Single-Slit Diffraction
When light passes through a narrow slit — one comparable in width to its wavelength — it does not travel in a straight line as simple ray optics would predict. Instead, it spreads out and produces an alternating pattern of bright and dark bands on a screen. This phenomenon is called single-slit diffraction, and it is one of the clearest demonstrations that light behaves as a wave. The central bright band is wide and intense, flanked by progressively dimmer secondary maxima separated by sharp dark minima called destructive interference fringes. The pattern arises because each point across the width of the slit acts as a separate source of Huygens wavelets. These wavelets travel different path lengths to reach a given point on the screen. At certain angles, wavelets from opposite halves of the slit arrive exactly half a wavelength out of phase and cancel each other out, producing a dark fringe. At other angles they partially reinforce, producing the fainter secondary bright bands. The mathematics of these path differences gives a precise formula for the positions of the dark fringes. Single-slit diffraction matters in everyday technology: it limits the resolution of cameras, telescopes, and microscopes. Understanding its pattern — a wide central maximum, dark minima at predictable angles, and weak secondary maxima in between — is therefore essential both for exam success and for appreciating why optical instruments have fundamental resolution limits.
- Huygens' Principle and Path Differences
- Condition for Dark Fringes (Minima)
- The Central Maximum
- Secondary Maxima
- Effect of Slit Width and Wavelength
Huygens' Principle and Path Differences
Imagine the slit of width a divided into a large number of infinitely thin strips. Each strip radiates a Huygens wavelet in every forward direction. For a point P on the screen at angle θ to the straight-ahead direction, the wavelet from the top of the slit travels a different distance than the wavelet from the bottom. The total path difference across the whole slit is a sin θ. When this path difference equals one full wavelength λ, you can pair every strip in the top half of the slit with a strip in the bottom half that is exactly λ/2 out of phase — so every pair cancels, producing the first dark fringe.
Condition for Dark Fringes (Minima)
Destructive interference occurs whenever the path difference a sin θ is an integer multiple of the wavelength: a sin θ = mλ, where m = ±1, ±2, ±3, … (m = 0 is excluded because that is the bright central maximum). Note carefully: in double-slit interference the same formula gives bright fringes, but in single-slit diffraction it gives dark fringes. The sign of m indicates whether the fringe is above or below the central axis; the pattern is symmetric.
The Central Maximum
The central maximum is centred at θ = 0 and extends to the first dark fringe on each side (m = ±1). Its angular half-width is given by sin θ = λ/a. Because it spans from m = −1 to m = +1, the central maximum is twice as wide as any of the secondary maxima. It also contains most of the light energy — roughly 84% of the total intensity. Making the slit narrower (smaller a) spreads the central maximum wider, which is why very narrow slits produce almost fully diffracted beams.
Secondary Maxima
Between consecutive dark fringes there are bright secondary maxima. They occur approximately (not exactly) midway between minima, at angles where a sin θ ≈ (m + ½)λ. For m = 1 this gives a sin θ ≈ 1.5λ, meaning the path difference is 1.5 wavelengths; one-third of the slit cancels in pairs and one-third remains unmatched, giving a weaker maximum. The intensity of the first secondary maximum is only about 4.5% of the central maximum's intensity — visibly much dimmer, which is consistent with what you see experimentally.
Effect of Slit Width and Wavelength
The angular width of the pattern scales as λ/a. Narrowing the slit (decreasing a) broadens the pattern — diffraction becomes more pronounced. Widening the slit compresses the pattern until, when a >> λ, the light travels almost in a straight line and ray optics is recovered. Similarly, using longer-wavelength light (red rather than violet) produces a wider pattern. This wavelength dependence means that white light creates a coloured diffraction pattern with red edges further from centre than violet edges.
The Key Equation
- a — Slit width (metres (m)): The physical width of the single slit through which light passes.
- \theta — Diffraction angle (degrees or radians): The angle between the straight-ahead direction and the line to the point of interest on the screen.
- m — Order number (dimensionless (integer)): An integer (±1, ±2, …) labelling each dark fringe; positive values are above centre, negative below.
- \lambda — Wavelength (metres (m)): The wavelength of the monochromatic light used.
Worked Example
Problem. Monochromatic light of wavelength 589 nm passes through a single slit of width 0.250 mm. A screen is placed 1.20 m away. Calculate (a) the angle to the first dark fringe and (b) the width of the central bright maximum on the screen.
Answer. The first dark fringe is at angle 0.135° (2.83 mm from centre on the screen). The central bright maximum has a total width of 5.66 mm.
Origins · 1 minAugustin-Jean Fresnel and how this was discovered
Fresnel developed the rigorous wave theory of diffraction in the early 19th century, deriving mathematical expressions for the intensity of light at any point in a diffraction pattern — including single-slit diffraction. His work, presented to the French Academy of Sciences in 1818, provided the theoretical foundation now used to calculate single-slit fringe positions and intensities. He extended Huygens' principle by adding the phase of each wavelet, creating what is now called the Huygens–Fresnel principle. When Fresnel submitted his wave theory to a prize committee, the mathematician Siméon Poisson tried to disprove it by showing it predicted a bright spot at the centre of the shadow of a circular disc — a seemingly absurd result. Dominique Arago promptly did the experiment and found the spot exactly where Fresnel's theory predicted, instantly vindicating the wave theory. The 'Poisson spot' or 'Arago spot' remains one of physics' most dramatic confirmations.
Related Concepts
- Double-Slit Interference — Double-slit interference and single-slit diffraction both arise from wave superposition; in practice every double-slit experiment is modulated by the single-slit diffraction envelope of each slit.
- Huygens' Principle — Single-slit diffraction is explained by treating every point in the slit as a secondary wavelet source, which is the direct application of Huygens' Principle.
- Diffraction Grating — A diffraction grating is a multi-slit system whose sharp maxima are shaped by the single-slit diffraction envelope; understanding single-slit patterns is a prerequisite.
- Resolving Power and the Rayleigh Criterion — The resolution limit of optical instruments such as telescopes and microscopes is set directly by single-slit (or circular aperture) diffraction spreading.