Round Holes, Sharp Edges & Resolution
No lens can focus a star to a point, which limits how much detail any instrument can see.
Lesson at a glanceShow the key points
- A circular hole of diameter b gives a bright disc with fainter rings; the first dark ring is at sin θ = 1.22λ/b (17.17).
- On a screen at distance D: R = 1.22λD/b (17.18); with a lens of focal length f: R = 1.22λf/b (17.19).
- A lens's rim diffracts too, so a point source is imaged as a small disc of radius 1.22λD/b, never a point.
- Straight edge: intensity falls gradually inside the geometrical shadow and oscillates with dying fringes outside it.
- Two points are just resolved (Rayleigh) when the centre of one disc lies on the first dark ring of the other: angular separation ≈ 1.22λ/b.
- Bigger lenses and mirrors resolve finer detail, which is why telescope objectives are large.
Even a perfect lens cannot focus starlight to a perfect point. Diffraction at its rim always blurs the image a little, and that sets the finest detail any telescope, microscope or eye can see.
Diffraction by a circular hole
Send a parallel beam through a small round hole in an opaque board and look at a distant screen. You see a bright disc surrounded by fainter dark and bright rings. The analysis (harder than for a slit) puts the first dark ring at
On a screen a distance D away (D ≫ b), or at the focal plane of a lens of focal length f placed after the hole, the radius of this ring is
Most of the light falls inside this ring, so R is called the radius of the diffraction disc.
Every lens is a hole
A lens only collects the part of a wavefront that falls within its rim, which is usually circular. So a lens diffracts light too, and then focuses that diffracted light. The image of a distant point is therefore a tiny bright disc with faint rings, never a point. The same is true for any point source: the image disc has radius 1.22λD/b, where D is the distance at which the light is focused.
Light of wavelength 550 nm is focused by a lens 5.0 cm across, 25 cm from the lens. Find the diameter of the image disc.
Diffraction at a straight edge
Now a Fresnel case. Light from a narrow slit meets an opaque sheet with a sharp straight edge, and falls on a screen. Ray optics predicts a sharp boundary between light and shadow. Instead:
- inside the geometrical shadow, the light fades away gradually rather than stopping abruptly;
- outside it, the brightness rises and falls in fringes, which get weaker further out until the illumination is uniform.
The limit of resolution
Two point sources in front of a lens produce two disc images. If the discs’ centres are much closer than their radii, they overlap into what looks like a single blob: the points are not resolved. Move the sources apart and at some point you can tell there are two discs: they are just resolved.
Rayleigh’s criterion
Whether two blurry discs look like two depends a little on who is looking. Rayleigh gave a definite rule: two images are just resolved when the centre of one disc falls on the first dark ring of the other. Then the combined intensity dips noticeably between the two peaks. The smallest angular separation the lens can resolve is therefore about
A car’s headlights are 1.4 m apart. With a pupil 3.0 mm wide and λ = 550 nm, how far away can you just see them as two?
- 1A circular hole gives a bright disc with rings; first dark ring at sin θ = 1.22λ/b.
- 2Disc radius 1.22λD/b on a screen, or 1.22λf/b at a lens's focal plane.
- 3A lens images a point as a disc, never a point.
- 4At a straight edge, light fades gradually into the shadow and makes dying fringes outside it.
- 5Rayleigh: just resolved when one disc's centre is on the other's first dark ring; θ_min ≈ 1.22λ/b, smaller for bigger apertures.
Check yourself
Keep exploring
Hand-picked videos, simulations and readings that explain this idea from a different angle.