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Diffraction Gratings

Physics: AQA A Level, topic 3.2.5

Diffraction Grating Equation

The pattern produced by a diffraction grating can be described by the diffraction grating equation.

Equation

Equation

  • Diffraction gratings produce a pattern which is described by:
  • dsin⁡θ=nλd\sin\theta = n\lambda
    • Where d is the distance between slits in the grating, θ is the angle between the maximum and the zero order line, λ is the wavelength of incident light, and n is the order of the maximum.
Slit spacing

Slit spacing

  • A diffraction grating is made of many slits.
  • If there are 1000 slits per metre, then slit spacing is 1/1000 metres.
  • In general, if there are x slits per metre, then the slit spacing is 1/x metres.
  • This gives the value of "d".
Orders

Orders

  • The variable "n" stands for the order of the maximum.
  • Knowing d and λ lets us predict the angle of the central maximum.
  • The central maximum is the zero order. "n" is zero.
  • This implies that θ is zero. This is what we expect.
  • The first order maximum will have n = 1, the second order maximum will have n = 2, and so on.
Not all orders exist

Not all orders exist

  • Remember that sin θ cannot be greater than 1.
  • If you are using the equation and find that sin θ is larger than 1, the order you are looking at must not exist.
Conclusions

Conclusions

  • As we increase λ, sin θ increases so θ increases. This means the pattern becomes more spread out.
  • If we increase the distance between slits, d, sin θ decreases so θ decreased. This means the pattern becomes less spread out.

Derivation of Grating Equation

You need to know how to derive the diffraction grating equation.

Step 1 - producing coherent sources

Step 1 - producing coherent sources

  • Light enters the grating as parallel rays.
  • It diffracts through each slit.
  • The slits then act as coherent (in phase) and monochromatic (same wavelength) sources of light.
  • These diffracted waves then interfere with each other to produce the pattern.
Step 2 - angle to 1st order

Step 2 - angle to 1st order

  • The 1st order maximum happens at an angle such that the path difference between two sources is one wavelength, λ (for constructive interference).
  • Let's call this angle θ.
Step 3 - create triangle

Step 3 - create triangle

  • We create a triangle as shown above.
  • We know the distance between the two slits is d.
  • We know the path difference is λ.
  • We know the angle θ by geometry.
Step 4 - use trigonometry

Step 4 - use trigonometry

  • Using trigonometry, we can see that:
  • path difference =λ=dsin⁡θ=\lambda = d \sin\theta
Step 5 - generalise for all n

Step 5 - generalise for all n

  • We know that maxima always occur for when the path difference is a whole number multiple of λ (constructive interference).
  • Therefore, we can generalise the equation so that path difference = nλ
  • nλ=dsin⁡θn\lambda = d \sin \theta

Applications of Diffraction Gratings

Diffraction gratings are very useful for physicists.

Analysing light

Analysing light

  • Diffraction gratings can be used to separate wavelengths in light from different substances.
  • The wavelengths present can help us learn new things about the substance being tested.
Atomic spacing in crystals

Atomic spacing in crystals

  • Atoms in a crystal can act like a diffraction grating, with little gaps for light to pass through.
  • We can find the spacing between the atoms in the crystal by looking at how X-rays diffract through the crystal.

Test yourself on Diffraction Gratings

  • What is the correct equation for a diffraction grating?

  • What is the equation for the path difference at the first order maximum?

  • Which of these roles can a diffraction grating perform?

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