Electromagnetic field transverse

Writing the electromagnetic field in terms of transverse electric (TE) and transverse magnetic (TM) components, the electric field has the form ... 

The last attribute of tire electromagnetic field we need to discuss is wave polarization. The nature of tire transverse field is such tliat tire oscillating field disturbance (which is perjDendicular to tire propagation direction) has a particular orientation in space. The polarization of light is detennined by tire time evolution of tire direction of tire electric field... 

Spatial Profiles. The cross sections of laser beams have certain weU-defined spatial profiles called transverse modes. The word mode in this sense should not be confused with the same word as used to discuss the spectral Hnewidth of lasers. Transverse modes represent configurations of the electromagnetic field determined by the boundary conditions in the laser cavity. A fiiU description of the transverse modes requires the use of orthogonal polynomials. 

Note that dra(t)/dt = [H,ra]=(l/ma)[pa-qaA(ra)] and, consequently, the first term in (69) represents the kinetic energy of the system of particles in the presence of the transverse electromagnetic field. Note the analogy between this representation and the dynamical solute-solvent coupling of section 2.6 where the optical phonons are equivalent to electromagnetic photons of low frequency (the acoustical phonons are related to sound waves). 

Surface plasmon-polaritons (SPP), also referred as to surface plasma waves, are special modes of electromagnetic field which can exist at the interface between a dielectric and a metal that behaves like a nearly-iree electron plasma. A surface plasmon is a transverse-magnetic mode (magnetic vector is perpendicular to the direction of propagation of the wave and parallel to the plane of interface) and is characterized by its propagation constant and field distribution. The propagation constant, P can be expressed as follows ... 

The Einstein coefficient of absorption of radiation for the longitudinal (along the z axis) and the transverse (perpendicular to the z axis) electromagnetic fields can be obtained from these matrix elements and are given by... 

In order to detect the NMR signal, it is necessary to have a radio frequency (r.f.) coil in the transverse plane, that is, perpendicular to the static magnetic field, B = B0k, which runs through the -axis, and with the help of this coil, an electromagnetic field is induced (Figure 1.39) [42],... 

PROBLEM 2.7.5. Show that, both in a dielectric insulator and in a vacuum, a plane-wave electromagnetic field solution propagating along x, whose amplitude depends only on the coordinate x and on the time f, can have no component along x, that is, show that it must be a transverse electric wave [13]. 

In order to proceed, we will accept that the transverse components of the electromagnetic field are the only ones that are relevant in the problem on the basis of the exact calculation that we have performed for the fundamental Gaussian beam. Instead, we will use trial functions for u that will lead to self-consistent expressions for the transverse components of Gaussian beams of arbitrary order when substituted into the vector Helmholtz equation. The derivation is clearest for the fundamental. We will redrive the transverse field components of the fundamental Gaussian beam here. The deviation of higher order modes is outlined in the Appendix. 

We have used scalar diffraction theory in this calculation, which is an approximation in two parts. The first part consists of approximating the electromagnetic field as a transverse field. We have derived the conditions under which it is permissible to do so. In the Appendix, we discuss the conditions under which it is possible to replace the vector Helmholtz equation by the scalar Helmholtz equation for transverse fields. In a sense, we have reduced the problem to a solution of the scalar Helmholtz equation. The second part of the approximation consists of exploiting the reduction of the vector Helmholtz equation to a scalar Helmholtz equation. Scalar diffraction theory is based on the scalar Helmholtz equation. Hence, when it is permissible to neglect the longitudinal and cross-polarized components of the Gaussian beam, we may use solutions of the scalar Helmholtz equation for transverse fields and may take over the results of scalar diffraction theory with confidence for this special case. 

We may now derive the electromagnetic field of higher order transverse Gaussian beam modes. In order to do so, we will use a technique developed for Cartesian coordinates described in Marcuse (1975), but adapted to cylindrical symmetry. For a system with cylindrical symmetry, we may take a trial solution of the form... 

The solution was interpreted in terms of the transverse vibrations of a string with a variable length. A few years later, these results were published [7], and were extended to the case of electromagnetic field. A similar treatment was reported by Havelock [8] in connection with the problem of radiation pressure. About 25 years later, the one-dimensional wave equation in the time-dependent interval interval 0 < x < a + bt was considered [9] under the name Spaghetti problem. ... 

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