“…For instance, the analysis of the ray matrix permits direct derivation of geometrical properties of the optical systems: the C parameter directly gives the optical power of the system, while the imaging condition (input and output planes are conjugated) is directly obtained by imposing the condition B = 0 [9]. We recently proposed the use of the ray matrix formalism also to analyse Fourier optics systems [11]. Optical systems providing an exact Fourier transform relation between the wavefront's amplitude at the input and output planes can be very easily identified since the ray matrix A and D parameters become null.…”

“…The ABCD ray matrix of an optical system performing an exact FRFT between the input and output planes is given by [10,11]…”

“…The comparison with equation (11) reveals that this system provides a real valued FRFT when the distance d is between 0 and 2f , the fractional order p being given by the relation…”