Related papers: Binary Representations of ABCD Matrices
The beam transfer matrix, often called the $ABCD$ matrix, is a two-by-two matrix with unit determinant, and with three independent parameters. It is noted that this matrix cannot always be diagonalized. It can however be brought by rotation…
With its three independent parameters, the $ABCD$ matrix serves as the beam transfer matrix in optics. If it is transformed to an equi-diagonal form, the matrix has only two independent parameters determined by optical devices. It is shown…
While the Lorentz group serves as the basic language for Einstein's special theory of relativity, it is turning out to be the basic mathematical instrument in optical sciences, particularly in ray optics and polarization optics. The beam…
We require decomposition methods for the ABCD-matrix formulation in rotationally symmetric paraxial geometric optics when designing a multi-component optical system from a given single paraxial specification (represented by an ABCD matrix)…
It is noted that two-by-two S-matrices in multilayer optics can be represented by the Sp(2) group whose algebraic property is the same as the group of Lorentz transformations applicable to two space-like and one time-like dimensions. It is…
Geometric optics is often described as tracing the paths of non-diffracting rays through an optical system. In the paraxial limit, ray traces can be calculated using ray transfer matrices (colloquially, ABCD matrices), which are 2x2…
Paraxial lens optics is discussed to study the continuity properties of the $ABCD$ beam transfer matrix. The two-by-two matrix for the one-lens camera-like system can be converted to an equi-diagonal form by a scale transformation, leaving…
We consider a construction of the fundamental spin representations of the simple Lie algebras $\mathfrak{so}(n)$ in terms of binary arithmetic of fixed width integers. This gives the spin matrices as a Lie subalgebra of a…
A binary matrix can be scanned by moving a fixed rectangular window (submatrix) across it, rather like examining it closely under a microscope. With each viewing, a convenient measurement is the number of 1s visible in the window, which…
The two-by-two Sp(2) matrix has three parameters with unit determinant. Yet, there are no established procedures for diagonalizing this matrix. It is shown that this matrix can be written as a similarity transformation of the two-by-two…
We find a simple expression for critical curves of a binary gravitational lens. On the basis of this, we present a parametric representation of such curves. The caustics can also be expressed with the same parameterization. The present…
It is not always possible to diagonalize the optical $ABCD$ matrix, but it can be brought into one of the four Wigner matrices by a similarity transformation. It is shown that the four Wigner matrices can be combined into one matrix with…
The two-by-two scattering matrix for one-dimensional scattering processes is a three-parameter Sp(2) matrix or its unitary equivalent. For one-dimensional crystals, it would be repeated applications of this matrix. The problem is how to…
There are many two-by-two matrices in layer optics. It is shown that they can be formulated in terms of a three-parameter group whose algebraic property is the same as the group of Lorentz transformations in a space with two space-like and…
We propose that the height-angle ray vector in matrix optics should be complex, based on a geometric algebra analysis. We also propose that the ray's 2x2 matrix operators should be right-acting, so that the matrix product succession would…
While modern optics is largely a physics of harmonic oscillators and two-by-two matrices, it is possible to learn about some hidden properties of the two-by-two matrix from optical systems. Since two-by-two matrices can be divided into…
It is shown that the one-lens system in para-axial optics can serve as an optical computer for contraction of Wigner's little groups and an analogue computer which transforms analytically computations on a spherical surface to those on a…
The proposed physical duality known as 3d mirror symmetry relates the geometries of dual pairs of holomorphic symplectic stacks. It has served in recent years as a guiding principle for developments in representation theory. However, due to…
We present a novel and effective binary representation for convex shapes. We show the equivalence between the shape convexity and some properties of the associated indicator function. The proposed method has two advantages. Firstly, the…
Let A,B denote binary forms of order d, and let C_{2r-1} = (A,B)_{2r-1} be the sequence of their linear combinants for r between 1 and (d+1)/2. It is known that C_1 and C_3 together determine the pencil generated by A and B, and hence…