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E. The Wave Nature of Light · The Wave Nature of Light

Resolution: How Diffraction Limits What We Can See

E1.2 E2.1 E2.4 E3.2 A1.12 A1.13About 65 minutes
Learning goals
  • I can explain why diffraction through a circular opening blurs every image into a small disc.
  • I can use the Rayleigh criterion θ = 1.22λ/D to find the smallest angle an instrument can resolve.
  • I can find the smallest separation that can be resolved at a given distance, using s = Lθ.
  • I can explain why telescopes are large, why radio telescopes are huge, and why shorter wavelengths store more data on a disc.

Words to know

resolution (resolving power)
The ability of an eye or instrument to show two close objects as separate.
diffraction limit
The finest detail an instrument can show, set by diffraction through its aperture.
Rayleigh criterion
Two point sources are just resolved when the centre of one diffraction pattern falls on the first dark ring of the other: θ = 1.22λ/D (in radians).
aperture (D)
The diameter of the opening (lens, mirror, pupil or dish) that collects the light.
angular separation
The angle between two objects as seen by the observer; for small angles, θ = s/L.

Two headlights or one?

On a dark highway, a car far ahead looks like a single light. As it comes closer, the single light splits into two headlights. Why can't your eyes separate them from far away? It is not only because your eyes are imperfect. Light passing through the round opening of your pupil diffracts, so even a perfect eye turns every point of light into a tiny blurred disc. When two discs overlap too much, they look like one. That limit on resolution applies to cameras, telescopes, microscopes and disc players too.

Diffraction through a round opening

You saw that light through a slit of width w spreads to a first dark fringe at sin θ = λ/w. A circular opening of diameter D (a pupil, a lens or a telescope mirror) makes a similar pattern in two dimensions: a bright central disc surrounded by faint rings. The first dark ring is at

sin θ = 1.22 λ / D

(the 1.22 comes from the round shape; you do not need to derive it). Angles are tiny, so sin θ ≈ θ in radians.

The Rayleigh criterion: two point sources are just resolved when the centre of one image falls on the first dark ring of the other. The smallest resolvable angle is therefore

θ_min = 1.22 λ / D (radians)

To resolve finer detail (smaller θ_min), use a larger aperture D or a shorter wavelength λ.

star Astar BDθimages: blurred diffraction discs
Light from two stars passes through an aperture of diameter D. Each star's image is a small diffraction disc, not a point. If the angle θ between the stars is too small, the discs overlap and the stars blur together (not to scale).
well resolvedseparation: largejust resolved (Rayleigh)separation: θ = 1.22λ/Dnot resolvedseparation: small
Brightness across the images of two point sources (dashed: each source; solid: total). Left: clearly separate. Centre: the Rayleigh criterion, where the peak of one falls on the first dark ring of the other, giving a visible dip. Right: one blurred blob. (Shapes are sketched, not exact.)
Worked example

Example 1: Headlights at night

At night your pupil has a diameter of about 5.0 mm. Using λ = 550 nm, find the smallest angle your eye could resolve if it were limited only by diffraction. How far away could you see a car's headlights (1.4 m apart) as two lights?

  1. θ_min = 1.22λ/D = 1.22(5.50 × 10⁻⁷) ÷ (5.0 × 10⁻³) = 1.34 × 10⁻⁴ rad.
  2. For small angles θ = s/L, so L = s/θ = 1.4 ÷ 1.34 × 10⁻⁴.
  3. L = 1.04 × 10⁴ m.

About 10 km. In reality, the spacing of light-sensitive cells in the retina, imperfections in the eye and the shimmering of air make the distance shorter.

Worked example

Example 2: Craters on the Moon

A backyard telescope has a mirror 0.20 m across. What is the smallest separation between two features on the Moon (3.84 × 10⁸ m away) that it can resolve in 550 nm light?

  1. θ_min = 1.22(5.50 × 10⁻⁷) ÷ 0.20 = 3.36 × 10⁻⁶ rad.
  2. s = Lθ = (3.84 × 10⁸)(3.36 × 10⁻⁶).
  3. s = 1.29 × 10³ m.

About 1.3 km. Features closer together than that blur into one.

Worked example

Example 3: Why radio telescopes are enormous

Radio astronomers often observe hydrogen at λ = 0.21 m. How large a dish would match the 0.20 m optical telescope's θ_min of 3.36 × 10⁻⁶ rad?

  1. Rearrange: D = 1.22λ/θ_min.
  2. D = 1.22(0.21) ÷ 3.36 × 10⁻⁶.
  3. D = 7.6 × 10⁴ m.

About 76 km, far too large to build as one dish. Radio astronomers instead link telescopes many kilometres (even continents) apart and combine their signals, which gives the resolution of a much larger aperture.

