The Maxwell Equations

The basic equations of electrodynamics are the Maxwell equations. For point charges in a vacuum, which is what we are mainly interested in, these equations take the form 

B is the magnetic field, E is the electric field, p is the charge density, j is the current density, and eo is the permittivity of the vacuum. The Maxwell equations are invariant under Lorentz transformations. In fact, it was the search for a transformation that would leave the Maxwell equations formally invariant that originally led to the Lorentz transformations. 

Current and charge densities are related through the continuity equation 

This is not the conventional definition D is often used for the d Alembertian. The current definition is appropriate for our purposes, in which we need a notation for both the four-vector and its square. 

A closer examination of the continuity equation reveals that this is just the scalar product of the gradient for the four-vector space from (3.1) and a vector 

Electromagnetic waves combine the propagation of two vector fields, E and B. These are the electric and magnetic induction fields, respectively, and in a vacuum are governed by the Maxwell equations 1,2,3]  

Equation (1.2) is not independent and is obtained by combining the first two equations in 

Constitutive relations connecting D, H and J to E and B arc required to close the full set of equations. When only linear interactions are important, the following empirical equations are used  

Conditions can arise when the material response to the imposition of electric fields is nonlinear. Under such circumstances, more complex constitutive relationships must be employed and it is most common to expand the electric displacement vector, D, as a power series in the electric field according to 

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We need to point out that, if the wavelengths of laser radiation are less than the size of typical structures on the optical element, the Fresnel model gives a satisfactory approximation for the diffraction of the wave on a flat optical element If we have to work with super-high resolution e-beam generators when the size of a typical structure on the element is less than the wavelengths, in principle, we need to use the Maxwell equations. Now, the calculation of direct problems of diffraction, using the Maxwell equations, are used only in cases when the element has special symmetry (for example circular symmetry). As a rule, the purpose of this calculation in this case is to define the boundary of the Fresnel model approximation. In common cases, the calculation of the diffraction using the Maxwell equation is an extremely complicated problem, even if we use a super computer. 

Strictly speaking, differentiation with respeet to a veetor quantity is not allowed. However for the isotropie spherieal samples for whieh equation (A2.1.8) is appropriate, the two veetors have the same direetion and eould have been written as sealars the veetor notation was kept to avoid eonfiision with other thennodynamie quantities sueh as energy, pressure, ete. It should also be noted that the Maxwell equations above are eorreet for either of the ehoiees for eleetromagnetie work diseussed earlier under the other eonvention A is replaeed by a generalized G.)... 

A fiill solution of tlie nonlinear radiation follows from the Maxwell equations. The general case of radiation from a second-order nonlinear material of finite thickness was solved by Bloembergen and Pershan in 1962 [40]. That problem reduces to the present one if we let the interfacial thickness approach zero. Other equivalent solutions involved tlie application of the boundary conditions for a polarization sheet [14] or the... 

Since div ( ) mid div 3 (x) commute with 8(x ) and 3 t (x ) for x0 —x, they have vanishing commutators with the hamiltonian and hence, they are time-independent operators. In fact, their constancy in tame implies that they commute with 3 (x) and S(x) at all times and hence they must be c-number multiples of the unit operator. If these c-numbers are set equal to zero initially, they will remain zero for all times. With this initial choice for div 8(x) and div 3tf(x), the operators S and satisfy all of the Maxwell equations (these now are operator equations ) ... 

By deriving or computing the Maxwell equation in the frame of a cylindrical geometry, it is possible to determine the modal structure for any refractive index shape. In this paragraph we are going to give a more intuitive model to determine the number of modes to be propagated. The refractive index profile allows to determine w and the numerical aperture NA = sin (3), as dehned in equation 2. The near held (hber output) and far field (diffracted beam) are related by a Fourier transform relationship Far field = TF(Near field). 

However, in Maxwell s days everyone assumed that there had to be a mechanical underpinning for the theory of EM. Many researchers worked on very detailed hidden variable theories for the EM field, in an attempt to prove that the laws of EM were in fact a theorem in NM, just like Kepler s laws are a theorem in NM. No one noticed that it was impossible to do this, since Maxwell s equations are not Galilei invariant and Newton s laws are. That includes Lorentz who discovered around 1900 that the Maxwell equations are invariant under another transformation that now bears his name. 

E and B are the fundamental force vectors, while P and H are derived vectors associated with the state of matter. J is the vector current density. The Maxwell equations in terms of E and B are... 

