36-3 Diffraction by a Circular Aperture

36-3 Diffraction by a Circular Aperture#

Prompts

  • Describe the diffraction pattern from a circular aperture. How does it differ from the single-slit pattern? What is the first minimum condition?

  • What is Rayleigh’s criterion for resolvability? Why does “central maximum of one at first minimum of the other” define the limit?

  • The pupil of your eye acts as a circular aperture. How does pupil diameter affect the minimum angular separation you can resolve? What happens in bright light (small pupil)?

  • How can we improve resolution—make the diffraction pattern smaller? Why do microscopes sometimes use ultraviolet light?

  • Pointillism: Why do the colored dots in a pointillistic painting blend when you stand far away? At what distance do they become irresolvable?

Lecture Notes#

Overview#

  • Circular aperture (lens, pinhole, pupil) produces a diffraction pattern: central bright disk plus concentric rings. The first minimum is at \(\sin\theta = 1.22\,\lambda/d\) (factor 1.22 from circular geometry).

  • Resolution: When two point sources are close, their diffraction patterns overlap. If they overlap too much, we cannot distinguish them.

  • Rayleigh’s criterion: Two objects are barely resolvable when the central maximum of one coincides with the first minimum of the other: \(\theta_R = 1.22\,\lambda/d\).

  • Applications: Limits on telescope/eye resolution; pointillism; microscopy with UV.

Diffraction by a circular aperture#

Light passing through a circular aperture of diameter \(d\) (e.g., a lens or pinhole) forms a diffraction pattern: a central bright disk (Airy disk) surrounded by progressively fainter concentric rings.

First minimum (angle from central axis to the first dark ring):

(303)#\[ \sin\theta = 1.22\,\frac{\lambda}{d} \]

Compare to a rectangular slit of width \(a\): \(a\sin\theta = \lambda\). The factor 1.22 arises from the circular geometry (derivation requires integration over the aperture).

For small \(\theta\): \(\theta \approx 1.22\,\lambda/d\). The radius of the first dark ring on a screen at distance \(D\) is \(y \approx D\theta = 1.22\,D\lambda/d\).

Resolvability#

When we image two point objects (e.g., two stars) through a lens, each produces its own diffraction pattern. If the objects are close, the patterns overlap.

Overlap

Result

Heavy overlap

Objects appear as one—not resolved

Central max of one at first min of other

Barely resolved (Rayleigh’s criterion)

Patterns well separated

Clearly resolved

Rayleigh’s criterion: Two point sources are on the verge of resolvability when the central maximum of one diffraction pattern falls on the first minimum of the other. The minimum angular separation is

(304)#\[ \theta_R = 1.22\,\frac{\lambda}{d} \]

where \(d\) is the diameter of the aperture (lens, pupil, etc.). If the actual angular separation \(\theta > \theta_R\), the objects can be resolved; if \(\theta < \theta_R\), they cannot.

Human vision

The pupil is a circular aperture. Rayleigh’s criterion is an approximation—actual resolvability depends on contrast, turbulence, and the visual system. The real limit is often somewhat larger than \(\theta_R\).

Improving resolution#

From \(\theta_R = 1.22\,\lambda/d\):

Change

Effect on \(\theta_R\)

Resolution

Larger aperture \(d\)

\(\theta_R\) smaller

Better

Shorter wavelength \(\lambda\)

\(\theta_R\) smaller

Better

  • Telescopes: Large diameter \(d\) reduces \(\theta_R\) \(\Rightarrow\) can resolve closer stars.

  • Microscopes: UV light (\(\lambda\) shorter than visible) improves resolution.

  • Bright light: Pupil constricts (\(d\) smaller) \(\Rightarrow\) \(\theta_R\) increases \(\Rightarrow\) worse resolution (considering diffraction only).

Pointillism#

Pointillism uses many small colored dots to form an image. The apparent color depends on viewing distance:

  • Close: Angular separation \(\theta\) of adjacent dots \(> \theta_R\) \(\Rightarrow\) dots are resolved \(\Rightarrow\) you see the true paint colors.

  • Far: \(\theta < \theta_R\) \(\Rightarrow\) dots blend \(\Rightarrow\) your brain combines light from several dots into a perceived color that may not exist in any single dot.

The painter exploits your eye’s diffraction limit to create colors through optical mixing.

Example: Smallest resolvable length (Rayleigh criterion)

You look at an object 40 m away. What is the smallest length (perpendicular to your line of sight) you can resolve? Assume pupil diameter 4.00 mm and \(\lambda = 500\) nm.

Solution: \(\theta_R = 1.22\lambda/d = 1.22(500\times10^{-9})/(4\times10^{-3}) \approx 1.5\times10^{-4}\) rad. Smallest resolvable length \(s = L\theta_R = 40\times1.5\times10^{-4} \approx 6\) mm.

Example: Viewing distance for pointillism

Dots separated by \(D = 2.0\) mm, pupil diameter \(d = 1.5\) mm. For \(\theta = D/L\) (small angle) and \(\theta_R = 1.22\lambda/d\), the least distance at which dots become irresolvable is \(L = Dd/(1.22\lambda)\). For blue light (\(\lambda = 400\) nm): \(L \approx 6.2\) m. Red dots (longer \(\lambda\)) blend first as you step back.

Summary#

  • Circular aperture: First minimum at \(\sin\theta = 1.22\lambda/d\); central disk + concentric rings.

  • Rayleigh’s criterion: \(\theta_R = 1.22\lambda/d\); two sources barely resolvable when one’s central max is at the other’s first min.

  • Better resolution: Larger \(d\) or shorter \(\lambda\).

  • Pointillism: Dots resolvable when close; blend when \(\theta < \theta_R\) (far away).