Chapter 4

Diffraction of Light Waves

Diffraction Huygen’s principle

requires that the waves spread out after they pass through slits

This spreading out of light from its initial line of travel is called diffraction In general, diffraction

occurs when waves pass through small openings, around obstacles or by sharp edges

A single slit placed between a distant light source and a screen produces a diffraction pattern It will have a broad, intense central band The central band will be flanked by a series

of narrower, less intense secondary bands Called secondary maxima

The central band will also be flanked by a series of dark bands Called minima

The results of the single slit cannot be explained by geometric optics Geometric optics

would say that light rays traveling in straight lines should cast a sharp image of the slit on the screen

Fraunhofer Diffraction

Fraunhofer Diffraction occurs when the rays leave the diffracting object in parallel directions Screen very far from the slit Converging lens (shown)

A bright fringe is seen along the axis (θ = 0) with alternating bright and dark fringes on each side

Single Slit Diffraction According to Huygen’s

principle, each portion of the slit acts as a source of waves

The light from one portion of the slit can interfere with light from another portion

The resultant intensity on the screen depends on the direction θ

All the waves that originate at the slit are in phase

Wave 1 travels farther than wave 3 by an amount equal to the path difference (a/2) sin θ

If this path difference is exactly half of a wavelength, the two waves cancel each other and destructive interference results

In general, destructive interference occurs for a single slit of width a when

sin θdark = mλ / a m = 1, 2, 3, …

Doesn’t give any information about the variations in intensity along the screen

The general features of the intensity distribution are shown

A broad central bright fringe is flanked by much weaker bright fringes alternating with dark fringes

The points of constructive interference lie approximately halfway between the dark fringes

Resolution of Single-Slitand Circular Apertures The resolution is the ability of optical

systems to distinguish between closely spaced objects, which are limited because of the wave nature of light

If no diffraction occurred, two distinct bright spots would be observed on the viewing screen. However, because of diffraction, each source is imaged as a bright central region flanked by weaker bright and dark bands.

If the two sources are separated enough to keep their central maxima from overlapping, their images can be distinguished and are said to be resolved.

If the sources are close together, however, the two central maxima overlap and the images are not resolved.

Rayleigh's criterion

To decide when two images are resolved, the following criterion is used:

When the central maximum of one image falls on the first minimum another image, the images are said to be just resolved.

This limiting condition of resolution is known as Rayleigh's criterion.

The diffraction patterns of two point sources (solid curves) and the resultant pattern (dashed curves) for various angular separations of the sources

From Rayleigh's criterion, we can determine the minimum angular separation, θmin , subtended by the sources at the slit so that their images are just resolved.

the first minimum in a single-slit diffraction pattern occurs at the angle for which

sin θ = λ / a

where a is the width of the slit. According to Rayleigh's criterion, this expression gives the smallest angular separation for which the two images are resolved.

Because λ « a in most situations, sin θ is small and we can use the

approximation sin θ ≈ θ . Therefore, the limiting angle

of resolution for a slit of width a is θmin = λ / a

where θmin is expressed in radians. Hence, the angle subtended by the two sources at the slit must be greater than λ / a if the images are to be resolved.

The diffraction pattern of a circular aperture consists of a central circular bright disk surrounded by progressively fainter rings. The limiting angle of resolution of the circular aperture is:

Where D is the diameter of the aperture.

Diffraction Grating

The diffracting grating consists of many equally spaced parallel slits A typical grating contains several thousand

lines per centimeter The intensity of the pattern on the screen is the

result of the combined effects of interference and diffraction

Diffraction Grating The condition for

maxima is d sin θbright = m λ

m = 0, 1, 2, … The integer m is the

order number of the diffraction pattern

If the incident radiation contains several wavelengths, each wavelength deviates through a specific angle

All the wavelengths are focused at m = 0 This is called the zeroth

order maximum The first order maximum

corresponds to m = 1 Note the sharpness of the

principle maxima and the broad range of the dark area This is in contrast to the

broad, bright fringes characteristic of the two-slit interference pattern

diffraction grating spectrometer.

The collimated

beam incident

on the grating is

spread into its

various wavelength

components with

constructive interference for a particular wavelength occurring at the angles that satisfy the equation

Resolving power of the diffraction grating The diffraction grating is useful for

measuring wavelengths accurately. Like the prism, the diffraction grating can be

used to disperse a spectrum into its components.

Of the two devices, the grating may be more precise if one wants to distinguish between two closely spaced wavelengths.

If λ1 and λ2 are the two nearly equal wavelengths between which the spectrometer can barely distinguish, the resolving power R is defined as

where λ = ( λ1 + λ2 ) / 2 , and

Δ λ = λ2 - λ1 a grating that has a high resolving power

can distinguish small differences in wavelength.

if N lines of the grating are illuminated, it can be shown that the resolving power in the mth order diffraction equals the product N m :

R = N m Thus, resolving power increases with

increasing order number. R is large for a grating that has a large

number of illuminated slits.

Consider the second-order diffraction pattern (m = 2) of a grating that has 5000 rulings illuminated by the light source.

The resolving power of such a grating in second order is: R = 5000 x 2 = 10,000.

The minimum wavelength separation between two spectral lines that can be just resolved, assuming a mean wavelength of 600 nm, is

Δλ = λ / R = 6.00 X 10-2 nm. For the third-order principal maximum,

R = 15 000 and Δλ = 4.00 x 2 nm, and so on.