the computations can be reduced to 2×2 matrix algebra. This is the spirit of LCT. Electromagnetic wave propagation Assuming the system looks like as depicted in the figure, the wave travels from the (, ) plane to the (, ) plane. The Fresnel transform is used to describe electromagnetic wave propagation in free space: where is the wave number, is the wavelength, is the distance of propagation, is the imaginary unit. This is equivalent to LCT (shearing), when When the travel distance () is larger, the shearing effect is larger. Spherical lens With the lens as depicted in the figure, and the refractive index denoted as , the result is where is the focal length, and Δ is the thickness of the lens. The distortion passing through the lens is similar to LCT, when This is also a shearing effect: when the focal length is smaller, the shearing effect is larger. Spherical mirror The spherical mirror—e.g., a satellite dish—can be described as a LCT, with This is very similar to lens, except focal length is replaced by the radius of the dish. A spherical mirror with radius curvature of is equivalent to a thin lens with the focal length (by convention, for concave mirror, for convex mirror). Therefore, if the radius is smaller, the shearing effect is larger. Joint free space and spherical lens The relation between the input and output we can use LCT to represent If , it is reverse real image. If , it is Fourier transform+scaling If , it is fractional Fourier transform+scaling Basic properties In this part, we show the basic properties of LCT Given a two-dimensional column vector we show some basic properties (result) for the specific input below: Example The system considered is depicted in the figure to the right: two dishes – one being the emitter and the other one the receiver – and a signal travelling between them over a distance D. First, for dish A (emitter), the LCT matrix looks like this: Then, for dish B (receiver), the LCT matrix similarly becomes: Last, for the propagation of the signal in air, the LCT matrix is: Putting all three components together, the LCT of the system is: Relation to particle physics It has been shown that it may be possible to establish a relation between some properties of the elementary fermion in the Standard Model of particle physics and spin representation