Vector wave equations sources

According to the linearity of the wave equation, the vector field of an arbitrary source can be represented as the sum of elementary fields generated by the point pulse sources. However, the polarization (i.e., direction) of the vector field does not coincide with the polarization of the source, F . For instance, the elastic displacement field generated by an external force directed along axis x may have nonzero components along all three coordinate axes. That is why in the vector case not just one scalar but three vector functions are required. The combination of those vector functions forms a tensor object G" (r, t), which we call the Green s tensor of the vector wave equation. 

In Chapter 11 we discussed the fundamental properties of modes on optical waveguides. The vector fields of these modes are solutions of Maxwell s source-free equations or, equivalently, the homogeneous vector wave equations. However, we found in Chapter 12 that there are few known refractive-index profiles for which Maxwell s equations lead to exact solutions for the modal fields. Of these the step-profile is probably the only one of practical interest. Even for this relatively simple profile the derivation of the vector modal fields on a fiber is cumbersome. The objective of this chapter is to lay the foundations of an approximation method [1,2], which capitalizes on the small... 

When there are no sources present, i.e. J = 0, the fields with the separable form of Eq. (30-4) satisfy the homogeneous vector wave equations... 

If we work with the cartesian field components of Eq. (30-16) and the separable fields of Eq. (30-4), then in source-free regions Eq. (30-17) reduces to the homogeneous vector wave equations with transverse and longitudinal components given by... 

Higher-order corrections to the modal fields Vector wave equation with sources Radiation modes... 

When current sources are present within a waveguide, the total fields are related to the currents through Maxwell s equations, or, equivalently, through the inhomogeneous vector wave equations. The cartesian components E, Ey and E of the total electric field E satisfy Eq. (30-17a)... 

A large class of molecular properties arise from the interaction of molecules with electromagnetic fields. As emphasized previously, the external fields are treated as perturbations and so one considers only the effect of the fields on the molecule and not the effect of the molecule on the field. The electromagnetic fields introduced into the electronic wave equation is accordingly those of free space. From (79) one observes that in the absence of sources the electric field has zero divergence, and so both the electric and magnetic fields are purely transversal. It follows that the scalar potential is a constant and can be set to zero. In Coulomb gauge the vector potential is found from the equation... 

Substituting (3.A. 11) into (3.A. 10), we see that the Hertz vector satisfies a wave equation, with a simple source term — (<5e) 4,... 

By this means the problem of evaluating the radiation field originating from the wave (3.56) is reduced to the problem of evaluating the radiation field of a polarization wave with a constant amplitude. This field can be found far from the surface in terms of the Hertz vector which satisfies a nonuniform wave equation with the polarization vector as a source (Bom and Wolf 1975). 

The electric and magnetic fields Ej and of a bound mode are source-free solutions of Maxwell s equations, Eq. (30-1), or, equivalently, the vector wave... 

So far we discussed the propagation of electromagnetic waves without external sources (i.e. when pe = 0 and Jg = 0). To solve the Maxwell s equations in presence of sources, we can introduce the magnetic vector potential A and the electric scalar potential (p so that ... 

In its simplest version the QPM technique for second harmonic generation (SHG) consists in alternatively reversing the polar axis direction in successive thin layers of the same thickness J of a crystalline material. This leads to a reversal of sign of in successive layers of the layered material and one can easily see that ii d = ml = mnl/ k then the conversion efficiency increases with the number of layers. This actually amounts to periodically modulating in the nonlinear optical source = yP )EE in Equation [9] with a wave vector that precisely matches and cancels Isk in the propagation direction note that the linear refractive index is unaffected and remains the same as in the bulk crystal while I2 grows as (Figure 3). In a certain sense the quasiphase-matched systems can also be viewed as a nonlinear photonic crystal. 

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