The Diffraction Grating (AQA A Level Physics): Revision Note

The diffraction grating

• A diffraction grating is a piece of optical equipment that also creates a diffraction pattern when it diffracts:

  • monochromatic light into bright and dark fringes

  • white light into its different wavelength components

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

The diffraction grating equation

• Diffraction gratings are useful because they create a sharper pattern than a double slit

  • This means their bright fringes are narrower and brighter while their dark regions are wider and darker

• Just like for single slit diffraction and double-slit interference, 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:

  • $d$ is the distance between the slits on the grating (m)

  • $\theta$ is the angle of diffraction of the light of order n from the normal as it leaves the diffraction grating (°)

  • $n$ is the order of the maxima, the number of the maxima away from the central (n = 0)

  • $\lambda$ is the wavelength of the light from the 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:

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

• 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

Angular separation

• The angular separation of each maxima is calculated by rearranging the grating equation to make $\theta$ the subject

• The angle $\theta$ is taken from the centre meaning the higher orders of $n$ are at greater angles

• 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 $n_1$ and $n_2$ is ${\theta }_2-{\theta }_1$

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 $\theta =90°$ and $\sin \theta =1$

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

$n=\frac{d}{\lambda }$

• 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

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, there is constructive interference, hence the path difference is $\lambda$

• Therefore, at the nth order maxima, the path difference is equal to $n\lambda$

• Using trigonometry, an expression for the first-order maxima can be written:

$\sin \theta =\frac{\lambda }{d}$

• Where:

  • $\theta$= the angle between the normal and the maxima

  • $\lambda$ = the wavelength of the light (m)

  • $d$ = the slit separation (m)

• This means, for $n=1$:

$\sin {\theta }_1=\frac{\lambda }{d}$

• Similarly, for $n=2$, where the path difference is $2\lambda$:

$\sin {\theta }_2=\frac{2\lambda }{d}$

• Therefore, in general, where the path difference is $n\lambda$:

$\sin {\theta }_n=\frac{n\lambda }{d}$

• A small rearrangement leads to the equation for the diffraction grating:

$d\sin \theta =n\lambda$

Applications of diffraction gratings

• 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 also used in monochromators:

  • To specifically analyse a wavelength emitted by molecules in diseased cells from a biopsy sample

  • To help excite molecules in a sample with a specific wavelength of light

• Diffraction gratings are used to select the optimum wavelength of light for use in an optical fibre