The Diffraction Grating (AQA AS Physics)

• A diffraction grating is a piece of optical equipment that also creates a diffraction pattern when it diffracts:
  • Monochromatic light into bright and dark fringes
  • White light into its different wavelength components

Diagram of diffraction grating used to obtain a fringe pattern

• A diffraction grating consists of a large number of very thin, equally spaced parallel slits carved into a glass plate

The Diffraction Grating Equation

• Diffraction gratings are useful because they create a sharper pattern than a double slit

  • This means their bright fringes are narrower and brighter while their dark regions are wider and darker

• Just like for single and double-slit diffraction the regions where constructive interference occurs are also the regions of maximum intensity
• Their location can be calculated using the diffraction grating equation

• Where:
  • n is the order of the maxima, the number of the maxima away from the central (n = 0)
  • d is the distance between the slits on the grating (m)
  • θ is the angle of diffraction of the light of order n from the normal as it leaves the diffraction grating (°)
  • λ is the wavelength of the light from the source (m)

Slit Spacing

• Diffraction gratings come in different sizes
  • The sizes are determined by the number of lines per millimetre (lines / mm) or lines per m
  • This is represented by the symbol N
• d can be calculated from N using the equation
  • If N is given in terms of lines per mm then d will be in mm
  • If N is given in terms of lines per m then d will be in m

Angular Separation

• The angular separation of each maxima is calculated by rearranging the grating equation to make θ the subject
• The angle θ is taken from the centre meaning the higher orders of n are at greater angles

Angular separation increases as the order of maxima increases

• The angular separation between two angles is found by subtracting the smaller angle from the larger one
• The angular separation between the first and second maxima at n₁ and n₂ is θ₂ – θ₁

Orders of Maxima

• The maximum angle of diffraction with which maxima can be seen is when the beam is at right angles to the diffraction grating
  • This means θ = 90^o and sin θ = 1

• The highest order of maxima visible is therefore calculated by the equation:

• Since n is an integer number of maxima, if the value obtained is a decimal it must be rounded down to determine the highest-order visible
  • E.g If n is calculated as 2.7 then n = 2 is the highest-order visible

• When light passes through the slits of the diffraction grating, the path difference at the zeroth order maximum is zero
• At the first order maxima, there is constructive interference, hence the path difference is λ
• Therefore, at the nth order maxima, the path difference is equal to nλ

Using this diagram and trigonometry, the diffraction grating equation can be derived

• Using trigonometry, an expression for the first order maxima can be written:

• Where:
  • θ = the angle between the normal and the maxima
  • λ = the wavelength of the light (m)
  • d = the slit separation (m)

• This means, for n = 1:

• Similarly, for n = 2, where the path difference is 2λ:

• Therefore, in general, where the path difference is nλ:

• A small rearrangement leads to the equation for the diffraction grating:

d sin θ_n = nλ

• Diffraction gratings are useful for separating light of different wavelengths with high-resolution
• They are used in spectrometers to:
  • Analyse light from stars
  • Analyse the composition of a star
  • Chemical analysis
  • Measure red shift / rotation of stars
  • Measure the wavelength / frequency of light from a star
  • Observe the spectra of materials
  • Analyse the absorption / emission spectra in stars

• Diffraction gratings also play a role in X-ray crystallography:
  • X-rays are directed at a thin crystal sheet which acts as a diffraction grating to form a diffraction pattern
  • This is because the wavelength of X-rays is similar in size to the gaps between the atoms
  • This diffraction pattern can be used to measure the atomic spacing in certain materials

• Diffraction gratings are also used in monochromators:
  • To specifically analyse a wavelength emitted by molecules in diseased cells from a biopsy sample
  • To help excite molecules in a sample with a specific wavelength of light
• Diffraction gratings are used to select the optimum wavelength of light for use in an optical fibre

Diffraction gratings are most commonly used in spectrometers. These devices play a crucial role in areas of physics such as atomic physics and astrophysics