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

128 2. INTERFERENCE

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PI1. Cos-Waves depending on Space and Time (see p. 80).

PI2. Superposition of two cos-waves with fixed Optical path Difference (see p. 82).

PI3. 3-D Graph of Maxima and Minima (see p. 84). PI4. Average( see p. 85).

PI5. Fresnel’s Mirrors (see p. 93).

PI6. Young’s and Lloyd Experiment (see p. 93).

PI7. Plane parallel Plate in two Beam Interferometry with different Refractive Indices (see p. 95).

PI9. Wedge shaped Film (see p. 99). PI10. Newton’s Rings (see p. 101).

PI11. Ring pattern and Michelson interferometer (see p. 107). PI12. Plane parallel plate (see p.108-111).

PI13. Interference of white light on a thin film (see p. 113). PI14. Reflection and Transmission coefficients (p. 111–112).

PI15. Plane parallel plate, graphs depending on wavelength (see p. 113). PI16. Dependence on angle of the Finesse (p. 117).

PI17. Plane parallel plate with mirror surface. PI18. Fabry-Perot (see p. 115).

PI19. Interference with an array of source points (see p. 122). PI20. Summation of random phase angles (see p. 125).

FIGURE 3.2 Conditions for diffraction on a single slit: (a) d λ, no appreciable diffraction; (b) d of the same order of magnitude of λ, diffraction is observed (fringes); (c) d λ, nonuniformly illuminated observation screen, but no fringes.

generated between adjacent light waves, are finite for the superposition process of interference and infinitely small for diffraction.

If we apply this division process to an open aperture, the incident wave generates new waves in the plane of the aperture, and these newly generated waves have fixed phase relations with the incident wave and with one another. We assume that all waves generated by the incident wave propagate only in the forward direction, and not backward to the source of light. Let us consider the diffraction on a slit (Figure 3.2). The observed pattern depends on the wavelength and the size of the opening. A slit of a width of several orders of magnitude larger than the wavelength of the incident light will give us almost the geometrical shadow (Figure 3.2a). A slit of width of an order or two larger than the wavelength will bend the light and fringes will occur; see Figure 3.2b. A slit smaller than the wavelength will show an intensity pattern with no fringes and decreasing intensity

for larger angles; see Figure 3.2c. All openings will show small deformations of the wavefront close to the edges of the slit (not shown in Figure 3.2).

The model we are using for the description of diffraction is called scalar wave diffraction theory and uses the Kirchhoff–Fresnel integral. All the waves we consider are solutions of the scalar wave equation, as used for the discussion of the interference phenomena in Chapter 2. Here we use spherical waves of the type Aeikr/r, where A is the magnitude of the wave, r the distance from the origin, and k 2π/λ. These spherical waves are solutions of the scalar wave equation

where we have not explicitly given the terms in θ and φ because we only use spherical symmetric solutions and they do not depend on the angular terms.

There is the question of why we should use a summation process based on the idea of Huygen’s Principle to describe diffraction theory. Why not solve Maxwell’s equations with the appropriate boundary conditions? The mathematical formulation of Huygens’ Principle was performed by Gustav Kirchhoff and Augustin Jean Fresnel before Maxwell’s theory was developed. It turned out that the use of the Kirchhoff–Fresnel integral for many applications is so much easier than solving Maxwell’s equations and applying boundary conditions, that one just continues to use the scalar wave diffraction theory. The wavelength is assumed to be smaller than the aperture opening under consideration.

3.2 KIRCHHOFF–FRESNEL INTEGRAL

3.2.1 The Integral

We assume for the summation process of the Huygens’wavelets, that the primary wave from the source S has amplitude A and travels distance R in the direction of the aperture (Figure 3.3). We disregard the time factor for all waves considered in this chapter. We recall that in Chapter 2 the time factor disappeared when calculating the intensity. At each point of the aperture a Huygens’ wavelet is generated, [(1/ρ) exp(ikρ)], and travels only in the “forward” direction (Figure 3.3). It has the amplitude of the incident wave, that is, {(A/R) exp(ikR)}. We have for a newly generated wavelet

where exp(iα) is a phase factor related to the generation process. However, it is set to 1.

FIGURE 3.3 (a) Maxima and minima of incident wave; (b) three newly generated Huygen’s wavelets are shown at the aperture.

