Two-Source Interference and Coherence
Two-source interference produces a stable pattern of maxima and minima only if the sources are coherent (same frequency and a constant phase difference); in Young’s double-slit experiment, bright fringes occur where the path difference is mλ and dark fringes where it is (m + ½)λ, with a fringe spacing w = λD ÷ s. In brief: for slits separated by s and a screen at distance D, s sin θ = mλ gives the maxima, and for small angles, the spacing of fringes is w = λD ÷ s. Using a shorter wavelength, a smaller slit separation or a larger screen distance changes the fringes accordingly. The experiment demonstrates the wave nature of light, and of electrons and atoms.
The colored fringes on a soap film and the pattern of loud and soft zones from two loudspeakers result from interference. This guide explains coherence, path difference and phase difference, derives the double-slit formula, discusses the intensity pattern, compares constructive and destructive interference in a table, works through numerical examples and corrects common mistakes.
What is coherence?
Two sources are coherent if they emit waves of the same frequency with a fixed phase relationship. Then the interference pattern at any point is stationary in time. Two separate light bulbs are incoherent, because each atom emits short wave trains with random phases (every 10−8 s or so), so the pattern changes too quickly to see. Coherent sources are obtained either by dividing the light of a single source (two slits, a double mirror, a thin film) or by using a laser, which has a long coherence length. For sound, two loudspeakers driven by the same signal generator are coherent. A monochromatic light source (a single wavelength) is also needed for a clear pattern: with white light, the fringes of different colors overlap and only a few colored fringes are seen.
How do path and phase differences work?
At a point P, the waves from the two sources have traveled different distances r1 and r2. The path difference is Δ = |r1 − r2|, and the phase difference is Δφ = 2πΔ ÷ λ (for sources in phase). The conditions:
- Constructive: Δ = mλ, Δφ = 2πm (m = 0, ±1, ±2, …): maximum intensity (bright, loud).
- Destructive: Δ = (m + ½)λ, Δφ = (2m + 1)π: minimum intensity (dark, quiet), zero for equal amplitudes.
For two equal sources, the intensity at a point varies as I = 4I0 cos2(Δφ ÷ 2), where I0 is the intensity from one source alone. The maximum is four times (not two times) the single-source intensity, and the minimum is zero: the energy is redistributed, not created, and the average intensity is 2I0. If the sources are in antiphase, the conditions are exchanged.
What is Young’s double-slit experiment?
Monochromatic light passes through two narrow parallel slits separated by a distance s, and falls on a screen at a distance D (D much larger than s). The central bright fringe (m = 0) is on the axis, where the path difference is zero. For a point at angle θ from the axis, the path difference is Δ = s sin θ. Bright fringes occur at s sin θ = mλ. For small angles, the position on the screen is ym = mλD ÷ s, so the fringes are equally spaced with w = λD ÷ s. Changes in the setup: red light gives wider fringes than blue; a smaller slit separation gives wider fringes; moving the screen away gives wider fringes. The finite width of each slit modulates the pattern with a single-slit diffraction envelope. The experiment gives a way of measuring the wavelength of light: λ = ws ÷ D. When it is performed with single electrons or atoms, an interference pattern builds up, one particle at a time, showing wave-particle duality.
Interference conditions compared
| Condition | Path difference | Phase difference | Result |
|---|---|---|---|
| Constructive | mλ | 2πm | Bright fringe, loud sound |
| Destructive | (m + ½)λ | (2m + 1)π | Dark fringe, quiet |
Worked example: double-slit fringes
- Fringe spacing. Light of wavelength 600 nm passes through slits 0.50 mm apart, and the screen is 2.0 m away. w = λD ÷ s = 6.0 × 10−7 × 2.0 ÷ 5.0 × 10−4 = 2.4 × 10−3 m = 2.4 mm.
- Position of the third bright fringe: y3 = 3w = 7.2 mm from the central maximum.
- Dark fringe. The first dark fringe is at (½)w = 1.2 mm from the center, where the path difference is λ/2 = 300 nm.
- Blue light. At 450 nm, w = 1.8 mm. In white light, the blue fringes are closer to the center than the red ones, so the outer fringes have a reddish edge.
- Finding the wavelength. Ten fringe spacings measure 2.5 cm on a screen at D = 1.50 m with s = 0.30 mm: w = 2.5 mm, λ = ws ÷ D = 2.5 × 10−3 × 3.0 × 10−4 ÷ 1.50 = 5.0 × 10−7 m = 500 nm.
- Two loudspeakers. Two speakers 2.0 m apart emit a 1,715 Hz tone in phase. λ = 343 ÷ 1,715 = 0.20 m. A listener on the central line is at a path difference of zero (loud). The first quiet position is at Δ = λ/2 = 0.10 m, where 2.0 sin θ = 0.10, θ = 2.9°.
How was it discovered?
Thomas Young demonstrated interference of light in 1801 with a card with two slits, in front of sunlight, and explained it in terms of waves in 1803, against Newton’s particle theory. Augustin Fresnel’s wave theory confirmed it in the 1810s to 1820s. In 1909, G. I. Taylor made the interference pattern with light so dim that photons passed one at a time, and in 1961 Claus Jönsson performed the double-slit experiment with electrons.
Which mistakes are common?
Mistake 1: Expecting interference from two independent light sources
They are not coherent, so no stable pattern forms.
Mistake 2: Mixing up the conditions for bright and dark fringes
Bright: whole number of wavelengths. Dark: half-integer.
Mistake 3: Thinking that the energy is destroyed at dark fringes
It is redistributed to the bright fringes.
Mistake 4: Using the small-angle formula for large angles
w = λD ÷ s is valid when the angles are small (D ≫ s and fringes near the axis).