Double-Slit Diffraction

When light passes through two narrow slits, two phenomena occur simultaneously: diffraction and interference. Diffraction spreads each slit's wavelet into a broad fan of light, while interference between the two fans creates a pattern of bright and dark fringes. The result is not a simple double-slit interference pattern — the envelope of the bright fringes is modulated by the single-slit diffraction pattern from each individual slit. Understanding this combined effect is essential for explaining real optical experiments and instruments. Young's double-slit experiment is often taught as producing equally bright fringes across the screen. In reality, the intensity of each interference maximum is weighted by the single-slit diffraction intensity at that angle. This means some interference maxima can be greatly reduced in brightness, or even completely absent — a phenomenon called a 'missing order'. The missing orders occur when an interference maximum coincides with a single-slit diffraction minimum, and their positions depend on the ratio of slit separation to slit width. This combined treatment of diffraction and interference represents the full physical picture of the double-slit experiment. It bridges the simpler models introduced earlier and prepares students for understanding the resolving power of instruments, diffraction gratings, and the wave nature of matter. The mathematics uses the sinc function and requires careful handling of two separate length scales: slit width and slit separation.

Key takeaways

Two separate length scales
The intensity envelope
Missing orders
Building intuition with an analogy
The combined intensity formula

Two separate length scales

The double-slit setup has two key dimensions: the width of each slit, a, and the centre-to-centre separation of the slits, d. Single-slit diffraction is governed by a, producing a broad central maximum with minima at angles where sin θ = mλ/a. The two-slit interference pattern is governed by d, producing narrow bright fringes at angles where d sin θ = nλ. Because d > a always, the interference fringes are more closely spaced than the diffraction envelope.

The intensity envelope

The full intensity pattern is found by multiplying the two-slit interference factor by the single-slit diffraction factor. Think of the diffraction pattern as a dimmer switch that controls the brightness of each interference fringe: fringes near the centre of the diffraction envelope are bright, while fringes near the diffraction minima are dim or absent. The single-slit diffraction factor is the sinc-squared function: I_diff ∝ [sin(β/2)/(β/2)]², where β = (2πa sin θ)/λ. The interference factor is cos²(δ/2), where δ = (2πd sin θ)/λ.

Missing orders

A missing order occurs when an interference maximum (d sin θ = nλ) coincides with a diffraction minimum (a sin θ = mλ). Dividing these conditions gives d/a = n/m. So if d/a = 4, the 4th, 8th, 12th… interference maxima are missing. This is a direct experimental signature of the slit width. In a real experiment, the missing orders appear as dark gaps in the otherwise regular fringe pattern — a beautiful consequence of two wave phenomena overlapping.

Building intuition with an analogy

Imagine two speakers (separated by distance d) each playing through a megaphone of width a. The megaphone shapes the sound into a forward lobe (diffraction envelope), while the two speakers interfere to create alternating loud and quiet directions (interference fringes). Only the directions that are both within the megaphone's lobe and at a constructive interference angle produce loud sound. Light behaves identically: the single-slit diffraction envelope is the 'megaphone', and the two-slit fringes are the 'interference pattern' within it.

The combined intensity formula

The total intensity on the screen is: I(θ) = I₀ · [sin(πa sin θ/λ)/(πa sin θ/λ)]² · cos²(πd sin θ/λ). The first factor (sinc²) comes from diffraction at a single slit; the second (cos²) comes from two-slit interference. At θ = 0, both factors equal 1, giving maximum intensity I₀. As θ increases, the cos² factor oscillates rapidly (interference fringes) while the sinc² factor slowly decreases (diffraction envelope). The product gives the observed pattern.

The Key Equation

I(\theta) = I_0 \left[\frac{\sin\!\left(\dfrac{\pi a \sin\theta}{\lambda}\right)}{\dfrac{\pi a \sin\theta}{\lambda}}\right]^{\!2} \cos^2\!\left(\frac{\pi d \sin\theta}{\lambda}\right)

I(\theta) — Intensity at angle θ (W m⁻²): The irradiance (power per unit area) arriving at the screen at angle θ from the centre.
I_0 — Peak intensity (W m⁻²): The maximum intensity, occurring at the centre of the pattern (θ = 0).
a — Slit width (m): The physical width of each individual slit; controls the diffraction envelope.
d — Slit separation (m): The centre-to-centre distance between the two slits; controls the interference fringe spacing.
\lambda — Wavelength (m): The wavelength of the incident monochromatic light.
\theta — Diffraction angle (rad (or degrees)): The angle from the central axis to the point of observation on the screen.

Worked Example

Problem. Monochromatic light of wavelength λ = 600 nm illuminates a double slit with slit width a = 0.10 mm and slit separation d = 0.40 mm. (a) At what angles do the first three interference maxima (n = 1, 2, 3) occur? (b) Which of these, if any, is a missing order?

\theta_1 \approx 0.086°,\quad \theta_2 \approx 0.172°,\quad \theta_3 \approx 0.258°;\quad \text{missing order at } n = 4

Answer. The first three interference maxima occur at approximately 0.086°, 0.172°, and 0.258°. None is a missing order; the first missing order is n = 4.

Figure

Diagram

Origins · 1 minThomas Young and how this was discovered

Thomas Young

1773–1829 · British

Thomas Young performed the famous double-slit experiment in 1801, demonstrating conclusively that light behaves as a wave. By observing the interference fringes produced when light passed through two close slits, he measured the wavelength of visible light for the first time and established the principle of superposition for light waves. His work laid the foundation for all subsequent wave optics, including the combined diffraction–interference analysis studied in this article. Young was a child prodigy who could read by age two and had read the Bible twice by age four. Beyond optics, he made major contributions to deciphering the Rosetta Stone and to understanding the elasticity of materials — Young's modulus is named after him.

Single-Slit Diffraction — The sinc² diffraction envelope in double-slit diffraction comes directly from single-slit diffraction; understanding one is prerequisite for the other.
Double-Slit Interference — The cos² interference factor in the combined formula is derived from standard two-slit interference; this concept provides the other half of the double-slit diffraction picture.
Diffraction Gratings — A diffraction grating extends the double-slit analysis to many slits, sharpening the maxima while the same diffraction envelope still modulates the pattern.
Wave Superposition and Interference — Both diffraction and interference are consequences of wave superposition; the principle introduced here underlies every calculation in double-slit diffraction.