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

Such surfaces are called wavefronts. Power flows in the direction of k which is perpendicular to the wavefront at a point r. The trajectory of a ray can be shown to be always perpendicular to the wavefront at every point. In other words, a ray follows the direction of power flow.

Equation (8.4-1) can be substituted in the Helmholtz equation

As l0 ! 0, the second term on the right-hand side above can be neglected so that

Equation (8.4-7) is known as the Eikonal equation. It can be used to determine wavefronts. Once the wavefronts are known, the ray trajectories are determined as being perpendicular to the wavefronts. Three types of wavefronts are shown in Figure 8.6.

The Eikonal equation can be interpreted as the limit of the Helmholtz equation as the wavelength approaches zero.

Once S(r) is obtained by using Eq. (8.4-7), A(r) can be determined by using Eq. (8.4-5). Thereby, the wave field U(r) is known.

Equation (8.4-7) can also be written as

Then, the index of refraction is given by

Rays

(a)

Rays

(b)

Rays

(c)

Figure 8.6. Three types of wavefronts: (a) a plane wave, (b) a spherical wave, (c) a general wave.

EXAMPLE 8.6 Show that the optical path length between two points A and B equals SðrBÞ SðrAÞ.

Solution: Since nðrÞ ¼ jrSðrÞj, the optical path length is given by

Note that the optical path length is similar to potential difference in electrostatics. Then, the phase S(r) is analogous to potential at a point.

8.5LOCAL SPATIAL FREQUENCIES AND RAYS

In previous chapters, an arbitrary wave was analyzed in terms of plane wave components with direction cosines. Optical rays can also be associated with direction cosines. In order to do so, it is necessary to consider the nonstationary characteristics of a wave field.

In general, a nonstationary signal h(t) is a signal whose frequency content changes as a function of t. Such signals are not well characterized by a single Fourier transform representation. Instead a short-time Fourier transform can be used by computing the Fourier transform of hðtÞoðtÞ, oðtÞ being a window function of limited extent.

The 1-D case outlined above can be generalized to the 2-D case in terms of nonstationary 2-D signals, which is relevant to wave fields. Thus, let us write a wave field at constant z as

fx0 and fy0 are also defined to be zero in regions where h(x,y) is zero.

In general, fx0 and fy0 are not the same as the Fourier transform spatial frequencies fx and fy discussed in previous chapters. Good agreement between the two sets occurs only when the phase fðx; yÞ varies slowly in the x–y plane so that fðx; yÞ can be approximated by three terms of its Taylor series expansion near any point (x0,y0).

It can be shown that these direction cosines are exactly the same as the direction cosines of a ray at (x,y) on a plane defined by constant z.

EXAMPLE 8.7 Determine the local spatial frequencies of a plane wave field. Solution: The spatial part of a plane wave field can be written as

hðx; yÞ ¼ ej2pðfx xþfyyÞ

Then, with fðx; yÞ ¼ 2p fxx þ fyy

fx0 ¼ 21p @@x fðx; yÞ ¼ fx fy0 ¼ 21p @@y fðx; yÞ ¼ fy

In this case, the local spatial frequencies and the Fourier transform spatial frequencies are the same.

EXAMPLE 8.8 A finite chirp function is defined by

(a)Determine the local spatial frequencies.

(b)Compare the local frequencies to the properties of the Fourier transform of

g(x,y).

Solution: (a) The phase is given by

fðx; yÞ ¼ paðx2 þ y2Þ

in a rectangular region around the origin of dimensions 2Dx by 2Dy. In this region, the local spatial frequencies are given by

Outside the rectangular region, fx0 and fy0 are zero by definition. Hence, they can be

Note that fx0 and fy0 vary linearly with x and y, respectively, inside the given rectangular region.

Dx