Diffraction through a Single Slit: Fourier Optics, Interference, and Resolution

In this MIT OpenCourseWare lecture, the instructor extends from two-slit interference to single-slit diffraction, showing how a slit of width D acts as a continuum of point sources. Using a two-dimensional Fourier transform, the total field on a distant screen is derived and the intensity pattern I(θ) ∝ (sin β / β)^2 with β = π D sin θ / λ. The talk emphasizes principal maxima and minima, the effect of slit width on the central peak, and an experimental wavelength measurement with a laser.

• Diffraction arises from interference of many sources within the aperture, linked to Fourier analysis.
• The diffraction envelope modulates the interference pattern and sets the resolution limit.
• Experiment shows λ for a red laser using a 0.16 mm slit at a 7.6 m distance.
• Rayleigh criterion and design implications for cameras and displays are discussed.

Introduction: From Interference to Diffraction

The lecture begins by recapping interference from two or more point-like sources and then demonstrates how a finite aperture turns this into diffraction. By viewing the slit as a collection of point sources, the instructor invokes Huygens principle to explain how their spherical waves superpose on a distant screen to form a characteristic diffraction pattern. A key message is that diffraction is just interference extended to a distributed, finite source.

"Diffraction is essentially interference from many sources within the slit." - MIT OpenCourseWare Lecturer

Mathematical Framework: The Two-Dimensional Fourier Transform

The core tool is a two-dimensional Fourier transform. The total field pattern on the screen is encoded in a C(kx, ky) function, which depends on the aperture shape F(x, y) and the phase factor e^{-i k·R}. The aperture acts as the integration domain for the source distribution, and the resulting C(kx, ky) maps directions of propagation to the sum of contributions from all points in the slit. This ties a physical optical problem to a well-known mathematical transform, giving a concrete physical meaning to the Fourier components.

"The F function describes the shape of the source, and the exponential encodes the spherical propagation from each point." - MIT OpenCourseWare Lecturer

Single Slit Calculation: From Aperture to Intensity

For a single slit of width D, the F(x, y) is 1 inside -D/2 < x < D/2 and 0 elsewhere, with the wall extending infinitely in the y direction. Evaluating the integral simplifies the problem: the ky part becomes a delta function due to the infinite extent in y, and the remaining x-integration yields a sine over sine form, leading to the familiar diffraction envelope I(θ) ∝ (sin β / β)^2 with β = (π D sin θ)/λ. This encapsulates how the slit’s finite size shapes the observed intensity pattern on the screen.

"The central maxima width scales with the slit width and the wavelength through sin θ and β." - MIT OpenCourseWare Lecturer

Diffraction Pattern and Resolution: From Angles to Images

Translating from k-space to real-space angles, the lecture shows kx = (2π/λ) sin θ, which becomes β = π D sin θ / λ. For small angles, sin θ ≈ θ, giving I(θ) ∝ sinc^2(β). The principal maxima occur where sin β is maximal, while minima occur at β = nπ (n ≠ 0). The central maximum width is controlled by D: larger D yields a narrower central peak and wider angular separation between fringes, while the spacing between maxima/minima scales with λ/D. This lays out the diffraction-limited resolution concept central to imaging systems.

"Rayleigh criterion sets the diffraction-limited resolution: sin θ = 1.22 λ / D, balancing aperture, wavelength, and angular separation." - MIT OpenCourseWare Lecturer

Applications: Wavelength Measurement and Imaging Technology

To connect theory with practice, the lecturer describes an experiment using a red laser and a slit width of 0.16 mm, with the screen placed several meters away. By measuring the distance to the screen and the separation between minima, the wavelength λ is extracted and found to be in the red region, approximately 7.37 × 10^-7 m. The discussion also ties diffraction to practical questions in camera design, phone screens, and human vision, showing how diffraction limits influence resolution in everyday technology from pocket cameras to display devices.

These ideas are also linked to larger-scale astronomy and concepts like telescope resolution, illustrating that diffraction is a universal limit across scales, from simple lab setups to interplanetary imaging.

"The experimental demonstration confirms the Fourier-based prediction and matches the expected red wavelength within measurement uncertainty." - MIT OpenCourseWare Lecturer