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# Diffraction and Interference Study Guide (page 3)

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Updated on Sep 27, 2011

#### Example 2

The first dark fringe on a screen is 50 cm away from an aperture and is at a height of 5 cm above the center of the bright fringe. Find the width of the aperture (if the wavelength is 600 nm) and the angle that the first dark fringe position makes with the horizontal.

#### Solution 2

First, all units need to be converted in SI, we will then draw the diagram, and we will set our equations and solve for the unknowns.

h = 5 cm = 0.05 m

d = 50 cm = 0.50 m

λ = 600 nm = 600 · 10–9m

w = ?

θ = ?

From Figure 20.9, one can determine the angle corresponding to the first fringe:

θ = sin–l 0.125

θ = 7.2°

Using this information and the condition for the first dark fringe (m = 1), we can set a new equation:

w = 4.8 μm

## Diffraction Gratings

A fringe pattern is not only produced when light encounters an aperture, but also when two or more parallel, thin slits are placed in front of an incident wave. A central bright fringe will form, followed by a first -order fringe on each side of the central one and so on. Both of the first -order fringes are at the same angle with respect to the central fringe, as shown in Figure 20.10.

The fringes where the result of the superposition of all the waves at a point and time is maximum are called the principal maxima (constructive interference). The fringes are less bright at other positions as well.

If the source is monochromatic (a single frequency), then the fringes are also monochromatic. If the source is a mixture of lights, then the central fringe will resemble the source, but the subsequent fringes will be split in multiple fringes, one for each of the composite colors. Therefore, a lamp producing a combination of blue and indigo as the output light after the diffraction grating will split into a higher-order, multicolor set of fringes with the blue one (λblue > λindigo) being spread farther since the corresponding angle is larger.

The condition to obtain a principal maximum is sin θ = , m = 0, 1, 2, 3,...

Diffraction gratings are characterized by the number of grooves per centimeter or per millimeter. If the number is known, then one can calculate the groove separation a.

## Polarization

We have studied the propagation of electromagnetic waves and should remember that they are transversal waves with an interdependent variation of electric and magnetic fields (E and B make a 90° angle and both are perpendicular at the direction of propagation). The direction of the electric field defines the direction of polarization. If a source produces waves that are all oriented such that E is directed in a single direction, we call those polarized waves (see Figure 20.11). When waves are produced such that an electric field is chaotically spread, the wave is unpolarized.

If an obstacle of a certain shape is placed in front of a source of electromagnetic waves, then the waves can be polarized. The alternation of the electrical field that sustains the magnetic field and, therefore the propagation of the wave, will not be possible in certain directions. Figure 20.12 shows a polarized wave that can propagate through the groove because the groove and the polarization are in the same plane.

And in the case where the polarization and the slit are at 90°, the light does not propagate farther than the slit (see Figure 20.13).

The light polarized can subsequently change polarization and intensity through a polarizer and an analyzer. Incoming unpolarized light will be absorbed by the polarizer in all directions other than the direction corresponding to the transmission axis of the polarizer. The average intensity of light leaving the polarizer is half the intensity of the incident light:

The light output will change further, and emerging light following the analyzer has an average intensity given by Malus' law (Etienne-Louis Malus, 1775-1812):

where is the average intensity of the outgoing light and is the average intensity of the incident polarized light, while θ is the angle between the incident and outgoing polarization of the wave.

The intensity of light is measured in watt/s2. It is proportional to the square of the electric field. Hence, an increase of E by two will increase the intensity by four (see Figure 20.14).

#### Example

Consider a set of a polarizer and an analyzer, and the incident light bringing in an average intensity of 800 W /m2. What is the average intensity after the polarizer and then after the analyzer if the analyzer's transmission axis makes a 90°-angle with the axis of the polarizer?

#### Solution

First, we define the data and then we set the equations needed to solve the problem:

θ = 90°

Therefore, no light will be coming out of the system.

Practice problems of this concept can be found at: Diffraction and Interference Practice Questions

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