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Journal of the Optical Society of America A

Use of Einstein’s addition law in studies of reflection by stratified planar structures

J. M. Vigoureux

Published Aug 1, 1992• 56 citations
Publisher

Abstract

I show that the reflection coefficient of any stratified planar structure can be obtained by using a complex generalization of Einstein's addition theorem for parallel velocities. This result also applies to multiple quantum wells. It provides a new mathematical tool in optics and in quantum theory and may lead to useful algorithms in computing. It may also give a new insight into special relativity. The composition law of velocities, in fact, no longer appears as a specific result of special relativity but rather as the expression, in the particular case of kinematics, of a more general law of physics. The possible use of the composition law of probability amplitude in quantum theory is also presented.

DOI
10.1364/josaa.9.001313

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I show that the reflection coefficient of any stratified planar structure can be obtained by using a complex generalization of Einstein's addition theorem for parallel velocities. This result also applies to multiple quantum wells. It provides a new mathematical tool in optics and in quantum theory and may lead to useful algorithms in computing. It may also give a new insight into special relativity. The composition law of velocities, in fact, no longer appears as a specific result of special relativity but rather as the expression, in the particular case of kinematics, of a more general law of physics. The possible use of the composition law of probability amplitude in quantum theory is also presented.
Mathematical methods in optics show that the overall amplitude resulting from the infinite summation over all the various alternative paths that the light can take inside a Fabry-P\'erot interferometer can directly be obtained by using a complex generalization of Einstein's composition law for parallel velocities. Moreover, this result appears to be quite general. In fact, the reflection coefficient for any number of interfaces or quantum wells can be easily obtained by using the composition law. This suggests that a more general way to obtain the probability amplitude for an event which usually needs infinite summation on virtual paths (as in the Fabry-P\'erot interferometer) is to compose amplitudes following Einstein's addition law. I explore this problem and show how the composition law avoids divergences and guarantees the modulus of any total wave function to be less than (or equal to) unity. By examining its physical meaning in the light of probability theory, I also show that it provides an interesting expression for nonlocality and nonseparability.
An alternative approach to the one-dimensional scattering problem is presented. We show that the reflection and the transmission coefficients of any number of quantum wells or barriers can directly be obtained by using two mathematical methods. The first method arises from a generalization of the so-called elementary symmetric functions used in the mathematical theory of polynomials; the second one consists of a complex generalization of Einstein's addition law for parallel velocities. These laws for the reflection and transmission amplitude coefficients permit a rapid evaluation of the transmittivity of any kind of potential barrier. Some examples are considered. Results are compared with the ones obtained with the WKB method. We also present a numerical study to test the stability of the method.
Using the well-known matrix formulation of the reflection and transmission of electromagnetic waves by a stratified planner structure, we show that the reflection and transmission coefficients of any number of isotropic media can be written by a simple general formula. This formula uses the so-called elementary symmetric functions that are extensively used in the mathematical theory of polynomials. The approach is then applied to the quantum scattering. We show that the reflection and transmission coefficients of any number of quantum wells or barriers can be written in the similar way. Finally, one-dimensional scattering from a series of delta-function barriers (a system that is called Dirac Comb) is studied. The computed numerical illustrations compared with the earlier results based on the transfer matrix and Chebychev polynomials reveal an excellent agreement.
This paper uncovers a surprising connection between special relativity and probability theory through simple competitive games. By modeling contests—first as draws from an urn and then as races between exponentially distributed events—we derive expectation values that mirror the relativistic velocity addition formula. Remarkably, these probabilistic outcomes obey Einstein’s rule for combining velocities, suggesting a structural analogy between relativistic kinematics and probabilistic competition. This finding blurs the traditional divide between the deterministic framework of relativity and the stochastic nature of quantum mechanics, hinting that probabilistic principles may underpin aspects of special relativity. We provide rigorous derivations and explore implications for interdisciplinary studies in physics and probability.
Often in optics we must deal with layered media when the optical properties change only in one specific direction (say, along the z-axis), while in any plane transverse to this direction, the optical properties are invariant. Such media with dielectric and magnetic responses given by ε(z) and µ(z) as functions of only z are also referred to as stratified media. There are many examples Wave Trends of such media. The most useful one refers to the optical interference coatings that are essential for most optical instruments. Reflection and transmission through these structures can be handled in terms of simple 2 × 2 matrices when the constituent layers are homogeneous and isotropic. In the case of an anisotropic layered medium, we can develop a 4 × 4 matrix formulation for uniaxial materials [61, 62]. In this chapter we deal with isotropic homogeneous layers and develop the characteristics matrix formalism to obtain the refection and transmission coefficients (see also Ref. [31]). We discuss how the dispersion in such structures can be engineered to lead to slow and fast light [7]. We apply the technique to investigate the modes of a structure. We probe the effects of finite temporal width of a pulse leading to the Wigner delay [63]. The space equivalent of Wigner delay, also known as the Goos-Ha¨nchen shift [64], for a spatially finite beam is then discussed. Finally, we show how someone using such structures can realize perfect transmission and coherent perfect absorption. We will define all the necessary notions and concepts as we go along.
Special relativity is foundation of many branches of modern physics, of which theoretical results are far beyond our daily experience and hard to realized in kinematic experiments. However, its outcomes could be demonstrated by making use of convenient substitute, i.e. squeezed light in present paper. Squeezed light is very important in the field of quantum optics and the corresponding transformation can be regarded as the coherent state of SU(1; 1). In this paper, the connection between the squeezed operator and Lorentz boost is built under certain conditions. Furthermore, the additional law of relativistic velocities and the angle of Wigner rotation are deduced as well.