Physical Optics, continued

15.1.1  Why Diffraction Occurs

Diffraction is another instance in which waves do something that is unique to their nature. The phenomenon of diffraction is easily explained qualitatively in terms of Huygen's theory. As a wavefront propagates along, it produces point sources which emit spherical waves. As long as all these point sources start at the same time and are allowed to emit waves without interruption, the wavefront along the direction of propagation is a straight line. If an obstruction or a barrier with a slit intrudes, however, then some of the spherical waves are blocked and cannot contribute to the wavefront. The response is that the wavefront becomes curved. To see this, consider the static images below that show a plane wave moving into a barrier with a slit or a just a barrier. As long as the obstruction or opening is much larger than the wavelength as shown in figure 15.1, the behavior of the waves is what we expect, namely the part of the wavefront that is allowed to continue does so with the wavefront remaining straight.

Figure 15.1: Plane wavefronts approach a barrier with an opening or an obstruction. Both the opening and the obstruction are large compared to the wavelength.

If, however, the size of the opening becomes comparable to the wavelength, the waves proceed to "bend through" or around the opening or obstruction as shown in figure 15.2.

Figure 15.2: Now the plane wavefronts impinge on a barrier with an opening or an obstruction which is not much larger than the wavelenth. The wavefront is not allowed to propagate freely through the opening or past the obstruction but experiences some retardation of some parts of the wavefront. The result is that the wavefront experiences significant curvature upon emerging from the opening or the obstruction.

Finally, as the obstruction or opening equals the wavelength, the diffraction becomes quite pronounced as shown in figure 15.3. We refer to this phenomenon as diffraction.

Figure 15.3: As the barrier or opening size gets smaller, the wavefront experiences more and more curvature.

15.1.2  Types of Diffraction

Given the mathematical complexity, only a part of what constitutes the theory of diffraction can be discussed in detail in the text. Just to introduce the nomenclature though, note that cases in which the source of radiation or the screen are close to the obstruction causing the diffraction are termed Fresnel diffraction. Cases in which the source and screen are far from the obstruction are termed Fraunhofer diffraction. The text describes only Fraunhofer diffraction in quantitative detail as Fresnel diffraction is beyond the mathematical scope of the text. To see the effect of Fraunhofer diffraction in action for visible light, look at the Java applet located here.

As described in the text, the minima of such diffraction is given by

a sinq = ml

(15.1.2.1)

where a is the size of the obstruction and q is the angle relative to the horizontal as shown in figure 15.4.

Figure 15.4: Defining the angle q for calculation of diffraction of light through an aperture of width a.

The condition for interference minima is given by considering, just as for Young's experiment, the "extra" distance traveled by waves from one part of the aperture versus waves from another part. If, for example, we divide up the aperture into two parts and look at a ray from the top half vs. a ray from the corresponding location in the bottom half in getting to point P in figure 15.5, then we see that the condition for destructive interference is

a 2 sinq = ± l 2 .

(15.1.2.2)

Figure 15.5: If the distance from the aperture to the screen, x, is much greater than the size of the aperture, a, then the distance from the top half of the aperture to point P on the screen is less than the distance from the bottom half of the aperture to point P by approximately (a/2)sinq.

So, we expect dark fringes at values of q  which satisfy

sinq = ml a    (m = ±1, ±2, ±3, ...)

(15.1.2.3)

since we could divide the slit into quarters, eighths, 16th's, etc. and repeat the argument of having interference minima for each adjacent pair of intervals.

This is the formula for single slit diffraction minima. Note that there is no central minimum. The center of a diffraction pattern is always a maximum just as it is for interference in Young's experiment. The first minimum for diffraction therefore corresponds to m = ±1 rather than zero as in Young's experiment. The vertical position of these minima is given approximately as

ym = xtanq @ x ml a

(15.1.2.4)

for y << x. The maxima or bright fringes for diffraction are approximately halfway in between the minima.

Intensity for Diffraction

We just state that the intensity for the many subdivions of the aperture that we can make can be expressed in terms of the following formula:

I = I0 é ê ë sin(b/2) (b/2) ù ú û 2

(15.1.2.5)

with

b = 2pa l sinq

(15.1.2.6)

and I₀ is the intensity at q = b = 0.

Circular Apertures

It should be remarked at this point that the Fraunhofer diffraction described up to this point assumes a rectangular obstruction or opening in a barrier. If the aperture is circular, then the pattern of maxima and minima on the screen is disk-shaped with a central bright spot surrounded by a series of bright and dark rings. The first minimum occurs at an angle q₁ which satisfies the equation

sinq = 1.22 l D

(15.1.2.7)

with D being the diameter of the obstruction or aperture.

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