9.3.3 Diffraction Grating Patterns

• A diffraction grating is a plate consisting of a very large number of parallel, identical, close-spaced slits
  • When monochromatic light is incident on a grating, a pattern of narrow bright fringes is produced on a screen

Diffraction grating with multiple slits, usually described in terms of 'slits per metre'

• The number of slits dictates both the interference and diffraction patterns which are seen

• On the screen a diffraction pattern is observed
  • The interference pattern is a measure of the intensity of the light at different angles of diffraction away from the central maxima
  • For a double slit arrangement this is shown in (9.3.2 Two-Slit Interference Patterns)

• On the diffraction pattern as the number of slits increases:
  • Between the maxima, secondary maxima appear
  • The central maxima and subsequent bright fringes become narrower

Diffraction patterns for light interfering through different numbers of slits

• When there are 3 slits, 1 secondary maxima can be seen between the primary maxima
• When there are 5 slits, 3 secondary maxima can be seen between the primary maxima
  • Therefore, with N slits (when N > 2), there are (N − 2) secondary maxima
• Once the number of slits increases to N > 20:
  • The primary maxima will become thinner and sharper (since slit width, )
  • The (N – 2) secondary maxima will become unobservable

• On a diffraction pattern, as the number of slits increases:
  • The intensity of the central and other larger maxima increases
  • Since the overall amount of light being let through each slit is increased, the pattern increases in intensity by a factor of N²I₀
    • Where I₀ is the intensity of the central maximum of a single slit diffraction pattern

• It is important to recognise the difference between interference and diffraction patterns

The diffraction pattern for light passing through a double-slit is the pattern observed on the screen

The interference pattern for light passing through a double-slit is different to the diffraction pattern

Investigating Interference by Diffraction Grating

The overall aim of this experiment is to calculate the wavelength of the laser light using a diffraction grating

• Independent variable = Distance between maxima, h
• Dependent variable = The angle between the normal and each order, θ_n (where n = 1, 2, 3 etc)
• Control variables:
  • Distance between the slits and the screen, D
  • Laser wavelength λ
  • Slit separation, d

Method

The setup of apparatus required to measure the distance between maxima h at different angles θ

1. Place the laser on a retort stand and the diffraction grating in front of it
2. Use a set square to ensure the beam passes through the grating at normal incidence and meets the screen perpendicularly
3. Set the distance D between the grating and the screen to be 1.0 m using a metre ruler
4. Darken the room and turn on the laser
5. Identify the zero-order maximum (the central beam)
6. Measure the distance h to the nearest two first-order maxima (i.e. n = 1, n = 2) using a vernier calliper
7. Calculate the mean of these two values
8. Measure distance h for increasing orders
9. Repeat with a diffraction grating with a different number of slits per mm

• An example table might look like this:

Analysing the Results

The diffraction grating equation is given by:

• Where:
  • n = the order of the diffraction pattern
  • λ = the wavelength of the laser light (m)
  • d = the distance between the slits (m)
  • θ = the angle between the normal and the maxima

• The distance between the slits is equal to:

• Where
  • N = the number of slits per metre (m^–1)

• Since the angle is not small, it must be calculated using trigonometry with the measurements for the distance between maxima, h, and the distance between the slits and the screen, D

• Calculate a mean θ value for each order
• Calculate a mean value for the wavelength of the laser light and compare the value with the accepted wavelength
  • This is usually 635 nm for a standard school red laser

Evaluating the Experiments

Systematic errors:

• Ensure the use of the set square to avoid parallax error in the measurement of the fringe width
• Using a grating with more lines per mm will result in greater values of h
  • This lowers its percentage uncertainty

Random errors:

• The fringe spacing can be subjective depending on its intensity on the screen, therefore, take multiple measurements of h (between 3-8) and find the mean
• Use a Vernier scale to record distances h to increase precision and therefore reduce percentage uncertainty
• Reduce the uncertainty in h by measuring across all visible fringes and dividing by the number of fringes
• Increase the grating to screen distance D to increase the fringe separation (although this may decrease the intensity of light reaching the screen)
• Conduct the experiment in a darkened room, so the fringes are clear

Safety Considerations

• Lasers should be Class 2 and have a maximum output of no more than 1 mW
• Do not allow laser beams to shine into anyone’s eyes
• Remove reflective surfaces from the room to ensure no laser light is reflected into anyone’s eyes