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

74 FRESNEL AND FRAUNHOFER APPROXIMATIONS

Choosing gðx; yÞ ¼ Uðx; y; 0Þ, it is observed that Eqs. (5.4-5) and (5.4-6) are the same. Hence, the Fresnel approximation is a solution of the paraxial wave equation.

5.5DIFFRACTION IN THE FRAUNHOFER REGION

then, U0ðx; y; 0Þ is approximately equal to U(x,y,0). When this happens, the Fraunhofer region is valid, and Eq. (5.2-13) becomes

Aside from multiplicative amplitude and phase factors, Uðx0; y0; zÞ is observed to be the 2-D Fourier transform of U(x,y,0) at frequencies fx ¼ x0=lz and fy ¼ y0=lz. It is also observed that diffraction in the Fraunhofer region does not have the form of a convolution.

EXAMPLE 5.8 Find the wave field in the Fraunhofer region due to a wave emanating from a circular aperture of diameter D at z ¼ 0.

Solution: The circular aperture of diameter D means

Uðx; y; 0Þ ¼ cylðr=DÞ

p where r ¼ x2 þ y2.

From Example 2.6, the 2-D Fourier transform of U(x,y,0) is given by

Figure 5.5. The intensity diffraction pattern from a square aperture in the Fraunhofer region.

The intensity distribution equal to U2ðx0; y0; zÞ is known as the Airy pattern. It is

The Fraunhofer diffraction intensity from the same square aperture of Figures 5.2 is visualized in Figures 5.5 and 5.6. Figure 5.5 is the Fraunhofer intensity diffraction pattern from the square aperture. Figure 5.6 is the same intensity diffraction pattern as a 3-D plot.

EXAMPLE 5.9 Determine when the Fraunhofer region is valid if the wavelength is 0.6 m (red light) and an aperture width of 2.5 cm is used.

Solution:

This large distance will be much reduced when the lenses are discussed in Chapter 8.

100

80

60

40

20

0

0.5

–0.5

–0.5

–1 –1

Figure 5.6. The same intensity diffraction pattern from the square aperture in the Fraunhofer region as a 3-D plot.

EXAMPLE 5.10 Consider Example 5.7. With the same definitions used in that example, the pseudo program for Fresnel diffraction can be written as

u1 ¼ fftshiftðu1Þ u2 ¼ fftðu1Þ

ejkz

u3 ¼ jlz ej2kzðx20þy20Þ u2

U¼ fftshiftðu3Þ

5.6DIFFRACTION GRATINGS

Diffraction gratings are often analyzed and synthesized by using Fresnel and Fraunhofer approximations. Some major application areas for diffraction gratings are spectroscopy, spectroscopic imaging, optical communications, and networking.

A diffraction grating is an optical device that periodically modulates the amplitude or the phase of an incident wave. Both the amplitude and the phase may also be modulated. An array of alternating opaque and transparent regions is called a transmission amplitude grating. Such a grating is shown in Figures 5.7 and 5.8.

wavelengths travel in different directions. The peak intensity at each wavelength depends on the number of slits in the grating.

The transmission function of a grating is defined by

where Uðx; y; 0 Þ and Uðx; y; 0þÞ are the wave functions immediately before and after the grating, respectively. If the grating is of reflection type, the transmission function becomes the reflection function.

t(x,y) is, in general, complex. If arg(t(x,y)) equals zero, the grating modifies the amplitude only. If jtðx; yÞj equals a constant, the grating modifies the phase only.

Gratings are sometimes manufactured to work with the reflected light, for example, by metalizing the surface of the grating. Then, the transmission function becomes the reflection function.

The phase of the incident wave can be modulated by periodically varying the thickness or the refractive index of a transparent plate. Reflecting diffraction gratings can be generated by fabricating periodically ruled thin films of aluminum evaporated on a glass substrate.

5.7 FRAUNHOFER DIFFRACTION BY A SINUSOIDAL AMPLITUDE GRATING

Assuming a plane wave incident perpendicularly on the grating with unity amplitude, the transmission function is the same as U(x,y,0) given by

where the diffraction grating is assumed to be limited to a square aperture of width D. U(x,y,0) can be written as

where