Fraunhofer Diffraction At Single Slit Derivation

Fraunhofer diffraction at a single slit is a fundamental concept in wave optics that provides deep insight into how light behaves when it encounters an obstacle. Unlike geometrical optics, which deals with light as rays, wave optics considers the wave nature of light, leading to phenomena such as interference and diffraction. Fraunhofer diffraction specifically refers to the far-field diffraction pattern, where both the source of light and the observation screen are effectively at infinite distances from the slit, or lenses are used to simulate this condition. Understanding the derivation of Fraunhofer diffraction at a single slit is essential for physics students, as it not only explains the intensity distribution of light on a screen but also lays the foundation for more complex diffraction studies involving multiple slits or gratings.

Introduction to Single Slit Diffraction

When a parallel beam of monochromatic light passes through a narrow slit, it spreads out and forms a pattern of bright and dark regions on a distant screen. This spreading, or diffraction, occurs because different parts of the wavefront emerging from the slit interfere with each other. The phenomenon was first analyzed in detail by Joseph von Fraunhofer, and the resulting patterns are called Fraunhofer diffraction patterns. For a single slit of width ‘a’, the diffraction pattern consists of a central maximum that is the brightest and widest, flanked by successive minima and secondary maxima that gradually decrease in intensity.

Assumptions for Fraunhofer Diffraction

• The slit width ‘a’ is comparable to the wavelength λ of the light.
• The light source is monochromatic and coherent.
• The screen is far away, or lenses are used to ensure parallel rays reach the observation point.
• Huygens-Fresnel principle is applied, treating each point on the slit as a secondary source of wavelets.

Derivation of the Fraunhofer Diffraction Pattern

To derive the intensity distribution of light in Fraunhofer diffraction at a single slit, we consider a slit of width ‘a’ illuminated by a plane wave of wavelength λ. According to Huygens’ principle, each point on the slit acts as a source of secondary wavelets. The light at a point P on a distant screen is the result of the superposition of all these wavelets, with phase differences determined by their path lengths.

Phase Difference Between Wavelets

Let the slit extend along the y-axis from y = -a/2 to y = +a/2. Consider a point P on the screen at an angle θ relative to the central axis. The path difference between wavelets from the top and bottom edges of the slit is given by

• Î = a sin θ

The corresponding phase difference is

• ϕ = (2π/λ) Î = (2π/λ) a sin θ

For an infinitesimal element dy of the slit located at a distance y from the center, the path difference relative to the center is

• δ = (2π/λ) y sin θ

Expression for Electric Field

The contribution dE of the infinitesimal element dy to the electric field at point P is proportional to dy and has a phase factor corresponding to δ

• dE = E₀ dy e^(iδ) = E₀ dy e^(i(2π/λ) y sin θ)

Integrating over the entire slit from y = -a/2 to y = +a/2 gives the total electric field at P

• E = ∫₋ₐ/₂^ₐ/₂ E₀ e^(i(2π/λ) y sin θ) dy

Solving the integral

• E = E₀ [e^(i(π a sin θ / λ)) – e^(-i(π a sin θ / λ))] / (i (2π/λ) sin θ)
• E = E₀ (λ / π a sin θ) sin(π a sin θ / λ)

Intensity Distribution

The intensity I of the diffraction pattern is proportional to the square of the magnitude of the electric field

• I(θ) = I₀ [sin(π a sin θ / λ) / (π a sin θ / λ)]²

Here, I₀ is the maximum intensity at θ = 0, which corresponds to the central maximum. The function sin(β)/β, where β = π a sin θ / λ, is called the sinc function. It describes the characteristic shape of the single slit diffraction pattern, with a central bright fringe that is the most intense and subsequent fringes that decrease in brightness.

Positions of Minima and Maxima

The minima in the diffraction pattern occur when the numerator of the sinc function equals zero, except at β = 0

• sin(π a sin θ / λ) = 0, for π a sin θ / λ ≠ 0
• π a sin θ / λ = m π, where m = ±1, ±2, ±3 …
• sin θ = m λ / a

These positions correspond to the dark fringes on the screen. The central maximum is located at θ = 0, and the width of the central maximum is twice the angular position of the first minimum

• Îθ = 2 λ / a

The secondary maxima occur between the minima, but their intensity is much lower, and their exact positions can be calculated numerically from the derivative of the sinc function.

Graphical Representation

The diffraction pattern can be represented graphically with intensity on the vertical axis and angle θ or position on the screen on the horizontal axis. The central maximum is the brightest and widest, with intensity gradually decreasing for successive maxima. This graphical representation is useful for visualizing the distribution of light and predicting the positions of bright and dark fringes in experimental setups.

Applications of Single Slit Diffraction

Understanding Fraunhofer diffraction at a single slit is not only crucial for academic purposes but also has practical applications

• Designing optical instruments such as spectrometers and telescopes.
• Analyzing the resolving power of optical systems.
• Studying wave behavior in other fields such as acoustics and water waves.
• Forming the basis for understanding more complex diffraction patterns like double slits and diffraction gratings.

Experimental Verification

Fraunhofer diffraction at a single slit can be observed using a monochromatic light source such as a laser, a narrow slit, and a distant screen or a lens to project the far-field pattern. By measuring the positions of minima and maxima, students can verify the relationship sin θ = m λ / a and confirm the theoretical derivation. This experiment strengthens the understanding of wave optics concepts and demonstrates the real-world relevance of mathematical formulations.

The derivation of Fraunhofer diffraction at a single slit provides a comprehensive understanding of how light spreads when passing through narrow apertures and interferes to form characteristic intensity patterns. Starting from Huygens’ principle, the calculation of phase differences and integration over the slit width leads to the sinc function, which describes the intensity distribution. The positions of minima and maxima, along with the central maximum’s width, can be predicted accurately, confirming the wave nature of light. This concept has far-reaching applications in optics and physics, making it an essential topic for students learning wave optics. By mastering this derivation, students gain a strong foundation for exploring more advanced phenomena such as multiple slit diffraction, diffraction gratings, and the resolving power of optical instruments.