9.3.4 The Diffraction Grating Equation

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

• The lines per m (or per mm, per nm etc.) on the grating is usually represented by the symbol N
• Therefore, the spacing between each slit, 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 = 90^o 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

Derivation of the Diffraction Grating Equation

• 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 (n = 1), 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