Reexamination of conservation of momentum and energy in partially coherent electromagnetic waves

Abstract

We reviewed the conservation of momentum and energy in partially coherent electromagnetic wave fields in a unified perspective. We find that there is interference between the radiation from polarization and magnetization sources in the momentum flow unlike the result proposed in the previous study. The interference effect can be probed in an example by observing the angular distribution.There has been significant progress in basic conservation relations in partially coherent wave fields recently [1-4] after first related studies [5][6][7]. The conservation laws of energy, momentum, and angular momentum of the wave fields were established and a few related examples were discussed in those studies.The Maxwell stress tensor [8] has been useful in expressing energy, momentum, and angular momentum and also in dealing with spectral densities of the electromagnetic waves [9] in the studies. A careful review of the tensor corresponding to spectral densities of the electromagnetic waves in the far zone reveals clearer views for the energy, the momentum, and the angular momentum of the waves.There were a couple of discussions [2,3] on the waves when there are both electric and magnetic sources. It is interesting to see the interference effect in the radiated fields from the sources. In this Brief Report, we review the interference effect on the momentum of the partially coherent electromagnetic waves previously studied in Ref.[2]. We will follow the conventions used in Refs. [2,3] throughout this Brief Report.For the study of radiated electromagnetic fields, it is often convenient to use the Hertz vectors [10] from sources consisting of polarization and magnetization. For simplicity, we will consider (quasi-)homogeneous sources. The vectors are represented by the spatial integration of the products of source densities and three-dimensional Green's functions for the corresponding Helmholtz equations,where e and m stand for polarization and magnetization, respectively, and G(R) is the free-space Green's function for the Helmholtz equation,* minntenn@yonsei.ac.krFor further mathematical simplification, we introduce two differential operators as in [2],where ∂ i is the partial derivative with respect to the ith Cartesian coordinate, ∂/∂x i . Here and throughout the remainder of this Brief Report, we will use the Einstein summation convention. By use of the differential operators, we may write the electromagnetic fields in the following forms:In the far zone (the source size is negligible, r r ), we may write the position vector as r = ru, where u is the unit vector in the direction of observation. Thus the Green function, G(R), can be expressed asThe resulting Hertz vectors can be written as the products of the outgoing waves and the Fourier transforms of the source densities,whereP(ku,ω) andM(ku,ω) are the three-dimensional Fourier transforms of the source polarization and the source magnetization, respectively, P(ku,ω) ≡ 1 (2π ) 3 D P(r,ω)e −iku·r d 3 r,M(ku,ω) ≡ 1 (2π ) 3 D M(r,ω)e −iku·r d 3 r.After the applications of the...

Statistical and strict momentum conservation

“…The entangled state concept has been addressed in many papers and some books [38][39][40][41][42][43][44][45][46][47][48][49] over the past two decades, but it has not been used to discuss the conservation of P. Some recent papers [25][26][27][28][29][30][31] have discussed the law of the conservation of P without considering the concept of an entangled state.…”

“…The neutron was discovered [19] and some experimental phenomena [20,21] were explained using the strict laws. Some authors [22][23][24][25][26][27][28][29][30][31] have suggested that the conservation laws are strict and have applied the laws in theoretical analyses. Different from many books, only Zeng's books [37,38] wrote about the strict conservation in the single processes.…”