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!