27.5 Single Slit Diffraction

Single Slit Diffraction

When light passes through a narrow opening, it doesn't just travel straight through. It spreads out and interferes with itself, creating a distinctive pattern of bright and dark bands on a screen. This behavior, called single slit diffraction, is one of the clearest demonstrations that light behaves as a wave.

Single slit diffraction matters beyond the physics classroom. It sets the fundamental limit on how much detail microscopes and telescopes can resolve, and it helps explain visual effects like the fringes of color you see around a bright light viewed through a narrow gap.

How Light Behaves Through a Single Slit

When a wave passes through a slit, every point across the width of that slit acts as its own tiny source of spherical waves. This idea comes from Huygens' principle. All of these mini-waves spread outward and overlap with each other.

Where the waves line up crest-to-crest (in phase), they add together and produce a bright region through constructive interference. Where they line up crest-to-trough (out of phase), they cancel and produce a dark region through destructive interference.

The result on a distant screen is the single slit diffraction pattern:

• A broad, bright central maximum at the center
• Alternating dark and bright fringes on either side
• The central bright fringe is twice as wide as any of the secondary bright fringes
• Each successive bright fringe is dimmer than the one before it, so the central fringe is by far the brightest

This type of pattern, observed when both the light source and the screen are effectively very far from the slit, is called Fraunhofer diffraction (far-field diffraction).

Finding the Angles of Destructive Interference

The positions of the dark fringes (minima) are found using this equation:

$D \sin\theta = m\lambda$

• $D$ = width of the slit
• $\theta$ = angle measured from the central axis to the dark fringe
• $m$ = order of the dark fringe ($m = 1, 2, 3, \ldots$; note that $m = 0$ is not used because that's the center of the bright central maximum)
• $\lambda$ = wavelength of the light

To solve for the angle directly:

$\theta = \sin^{-1}\!\left(\frac{m\lambda}{D}\right)$

Example: Suppose light of wavelength $\lambda = 550 \text{ nm}$ passes through a slit of width $D = 2.0 \times 10^{-5} \text{ m}$. For the first dark fringe ($m = 1$):

1. Plug into the formula: $\sin\theta = \frac{(1)(550 \times 10^{-9})}{2.0 \times 10^{-5}} = 0.0275$
2. Take the inverse sine: $\theta = \sin^{-1}(0.0275) \approx 1.58°$

A few trends to keep in mind:

• As $m$ increases, $\theta$ increases, so higher-order dark fringes appear farther from the center.
• A narrower slit (smaller $D$) produces a wider diffraction pattern because $\theta$ increases when $D$ decreases.
• A longer wavelength also spreads the pattern out more.

Comparing Intensity Patterns Across Diffraction Types

Wave Interference and the Diffraction Limit

Wave interference is the core principle behind all diffraction patterns. Waves from different parts of the slit combine, and the geometry of the setup determines where they add up or cancel out.

One practical consequence is the diffraction limit: the smallest angular separation between two points that an optical system can distinguish. A larger aperture (or shorter wavelength) allows finer detail to be resolved. This is why telescopes need large mirrors and why electron microscopes, which use much shorter wavelengths than visible light, can image far smaller structures.

Fraunhofer vs. Fresnel diffraction: The patterns discussed in this unit are Fraunhofer (far-field) diffraction, where the screen is far enough away that incoming and outgoing light rays are nearly parallel. When the screen is close to the slit, you get Fresnel (near-field) diffraction, which produces more complex patterns that are harder to analyze mathematically.