The Diffraction Grating (Cambridge International 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

$d \sin ( \\theta ) = n \\lambda$

• Where:

  • d = spacing between adjacent slits (m)
  • θ = angle between the zero-order and nth-order 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

$d=\frac{1}{N}$

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 diffracted light travels along the plane of the grating

  • This means θ = 90° and sin θ = 1

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

$n=\frac{d}{λ}$

• 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

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.

Answer:

Step 1: List the known quantities

• Order of maxima, n = 2
• Diffraction slit spacing, d = 1.7 µm = 1.7 × 10^–6 m
• Wavelength, *λ *= 550 nm = 550 × 10^–9 m

Step 2: Rearrange for β and substitute in the values

$d \sin ( \\theta ) = n \\lambda$

$\sin ( \\beta ) = \frac{n \\lambda }{d}$

$\sin ( \\beta ) = \frac{2 \times ( 550 \times 10^{- 9} ) }{1.7 \times 10^{- 6} } = 0.64706$

$\beta = \text{sin} ^{ - } 1 ( 0.64706 ) = 40.32^{ ^{\circ} }$

Step 3: Calculate α

• *β *is the angle from the centre to the second-order line

$\\alpha = 2 \\beta = 2 \times 40.32 = 81^{\circ} ( 2 \mathrm{s} . \mathrm{f} )$

Exam Hint Take care that the angle θ is the correct angle taken from the centre and not the angle taken between two orders of maxima.

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

$d \sin ( \\theta ) = n \\lambda \rightarrow \\lambda = \frac{d \sin ( \\theta )}{n}$

• 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 $\mathrm{tan}θ=\frac{\mathrm{opposite}}{\mathrm{adjacent}}=\frac{h}{D}$

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 $\\theta = \tan ^{- 1} ( \frac{h}{D} )$
• 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 between the two nth-order maxima and halving
• 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