# Diffraction Gratings

A diffraction grating is an optical plate that divides or disperses white light. As you know, white light is composed primarily of seven different colours, each with a different wavelength. The simplest type of grating is a structure with evenly-spaced identical slits.

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Refraction gratings are based on the principle of refraction of light, which states that when a light beam passes through an opening, it spreads out around the opening in a wave pattern.

When light passes through several openings, the light will refract around the openings. The waves that are created behind the openings will interfere with one another, merging together where two peaks meet to create a new peak of higher amplitude; this is also known as constructive interference. They interfere destructively when a trough and a peak meet. This creates an interference pattern, which is shown below.

This is the principle of a diffraction grating. When a parallel beam of light is directed at a diffraction grating with several identical openings, this will result in an interference pattern of bold and faint points of light.

## What is a diffraction grating pattern?

When white light is incident on a parallel grating plate with several or even hundreds of evenly-spaced identical slits, it is diffracted creating spherical waves around the openings that interfere with one another. This creates an interference pattern, where each wave interacts with another. This further creates a pattern of maximums and minimums, as seen below.

Diffraction grating pattern.

The light that is shown on the back screen is a series of dots called maximums. The empty space in between the maximums is called the minimum. The maximum that is parallel to the light beam is the zero-order maximum, while the dots on the sides are the first and the second order maximums going outwards from the middle. The visible points are those points at which many different rays of light interfered.

The angles at which the maximum intensity points occur are known as fringes, and can be calculated using the grading equation below.

$d \cdot \sin\theta = n \cdot \lambda$

where d is the spacing between the slits in metres, θ is the separation angle between the order of maximum in degrees, n is the order of maximum, and λ is the wavelength of the source in metres.

Therefore, $$\sinθ$$ is proportional to the wavelength, which means the longer the wavelength of light (red light has the longest wavelength), the greater the angle. It can also be derived from the above equation that the larger the number of slits per metre (hence, the smaller the d component), the bigger the angle of diffraction.

### Diffraction grating diagram

When a light beam impacts the diffraction grating plate, the white light is separated into the seven different light colours of which it is comprised, each one with its own different wavelength. It can be derived from the equation that the longer the wavelength, the greater the separation angle, and the shorter the wavelength, the smaller the angle. Hence, in the middle, where the angle is zero, the maximum spot will be white. In the first-order maximum points, the blue light will be closest to the white spot while red light will be the light that has the greatest angle. Hence, the light furthest away from the zero-order is the white light spot. This pattern will be then repeated for each order point as seen below.

Diffraction grating diagram.

### Angular separation

The angular separation θ1 (as seen below) of each maximum is calculated by solving the grating equation for θ. Using the equation below and substituting the order of maximum n, we can find the angle between that order of maximum and the zero-order. For example, if we want to estimate angle θ2, we need to replace n with 2 to find the angle between the zero-order and the second-order maximum.

$\sin \theta = \frac{n \cdot \lambda}{d} \theta = \sin^{-1}(\frac{n \cdot \lambda}{d})$

The maximum angle required for orders of maxima to be created is when the beam is at a right angle to the diffraction grating. Hence θ = 90o and sin(θ) = 1.

Separation angle diagram.

Αn experiment was conducted using a diffraction grating with an opening of 1.9 μm. The wavelength of the light beam is 570 nm. Find the angle x between the two second-order lines.

Solution

Use the equation solved for θ and substitute the given values. Use n = 2 as second-order maximum angle required.

$$\sin \theta = \frac{n \cdot \lambda}{d} \theta = \sin^{-1}(\frac{2 \cdot 570 \cdot 10^{-9}}{1.9 \cdot 10^{-6}})\theta = \sin^{-1}(0.6) = 36.8 ^{\circ}$$

However, the diffraction grating equation gives the separation angle, which is the angle between the central zero maximum. But the question requires the angle between the two angles as seen in the diagram below. Hence angle θ is doubled to find angle x.

$$x = \theta \cdot 2 = 73.6^{\circ}$$

Finding the angle between two second-order lines

A light with a wavelength of 480 μm passes through a diffraction grating. The separation angle is 40.85° and the diffraction creates the first-order maximum. Find the opening of the slits.