From experiments we know that there is an angular dependence of the intensity in the direction of propagation. Therefore we multiply by cos θ, where θ is the angle to the normal of the aperture, pointing into the forward direction. Integration over all points of the aperture results in the Kirchhoff–Fresnel integral,

opening of aperture

where dσ is the surface element for integration over the opening of the aperture. This integral may be derived from the scalar wave equation and Green’s Theorem. The derivation yields the factor cos θ and shows that the diffracted light is only traveling in the forward direction. However, there are some problems with the boundary conditions. A formulation using Green’s function avoids this problem, but is not necessarily better. For more information, see Goodman, 1988, p. 42.

In the following two sections we apply Eq. (3.4) to a special symmetric arrangement of source and observation points, both at large distances from the aperture. We calculate the diffracted intensity only at one observation point on

FIGURE 3.4 Coordinates for the circular opening. The source point and the observation point have the same distance from the aperture.

the axis of the system for the diffraction on a round aperture and a round stop. These calculations are taken from Sommerfeld’s book on theoretical physics.1

3.2.2 On Axis Observation for the Circular Opening

The diffraction on a round opening is important since most lenses, spherical mirrors, and optical instruments have circular symmetry. We consider a round aperture of radius a with equal distance to the aperture of the source point S and the observation point O. In Figure 3.4 we show the coordinates and have R0 ρ0, R ρ, cos θ ρ0/ρ, and for the surface element dσ 2πrdr. The amplitude at the observation screen is then obtained by the integral over the opening

Integration by parts with u 1/ρ2, dv eik2ρ , and v (1/2ik)eik2ρ results in

Since ρ is large, we retain only the first term with the power of 1/ρ2 and have

√

u (2πAρ0/2ik) ei2k a2+ρ02 /(a2 + ρ02) − (exp{i2kρ0})/ρ02 . (3.7)

1Vorlesungen uber theoretische Physik, Band IV, by A. Sommerfeld. Dieterich’sche Verlagsbuchhandlung, Wiesbaden, 1950.

FIGURE 3.6 Coordinates for the circular stop. The source point and the observation point have the same distance from the aperture.

3.2.3 On Axis Observation for Circular Stop

In optics, it is often of interest to study complementary screens or arrays. We apply the Kirchhoff–Fresnel integral, Eq. (3.4), to a circular stop, as shown in Figure 3.4. Similar to the “opening,” we must now evaluate

Multiplication of u in Eq. (3.13) by u yields the intensity, and taking for the normalization I0 A2ρ02/(a2 + ρ02), we have

The intensity of Eq. (3.14) depends only on the wavelength, not on the diameter of the aperture or the distance from it. We have the result that at any point in the shadow of an aperture stop, we will observe a bright spot.

There is the story that Fresnel presented his wave theory of light to the French Academy of Sciences. The famous Poisson questioned the validity and argued that there should be a light spot in the shadow of an illuminated sphere, for example, a steel ball. Another scientist of the Academy, Arago, made the experiment, observed the spot and presented his finding to the Academy in support of Fresnel’s theory, but the spot remains the “Poisson spot.” A photograph of the Poisson spot and an experimental setup for observation are shown in Figure 3.7.

FIGURE 3.7 (a) Photograph of the diffraction pattern produced by a round stop. The Poisson spot appears in the middle (from Cagnet, Francon, Thrierr, Atlas of Optical Phenomena, SpringerVerlag, Heidelberg, 1962); (b) parameters for the observation of the Poisson spot (after R. Pohl. Einf¨uhrung in die Optic, R.W. Pohl, Springer-Verlag, Heidelberg, 1948).

3.3FRESNEL DIFFRACTION, FAR FIELD APPROXIMATION, AND FRAUNHOFER OBSERVATION

In the first two applications of the Kirchhoff–Fresnel integral, we have assumed that the source of light and the observation point are at large distances from the aperture. How large was not specified. When assuming that the distance is “infinitely large,” so large that we essentially have plane waves incident on the aperture, we are at the approximation used in Chapter 2. When observing at a screen similarly far away from the aperture, the waves arriving there are also considered plane waves and are also parallel for their superposition. This is called far field approximation. When we use a lens and observe the diffraction pattern in the focal plane, we have Fraunhofer diffraction. The mathematical presentation of far field approximation and Fraunhofer diffraction is the same. In contrast, when the distance from the aperture to the observation screen is large but finite,