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UNDERSTANDING HALF-WAVE LOSS, PHASE SHIFT IN A SYMMETRIC BEAM SPLITTER, AND SECOND-ORDER INTERFERENCE FROM THE PERSPECTIVE OF THE BARRIER MODEL

Chunguang Du

Published Aug 1, 2026• 0 citations
Publisher

Abstract

Abstract When a light wave is reflected at the interface between two semi-infinite media, the relative phase between the reflected and incident amplitudes is a fundamental optical effect. Another fundamental phase relation arises in a symmetric beam splitter between its reflection and transmission amplitudes. Both effects originate from the phase structure of wave scattering, although their underlying connection is not always clearly understood. As a pedagogical investigation, this paper systematically analyzes these phase relations using simple one-dimensional potential models in quantum mechanics and provides intuitive physical explanations. For a symmetric square barrier, the reflected and transmitted amplitudes necessarily differ in phase by ±π/2, with the reflection phase referenced at the interface on the incident side and the transmission phase referenced at the interface on the exit side. The result is further extended to general symmetric barriers, for which the detailed form of the barrier determines only whether the phase difference is +π/2 or -π/2. The role of this phase structure in the second-order interference of two identical particles is then examined. Through this sequence of analysis, half-wave loss at a reflecting interface, the phase relation in a symmetric beam splitter, and multiparticle quantum interference are incorporated into a unified wave-scattering framework. This treatment provides a useful pedagogical reference for related topics in quantum mechanics and optics courses.