The polarization vectors vanish in free space, so that in the absence of charge and matter D = e0E, H = —B and the Maxwell equations are ... 

To transform the Maxwell equations into k space the field is considered as a function of a space coordinate r measured along a line whose direction... 

Any point on either branch of the dispersion surface is an equally good solution of the Maxwell equations. However, the only points that will be selected are... 

Thus, the perpendicular conductivity is always less than the parallel conductivity. If the second component is a number of spheres embedded in a matrix of the first component, then the composite conductivity is given by the Maxwell equation when the volume fraction of spheres is very small ... 

So far, we have used the Maxwell equations of electrostatics to determine the distribution of ions in solution around an isolated, charged, flat surface. This distribution must be the equilibrium one. Hence, when a second snrface, also similarly charged, is brought close, the two surfaces will see each other as soon as their diffuse double-layers overlap. The ion densities aronnd each surface will then be altered from their equilibrinm valne and this will lead to an increase in energy and a repulsive force between the snrfaces. This situation is illustrated schematically in Fignre 6.12 for non-interacting and interacting flat snrfaces. 

As stated in the introductory chapter, we adopt a macroscopic approach to the problem of determining absorption and scattering of electromagnetic waves by particles. Therefore, the logical point of departure is the Maxwell equations for the macroscopic electromagnetic field at interior points in matter, which in SI units may be written ... 

If we Fourier analyze the Maxwell equations (2.1)-(2.4), with = 0, and assume that the operations of integration and differentiation may be interchanged, we obtain... 

Let us look for plane-wave solutions to the Maxwell equations (2.12)- (2.15). What does this statement mean We know that the electromagnetic field (E, H) cannot be arbitrarily specified. Only certain electromagnetic fields, those that satisfy the Maxwell equations, are physically realizable. Therefore, because of their simple form, we should like to know under what conditions plane electromagnetic waves... 

Our analysis shows that plane waves (2.39) are compatible with the Maxwell equations provided that k, E0, and H0 are perpendicular ... 

Consider a plane wave propagating in a nonabsorbing medium with refractive index N2 = n2, which is incident on a medium with refractive index A, = w, + iky (Fig. 2.4). The amplitude of the incident electric field is E(, and we assume that there are transmitted and reflected waves with amplitudes E, and Er, respectively. Therefore, plane-wave solutions to the Maxwell equations at... 

The electromagnetic field is required to satisfy the Maxwell equations at points where e and ju, are continuous. However, as one crosses the boundary between particle and medium, there is, in general, a sudden change in these properties. This change occurs over a transition region with thickness of the order of atomic dimensions. From a macroscopic point of view, therefore, there is a discontinuity at the boundary. At such boundary points we impose the following conditions on the fields ... 

Our fundamental task is to construct solutions to the Maxwell equations (3.1)—(3.4), both inside and outside the particle, which satisfy (3.7) at the boundary between particle and surrounding medium. If the incident electromagnetic field is arbitrary, subject to the restriction that it can be Fourier analyzed into a superposition of plane monochromatic waves (Section 2.4), the solution to the problem of interaction of such a field with a particle can be obtained in principle by superposing fundamental solutions. That this is possible is a consequence of the linearity of the Maxwell equations and the boundary conditions. That is, if Ea and Efc are solutions to the field equations,... 

If we assume harmonic time dependence e lut, the Maxwell equations (2.1)—(2.4) may be written... 

In Chapter 4 a plane wave incident on a sphere was expanded in an infinite series of vector spherical harmonics as were the scattered and internal fields. Such expansions, however, are possible for arbitrary particles and incident fields. It is the scattered field that is of primary interest because from it various observable quantities can be obtained. Linearity of the Maxwell equations and the boundary conditions (3.7) implies that the coefficients of the scattered field are linearly related to those of the incident field. The linear transformation connecting these two sets of coefficients is called the T (for transition) matrix. I f the particle is spherical, then the T matrix is diagonal. 

To give physical meaning to the principal dielectric functions, we consider propagation of plane waves E0exp(/k x — ioot) in an anisotropic medium that is, we ask What kind of plane waves can propagate in such a medium without change of polarization If we follow the same reasoning as in Section 2.6, we obtain from the Maxwell equations... 

V. Symmetry Reduction and Exact Solutions of the Maxwell Equations... 

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