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

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

The inverse Fourier transform yields

Uðx0; y0; zÞ ¼ 12 þ m4 e jplzf02 ½ej2pf0x0 þ e j2pf0x0 &

ð5:8-3Þ

¼ 12 ½1 þ m e jplzf 02 cosð2pf0x0Þ&

The intensity of the wave is given by

EXAMPLE 5.11 Determine the intensity of the diffraction pattern at a distance z that satisfies

2n

z ¼ n integers

lf02

Solution: Iðx0; y0; zÞ is given by

Iðx0; y0; zÞ ¼ 14 ½1 þ mcosð2pf0x0Þ&2

This is the image of the grating. Such images are called Talbot images or selfimages.

EXAMPLE 5.12 Repeat the previous example if

Solution: Iðx0; y0; zÞ is given by

Iðx0; y0; zÞ ¼ 14 ½1 mcosð2pf0x0Þ&2

This is the image of the grating with a 180 spatial phase shift, referred to as contrast reversal. Such images are also called Talbot images.

EXAMPLE 5.13 Repeat the previous example if

This is the image of the grating at twice the frequency, namely, 2f0. Such images are called Talbot subimages.

5.9 FRAUNHOFER DIFFRACTION WITH A SINUSOIDAL PHASE GRATING

As with the sinusoidal amplitude grating, with an incident plane wave on the grating, the transmission function for a sinusodial phase grating is given by

The analysis can be simplified by use of the following identity:

where Jkð Þ is the Bessel function of the first kind of order k. Using the above identity, the FT of f(x,y) is given by

where is the difference of the optical path lengths between the rays of two adjacent slits and I0 is the initial intensity of the beam.

When there are N parallel slits, the intensity in the Fraunhofer region can be shown to be

The second factor above equals N cosðpNmÞ= cosðpmÞ when the grating equation d sin f ¼ ml is satisfied.

Figure 6.1. Geometry for inverse diffraction.

where zr and z0 are the input and output z variables, and z0r ¼ z0 zr.

The Fraunhofer diffraction is governed by Eq. (5.4-2). Its inversion is given by

6.3 INVERSION OF THE ANGULAR SPECTRUM REPRESENTATION

Below the method is first discussed in 2-D. Then, it is generalized to 3-D. Equation (4.3-12) in 2-D can be written as

As this is a Fourier integral, recovery of Aðfx; zrÞ involves the computation of the Fourier transform of Uðx0; z0Þ:

Uðx; zrÞ is the inverse Fourier transform of Aðfx; zrÞ:

With F[.] and F 1[.] indicating the forward and inverse Fourier transforms, Eq. (6.3-3) can be expressed as

Note that zr can be varied above to reconstruct the wave field at different depths. In this way, a 3-D wave field reconstruction is obtained.

If measurements are made with nonmonochromatic waves, resulting in a wave field Uðx0; y0; z0; tÞ, where t indicates the time dependence, a single time frequency component Uðx0; y0; z0; f Þ can be chosen by computing

where v is the phase velocity. The technique discussed above was used with ultrasonic and seismic image reconstruction [Boyer, 1971; Boyer et al., 1970; Ljunggren, 1980]. It is especially useful when z0r is small so that other approximations to the diffraction integral, such as Fresnel and Fraunhofer approximations discussed in Chapter 5, cannot be used.

6.4 ANALYSIS

Interchanging orders of integration in Eq. (6.4-3) results in

1ð

1

where

p

integration are restricted such that k > 2pfx [Lalor, 1968]. Then, fx is restricted to the range

Suppose that the exact wave field at z ¼ zr is T(x,zr). How does U(x,zr) as computed above compare to T(x,zr)? In order to answer this question, we can first determine Uðx0; z0Þ in terms of T(x,zr) by forward diffraction. This is given by