Subsections

• Diffraction
• Single Slit Diffraction
• Diffraction by a Double-slit
• Diffraction Grating

Theory

Diffraction

Figure 1 shows diffraction of a wave through a narrow aperture. The degree of spreading in the outgoing wavefronts is large when the aperture size is comparable to or smaller than the wavelength, and is small when the opening is much bigger than the wavelength. The shape of the wavefronts may be derived using Huygens principle; each point in a wavefront is treated as the source of spherical wave-lets whose superposition generates the next wavefront.

Figure 1: Diffraction through a narrow aperture

Single Slit Diffraction

Consider the situation shown in Figure 2. The intensity pattern seen on the projection screen is the sum of the contributions from each point in the wavefront in the slit. These contributions will differ in phase at the screen because the wave-lets travel different distances to reach a common point on the screen. Consider first a point in the slit a distance x from the midpoint of the slit. Assuming that the distance D to the projection screen is much greater than the slit width a, the extra distance is approximately equal to $$\Delta L\approx x\sin \alpha$$

Figure 2: Path difference  for a single slit

Let  equal the amplitude of wavefront in the slit. If this is constant over the entire width of the slit, then a small section of width dx will contribute an amount  to the total. The wave-let originating a x will be out of phase with the one from the midpoint by a phase angle . Hence this wave-let's contribution is $$\frac{A_{0}dx\cos \left( \frac{2\pi x\sin \alpha }{\lambda }\right) }{a}$$

The amplitude of the wave at any point on the projection screen is then found by integrating this expression over the whole slit. If we define a new variable

$$\theta =\frac{\pi a\sin \alpha }{\lambda }$$

this expression will simplify to

 

$$A=A_{0}\frac{\sin \theta }{\theta }$$ (1)

The intensity of the light is proportional to the square of the amplitude. A plot of the intensity pattern is shown in Figure . Note that the y-axis scaling is logarithmic. The zeros in the intensity pattern occur at the locations for which  (other than ). This in turn requires that $$\frac{\pi a\sin \alpha }{\lambda }=m\pi \; \; (m=1,2,3\ldots )$$

and hence

 

$$a\sin \alpha =m\lambda$$ (2)

Figure 3: Intensity vs. angle plot for a single slit

Since the intensity is proportional to the square of the amplitude, Eq. 1 tells us that

 

$$I=\frac{I_{0}\sin ^{2}\theta }{\theta ^{2}}$$ (3)

Diffraction by a Double-slit

The double-slit diffraction pattern may be thought of as the result of two single-slit patterns that are  superimposed so that they nearly coincide. To compute the sum, note that the light from one slit lags the other in phase as a result of the difference in path lengths between them (see Figure 4). The extra distance is  where d is the slit separation. The phase angle between the two is $$\phi =\frac{2\pi d\sin \alpha }{\lambda }$$

The overall amplitude is found by multiplying the amplitude of the lagging contributor by  and adding it to the first

$$A=\frac{A_{0}\sin \theta }{\theta }\left( 1+\cos \left( \frac{2\pi d\sin \alpha }{\lambda }\right) \right)$$

The condition for a maximum is

 

$$\frac{2\pi d\sin \alpha }{\lambda }=2n\pi \quad (n=1,2,3\ldots )$$ (4)

while the condition for a minimum is

 

$$\frac{2\pi d\sin \alpha }{\lambda }=(2n+1)\pi \quad (n=1,2,3\ldots )$$ (5)

The intensity is again proportional to the square of the amplitude.

 

$$I=I_{0}\left( \frac{\sin \theta }{\theta }\right) ^{2}\left(......\left( \frac{2\pi d\sin \alpha }{\lambda }\right) \right) ^{2}$$ (6)

Figure 4: Path difference in terms of the angle 

The typical form of a double-slit intensity pattern is shown in Figure 5.

Figure 5: Intensity graph for a double slit

Diffraction Grating

The diffraction grating pattern can be understood using the above results for the double slit. A typical grating has hundreds of small openings through which light is passed. Diffraction causes light from different openings to overlap in the region behind the grating. Since light from different openings travels different distances, the wave-lets will have different relative phases when they arrive at a particular location.

Figure 6: Path lengths for adjacent slits in a grating

The path lengths for each are related; if adjacent slits have a difference, the next one will differ from the first by , and so on (see Figure 6). If the adjacent slits are separated by a, the intensity at a particular point will have the form $$A=\frac{A_{0}\sin \theta }{N\theta }\sum _{n=0}^{N}\cos \left( \frac{2n\pi d\sin \alpha }{\lambda }\right)$$

The sum in this expression will in most cases essentially averages the cosine function over several arguments. For most viewing angles, this will result in virtually complete cancellation, so the region appears dark. If, however,  satisfies the condition that the distances traveled by light from adjacent slits differ by a whole number of wavelength, constructive interference will occur. The  that satisfy this condition are the same as those that produce constructive interference for the double slit.