Diffraction of Light - Young’s Single Slit Experiment

Diffraction is the line of study that helped in the development of precise spectrometers, hence helping Astronomy take great leaps. This was possible in the early nineteenth century when two great scientists were struggling in their lives separately in Italy and Germany. These two weak youngsters were still struggling to get educated. Fraunhofer was the son of a maker of decorative glasses and mirrors in Bavaria, while Fresnel was the son of an architect in Normandy.

What is Diffraction of Light? 

Diffraction is the bending of light around the sharp corner of an obstacle. When light is incident on a slit, with a size comparable to the wavelength of light, an alternating dark and bright pattern can be observed. This phenomenon is called the single slit diffraction. According to Huygens’ principle, when light is incident on the slit, secondary wavelets generate from each point. These wavelets start out in phase and propagate in all directions. Each wavelet travels a different distance to reach any point on the screen. Due to the path difference, they arrive with different phases and interfere constructively or destructively.

Diffraction Due to Single Slit

When light is incident on the sharp edge of an obstacle, a faint illumination can be found within the geometrical shadow of the obstacle. This suggests that light bends around a sharp corner. The effect becomes significant when light passes through an aperture having a dimension comparable to the wavelength of light.

If light is incident on a slit having a width comparable to the wavelength of light, an alternating dark and bright pattern can be seen if a screen is placed in front of the slit. This phenomenon is known as single slit diffraction.

(Image will be uploaded soon)

Young’s Single Slit Experiment

Thomas Young’s double-slit experiment, performed in 1801, demonstrates the wave nature of light. In this experiment, monochromatic light is shone on two narrow slits. The waves, after passing through each slit, superimpose to give an alternate bright and dark distribution on a distant screen. All the bright fringes have the same intensity and width.

In a single slit experiment, monochromatic light is passed through one slit of finite width and a similar pattern is observed on the screen. Unlike the double-slit diffraction pattern, the width and intensity in the single-slit diffraction pattern reduce as we move away from the central maximum.

Explanation of The Phenomenon and Diffraction Formula

According to Huygens’ principle, when light is incident on the slit, secondary wavelets generate from each point. These wavelets start out in phase and propagate in all directions. Each wavelet travels a different distance to reach any point on the screen. Due to the path difference, they arrive with different phases and interfere constructively or destructively.

If a monochromatic light of wavelength λ falls on a slit of width a, the intensity on a screen at a distance L from the slit can be expressed as a function of θ.

Here, θ is the angle made with the original direction of light. It is given by,

I\[\left ( \theta  \right )\] = I\[_{0}\] \[\frac{sin^{2}\alpha }{\alpha ^{2}}\]

Here, \[\alpha\] = \[\frac{\Pi }{\lambda }\] sin\[\theta\]   and I₀ is the intensity of the central bright fringe, located at θ =0.

(Image will be uploaded soon)

Diffraction Maxima and Minima 

Bright fringes appear at angles,

\[\theta\] \[\rightarrow\] 0,\[\theta\]\[\rightarrow\]  sin-1 \[\left ( \pm \frac{3\lambda }{2} \right )\], \[\theta\] \[\rightarrow\] sin-1 \[\left ( \pm \frac{5\lambda }{2} \right )\]

\[\theta\] \[\rightarrow\] 0 is the central maximum

Dark fringes correspond to the condition,

a sin\[\theta\] = m\[\lambda\] with m = ±1, ±2, ±3…

In a double slit arrangement, diffraction through single slits appears as an envelope over the interference pattern between the two slits.

(Image will be uploaded soon)

Fringe width  

The angular distance between the two first order minima (on either side of the center) is called the angular width of central maximum, given by

2\[\theta\] = \[\frac{2\lambda }{a}\]

The linear width is as follows,

\[\Delta\] = L.2\[\theta\] = \[\frac{2L\lambda }{a}\]

The width of the central maximum in the diffraction formula is inversely proportional to the slit width. If the slit width decreases, the central maximum widens, and if the slit width increases, it narrows down. It can be inferred from this behaviour that light bends more as the dimension of the aperture becomes smaller.

Conditions for Diffraction

• The incident light should be monochromatic.

• The slit width should be comparable to the wavelength of incident light.

Types of Diffraction

• Fresnel Diffraction: The light source and the screen both are at finite distances from the slit. The incident waves are not parallel. 

• Fraunhofer Diffraction: The light source and the screen both are infinitely away from the slit such that the incident light rays are parallel.

Solved Examples

1. Fraunhofer diffraction at a single slit is performed using a 700 nm light. If the first dark fringe appears at an angle 30°, find the slit width.

Solution: Using the diffraction formula for a single slit of width a, the nth dark fringe occurs for,

a sin\[\theta\] = nλ

At angle θ=30°, the first dark fringe is located. Using n=1 and

λ = 700 nm=700 X 10⁻⁹m,

a sin 30°=1 X 700 X 10⁻⁹m

a=14 X 10⁻⁷m

a=1400 nm

The slit width is 1400 nm.

2. Find the angular width of central maximum for Fraunhofer diffraction due to a single slit of width 0.1 m, if the frequency of incident light is 5 X 10¹⁴ Hz.

