Diffraction and Interference Explaining the wave nature of light.

Christiaan Huygens Christiaan Huygens improved upon the first Wave Model of Light proposed by Robert Hooke Huygens’ Principle: A wave consists of many small point sources of tiny secondary waves, called wavelets, which propogate outward in a concentric circle at the same speed as the wave itself. The line tangent to all the wavelets constitutes the wave front.

Diffraction Diffraction is defined as the spreading out of the wave as it passes through a small opening or around an obstacle Particles do not experience diffraction If light diffracts, then clearly it is a wave

Diffraction Based on experimental data using water waves in a ripple tank, it has been found that the extent of diffraction depends on the wavelength and the width of the opening. More specifically, it depends on the ratio of the two: The longer the wavelength, the greater the diffraction. The smaller the opening, the greater the diffraction Since light has a very small wavelenth, you need to have a very small opening to experience diffraction

Interference

Interference

Interference

Young’s Experiment Bright areas on the interference pattern represent constructive interference and are called maxima and dark areas represent destructive interference and are called minima.

Young’s Experiment Antinodes are defined as positions in an interference pattern where there is only constructive interference Nodes are defined as positions in an interference pattern where there is only destructive interference

The Central Antinode The central antinode occurs along the perpendicular bisector Both waves travel the same distance from the slits to the screen and therefore arrive at the screen in phase .

The First Node The first node is also called the first minimum or first dark fringe. One wave will travel half a wavelength more than the other, causing them to be out of phase (destructive interference)

The First Antinode The first antinode is also called the first maximum or first dark fringe. One wave will travel one wavelength more than the other, causing them to be in phase (constructive interference)

The Mathematics Recall that antinodes have a path difference of one wavelength The distance S1 – X represents the path difference. If Pn is a large distance from the slits, then the two paths are approximately parallel near the slits and as such, we can make a right triangle connecting S2 to X. As a result of this right triangle, we have the following relationship:

The Mathematics For nodes (destructive interference), we could derive the relationship the same way with one difference: the path difference is half a wavelength. As such our formula for nodes is:

Finding Wavelength Experimentally In the laboratory, it is difficult to measure the angle of diffraction because it is extremely small. It is much easier to find sinθ by determining where l is much larger than x allowing us to use sinθ rather than tan θ

Finding Wavelength Experimentally If we substitute into our relationship in place of sinθ and solve for wavelength, we get a new relationship for antinodes: and for nodes: Remember this equation is an approximation and only suitable for θ <10o

Diffraction Gratings Young’s experiment only used two small openings. A diffraction grating has a very large number of equally spaced, parallel lines that act as individual light sources The pattern that is created is similar to the double slit apparatus, but delivers more energy increasing the brightness of the fringes. The fringes are also narrower. If light with multiple wavelengths is used, a diffraction grating will separate the wavelengths and a spectrum will be produced. Diffraction gratings usually are described in terms of number of lines per unit of measurement Ex) 1000 lines/cm To determine the value of d (distance between slits), find the inverse of number of lines per unit of measurement Ex) d = 1/1000 lines/cm = 0.001 cm

In Summary If the angle is > 10o, the second equation must be used. It is helpful to determine the angle at the beginning of the problem in order to determine which equation is appropriate to use.

Example 1 Monochromatic light is incident on two slits separated by 0.30 mm, and the first bright fringe (n = 1 antinode) is located at an angle of 0.080 from the central antinode. What is the wavelength of the light?

Example 2 A student measuring the wavelength of light emitted by a krypton gas sample directs the light through two slits separated by 0.264 mm. An interference pattern is created on a screen 3.0 m from the slits and the distance between the second bright (n = 2 antinode) fringe and the central antinode is measured to be 1.18 cm. What is one of the wavelengths of light emitted by the krypton gas sample?

Example 3 A diffraction grating has 1000 lines/1 cm. When light passes through the grating, an interference pattern is produced on a screen 4.00 m away. The first-order bright fringe is 19.2 cm away from the central antinode. What is the wavelength and colour of the light?