Overview#

• Single-slit diffraction: Light passing through a narrow slit produces a diffraction pattern—a broad central maximum, narrower secondary maxima, and minima in between. This is interference of Huygens wavelets from different points within the slit.

• Minima occur when \(a\sin\theta = m\lambda\) (\(m = 1, 2, 3, \ldots\)). Maxima lie approximately halfway between adjacent minima.

• Narrower slit \(\Rightarrow\) more diffraction (wider pattern). Diffraction limits geometrical optics when apertures are comparable to \(\lambda\).

Diffraction and the diffraction pattern#

When waves pass through an aperture or past an edge comparable in size to the wavelength, they spread (diffract) and interfere with themselves. The result is a diffraction pattern.

For a long narrow slit of width \(a\) illuminated by monochromatic light:

• Central maximum: broad, intense. All Huygens wavelets from the slit travel nearly the same distance to the center \(\Rightarrow\) in phase.

• Secondary maxima: narrower, less intense, on both sides.

• Minima: dark fringes between the maxima.

Geometrical optics would predict a sharp slit image; the actual pattern shows that light is a wave.

Locating the minima: the zone method#

To find the first minimum, divide the slit into two zones of width \(a/2\) each. Consider rays from corresponding points in the two zones (e.g., top of each zone) reaching a point \(P_1\) at angle \(\theta\).

Path length difference between the two rays: \((a/2)\sin\theta\) (from the geometry when \(D \gg a\)).

For destructive interference, this difference must be \(\lambda/2\)—so the two rays cancel. Every such pair cancels; the whole slit contributes zero at \(P_1\):

Second minimum: Divide the slit into four zones. Pairs from adjacent zones cancel when \((a/4)\sin\theta = \lambda/2\) \(\Rightarrow\) \(a\sin\theta = 2\lambda\).

All minima:

(\(m = 0\) gives \(\theta = 0\), the center—a maximum, not a minimum.)

Maxima: Approximately halfway between minima (e.g., first side maximum near \(a\sin\theta \approx 1.5\lambda\)). Exact positions require the intensity analysis of section 36-2.

Slit width and wavelength#

From \(a\sin\theta = m\lambda\):

When \(a = \lambda\), the first minimum is at \(\theta = 90°\)—the central maximum fills the entire screen. Diffraction is strongest when the aperture is on the order of the wavelength.

Distance on the screen#

When \(\theta\) is small, \(\sin\theta \approx \theta\). The distance \(y\) from the center to the \(m\)th minimum is

where \(D\) is the slit-to-screen distance. The width of the central maximum (distance between first minima) is \(\approx 2D\lambda/a\).

The Fresnel bright spot#

In 1819, Poisson (a Newtonian) argued that Fresnel’s wave theory predicted a bright spot at the center of the shadow of a sphere—seemingly absurd. The experiment showed the spot was there. This Fresnel bright spot supported the wave theory: light diffracts around the edge and interferes constructively at the center of the shadow.

Summary#

• Single-slit diffraction: central maximum + secondary maxima separated by minima.

• Minima: \(a\sin\theta = m\lambda\), \(m = 1, 2, 3, \ldots\) (zone method: pair canceling waves).

• Narrower slit or longer \(\lambda\) \(\Rightarrow\) pattern expands.

• Fresnel bright spot: bright spot in shadow of disk/sphere; supports wave theory.