Single-slit diffraction

Diffraction from one opening

Even a single slit produces a pattern of bright and dark regions. This happens because different parts of the slit act as sources of wavelets. Their contributions interfere at the screen.

The result is a broad central maximum with weaker side maxima. This cannot be explained by ray optics alone.

Destructive interference condition

Let the slit width be $a$. For light of wavelength $lambda$, dark minima occur at angles satisfying

$asin heta=mlambda$

where

$m=pm1,pm2,pm3,ldots$

There is no $m=0$ minimum because the center is the main maximum.

Why the minima occur

For the first minimum, divide the slit into two halves. At an angle where the path difference between light from corresponding points in the two halves is $lambda/2$, every contribution from the top half cancels a contribution from the bottom half.

This gives

$asin heta=lambda$

For higher minima, the slit can be divided into more pairs or groups that cancel similarly.

Central maximum width

The central maximum extends between the first minima on either side:

$asin heta=pmlambda$

For small angles,

$sin hetaapprox heta$

so the angular half-width is approximately

hetaapprox rac{lambda}{a}

The full angular width of the central maximum is approximately

Delta hetaapprox rac{2lambda}{a}

A narrower slit produces a wider diffraction pattern.

Intensity pattern

The full intensity pattern is

ight)^2$$ where $$eta=rac{pi asin heta}{lambda}$$ This function has a strong central maximum and weaker side lobes. The minima occur when $sineta=0$ except at $eta=0$. ## Diffraction and resolution Diffraction limits the ability of optical systems to form sharp images. A point source imaged through a circular aperture forms an Airy pattern, not a perfect point. For a circular aperture of diameter $D$, the angular resolution limit is approximately $$ heta_{min}=1.22rac{lambda}{D}$$ Although this formula is for circular apertures rather than slits, the same idea applies: finite aperture causes spreading. ## Slit width tradeoff If the slit is very wide compared with wavelength, diffraction is small and ray optics works well. If the slit is comparable to wavelength, diffraction dominates. This is why visible light, with wavelengths around hundreds of nanometers, diffracts noticeably through very small apertures but not strongly through ordinary doorways. ## Diffraction versus interference Single-slit diffraction and double-slit interference are related. In a real double-slit experiment, each slit has finite width, so the observed pattern is a two-slit interference pattern modulated by a single-slit diffraction envelope. ## The big idea Single-slit diffraction arises from interference among wavelets from different parts of the same slit. Minima occur at $asin heta=mlambda$, and a narrower slit creates a wider diffraction pattern. Diffraction sets fundamental limits on optical resolution.