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

orthogonal and normalized within a circle of unit radius [Kim and Shannon, 1987]. In this process, the phase function fAðx; yÞ is represented in terms of an expansion in Zernike polynomials zkðr; yÞ, where r is the radial coordinate within the unit circle, and y is the polar angle.

Table 10.1 shows the Zernike polynomials for 1 k 37. Note that each polynomial is of the form

where n and m are nonnegative integers. Rmn ðrÞ is a polynomial of degree n and contains no power of r less than m. In addition, Rmn ðrÞ is even (odd) when m is even (odd), respectively. The representation of fAðx; yÞ ¼ fAðr; yÞ can be written as [Born and Wolf, 1969]

The coefficients Anm are determined for finite values of n and m by least squares. In turn, fAðr; yÞ can also be written as

XK

where K is an integer such as 37. The coefficients wk are found by least squares. As each successive Zernike term is normal with respect to every preceding term, each term contributes independently to the mean-square aberration. This means the rootmean square error fA due to aberrations can be written as

" #1 X1 2

Note that the Zernike representation of aberrations is valid when the exit pupil is circular. Otherwise, the Zernike polynomials are not orthogonal. In some cases, such as aberrations due to atmospheric phase disturbances, the Zernike polynomial representation does not easily give a satisfactory representation with a finite number of terms.

emulsion is sandwiched between a protective layer and a base consisting of acetate or mylar for films and glass for plates. In black-and-white photographic film, there is typically one layer of silver grains. In color film, there are at least three such layers. Dyes added to the silver grains make the crystals sensitive to different colors.

Modulation of light occurs as follows:

1.If a photon is absorbed by a silver halide grain, an electron–hole pair is generated within the grain.

2.A generated electron is in the conduction band and hence moves around in the crystal and may become trapped at a crystal dislocation.

3.A trapped electron attracts a mobile silver ion. This results in a single atom of metallic silver with a lifetime of the order of a few seconds.

4.Several additional silver atoms may be formed at the same site. At least four silver atoms called silver speck are needed for the development process.

The development is done in a chemical bath. The developer acts on the silver specks and causes the entire crystal to be reduced to metallic silver in regions where enough light has been absorbed. Those regions that did not turn to metallic silver will eventually do so and hence must be removed by a process called fixing. This involves immersing the transparency in a second chemical bath.

The terms used in connection with the usage of photographic films and their definitions are discussed below.

where

Iðx; yÞ ¼ intensity transmitted by the transparency after development Iiðx; yÞ ¼ incident intensity

Photographic Density

Silver mass per unit area is approximately proportional to the photographic density D, which is defined as the logarithm of the reciprocal of the intensity transmittance of a photographic transparency. Thus, D is given by

Figure 11.1. The Hurter–Driffield curve for photographic emulsion.

This can also be written as

Hurter–Driffield (H & D) Curve

The H & D curve is the plot of D versus log E. A typical H & D curve for a negative film is shown in Figure 11.1. Note that the density is essentially constant for very low exposures and very high exposures. In between, there is the linear region, which is the most commonly used region in photography.

Gamma

The gamma g of the emulsion is the slope of the H & D curve in the linear region. A high-contrast film has a large gamma (typically 2 or 3), and a low-contrast film has a small gamma (1 or less). g is positive for a negative film and negative for a positive film.

11.3TRANSMITTANCE OF LIGHT BY FILM

In incoherent and coherent optical systems, film is often used to modulate the transmittance of light. How this is achieved depends on whether coherent or incoherent light is used. Both these cases are discussed below.

Incoherent Light

In the linear region of the H & D curve, which is assumed to be used, the density can be written as [Goodman]

where K ¼ 10D0 T g is a positive constant. Equation (11.3-3) shows that transmittance depends nonlinearly on the incident intensity and decreases as increases for positive g.

EXAMPLE 11.1 Show that two negative films in tandem results in transmittance I increasing with incident intensity.

Solution: The first film has transmittance given by

a1 ¼ K1I1 g1

where I1 is the intensity of the illuminating beam.

Suppose the first film is placed in contact with the second unexposed film and illuminated with intensity I2. The intensity incident on the second emulsion equals a1I2. The intensity transmittance a2 can be written as

a2 ¼ K2ða1I2Þ g2 ¼ K I1g1g2

where

K ¼ K1K2

It is seen that the overall transmittance a2 increases with I1.

Coherent Light

With coherent light, phase modulation also becomes important. Phase modulation is caused by variations of the film or plate thickness. The amplitude transmittance of the film can be written as

where fðx; yÞ is the phase due to thickness variation.

The phase term ejfðx;yÞ is usually undesirable. It is possible to remove it by using a liquid gate. This is schematically shown in Figure 11.2.