PhysicsLAB: Physical Optics - Interference and Diffraction Patterns

Background Information

The central region of constructive interference is known as the central maximum, or A_o. On either side of the central maximum are the first order nodes, N₁. These are regions of destructive interference. On either side of N₁ are the next antinodes, A₁. This alternating pattern of nodes and antinodes continues throughout the construction.

Utilize this ripple tank wave physlet to assist you in understanding the following sets of interference properties:

When two in-phase point sources are moved closer together, there is less interference produced, as evidenced by fewer nodes. When the amount of interference decreases, the width of any given antinode increases.
When two in-phase point sources are moved further apart, there is a greater amount of interference produced, as evidenced by a larger number of nodes. When the amount of interference increases, the width of any given antinode decreases.

Another way to change the amount of interference produced by two in-phase point sources is to change their frequency but leave their separation distance unchanged.

When the frequency is decreased, less interference is produced since the wavelengths will increase, generating fewer wavefronts between the two sources (the equivalent to moving the point sources closer together).
When the frequency is increased, more interference is produced since the wavelengths will decrease, generating more wavefronts between the two sources (the equivalent to moving the point sources further apart).

Young's Double Slit Experiment

Historically, why was Young’s discovery of the interference of light important? Prior to his experiment, the controversy regarding the nature of light was centered in two camps:

Newton’s corpuscular theory
Huygen’s wavelet theory

Both groups had developed theories that explained the reflection and refraction of light. Since Newton was the most eminent scientist of his day, his corpuscular theory [light rays were composed of tiny bullets of finite mass that traveled at extremely high speeds] received wider acceptance, even though his dynamics erroneously demanded that light would travel faster in denser media. Huygen’s wave theory supported the phenomena of interference and diffraction of light - it had just never been observed.

Young’s interference experiment, along with the diffraction effects seen in Fresnel's prediction and the subsequent demonstration by Dominique Arago of Poisson's Spot, showed that light DID have a wave nature. The reason these effects had not be seen previously was a result of light’s extremely small wavelengths.

Newton’s corpuscular theory was completely abandoned after Fizeau and Foucault showed that light SLOWED down while it traveled through water. Einstein’s explanation of the photoelectric effect re-instituted the particle nature of light - now called a photon which represents, not a particle with mass, but a bundle of radiant energy (E = hf) that interacts with matter, principally electrons.

Young began his experiment by sending waves of coherent light towards a barrier with two slits. The two slits are separated by a distance d. On a screen, a perpendicular distance L from the slits, a series of interference fringes were viewed. The formulas we will now develop will allow us to determine if a point P in the interference pattern a distance y from its center will fall into a bright (maximum, constructive interference) zone or into a dark (minimum, destructive interference) zone.

EPD represents extreme path difference, or the effective difference in the distances the light must travel to reach a given position on the screen from each of the slits. For bright fringes, this path difference must be a multiple of a wavelength to insure constructive interference. When this overall path difference equals an odd-multiple of a half wavelength, then destructive interference is insured and a dark fringe is formed on the screen.

Thomas Young's equation for double slit interference is

where

For bright fringes (constructive interference)

where

For dark fringes (destructive interference)

where

	How is an interference pattern affected when the distance between the slits is decreased but the wavelength remains unchanged?

	How is an interference pattern affected when the distance between the slits is increased but the wavelength remains unchanged?

	How is an interference pattern affected when the distance between the slits remains constant but the wavelength of the impinging light decreases?

Diffraction Gratings

Many times these interference properties are investigated through the use of diffraction gratings. A grating is a huge collection of tightly spaced slits - usually upwards of 5 x 10⁵ slits per meter. The distance between the slits in a grating is calculated as the reciprocal of its grating spacing. That is

d = 1/[number of slits/meter]

The principal difference between an interference pattern caused by two slits and that caused by a grating is that a grating has more intense bright fringes that are more widely-spaced. These two effects should make sense.

The brightness would increase since more light can reach the screen (there are more openings, or slits, in the barrier).
The greater spacing can be accounted for by the fact that since d is getting increasingly smaller, y would get increasing larger.

	A common commercial grating has 13400 lines per inch. What is the distance between each slit in meters?

	Light of wavelength 500 nm is shown through this grating onto a screen located on a wall 3 meters away. How wide is the central maximum?

	What is the maximum number of interference fringes that could be viewed on the screen?

Diffraction

Diffraction is the bending of a wave around an obstacle or through an opening. Diffraction fringes are caused by the interference between the component rays as they pass through the same slit.

The formula used to calculate diffraction fringes for a single slit of width w is very similar to that used for double-slit interference.

Notice how similar this is to the formula for interference. The distance between the slits, d, is replaced with the width of a single slit, w. Since diffraction represents the interference between component rays WITHIN a single slit, dark fringes occur when the

. This results from the fact that the slit can be divided into a even number of half-wavelength regions which cancel out each other. The location of bright fringes is not calculated.

In the "zoomed-in" diagram below, point P would be located in D₁ since the EPD between the top ray and the bottom ray emanating from the slit equals 1l. The slit can be effectively divided into two regions where all of the component rays reaching the screen from the top half of the slit have counterparts in the bottom half that are exactly a ½λ out-of-phase with each other. This results in complete destructive interference - or a dark fringe on the screen.

The width of the central maximum can be calculated by letting m = 1, finding the value for y and then doubling it since the diffraction pattern is symmetric. Remember in this formula that y represents the linear deviation along the screen from the center of the central maximum to the center of a specified dark fringe.

If the EPD equals 1½ λ, then are three ½λ-regions and P would be located in B₂ where the intensity would be 1/3^rd that of the central maximum. Remember that only light from one of the three sections reaches the screen.

If the EPD equals 2λ then there are four ½λ-regions and P would be located in D₂.
If the EPD equals 2½ λ, then are five ½λ-regions and P would be in B₃ where the intensity would be 1/5^th that of the central maximum.

	How does the number of fringes change when the width of the slit is decreased?

	What happens when the width of the slit remains unchanged, but the wavelength is decreased?

Take a moment and investigate these relationships with this physlet on single slit diffraction.

Further differences between interference and diffraction patterns.

Note that the intensity of the bright fringes in an interference is equally bright, equally spaced. In a diffraction pattern, the central maximum has the greatest brightness, with each successive bright fringe getting narrower and dimmer.

Given below is a sketch of an intensity pattern that shows the combined effects of both diffraction and interference when light passes through multiple slits.

Notice that the diffraction pattern forms "an envelope" around the interference pattern that defines the intensity of the interference fringes. Also notice that since the interference fringes REMAIN equally spaced, some of the bright fringes are "squelched" by diffraction minima.