Understanding diffraction in classical physics

Understanding diffraction in classical physics

Diffraction, which was first observed by Leonardo da Vinci, is often considered to be the signature of a wave. Diffraction occurs when ripples in a pond encounter a pair of logs that are close together, when light passes through a narrow slit in a window shade, or when x-rays scatter from a crystal. In each case, we can explain the distinctive pattern that forms using classical wave theory. Understand what happens to a wave when it passes through a small aperture-that is, understand diffraction-and you are well on your way to understanding the behavior of waves in more complicated situation.

A schematic of a single-slit diffraction experiment with light is shown in Fig. 2.1. Monochromatic light of frequency v is incident on a diaphragm in which there is a single slit of width w.

 We'll assume that the light source is far to the left of the diaphragm, so the incident radiation can be represented by a plane wave. The width of the slit must be small compared to the wavelength  = c/v of the radiation if the slit is to appreciably diffract the light; for example, to diffract visible light enough to observe this phenomenon, we require a slit of width w=10^-4 cm. Light scattered by the diaphragm falls on a detector, such as a photographic plate or a photocell, located at a distance D far to the right of the slit. The detector measures the energy delivered by the scattered wave as a function of the distance x in Fig. 2.1.

Light scattered by the single-slit diaphragm forms a beautiful diffraction pattern at the detector. This pattern is characterized by a very bright central band located directly opposite to the slit, surrounded by a series of alternating light and dark regions. The light regions on either side of the central band are called secondary hands, because they are much less intense than the central band. Indeed, the intensity of the secondary bands drops off so dramatically on either side of the central band that only one pair of secondary bands is visible. Additional weak secondary bands exist, though, as you can see in Fig. 2.2, which is a graph of the intensity measured by the detector. If we play around with the frequency control on the light source and study the resulting diffraction patterns, we discover that the separation between adjacent bright bands is proportional to the wavelength of the incident radiation.

A classical physicist would call upon Maxwell's electromagnetic wave theory to explain the pattern in Fig. 2.2. To understand qualitatively what happens when a plane wave passes through a diaphragm we invoke Huygens's Principle, which lets us replace the plane wave and the slit (in our mind's eye) by a large number of closely-spaced , discrete radiating charges that fill the region of space where the slit is located."Scattered waves" radiated by different oscillators-i.e., waves that emerge from different locations in the slit region-have different phases, so the superposition of these scattered waves exhibits regions of high and low intensity, as in Fig. 2.2.

to derive an expression for the electric field at a point (r, ) on the detector (see Fig. 2.1). But the quantity measured in this experiment is not the electric field; it is the intensity the rate at which the scattered radiation delivers energy to the detector. This quantity is proportional to the time-averaged energy flux-i.e., to the average over one period of the square of the modulus of the electric field. I'll denote by  the intensity at a fixed value of r due to radiation scattered by a single slit.

Omitting the details of the derivation of , I'll just quote the result. For convenience, I'll write the intensity in terms of its value  at the principal maximum i.e, at the point r=D, ( = 0, in the central peak in Fig. 2.2-and the handy intermediate quantity

the single-slit intensity at fixed r is

The intensity (2.4) is graphed in Fig. 2.2 as a function of w sin(); plotted this way, exhibits a characteristic pattern of equally-spaced nodes [where  = 0], which occur at

The principal (zeroth-order) maximum of the intensity pattern and higher order maximum occur (approximately) at

Equation (2.4) fully accounts for the properties of patterns such as the one in Fig. 2.2. Thus does classical electromagnetic theory rend the veil of mystery from the phenomenon of diffraction.

The source:

Michael A. Morrison - Understanding Quantum Physics.

By. Fady Tarek