The Diffraction Grating (Cambridge (CIE) AS Physics)

The diffraction grating equation

• A diffraction grating is a piece of optical equipment that creates a diffraction pattern when it diffracts monochromatic light into bright and dark fringes

A laser light is diffracted using a diffraction grating

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

A diffraction grating consists of many parallel equally spaced slits cut into the glass plate

• 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:

  • d = spacing between adjacent slits (m)

  • θ = angular separation between the order of maxima (degrees)

  • n = order of maxima (n = 0, 1, 2, 3...)

  • λ = wavelength of light 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

Diffraction gratings come in different sizes according to the number of lines per mm

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

Determining the wavelength of light

Method

• The wavelength of light can be determined by rearranging the grating equation to make the wavelength λ the subject

• The value of θ, the angle to the specific order of maximum measured from the centre, can be calculated through trigonometry

• Create a right angled triangle to determine the angle of diffraction, θ

  • The distance from the grating to the screen is marked as D

  • The distance between the centre and the order of maxima (e.g. n = 2 in the diagram below) on the screen is labelled as h, the fringe spacing

• Measure both of these values with a ruler

• Obtain the ratio

Order of maxima and wavelength

The wavelength of light is calculated by the angle to the order of maximum

• Calculate the inverse of tan to find

• Substitute for θ back into the diffraction grating equation to find the value of the wavelength (with the corresponding order n)

Improving the experiment and reducing uncertainties

• The fringe spacing can be subjective depending on its intensity on the screen. Take multiple measurements of h (between 3-8) and find the average

• Use a Vernier scale to record h, in order to reduce percentage uncertainty

• Reduce the uncertainty in h by measuring across all 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 clearer

• Use grating with more lines per mm, so values of h are greater to lower percentage uncertainty