Solution: wavelength of the incident light is,

\[\lambda\] = \[\frac{c}{\nu }\]

Here, c=3 X 10⁸m/s is the speed of light in vacuum and =5 X 10¹⁴Hz  is the frequency.

The angular width of the central maximum is,

2\[\theta\] = \[\frac{2\lambda }{a}\]

2\[\theta\] = \[\frac{2c}{\nu a}\]

Using c = 3×10⁸m/s, = 5×10¹⁴Hz a = 0.1m,

2\[\theta\] = 1.2×10⁻⁴rad

The angular width is 1.210⁻⁴rad

About Fraunhofer and Fresnel

Fraunhofer

Fraunhofer and Fresnel, both kids faced immense hardships in life, partly because of their weak constitutions. At the end of the first decade of the nineteenth century, the building where Fraunhofer lived collapsed. Fraunhofer was the only one who survived. Maximilian IV Joseph, Prince-Elector of Bavaria, who was busy in the rescue efforts, noticed Fraunhofer and decided to take him under his wing. With the entry of Maximilian into his life, Fraunhofer’s life took a turn for the better.

Fresnel

In the neighbouring country of Italy simultaneously, Fresnel was still struggling to read at the age of eight. Surrounded by his three siblings, Fresnel had toys aplenty. At the age of nine, he managed to transform a harmless toy into such a lethal weapon that elders in his society were forced to ban the use of the now-dangerous toy. This caused no end of worry and consternation to his parents and grown-ups, but his genius was finally recognized. 

He was taught by fine tutors and by the age of 16, he was admitted to the École Polytechnique. Here, an eminent mathematician, Adrien-Marie Legendre noticed Fresnel’s answer paper of striking ingenuity. Fresnel was encouraged in his studies of Maths and Science. He then graduated from École des Ponts et Chaussées as an Engineer.

Fraunhofer, The Scientist

Somewhere down a few longitudes, Maximilian took charge of the education of Fraunhofer and put him under the care of Joseph von Utzschneider, an attorney with a unique business mind. In those days industrious Utzschneider was busy partnering with Georg Reichenbach and J. Leibherr to open the Mathematical Mechanical Institute Reichenbach Utzschneider Liebherr. This institute was established to produce surveying instruments with high-quality lenses – incidentally, just the thing that Fraunhofer had a knack for.

After a failed stint at establishing his own business, Fraunhofer was offered a position in the Institute by Utzschneider. Here, under the guidance of expert glassmaker Pierre Guinand, Fraunhofer shined. Within two years he was offered a junior partnership in the Institute. Within a decade, Fraunhofer had turned the Institute around and was producing quality refractor lenses with a diameter of seven inches. This was an achievement in those times and was ably equipped with a decent-sized telescope.

Fresnel, The Scientist

On the other side, Fresnel was following the footsteps of his father, the architect. Having a few false starts mostly because he supported the monarchy that kept changing at that point in history, Fresnel settled as an engineer for the roads of Paris.

By now the careers of both scientists were converging as both were taking a keen interest in optics. By the middle of the second decade, both Fresnel and Fraunhofer were knee-deep in their research, resulting in scintillating findings that are studied in Optics for class 12 and higher classes.

Fraunhofer had a knack for handling glass and optics. A major challenge at his time was accurately polishing spherical surfaces of large glass objects. With his ingenuity, Fraunhofer improved upon and also invented machines that did more accurate grinding and polishing of glass surfaces. This greatly improved the kinds of lenses being manufactured and thus improved the quality of optical instruments being used.

In 1811, he constructed the furnace that would make the perfect large-sized lenses required in larger telescopes. With this new furnace, Fraunhofer created his own crown glass that had no irregular refractive power as larger crown glasses of that time tended to have.

In 1814, Fraunhofer invented the modern spectroscope through a series of experiments. By observing the orange fixed light in the spectrum of fire, he researched the dark lines of the solar spectrum, as well as in several bright stars, thus initiating research in stellar spectroscopy.

In 1821, he invented the diffraction grating and became the first scientist to get line spectra and then measure the wavelengths of spectral lines. His passion always remained practical optics so he kept developing and bettering optical instruments. He also created the Dorpat Refractor ad and Bessel Heliometer.

The most practical and popular invention of Fresnel was Fresnel Lens. He created this for Napoleon. It is used in theatrical and cinematic lighting fixtures and lighthouses to shine a narrow beam in all directions simultaneously. These are made of a lot of complicated wedge-shaped parts. Fresnel lenses magnify a light beam about 150 times and shine it, thus making it very practical in lighthouses in his time. Today these lenses are popular in film-making and theatre.

Did you know?

• In the diffraction pattern of white light, the central maximum is white but the other maxima becomes colored with red being the farthest away.

• Diffraction patterns can be obtained for any wave. Subatomic particles like electrons also show similar patterns like light. This observation led to the concept of a particle’s wave nature and it is considered as one of the keystones for the advent of quantum mechanics.

• The interatomic distances of certain crystals are comparable with the wavelength of X-rays. Using X-ray diffraction patterns, the crystal structures of different materials are studied in condensed matter physics.