Transverse magnetic mode propagation

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 ... 

TM (transversal magnetic) mode Electromagnetic field distribution in which magnetic field vector lies in the plane perpendicular to the propagation direction. 

The following equations define the requirements for the transverse-electric (TE) and the transverse-magnetic (TM) modes of light propagation, respectively ... 

Fig. 23. The geometry of the propagation (k), polarization (e) and magnetic field (5) vectors for the Voigt or Cotton-Mouton (CM) and Faraday (FA) effects. The change of polarization state of transverse C44 modes is indicated. For clarity the polarization is not shown in perspective. 0 is the CM or FA rotation angle.

In a rectangular cavity, electromagnetic waves are classified as transverse electric (TE) or transverse magnetic (TM) modes. AU the field combinations can be obtained by the superposition of TE and TM modes. TE modes are defined as the waves that have no electric field component in a defined propagation direction. In this discussion, the propagation direction is assumed to be the f-direction. Similarly, TM modes have no magnetic field component in the f - direction. By assuming a cavity with dimensions, a X b X d in the x-,y-, and z-directions, respectively, the frequencies at which nontrivial solutions of the Helmholtz Equation occur are... 

The effective index represents the dimensionless in-plane component of the propagation vector of the mode (the propagation vectors are in units of A being the vacuum wavelength). The optical modes can be characterized as transversal electric (TE the electric field is polarized in-plane) and transversal magnetic (TM the magnetic field is polarized in-plane). For unsymmetric slab waveguides, a minimum thickness (cutoff) exists for each mode to appear [48]. 

Magnetic fields introduce hydromagnetic waves, which are transverse modes of ion motion and wave propagation that do not exist in the absence of an apphed B field. The first of these are Alfven, A, waves and their frequency depends on B and p, the mass density. Such waves move parallel to the apphed field having the following velocity ... 

Here e1 is the fundamental transverse microscopic electric held operator and b is the corresponding magnetic held operator. The superscript on the electric held operator designate its transverse character with respect to the direction of propagation, redundant in the case of the magnetic held as it is intrinsically transverse, namely, divergence-free, since it arises from the curl of a vector potential held a(r). Since the electric held also derives from a(r), we concentrate first on the second-quantized form of this vector potential, which is cast in terms of a summation over radiation modes as follows ... 

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