Slide-rule-like property of Wigner’s little groups and cyclicSmatrices for multilayer optics

Abstract

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 noted also that Wigner's little groups have a sliderule-like property which allows us to perform multiplications by additions. It is shown that these two mathematical properties lead to a cyclic representation of the S-matrix for multilayer optics, as in the case of ABCD matrices for laser cavities. It is therefore possible to write the N -layer S-matrix as a multiplication of the N single-layer S-matrices resulting in the same mathematical expression with one of the parameters multiplied by N. In addition, it is noted, as in the case of lens optics, multilayer optics can serve as an analogue computer for the contraction of Wigner's little groups for internal space-time symmetries of relativistic particles. 42.25.Gy, 42.15.Dp, 02.20.Rt, 11.30.Cp

ABCD matrices as similarity transformations of Wigner matrices and periodic systems in optics

“…The computation of this process was started in reference [5]. There, the cycle had to start from the midway in one of the media, but no explanation was given why.…”

“…In 2003, Georgieva and Kim used a method based on the Lorentz group to study multilayer optics [5]. These authors also end up with the inconvenience of starting their cycle from the midway in one of the layers.…”

“…This was called "slide-rule" property in reference [5], but it is more appropriate to call it "logarithmic property." The remaining three matrices in equation (1) have this logarithmic property.…”

“…Indeed, the Sp(2) group has been playing a pivotal role both in classical and quantum optics. It is the basic language for lens optics [2,3], multilayer optics [4,5], laser cavities [6], as well as squeezed states of light in quantum optics [7,8].…”

Pauli algebraic forms of normal and nonnormal operators

“…Well known, especially in the past decade, the group theory of SL͑2,c͒ and of some of its subgroups was extensively applied in various fields of classical and quantum optics, e.g., ray optics, 40 beam propagation through firstorder systems, 41 analysis of the states of light with orbital angular momentum, 42 polarization optics, 43 multilayer optics, 44,45 interferometry, 46 and coherent and squeezed states of light. 47 Bearing in mind that SL͑2,c͒ is locally isomorphic to the six-parameter Lorentz group SO͑3,1͒, a physical system that can be analyzed in terms of SL͑2,c͒ language can be equally explained in the language of the Lorentz group.…”