Resolution in technology and society

  • Telescopes: larger mirrors collect more light and resolve finer detail. Space telescopes avoid the blurring caused by the moving air of the atmosphere, so they can reach their diffraction limit.
  • Microscopes: the finest detail an ordinary light microscope can show is a few hundred nanometres, about half the wavelength of light. Using blue or ultraviolet light helps a little; electron microscopes, which use the wave nature of electrons, do far better (unit Quantum Mechanics).
  • Optical discs: each generation used a shorter wavelength laser (CD about 780 nm, DVD 650 nm, Blu-ray 405 nm) focused through a lens to a smaller spot, so pits and tracks could be packed more tightly. A smaller spot meant more data on the same size disc.
  • Screens and printing: if pixels or dots are closer together than your eye can resolve at normal viewing distance, the image looks smooth. Pointillist paintings, made of separate dots of colour, blend into smooth colours when you step back.
  • Surveillance and privacy: satellite cameras have a resolution limit set by their aperture and distance. The diffraction limit is one reason that even very powerful cameras cannot read small print from orbit; it is also why improving camera resolution raises questions about privacy.
Just for fun

Hold a page with two pencil dots 2 mm apart at arm's length: easy to see two dots. Walk across a large room and they merge into one. Your eyes have officially hit the Rayleigh criterion, which is a much more scientific excuse than "I need new glasses."

Hands-on activity

Find your eye's resolution

You need: white paper, a fine black marker, a ruler, a tape measure (or count your steps and measure one step), sticky tape, and a helper. Optional: a piece of aluminium foil with a pinhole about 1 mm across.

  1. Draw two dots, each about 1 mm wide, with their centres 2.0 mm apart (s). Draw a second pair 4.0 mm apart.
  2. Tape the paper to a well-lit wall. Walk backward until the 2.0 mm pair just looks like one dot. Measure that distance L. Repeat 3 times and average.
  3. Calculate your measured resolution θ = s/L (radians).
  4. Compare with the diffraction limit 1.22λ/D for λ = 550 nm and a pupil of about 3 mm in a bright room.
  5. Optional: look through the pinhole (D ≈ 1 mm) and repeat. Do not look at bright lights or the Sun through the pinhole.

What to look for: many people find θ around 3 × 10⁻⁴ rad (for example, 2.0 mm dots merging at about 6 to 7 m). The diffraction limit for a 3 mm pupil is about 2.2 × 10⁻⁴ rad, so your measured value is similar but somewhat larger, because the retina and the eye's lens also limit sharpness. With the 1 mm pinhole, the dots merge at a shorter distance: a smaller aperture means more diffraction and worse resolution.

Your turn

1

State the Rayleigh criterion.

2

Find θ_min for a 2.4 m telescope mirror at λ = 500 nm.

3

A pupil is 3.0 mm across. Find θ_min for 600 nm light.

4

Using the last answer, how far away can two dots 2.0 mm apart be resolved?

5

Give two ways to improve the resolution of a telescope.

6

Why do radio telescopes need very large dishes?

7

Why did Blu-ray discs need a violet laser?

More practice: warm-up

1

Does a larger aperture give better or worse resolution?

2

Does blue or red light give better resolution with the same lens?

3

For θ_min = 1.22λ/D, what unit is θ in?

4

Write the small-angle link between separation s, distance L and angle θ.

5

What is a diffraction-limited image of a star?

More practice: on your own

1

A radio dish 100 m across observes at λ = 3.0 cm. Find θ_min.

2

A camera on a satellite 400 km above Earth has an aperture of 0.30 m. What is the smallest separation it could resolve on the ground in 550 nm light (diffraction limit only)?

3

Compare θ_min for a pupil 7.0 mm wide (dim light) and 2.0 mm wide (bright light) at 550 nm.

4

By what factor does the diffraction-limited spot size shrink going from a DVD laser (650 nm) to a Blu-ray laser (405 nm), with the same lens?

5

Explain why a pointillist painting looks smooth from across a gallery.

Challenge

1

What diameter of radio telescope would be needed to match the 2.4 m telescope's resolution at 500 nm (2.54 × 10⁻⁷ rad) when observing at λ = 6.0 cm? What do astronomers do instead?

2

Two stars are 4.0 × 10¹¹ m apart and 1.0 × 10¹⁷ m from Earth. Can a 0.20 m telescope (550 nm) resolve them? Show your reasoning.

3

A student says a microscope with a high enough magnification can show any detail, no matter how small. Respond using the wave model of light.

Check your understanding

Quiz

1.

The smallest angle an aperture of diameter D can resolve is about…

2.

Which change improves a telescope's resolution?

3.

A 0.10 m lens, λ = 500 nm. θ_min is…

4.

Radio telescopes are built much larger than optical ones mainly because…

5.

At the Rayleigh criterion, the centre of one star's image falls on…

6.

Why could Blu-ray discs hold more data than DVDs?

For parents and teachers

Big idea: expectation E3.2 names "reduced resolution caused by diffraction". This lesson connects single-slit diffraction to the eye, telescopes and data storage, and touches on social impacts (E1.2) such as privacy and astronomy.

How to help: the dot test is easy and gives real numbers. Help by measuring the distance and checking the radians calculation.

Common difficulties:

  • Using degrees instead of radians: θ = 1.22λ/D comes out in radians.
  • Forgetting to convert mm to m for the aperture.
  • Thinking more magnification always reveals more detail. Diffraction sets a hard limit.