Transverse field components

As has already been mentioned, at the interface between two neighboring seetions, two transversal field components are continuous. Their distributions ean be expressed by two expansions corresponding to modes of each section. From Eq. (11) we get for the interface between sections s and t... 

For a complete suppression of the local magnetic field effects, however, one should also compensate its transversal component perpendicular to the magnet axis. This requires a pair of auxiliary coils, oriented in a correct direction in the magnet s azimuth plane, and capable of generating a transversal field component of the correct magnitude and sign (for more details, see Ref. 70)). 

Thus the role of the high-frequency oscillators is to suppress the transverse field component (in other words, the transverse g-factor). If we are interested only in the contribution to the level spacing (the Lamb shift), one should consider only the longitudinal ( B) part of the renormalization, i.e. multiply the result by sin0, to obtain Eq. (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 express the index profile of a GI medium by Eq. (6). If we express the transverse field component by Qxpijcot — jPz), the function xj/ for the index profile is given approximately by the scalar wave equation. 

We have shown that the modes of weakly guiding waveguides are approximately TEM waves, with transverse field components e, and h,. However, the exact modal fields have longitudinal components. For the weakly guiding waveguide these components are very small, and are expressible approximately in terms of e, and h,. From Eq. (32-18) we have... 

We begin with a brief review of the weak-guidance approximation for fundamental modes on circular fibers. In Sections 13-2 and 13-4, we showed that the two fundamental modes are virtually TEM waves, with transverse fields that are polarized parallel to one of a pair of orthogonal directions. The transverse field components for the x- and y-polarized HEj j modes are given by Eq. (13-9) relative to the axes of Fig. 14-1. The spatial variation Fq (r) is the fundamental-mode solution (/ = 0) of the scalar wave equation in Table 13-1, page 288. Hence... 

We use Eq. (30-9) for the transverse-field components and determine the constants from continuity of e j, h j, e j and h j at the interface. With the aid of the Wronskian of Eq. (37-77) this leads to the expressions in Table 25-3 for the ITE and ITM modes. The orthogonality and normalization of each radiation mode is identical to the corresponding free-space normalization of Table 25-2 for reasons given above. Alternatively, we can parallel the derivation of Nj(Q) in Section 25-7 using the radiation-mode fields. 

The transverse field components can be expressed in terms of and by replacing n with n, in Eq. (30-6), as may be readily verified. Thus, for example, the radial and... 

In continuous-flow zone electrophoresis the solute mixture to be separated is injec ted continuously as a narrow source within a body of carrier fluid flowing between two electrodes. As the solute mixture passes through the transverse field, individual components migrate sideways to produce zones which can then be taken off separately downstream as purified fractions. 

The transverse electric field component, with azimuthal order v, is given by ... 

For weakly guiding structures, the second term can be neglected, and we obtain the standard Helmholtz equation in which individual components of the electric field intensity vector E remain uncoupled. For high contrast waveguides this is clearly not the case. The second term in Eq. (2) in which the transversal electric field components are mutually coupled must be retained. 

The term immittanee is used for the sake of generality for TE modes, the physical meaning of the matrix U is namely the transversal admittance (it mutually relates the field components and ) while for TM modes it is impedanee (the field components and Ely are related). It ean be derived from Eq. (13) by algebraic manipulations that upon translation from C to C + AC, the immittance matrix is transformed as follows ... 

The FDTD approach is based on direct numerical solution of the time dependent Maxwell s curl equations. In the 2D TM case the nonzero field components are E Hy and E, the propagation is along the z direction and the transverse field variations are along x. In lossless media. Maxwell s equations... 

In this final section, it is shown that the three magnetic field components of electromagnetic radiation in 0(3) electrodynamics are Beltrami vector fields, illustrating the fact that conventional Maxwell-Heaviside electrodynamics are incomplete. Therefore Beltrami electrodynamics can be regarded as foundational, structuring the vacuum fields of nature, and extending the point of view of Heaviside, who reduced the original Maxwell equations to their presently accepted textbook form. In this section, transverse plane waves are shown to be solenoidal, complex lamellar, and Beltrami, and to obey the Beltrami equation, of which B is an identically nonzero solution. In the Beltrami electrodynamics, therefore, the existence of the transverse 1 = implies that of , as in 0(3) electrodynamics. 

For the purely transverse field, the spacial components of the propagator are... 

Figure 2. Typical field distributions for waveguides. For the channel case, the transverse distribution f(x,y) is approximated by f(x)f(y). The arrows indicate the dominant field component, (a) TEmn channel modes, (b) TMmn channel modes, (c) TEm slab field distributions and, (d) TMm slab waveguide fields.

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