Schwinger's variational method for the scattering phase shift produced by a central potential is adapted to reflection by a planar potential barrier (or well). The formulation is general, for an arbitrary transition between any two media, but the application here is limited to reflection at a barrier between media of equal potential energy. The simplest variational estimate for the reflection amplitude correctly tends to -1 at grazing incidence, as it must for any finite barrier. This is in contrast to the first order perturbation reflection amplitude, which diverges at grazing incidence. The same variational estimate is also correct to second order in the ratio of the interface thickness to the wavelength of the incident wave. The theory applies also to the reflection of the electromagnetic s (or transverse electric) wave at an interface between two media.
Schwinger's variational method for the scattering phase shift produced by a central potential is adapted to reflection by a planar potential barrier (or well). The formulation is general, for an arbitrary transition between any two media, but the application here is limited to reflection at a barrier between media of equal potential energy. The simplest variational estimate for the reflection amplitude correctly tends to -1 at grazing incidence, as it must for any finite barrier. This is in contrast to the first order perturbation reflection amplitude, which diverges at grazing incidence. The same variational estimate is also correct to second order in the ratio of the interface thickness to the wavelength of the incident wave. The theory applies also to the reflection of the electromagnetic s (or transverse electric) wave at an interface between two media.
At total reflection, beams are not reflected ideally from their point of incidence at the reflecting boundary but shifted by a length D (Goos–Haenchen effect). Based on the model of wave bundles consisting of a composite of totally reflected partial waves with varying angles of incidence, we treat the case of matter waves for a vanishing potential step at the boundary (the critical angle of total reflection tending to π/2). It is well known from Renard’s formula that the partial wave for grazing incidence has a vanishing shift D, with an angle α0 of incidence midway between the critical angle and π/2. However, the Goos–Haenchen shift of a wave bundle goes to infinity at α0 → π/2. The width W of the bundle also goes to infinity because the angular interval of the partial waves goes to zero. But the quantity D/W remains constant. D/W has a value of 18% if Artmann’s formula for D is used and is 4.5% with Renard’s more refined formula for D.
The phase-shift equations are derived in a form that is convenient for the analysis of both the radial Schr\"odinger equation and the problem of wave propagation in nonhomogeneous media. In the case of bound quantum states, the conservation condition is discussed for the sum of the classical action (or eikonal) and the phase. The expression for the reflection coefficient describes exactly two opposite limiting cases: the Fresnel reflection and the short-wavelength limit. In the case of wave propagation in symmetric media, both transmission and reflection coefficients are given as elementary functions of the phase shifts only.
We analyze transverse-electric (TE) wave reflection at a planar interface between a linear medium and a Kerr nonlinear half-space. Using an amplitude–phase formalism, we derive exact expressions for the reflection coefficient and the incident intensity as functions of the interface permittivity ε . For total internal reflection, we obtain the spatial evolution equation for the permittivity, which decays monotonically toward the linear value ε l . The shape of S inc ( ε ) is examined, yielding the condition for a maximum on the evanescent branch: 2 β 2 > ε 1 + ε l , where β is the conserved tangential wavenumber. When this condition holds, an intensity interval ( S c , S max ) exists where three steady states coexist (two evanescent and one propagating), giving rise to optical bistability. The propagating branch exhibits a square-root singularity at ε = β 2 , leading to a non-analytic derivative d S inc / d ε ; the reflectivity slope d R / d S inc remains finite. An alternative representation in terms of the normalized decay constant μ reveals a folded curve, indicating internal bistability within the total internal reflection regime. Both self-focusing ( α >0) and self-defocusing ( α <0) Kerr nonlinearities are treated. The propagation constant β appears as a free design parameter, allowing the switching intensity to be tuned via waveguide geometry. The two switching thresholds exhibit distinct critical behavior: a first-order jump at S max and a second-order (divergent derivative) transition at S c , the latter suggesting analogies with thermodynamic critical points. A Landau-type expansion of the effective potential is presented heuristically to support this connection. The results may be useful for all-optical switching and sensing applications.
AMPLITUDE METHOD.In this work, a simple derivation of the variable amplitude method using the variation of parameters to solve a differential equation is presented.The variable amplitude method was originally devised by Tikochinsky in 1977, using the quantum theory of scattering.The method is applied to two model potentials, the rectangular potential barrier and the Eckart potential, both with analytical solutions for the reflection coefficient.Numerical results will be compared with the exact values for several energies.The problem of calculating the reflection coefficient, usually involving extensive algebra as described in several textbooks, is reduced to solving a first order differential equation with initial condition.The method is very simple to apply, representing an attractive tool for teaching introductory quantum mechanics.A simple computer code is available from which reflection coefficients for the Eckart potential can be calculated.
A recurrence method for analytical and numerical evaluation of tunneling, transmission, and reflection amplitudes is developed. As the first step, a rule for composition of two arbitrary scatterers separated by a region of constant potential is obtained. Transmission and reflection amplitudes for this double-barrier potential are expressed in terms of transmission and reflection amplitudes for its subparts. As the length of the constant-potential region can be arbitrary and the subparts of a potential may, in turn, be arbitrary segmented potentials, one obtains formulas recurerence formulas which express the scattering amplitudes for the arbitrary segmented potential via the scattering amplitudes for the subparts into which the complete potential can be divided. The efficiency of the method is demonstrated by solving analytically the problem of scattering from locally periodic potentials. Since an arbitrary potential can be approximated by a set of infinitely narrow rectangular barriers, the recurrence formulas can be applied to any potential, giving, in the limit of zero-width segments, differential equations for transmission, and reflection amplitudes.
We analyze transverse-electric (TE) wave reflection at a planar interface between a linear medium and a Kerr nonlinear half-space. Using an amplitude-phase formalism, we derive exact expressions for the reflection coefficient and the incident intensity as functions of the permittivity at the interface. Under total internal reflection, the spatial evolution of the permittivity is described by a first-order differential equation and decays monotonically to the linear value. The incident intensity on the evanescent branch is shown to possess a maximum when 2𝛽2 > 𝜀1 +𝜀𝑙, where 𝛽 is the conserved propagation constant, 𝜀1 the linear permittivity, and 𝜀𝑙 the low-intensity permittivity of the Kerr medium. For intensities between the threshold 𝑆𝑐 and the maximum 𝑆max, three steady states coexist: two evanescent (TIR) solutions (one stable, one unstable) and one propagating solution. The propagating branch rises with an infinite slope at 𝜀 = 𝛽2 (square-root singularity), leading to an ultra-sharp switching transition. The relation between the normalized decay constant 𝜇 and the incident intensity exhibits a folded curve, revealing an “internal” bistability in the TIR regime. Both self-focusing (𝛼 > 0) and self-defocusing (𝛼 < 0) Kerr nonlinearities are treated. The propagation constant 𝛽 appears as a free design parameter, allowing the switching intensity to be tuned through waveguide geometry. These results provide a unified picture of optical bistability at a nonlinear interface, with potential applications in all-optical switching and sensing.