Solution

Use the diffraction grating equation but rearrange for d. Substitute the given values.

$$d \cdot \sin \theta = n \cdot \lambda d \cdot \sin \theta = 1 \cdot 480 \cdot 10^{-9}d = \frac{1 \cdot 480 \cdot 10^{-9}}{\sin(40.85)} = 7.33 \cdot 10^{-7} m \text{ or } 0.73 \space \mu m$$

## What is the diffraction grating experiment?

The aim of the experiment is to calculate the wavelength of light.

### Materials

• Diffraction grating
• Laser beam
• Ruler
• Binder clips
• Tape
• Colour filter

For conducting the experiment, position a white light source opposite a diffraction grating. A wall behind the grating will be used as a projection screen. Secure the light source with tape and the diffraction grating with binder-type clips. Position a piece of coloured plastic or colour filter between the source and the diffraction grating as needed.

### Methodology

1. Direct the white light beams through the diffraction grating and observe the pattern projected on the wall. Adjust the angle between the beam of light and the glass as needed to achieve the diffraction grating pattern required.

2. Identify the zero-order beam and the diffracted beams by the intensity in the spots illustrated on the wall.

3. Using a ruler, measure distance between the glasses and the white spot on the screen.

4. Repeat the experiment with several laser pointers.

5. For each different light beam, measure the distance between the straight unbent beam and the diffracted beams, also known as h.

6. Calculate the wavelength and compare it to the manufacturer's wavelength, which is the wavelength of the laser used.

7. Insert a piece of coloured cellophane plastic or filter, between the white light beam and the diffraction grating. Record any observations.

### Observations

• The wavelength is calculated by rearranging the equation so that λ is the subject. Using trigonometry angle θ can be found.
• The distance D is shown below, which can be used to find the separation angle.
• The wavelength can be determined using the equation below, where a triangle is shaped between the distance from the grating to the wall and the fringe spacing shown in figure 7 as h.

$\tan \theta = \frac{h}{D} \lambda = \frac{d \cdot \sin \theta}{ n}$

Experimental pattern diagram.
• The filter or coloured plastic filters out colours from the spectrum and only allows one wavelength of light to pass through, hence only colour appears.

### Errors and uncertainties

• Multiple measurements of h should be taken to find the average.

• Use a Vernier scale to record h to minimise uncertainty.

• Conduct the experiment in a darkened room, so the fringes and measurements are clearer.

• Use a grating with lots of slits so that magnitudes of h are greater to minimise uncertainty.

### Applications

Diffraction gratings are used in multiple optical devices such as:

• Spectrometers.

• Lasers.

• CD and DVDs.

• Monochromators.

• Optical pulse compression devices.

## Diffraction Gratings - Key takeaways

• A diffraction grating is an optical plate that divides or disperses white light, which is composed of seven different colours, each with its own different wavelength.
• A diffraction grating pattern is an interference pattern consisting of maximums and minimums when light is diffracted.
• The angular separation is the angle between the unbent and bent light beams.

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How does a diffraction grating work?

By refraction of light around openings. This forces the waves to interfere with one another either constructively or destructively, creating an interference pattern.

How do you make a spectroscope?

By using diffraction gratings which split light into its components, allowing accurate wavelength measurement of spectrum-emitting substances.

What is diffraction grating used for?

A diffraction grating is used for optical devices such as CDs, DVDs, monochromators, lasers, spectrometers, etc.

How does diffraction grating separate colours?

The light passes through several slits in the grating and is separated into different colours based on speed and angles of diffraction.

What is a diffraction grating?

It is an optical plate that divides white light into an interference pattern composed of all colours of the light spectrum, in a dispersed manner.

## Test your knowledge with multiple choice flashcards

What is the relation between the wavelength and angle of separation?

Which of the following is true?

Which of the following is not an application of diffraction grating?

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