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

FileFig 3.20 (D20FEEDGES)

The intensity I (Y ) for the diffraction on an edge is shown for the range from Y −5 mm to Y +15 mm. To show that this is one side of a large slit, the diffraction pattern of a large slit is shown as I I (Y ).

D20FEEDGES

We treat the diffraction at an edge as diffraction on a large slit. One side is set at d 0, the other at d −∞. This translates into

For p(Y ) −infinte; we have Cp(Y ) Sp(Y ) −.5

We take q(Y ) equal Y , square root is for scaling q(Y ) : Y .

I (Y ) : (Cq(Y ) − (−.5))2 + (Sq(Y ) − (−.5))2

X ≡ 4000 λ ≡ 5 · 10−4.

To see that we actually derived this from a large slit, we treat a large slit with positions at 0 and 10.

p(Y ) : (Y − (10))

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

Application 3.20. Vary the width of the slit by choosing values other than 10 in P (Y ) Y − 10 and compare the diffraction pattern of the slit with the pattern of the edge.

APPENDIX 3.1

A3.1.1 Step Grating

A grating with a rectangular reflecting surface is called a step grating. The grating has the periodicity constant a, and the reflecting surfaces of length a/2 are positioned in two planes. Such a grating may be produced by using two sets of interpenetrating gratings, shown schematically in Figure A3.29. The distance H between planes I and II may be varied, and the corresponding interference diffraction pattern, depending on the height of H, is called an interferogram. Assuming that the incident light contains many wavelengths, the application of a Fourier transformation to the interferogram results in a spectrum.

We discuss here the step grating as shown in Figure A3.29b for a fixed step height H. The incident light is reflected at planes I and II and travels in direction θ. At a faraway screen, one observes an interference diffraction pattern. We calculate the diffraction pattern on the array of N steps, having width d, step height H, and periodicity constant a.

Each of the two interpenetrating gratings produces the pattern of an amplitude grating and, in addition, we have the interference of the light from planes I

FIGURE A3.29 (a) Schematic of a step grating; (b) coordinates for the calculation of the diffraction on a step grating. Only the emerging diffracted light is shown for the calculation of the path difference. The two gratings in planes I and II interfere with each other in the X direction.

and II in the direction of the observation screen. The optical path difference is δ d sin θ + H and in order to get the intensity of the interference (see Chapter 2, Eq. (2.8)), we have to insert this path difference δ into [cos(πδ/λ)]2. The intensity P uu of the diffraction and interference of each grating and the interference of the two gratings with each other is then the product of the diffraction factor

The diffraction pattern depends not only on d and a, but also on H. The graph of FileFig A3.21 shows P (θ) for several values of H and also the diffraction factor D(θ). The diffraction factor supplies the envelope and the interference factor supplies the lines. The zeroth order is at the center and the first order is within the envelope. Since we necessarily have 1:2 for the ratio d/a, the second order is suppressed by the zeros of the diffraction factor. Variation of H will redistribute the intensity between the zeroth and first order. This may be understood from energy conservation. We assume that the third and higher orders may be neglected. Then for H 0, the incident light is reflected into the zeroth order, that is, in the direction of reflection on the mirror surface. When H λ/4, no light can travel in the direction of the zeroth order. In other words, no light may be reflected on the surface of the grating facets and all light travels into the first order. For H λ/2 we have again reflection on the grating facets.

For H being a multiple of the wavelength, all light is diffracted into the zeroth order and for H being a multiple of half a wavelength, all light is diffracted into the first order. The graph in FileFig A3.21 shows the intensity pattern P (θ) for two different wavelengths. One observes only one peak for the zeroth order, but two peaks for the first order. The zeroth order changes its intensity depending on H . Recording the intensity depending on H will reveal the interferogram.

FileFig A3.21 (DA1FAGSTEP1S)

There are four intensity patterns P 1 to P 4, each for a different value of H . The values of H are presented as H nλ. When n is an integer, we have all light in the zeroth order. For noninteger values of n, we subtract from λ all full wavelengths and look at the remaining fraction n . For example, if n 10.5, we look at n 0.5. The optical path difference is now half a wavelength and all the light is diffracted in the first orders. For values such as 0.125, 0.25, and 0.375 there is light partly diffracted into the zeroth and the first order.

DA1FAGSTEP1S is only on the CD.

Application A3.21.

1.Find the values of n for which the patterns of constructive and destructive interference are repeating for values of n from 0 to 2. Observe that the path difference between constructive and destructive interference is λ/2, and the successive constructive or destructive interference pattern is λ. These length differences correspond to the length differences produced by H .

an example we discuss the diffraction on a slit. The intensity is given by

and depends on the points η1(Y ) and η2(Y ), assuming that d, λ, and X are given. I (Y ) is the square of the geometrical distance on the Cornu spiral between the points η1 and η2. The diffraction pattern is obtained by plotting (distance)2 as a function of Y for η1 and η2. By the division of all values by (distance)2 for Y 0 one can obtain a normalized pattern.

FileFig A3.23 (DA3FECOR)

Graph of S(Y ) as function of C(Y ). This graph is called Cornu’s spiral.

DA3FECOR is only on the CD.

