Ординатура / Офтальмология / Английские материалы / Optics Learning by Computing with Examples using MATLAB_Dieter Moller_2007

FIGURE 3.23 Examples of complementary screens.

The general appearance of the diffraction pattern on the observation screen is the same, regardless of one screen having larger openings than the other.

Let us consider two gratings, both having periodicity constant a. One has a width of opening d1 and the other of d2. These gratings are complementary screens when d1 +d2 a. To get the diffraction pattern, we apply the Kirchhoff– Fresnel integral to the openings of each screen. The two integrals must add up to zero since there are no openings to integrate. We assume there is a single wave incident on all openings and use far field approximation,

The diffraction pattern of the two screens has the same overall appearance. The first equation tells us that the two diffraction patterns are “out of phase” by 180

FIGURE 3.24 Diffraction patterns of complementary screens show the same intensity pattern.

degrees and the second one tells us that the diffraction patterns of the two screens are similar.

In Figure 3.24 we show schematically the diffraction pattern of two complementary screens. One screen is made of open squares and the other of black squares. In FileFig 3.15 we consider two amplitude gratings with complementary open areas as complementary screens. The appearance of the diffraction pattern is the same, but the heights of the peaks are different. We assumed that d1 is different from d2 and therefore the open areas are different.

FileFig 3.15 (D15FABAGRS)

Two complementary screens are considered. Both are amplitude gratings of periodicity constant a. One has the width d1 of open strips, the other the width d2, and since we assume that a d1 + d2 the screens are complementary. The diffraction patterns are shown as P1 for the grid with d1 and P2 for the grid with d2. One observes that the diffraction patterns are similar. Both patterns have peaks at the same location. However, they have different intensities. The different intensities result from the different d1 and d2 values and consequently different areas of integration.

D15FABAGRS

Babinet’s Theorem

Diffraction on two amplitude gratings, one with width of openings d1, the other with width of opening d2, and both having center-to-center distance of strips a d1 + d2. Wavelength λ, distance from gratings to screen X, and coordinate on screen Y . All distances and wavelengths area in mm; both have number of lines N. Normal incidence. D1 and D2 are the diffraction factors, I is the

FileFig 3.17 (D17ARAYRA3DS)

The product f (x, y) of intensity of diffraction and interference factor for a twodimensional grating is shown. Three-dimensional plots are shown for regular and random arrays.

D17ARAYRA3DS

3-D Graph of Diffraction Pattern

Periodic array of rectangular apertures compared to the diffraction pattern of rectangular apertures in a random array.

1. Periodic array

N ≡ 40.

3.7 FRESNEL DIFFRACTION

At the beginning of this chapter we used Fresnel diffraction for the calculation of the Poisson spot. The Kirchhoff–Fresnel integral was applied to a round stop and we found a constant illumination in the shadow of the round aperture on the axis of the system. We are now interested in discussing the diffraction on an edge, which was done by Fresnel using the integrals named after him. The first steps are to study the Fresnel diffraction on a slit and give the definitions of Fresnel’s integrals.

3.7.1Coordinates for Diffraction on a Slit and Fresnels Integrals

We consider the Kirchhoff–Fresnel diffraction integral in small angle approximation; see Eq. (3.20).

y1 −d/2

and set the constant C equal to 1. To get to the definition of Fresnel’s integrals, we change coordinates

In FileFig 3.18 we show graphs of Fresnel’s integrals.

FileFig 3.18 (D18FEFNCS)

The integrals C(η ) and S(η ) are plotted as C(Y ) and S(Y ) for Y 0 to 5.

D18FEFNCS is only on the CD.

3.7.2 Fresnel Diffraction on a Slit

Fresnel diffraction on a slit is calculated using the coordinates of a slit of width d as shown in Fig. 3.26. From Eq. (3.73) and (3.76) we have for the amplitude at the observation point

FIGURE 3.26 Coordinates for the calculation of the Fresnel diffraction pattern of a single slit of width d.

The graph in FileFig 3.19 shows Fresnel diffraction on a slit. One observes that for the parameters used, the first minimum of the pattern is not zero. By changing d to a smaller value or X to a larger value, one may get to a zero value for the first minimum.

Fresnel diffraction I (Y ) is plotted for a slit of width d at distance X 4000 mm for λ 0.0005 mm. These are the same values as used in FileFig 3.3 for far field approximation. For a small slit width, there is no difference. For a larger slit width, the Fresnel diffraction is not zero for the first minimum.

D19FESLITS

Fresnel’s Integrals for Calculation of Diffraction on a Slit

All units are in mm, global definition of parameters.

We call η1

I (Y ) : (Cp(Y ) − Cq(Y ))2 + (Sp(Y ) − Sq(Y ))2.

2

λ · X

1

TOL ≡ .1

λ ≡ 5 · 10−4 X ≡ 4000 d ≡ 1.5.

Application 3.19.

1.Normalize the pattern of the diffraction on a slit using Fresnel diffraction. Make it 1 at the center, and extend the Y range to negative and positive values.

2.Add to the graph the diffraction on a slit using far field approximation. Use the same slit width, wavelength, and distance from aperture to observation screen.

3.For what values of d do we have close agreement?

3.7.3 Fresnel Diffraction on an Edge

Fresnel diffraction on an edge is treated as the diffraction on a large slit with one edge at y 0 and the other at y ∞ (Figure 3.27). For the slit we had the limits for −d/2 to d/2. Note the negative sign in Eq. (3.74)

FIGURE 3.27 Coordinates for the calculation of the Fesnel diffraction pattern of an edge treated as a large slit, position of slit from y 0 to y (∞). For the value of the integral from 0 to −∞ we use the value −.5. The dependence on Y is now only contained in η1.

For the edge presented by a slit with one side at y 0 and the other side at y ∞, we have

The integrals C(η) and S(η) have to be taken from η1 to −∞. At −∞, both are −.5, and we get for the intensity (not normalized)

The first graph in FileFig 3.20 shows the intensity diffraction pattern (Eq. (3.86)) and Figure 3.28 shows a photograph of the diffraction on an edge. The second graph in FileFig 3.20 shows how the diffraction on an edge is derived from the diffraction on a large slit.

FIGURE 3.28 Photograph of the Diffraction on an edge. (From M. Cagnet, M. Francon, and J.C. Thrieer, Atlas of Optical Phenomena, Springer-Verlag, Heidelberg, 1962.)