Файл: Ersoy O.K. Diffraction, Fourier optics, and imaging (Wiley, 2006)(ISBN 0471238163)(427s) PEo .pdf

v is also known as the phase velocity and can be larger than the vacuum speed of light in the case of light waves.

More generally, a 1-D wave given by Eq. (3.2-8) may satisfy another partial differential equation with respect to x and t, in addition to Eq. (3.2-5). A general wave equation that has a simple harmonic solution can be written as

Such an equation relating o and k is known as the dispersion equation. The phase velocity v in this case is given by

A wave with a phase velocity, which is a function of wave number or wavelength other than Eq. (3.2-10), is known as a dispersive wave. Hence, Eq. (3.2-14) is a dispersive wave equation.

Some examples of waves are the following:

A.Mechanical waves, such as longitudinal sound waves in a compressible fluid, are governed by the wave equation

where is the fluid density, and Kc is the compressibility. Hence, the phase velocity is given by

s

Kc

v ¼ ð3:2-16Þ

Aðfx; fy; f Þ cosð2pðfxx þ fyy þ ftÞ þ ðfx; fy; f ÞÞ is a 2-D plane wave with amplitude Aðfx; fy; f Þ, phase ðfx; fy; f Þ, spatial frequencies fx and fy, and time frequency f. In this case, the wave vector k is given by

where ex and ey are the unit vectors along the x- and y-directions, respectively. The components kx and ky are the radian spatial frequencies along the x-direction and y-direction, respectively. fx and fy are the corresponding spatial frequencies.

The 2-D plane wave can be written as

where

If we assume uðr; tÞ moves in the direction k with velocity v, uðr; tÞ can be shown to be a solution to the 2-D nondispersive wave equation given by

where

In conclusion, time-varying imagery can be represented as the sum of a number of 2-D plane waves. This conclusion can be extended to any multidimensional signal in which one of the dimensions is treated as time.

Equation (3.2-26) is the generalization of Eq. (3.2-5) to two spatial coordinates x and y. If three spatial coordinates x, y, and z are considered with the same wave property, Eqs. (3.2-23) and (3.2-25) remain the same, with the following definitions of r and r2:

where ez is the additional unit vector along the z-direction, and

The r2 operator is known as the Laplacian. The 3-D wave vector used in Eq. (3.2-24) is given by

Thus, kz is the additional radian frequency along the z-direction, with fz being the corresponding spatial frequency. The plane for which uðr; tÞ is constant is called the wavefront. It moves in the direction of k with velocity v. This is discussed further in Section 3.5.

In what follows, we will assume linear, homogeneous, and isotropic media unless otherwise specified. We will also assume that metric units are used.

EXAMPLE 3.1 Consider uðx1; t1Þ ¼ uðx2; t2Þ; x2x1; t2t1 for a right-traveling wave. Show how x1 and x2 are related.

Solution: uðx1; t1Þ can be considered to be moving to the right with velocity v. Letting t ¼ t2 t1, uðx1; t1Þ and uðx2; t2Þ will be the same when

x2 x1 ¼ v t

3.3ELECTROMAGNETIC WAVES

Electromagnetic waves are 3-D waves, with three space dimensions and one time dimension. There are four quantities D, B, H, and E, which are vectors in space coordinates and whose components are functions of x, y, z, and t. In other words, D, B, H, and E are directional in space. D is the electric displacement (flux) (vector)

field (C=m2), E is the electric (vector) field (V=m), B is the magnetic density (vector) field (Wb/m2), H is the magnetic (vector) field ðA=mÞ, is the charge density (C/ m3), and J is the current density (vector) field (A/m2).

Electromagnetic waves are governed by four Maxwell’s equations. In metric units, they are given by

where the divergence and curl of a vector A with components Ax, Ay, and Az are given by

where e is the permittivity, and m is the permeability. Their values in free space or vacuum are given by

10 9

e0 ¼ 36p F=m

m0 ¼ 4p 10 7 H=m

In dielectrics, e and m are greater than e0 and m0, respectively. In such media, we also consider dipole moment density P (C/m2), which is related to the electric field E by

where w is called the electric susceptibility. w can be considered as a measure for electric dipoles in the medium to align themselves with the electric field.

We also have

In a uniform isotropic dielectric medium in which space charge and current density J are zero, Maxwell’s equations become