A3.2.2 Babinet’s Principle and Cornu’s Spiral

We consider two complementary screens, I and II. Screen I may just have a hole and screen II has a stop of the same size as the hole. If added together, no light may pass. Babinet’s principle tells us that complementary screens generate similar diffraction patterns. In FileFig 3.A3 we have plotted one-half of the Cornu spiral for η from 0 to ∞. Let us consider a slit and the complementary screen, a stop of the same width. The diffraction pattern of screen I for one point Y is obtained by measuring the length between the corresponding points η1 and η2 (as discussed

FIGURE A3.30 Cornu’s spiral. The line(b) corresponds to the diffraction pattern of a slit and (a) and (c) of a stop.

above) which we call length (b) (see Figure A3.30). The diffraction pattern of screen I I may be obtained by using the distances (a) from −∞ to η1 and (c) from η2 to ∞. If we want to get the diffraction pattern on a different point Y , the points η1 and η2 are displaced in correspondence with the choice of the Y -value and will affect the diffraction pattern for both screens in the same way.

The distances (a) and (c) and the distance (b) from η1 to η2 represent the diffraction pattern for the addition of the two complementary screens for which no diffraction pattern can be observed.

See also on CD

PD1. Diffraction on a Slit and Width of Diffraction Pattern (see p.1 40). PD2. Diffraction on a Slit and the first Maxima (see p. 141).

PD3. Rectangular Aperture (see p. 145).

PD4. Circular Aperture, first Minimum (see p. 149).

PD5. Circular Aperture, Comparison with Slit (see p. 149). PD6. Double Slit (see p. 153).

PD7. Grating at Normal Incidence (see p. 153). PD8. Amplitude Grating (see p. 156).

PD9. Echelette Grating (see p. 158).

PD10. Resolution depending on N and d/a ratio (see p. 162). PD11. Grating Resolution (see p. 163).

PD12. Babinet’s Principle (see p. 166).

PD13. Fresnel and Far Field Diffraction of a Slit and round Aperture (see p. 173).

PD14. Fresnel Diffraction on an Edge (see p. 175).

FIGURE 4.1 (a) Superposition of two of Young’s experiments using separate sources S1 and S2. The distance between the sources is s and both “use” the same separation a of the double slit. The distance from the sources to the aperture is Z, and to the observation screen is X; (b) light from S2 has the angle ψ with the axis and is diffracted into the angle θ.

For each source point we have an intensity diffraction pattern and for the two source points we look at the superposition of two intensity patterns. We assume in our model calculation that each source point generates a monochromatic wave of wavelength λ and both are incident on the double slit aperture. The light from source point S1 produces on the observation screen the intensity

I (θ, 0) I0{sin[(πd/λ) sin θ]/[(πd/λ) sin θ]}2{cos[(πa/λ) sin θ]}2, (4.1)

where θ is the diffraction angle and Io is the normalized intensity. Equation (4.1) is obtained from the discussion of the double slit (Chapter 3, FileFig 3.10). For source point S2, the axis of the double slit experiment is rotated by the angle ψ around the point at the center between the two openings. As a result, the diffraction angle is counted from the new axis and we have to use d(sin θ −sin ψ) and a(sin θ − sin ψ) in the diffraction and interference factors of Equation (4.1), respectively. For the intensity of the light from point S2 we have

I (θ, ψ) I0{sin[(πd/λ)(sin θ − sin ψ)]/[(πd/λ)(sin θ − sin ψ)]}2

The calculation of the optical path difference is similar to the calculation of the optical path difference for a grating, illuminated under an angle to the normal (see Chapter 3, Eq. (3.61)). In this model of spatial coherence, monochromatic light from each point source uses the same double slit aperture and generates an intensity fringe pattern. The waves from each source point have no fixed phase relation between each other and each produces an intensity fringe pattern of its own. The intensities of these two fringe patterns are superimposed. Whether fringes can be observed depends not only on the separation s (and consequently on ψ) and the wavelength, but also on the separation a of the two slits in the double slit arrangement. However, this separation “a” is assumed to be constant.

In FileFig 4.1 we calculate the superposition of the intensity pattern depending on the separation of the two source points. The separation s is taken in “common length units” as discussed in Chapter 1. The first graph shows the intensity interference pattern for both source points at the same spot, that is, for s 0. The second graph shows the reduced interference pattern for the distance between the two source points of s 1.5. The third graph shows the disappearance of the intensity pattern at the distance of s 2.25 and the fourth graph shows the reappearance at s 2.6. We see that the superposition of the intensity pattern, produced by the two sources with incoherent light, cancel for the specific distance between the two source points of s 2.25.

When fringes are observed of the superposition of the two intensity fringe patterns, one calls the light producing the fringe pattern spatially coherent. When no fringes are observed, the light is called spatially incoherent.

FileFig 4.1 (C1COH2S)

Graphs are shown for the superposition of the intensities I (θ, 0) and I (θ, ψ) for two point sources at variable distances s as a function of the angle θ. Parameters used are the separation of the two openings a 1 mm, opening of the slits d 0.05mm, wavelength λ 0.0005 mm, distance from source to double slit Z 9000 mm, and distance from aperture to observation screen X 4000 mm. Four distances are used, s 0, s 1.5mm, s 2.25mm, and s 2.6mm, of separation s Zψ of the two source points, corresponding to four values of ψ. Fringe patterns are observed for separations s smaller than 2.25 mm. For a · ψ λ/2, that is for s 2.25 mm, we have for the first time disappearance of fringes; that is, the maxima of I (θ, ψ) are at the minima of I (θ, 0). For s larger than 2.25 mm the fringe pattern reappears.