Photon Flux Density in the Diffraction Pattern During Scattering of H-polarized Photons by the Infinite Grating of Metallic Strips

Abstract

The problem of diffraction of H-polarized light at normal incidence on an unlimited sequence of infinitely thin metallic strips is solved. A quantum-mechanical approach to the problem of diffraction is applied. Light wave is represented as a flow of particles-photons. Probability of a photon in front of and behind the grating is described by two-dimensional psi-function-single-valued, continuous and restricted, which satisfies the two-dimensional Schrödinger equation. A strict solution of the problem about determination of psi-function of a photon dissipated by the grating is led down to the boundary Riemann-Hilbert problem. The solution is obtained in the view of the convergent infinite system of linear algebraic equations. The system is suitable for any relation between the wavelength and the period of the structure and any relation between the width of a slit and the width of a strip and is convenient for numerical calculations with the help of PC. Analysis of the expressions obtained for psi-function gives the possibility to conclude the following. Photons passed through or reflected by the grating get the discrete values of momentum during the interaction with the grating and deviate at discrete angles which are determined by the obtained expressions. There are intensity maxima in the points where photons come and minima-in the points where photons do not come. As follows from the analysis of the values of the photon momentum, the possible values of the constituent of the photon momentum perpendicular to its initial direction of motion are determined by even values of the "quantum" of momentum which magnitude is determined by the grating period. This result may be examined as a rule of selection of possible values of the perpendicular constituent of the photon momentum. As follows from numerical calculations, the diffraction maxima are located in front of a slit and have some internal structure, depending on the relation between the wavelength and the grating period. When the ratio of the wavelength to the grating period decreases, the diffraction peak turns out to be slightly modulated. When the ratio of the wavelength to the grating period is more than one, the diffraction pattern vanishes and we have homogeneous illuminance.

Diffraction of E-polarized Photons on Periodic Grating of Metal Strips

“…The final one is determined by the even values of the "quantum" of the momentum ħ/l of an electron moving in the metal strip. It should be noted that in [4] it was assumed that the width of the potential well is equal to the width of the metal strip l -d. But the results obtained in this 03032-3 work and in [2] clarify this assumption. The width of the potential well is determined by the grating period.…”

“…Here we must make a reservation comparing the representations of the -function for the E-and Hpolarizations [2]. They have the same form, and the same conclusions follow from their analysis.…”

“…In this case, the approach was based on application of laws of conservation of energy and momentum at the elastic collision of a photon with an electron moving in the strip (what can be considered as practically free movement in an infinitely deep potential well [5]). As noted in [2], the result follows from the analysis of the ycomponent of the momentum py, which can be examined as some rule of selection of possible values of the y-component of the photon momentum acquired as a result of collision with an electron. The final one is determined by the even values of the "quantum" of the momentum ħ/l of an electron moving in the metal strip.…”

“…To obtain a distinguishable interference pattern in the Young installation, the distance L between the screen and the sources must be much greater than the distance b between the sources (two slits in the nontransparent screen). In this case, the position of the intensity maxima is observed at the value of the x coordinate, according to [2], equal to…”

“…Equalizations ( 7), (8), are reduced to the Riemann-Hilbert problem, the exact solution of which is represented as an infinite system of linear algebraic equations relative to the coefficients bn [2]  …”