Objectives:

  • To learn about and understand the Young double-slit experiment using visible light
    (you should understand that this experiment gave a classical confirmation of the wave-nature of light) (Internet research on the ideas of Newton and Huygens about the nature of light);
    (ii) techniques and procedures used to determine the wavelength of light using a diffraction grating and using the equation  d \sin \theta = n \lambda

A transmission diffraction grating is similar to the Young’s slit experiment except instead of having two slits there are many equally spaced lines ruled on a glass slide. Each line diffract the light source causing a diffraction pattern. The diffraction pattern caused by the grating pattern is usually much more widely spaced, while the double slit pattern normally is a series of closely spaced bright fringes. The grating pattern therefore is easier to use as it can give more accurate results in most occasions.

When a narrow beam of monochromatic light is directed normally at a transmission grating, the beam passes through and is diffracted into well-defined directions.

The result that we get is a series of bright spots on a screen, which are called maxima. The one in front of the light source is called 0^{ \text{th} } maxima and it is the brightest, the others are in numerical order (the first maxima is on either sides of the zeroth one). Let’s assume we have a line that goes from the light source to the zeroth maxima and a line that goes from the source to the first maxima, and let’s call \theta the angle between them.

These directions are indicated and can be calculated by the following equation:

d \sin \theta = n \lambda

where $d$ is the grating spacing (the space between two lines in the grate), \theta is the angle between the line that connects the source to the zeroth maxima and the line that connects the source with the chosen n^{ \text{th} } maxima, n is the number of the maxima that is considered and \lambda is the wavelength of light that is shining on the grating.

In an examination a key component you will likely be given is that the grating has somewhere between 100 and 1000 lines per millimetre, but this is not exactly the value of d .

If there are let’s say 100 lines per millimetre, how many lines there are in a metre? The answer is 100,000 . Then we want to know how much space there is between a line and the next so:

d = \frac{1}{100,000} = 0.00001 \ m

This is the distance betweentwo adjacent slits that can then be used in the equation $latex d sin \theta = n \lambda $.

Note:
The Young’s slit experiment uses the letter a for the slit separation, whereas frequently diffraction gratings use the letter d for two adjacent slit separations. This is not always the case but is so if you study OCR A Physics. Study OCR B Physics and fortunately for you they use the same letter for both equations.

Example question:
If a red laser light, with a wavelength of 700 \ nm is shone through a diffraction grating with 400 lines per mm, how what will be the angle that separates the zeroth maxima and the n = 1 maxima?

Solution:
Determining slit separation
d \rightarrow 400,000 lines per metre
d = \frac{1}{400,000} = 2.5 \times 10^{-6} m

Determining the angle
d \sin \theta = n \lambda
\sin \theta = \frac{1 \times 700 \times 10^{-9}}{2.5 \times 10^{-6}} = 0.28
\theta = \sin^{-1}{0.28} = 16.3^{ \circ}

This tells us that the light that the transmits through the diffraction grating perpendicularly to its surface will have a fringe at an angle of 16.3^{ \circ} to the normal.

Similarly, green light with a wavelength of 550 \ nm transmitting through the same diffraction grating will have a fringe at an angle of 12.7^{ \circ} to the normal.

From this, and further calculations, it can be seen that different colours will diffract through a larger or smaller angle depending on their wavelength (or more accurately their frequency). So white light that is shone through a diffraction grating will have the colours separated, as shown in the diagram below:

Page co-written with Luca Quinci – Thank you!


Further reading:

  • Isaac Physics – Diffraction – This is a reading resource
  • Waveguide Diffraction Gratings and Applications by Nahum Izhaky
  • Diffraction Gratings and Applications by Erwin G. Loewen and Evgeny Popov – preview