Main equations of diffraction theory

1.1 Maxwell equations

1.1.1 Mathematical concepts and notations

The Hamiltonian operator in the Cartesian coordinate system is determined as

follows:

where ex, e y, ez are the unit vectors of the Cartesian coordinate system. The operators grad, div, rot and ? are defined as follows:

Where f, F = (Fx, Fy, Fz) are the scalar and vector functions, (.,.) and [.,.] are the operations of the scalar and vector products, det A is the determinant of the matrix A.

The most important integral relationships of vector analysis

The Ostrogradskii-Gauss theorem:

where n is the unit vector of the external normal, V is the domain of the space, restricted by the surface S.

The Stokes theorem

Here L is the contour, restricting the surface S.

1.1.2 Maxwell equations in the differential form

The electromagnetic theory of light is based on the system of Maxwell equations [1] (in the Gauss unit system):

(1.1)

(1.2)

(1.3)

(1.4)

The notations used here and in the rest of the book are presented in Table 1.1.

Table 1.1

Electromagnetic quantities

The functions E = E (r, t), H = H (r, t), D = D (r, t), B = B (r, t) describe the electromagnetic field in the medium characterised by the parameters e = e (E, r, t), µ = µ (H, r, t), ? = (r, t), j = j (E, r, t) (r are the spatial coordinates, t is time) and secondary current jsec which will be described separately.

Assuming that the processes in the medium are local and inertialess (the state at every point is independent of the adjacent points and at every moment of time is independent of 'prior history'), the characteristics of the field and the medium are be linked by material equations [1]:

(1.5)

(1.6)

(1.7)

and by the law of charge conservation

(1.8)

Further, it is assumed that the parameters of the medium are independent of the vectors of the field and do not change with time: e = e (r), µ = µ (r) (linear medium), and are scalar (isotropic medium).

If the strength of the electrical and magnetic fields can be described in the form: E = Re(E exp (-i?t)), H = Re (H exp(-i?t)), where E = E (r), H = H(r) are the complex-valued functions [1], ? is cyclic frequency, i is the apparent unity, we are concerned with a monochromatic field for which the equations (1.1), (1.2) take the following form:

(1.9)

(1.10)

where is the wavenumber.

1.1.3 Maxwell integral equations

Integrating the equations (1.1), (1.2) with respect to the surface S restricted by contour L and accepting the Stokes theorem, we obtain the following equations:

(1.11)

(1.12)

Equations (1.3), (1.4) are integrated with respect to volume V restricted by the surface S. Subsequently, using the Ostrogradskii-Gauss theorem:

(1.13)

(1.14)

The system (1.11)-(1.14) is referred to as the Maxwell integral equations in the integral form.

1.1.4 Boundary conditions

Using the Maxwell integral equations for the infinitely small contours and volumes at the interface of two media, we obtain the following boundary conditions [1) for the characteristics of the electromagnetic field:

(1.15)

(1.16)

(1.17)

(1.18)

where is the density of the surface charge, is the density of surface current (plane, separating the media 1 and 2, is normal to the vector ey).

1.1.5 Poynting theorem

The equation (1.1) is multiplied by E and equation (1.2) by H and, consequently, we obtain:

Deducting the second equation from the first one, we obtain the Poynting theorem [1] according to which:

(1.19)

In the integral form

(1.20)

we have the balance equation of the energy of the electromagnetic field in the volume V. The energy in the volume V is , the consumed power is the Umov-Poynting vector, indicating the direction of movement of energy and is equal to the density of the energy flux.

The monochromatic field can be described by the Umov-Poynting complex vector

where the symbol * indicates the complex conjugation, and the mean value of the Umov-Poynting vector is equal to the real part of the complex vector.

1.2 Differential equations in optics

1.2.1 Wave equations

The current and charges, usually not found in optics problems, are excluded from the Maxwell equations. Consequently, equations (1.1) and (1.2) take the following form:

(1.21)

(1.22)

Dividing both parts of equation (1.22) by µ and using the rot operator:

(1.23)

Equation (1.21) is differentiated with respect to time to exclude the second term from equation (1.23):

Consequently, taking into account that

rot au = a rot u + [grad a, u] and rot rot u = grad div u - V2u

we obtain:

(1.24)

For the equation div (eE) = 0 we used the identity div au = a div u + (u, grad a) and obtain e div E + (E, grad e) = 0. Expressing div E from the last equality, we substitute the result into (1.24), writing the wave equation [1] for the strength of the electrical field in an inhomogeneous dielectric medium:

(1.25)

The same procedure is used for deriving the wave equation for the strength of the magnetic field H:

(1.26)

For a homogeneous medium, electrical and magnetic µ permittivitties are constant and the wave equations take the form:

(1.27)

(1.28)

1.2.2 Helmholtz equations

The wave equations, written for the complex amplitudes...