DOI
10.26599/phys.2026.9320511

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Abstract When a light wave is reflected at the interface between two semi-infinite media, the relative phase between the reflected and incident amplitudes is a fundamental optical effect. Another fundamental phase relation arises in a symmetric beam splitter between its reflection and transmission amplitudes. Both effects originate from the phase structure of wave scattering, although their underlying connection is not always clearly understood. As a pedagogical investigation, this paper systematically analyzes these phase relations using simple one-dimensional potential models in quantum mechanics and provides intuitive physical explanations. For a symmetric square barrier, the reflected and transmitted amplitudes necessarily differ in phase by ±π/2, with the reflection phase referenced at the interface on the incident side and the transmission phase referenced at the interface on the exit side. The result is further extended to general symmetric barriers, for which the detailed form of the barrier determines only whether the phase difference is +π/2 or -π/2. The role of this phase structure in the second-order interference of two identical particles is then examined. Through this sequence of analysis, half-wave loss at a reflecting interface, the phase relation in a symmetric beam splitter, and multiparticle quantum interference are incorporated into a unified wave-scattering framework. This treatment provides a useful pedagogical reference for related topics in quantum mechanics and optics courses.
The familiar input-output relations for an optical beam splitter are generalized to allow for linear absorption by the medium forming the mirror. Beam-splitter losses generally affect the noise levels detectable in experiments involving nonclassical light. When employed to investigate two-photon interference effects, a lossy beam splitter can lead to apparent nonlinear absorption, which, in the most extreme case, leads to either both or neither of the photons being absorbed. The degree of second-order coherence of antibunched light can be maintained on transmission through the beam splitter but any amplitude squeezing in the incident light is degraded.
The phase changes experienced by a wave reflected from an interface are a familiar fixture in introductory physics courses. Examples include the ±π phase shift acquired upon reflection from a hard mirror (for example, in thin film interference patterns and Newton’s rings) and the −π∕2 phase shift studied in soft mirror reflections (for example, in mirage ray paths and the WKB connection formulas). We focus on the transition between these two limits, where the phase change upon reflection evolves continuously between the limiting cases of −π and −π∕2. We study a simple quantum system that exhibits this transition: a one-dimensional free electron reflected from a linear potential.
In the propagation of an electron through a one-dimensional asymmetric complex potential, it is known that while the conventional Green function reciprocity symmetry will ensure transmission to be symmetric between a “left-incident” and a “right-incident” beam, no such symmetry exists for the case of reflection. Here we derive generalized reciprocity relations for both the amplitude and phase of the reflected waves as constraints on the left- and right-incident beams, in complete analogy to what was established in optics. We further provide illustrations of these relations via direct analytical calculations in the case of a real potential, and via numerical studies in the case of a complex potential.
The stationary phase method is often employed for computing tunneling phase times of analytically-continuous gaussian or infinite-bandwidth step pulses which collide with a potential barrier. The indiscriminate utilization of this method without considering the barrier boundary effects leads to some misconceptions in the interpretation of the phase times. After re-examining the above barrier diffusion problem where we notice the wave packet collision necessarily leads to the possibility of multiple reflected and transmitted wave packets, we study the phase times for tunneling/reflecting particles in a framework where an idea of multiple wave packet decomposition is recovered. To partially overcome the analytical incongruities which rise up when tunneling phase time expressions are obtained, we present a theoretical exercise involving a symmetrical collision between two identical wave packets and a one-dimensional squared potential barrier, where the scattered wave packets can be recomposed by summing the amplitudes of simultaneously reflected and transmitted waves.
Exact numerical calculations are made for the scattering of quantum mechanical particles from a square two-dimensional potential barrier. This treatment is an exact analog of both frustrated total reflection of perpendicularly polarized light and the longitudinal Goos-Hänchen shift. Quantum mechanical streamlines (which are analogous to either classical mechanical trajectories or optical rays) are plotted. These streamlines are smooth, continuous, and have continuous first derivatives even through the classically forbidden region. The streamlines form quantized vortices surrounding each of the nodal points (which result from interference between the incident and reflected waves). Similar vortices occur in reactive collisions of H + H2 (McCullough and Wyatt; Kuppermann, Adams, and Truhlar) and undoubtedly play an important role in molecular collision dynamics. The theory for these vortices is given in a companion paper. A comparison is given between our numerical calculations and the stationary phase approximation (SPA). Although our incident wave packet has a half-width of more than one de Broglie wavelength in contrast to the SPA which replaces the diffuse beams by rays with delta function profiles, the agreement was surprisingly good for both the Goos-Hänchen shifts and for the reflection coefficients. However, we found that the Goos-Hänchen shift for the transmitted beam is significantly smaller than for the reflected beam, although in the stationary phase approximation these two shifts are equal. Furthermore, we found that the scattering from the potential barrier has very little effect on the shape of the wave packets. The power series expansion of the incident Debye-Picht wave packet ψI has an extremely small radius of convergence, whereas the power series for ψI*ψI has a radius of convergence of more than two de Broglie wavelengths. The imaginary velocity viis introduced into the Madelung-Landau-London hydrodynamical formulation of quantum mechanics. The corresponding imaginary streamlines will be considered in a forthcoming paper. The time-independent Schrödinger equation for real wavefunctions is reduced to solving the nonlinear first order partial differential equation: ℏ▿·vi = 2(E − V) + vi2. Here, vi is irrotational. This equation may lead to interesting new methods of solving the Schrödinger equation. It does lead to a generalization of the Prager-Hirschfelder perturbation scheme which invokes an electrostatic analogy. The philosophical implications of the hydrodynamical formulation of quantum mechanics are discussed. Penetration of a one-dimensional square potential barrier is used to demonstrate exactly how tunneling occurs by particles ``riding over the barrier'' à la Bohm. Cases are cited where quantum and classical mechanical motions are identical.
A dissipative beam-splitter (BS) has been analyzed by modeling the losses in the BS due to the excitation of optical phonons. The losses are obtained in terms of the BS medium properties. The model simplies the picture by treating the loss mechanism as a perturbation on the photon modes in a linear, non-lossy medium in the limit of small losses, instead of using the full field quantization in lossy, dispersive media. The model uses second order perturbation in the Markoff approximation and yields the Beer's law for absorption in the first approximation, thus providing a microscopic description of the absorption coecient. It is shown that the fluctuations in the modes get increased because of the losses. We show the existence of quantum interferences due to phase correlations between the input beams and it is shown that these correlations can result in loss quenching. Hence in spite of having such a dissipative medium, it is possible to design a lossless 50-50 BS at normal incidence which may have potential applications in laser optics and dielectric-coated mirrors.