• A diffraction grating is a plate on which there is 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

Diagram of diffraction grating used to obtain a fringe pattern

• The angles at which the maxima of intensity (constructive interference) are produced can be deduced by the diffraction grating equation:

• Exam questions sometime state the?lines per m?(or per mm, per nm etc.) on the grating which is represented by the symbol?N
• d?can be calculated from?N?using the equation

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 are at greater angles

Angular separation

• 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 n1?and n2?is?θ2?–?θ1

Orders of Maxima

• The maximum angle to see orders of maxima is when the beam is at right angles to the diffraction grating
  • This means?θ?= 90o?and sin?θ?= 1

   

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

• Note that since?n?must be an integer, if the value is a decimal it must be rounded?down
  • E.g If?n?is calculated as 2.7 then?n?= 2 is the highest order visible

   

Worked Example

An experiment was set up to investigate light passing through a diffraction grating with a slit spacing of 1.7 μm. The fringe pattern was observed on a screen. The wavelength of the light is 550 nm. Calculate the angle α between the two second-order lines.

Exam Tip

• 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 θn?= 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 most commonly used in spectrometers. These devices play a crucial role in areas of physics such as atomic physics and astrophysics

Exam Tip