Diffraction Limits

Diffraction and Resolution

Even a perfectly manufactured optical instrument cannot produce unlimited detail. There is a fundamental limit caused by the wave nature of light.

When light passes through an opening, such as the aperture of a telescope or the objective of a microscope, the light spreads out slightly. This spreading is called diffraction.

The smaller the opening compared with the wavelength of the light, the more noticeable the diffraction becomes.

Diffraction causes light from a point object to form a small pattern rather than a perfectly sharp point. When two objects are close together, their diffraction patterns may overlap. Eventually, it becomes impossible to distinguish the objects as separate.

This creates the diffraction limit of an optical instrument.

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The Airy Pattern

When light from a point source passes through a circular aperture, such as a telescope lens or mirror, it forms a diffraction pattern called an Airy pattern.

The pattern consists of:

• a bright central region called the Airy disk
• alternating dark and bright rings surrounding the central disk

The object itself may be extremely small, but diffraction causes its image to spread across this pattern.

A larger Airy disk means that nearby objects are more likely to overlap and become difficult to distinguish.

A smaller Airy disk allows finer details to be resolved.

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Two Objects and Their Diffraction Patterns

Imagine two distant stars.

Each star produces its own Airy pattern in a telescope.

If the stars are widely separated, their central maxima are clearly separated.

As the stars become closer together, their diffraction patterns begin to overlap.

There are three useful situations:

Clearly resolved — two distinct peaks can easily be identified.

Just resolved — the two objects can still be distinguished, but their diffraction patterns overlap significantly.

Unresolved — the patterns overlap so much that the objects appear to be a single source.

This leads to an important rule for deciding when two objects can be considered resolved.

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Rayleigh's Criterion

Rayleigh's criterion states that two point sources are just resolved when the central maximum of one diffraction pattern falls on the first minimum of the other.

For a circular aperture:

θ = 1.22λ/D

where:

• θ = minimum angular separation that can be resolved, in radians
• λ = wavelength of light, in metres
• D = diameter of the aperture, in metres
• 1.22 = constant resulting from diffraction through a circular aperture

The quantity θ represents the diffraction-limited angular resolution.

A very important point is:

Smaller θ = better resolution

An instrument capable of resolving an angular separation of 0.000001 rad has better resolving power than one limited to 0.00001 rad.

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Aperture Size and Resolving Power

From:

θ = 1.22λ/D

we can see that θ is inversely proportional to aperture diameter.

Therefore:

larger aperture → smaller θ → better resolution

Increasing the aperture allows the instrument to distinguish objects separated by smaller angles.

For example, suppose two telescopes observe the same stars using the same wavelength.

A large telescope therefore has two major advantages:

• it collects more light
• it has better diffraction-limited resolution

This is one reason professional astronomical telescopes use extremely large mirrors.

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Why Does a Larger Aperture Improve Resolution?

A smaller aperture produces greater diffraction.

The light spreads over a larger angle, producing a larger diffraction pattern.

A larger aperture produces less angular spreading, resulting in a smaller Airy pattern.

Therefore, two nearby objects are less likely to have diffraction patterns that completely overlap.

This means:

small aperture → more diffraction → poorer resolution

large aperture → less diffraction → better resolution

This relationship applies even if both optical instruments are perfectly manufactured.

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Wavelength and Resolution

Rayleigh's equation also shows that:

θ ∝ λ

Therefore:

shorter wavelength → smaller θ → better resolution

and:

longer wavelength → larger θ → poorer resolution

For example, blue light has a shorter wavelength than red light. Under otherwise identical conditions, an optical instrument can theoretically achieve better resolution using blue light.

This principle becomes extremely important when comparing different types of imaging technology.

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Worked Example: Telescope Resolution

A telescope has an aperture diameter of:

D = 0.50 m

It observes light with a wavelength of:

λ = 600 nm

Find its diffraction-limited angular resolution.

First convert nanometres to metres:

600 nm = 600 × 10⁻⁹ m

Use:

θ = 1.22λ/D

Substitute:

θ = (1.22 × 600 × 10⁻⁹) / 0.50

θ = 1.464 × 10⁻⁶ rad

Therefore:

θ ≈ 1.46 × 10⁻⁶ rad

Two point sources must therefore be separated by approximately this angle or more to satisfy Rayleigh's criterion.

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Comparing Two Telescopes

Suppose Telescope A has:

D = 0.25 m

and Telescope B has:

D = 1.00 m

Both observe the same wavelength.

Because:

θ ∝ 1/D

Telescope B has an aperture four times larger.

Its minimum resolvable angle will therefore be:

four times smaller

So Telescope B has better resolving power.

Notice that we can make this comparison without calculating the actual wavelength.

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Worked Example: Finding Aperture Diameter

A telescope must resolve two objects separated by:

θ = 2.0 × 10⁻⁶ rad

It observes light of wavelength:

λ = 500 nm

What minimum aperture diameter is required?

Start with:

θ = 1.22λ/D

Rearrange:

D = 1.22λ/θ

Convert the wavelength:

500 nm = 500 × 10⁻⁹ m

Substitute:

D = (1.22 × 500 × 10⁻⁹) / (2.0 × 10⁻⁶)

D = 0.305 m

Therefore:

D ≈ 0.31 m

The telescope requires an aperture of approximately 31 cm or greater.

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Worked Example: Finding Wavelength

An optical instrument has:

D = 0.40 m

and a diffraction-limited angular resolution of:

θ = 1.5 × 10⁻⁶ rad

Find the wavelength.

Start with:

θ = 1.22λ/D

Rearrange:

λ = θD/1.22

Substitute:

λ = (1.5 × 10⁻⁶ × 0.40) / 1.22

λ ≈ 4.92 × 10⁻⁷ m

Convert to nanometres:

λ ≈ 492 nm

This wavelength lies within the visible region of the electromagnetic spectrum.

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Diffraction Limits in Telescopes

Telescopes usually observe objects that are extremely far away.

For this reason, telescope resolution is commonly expressed as an angular resolution.

Two distant stars may be physically separated by an enormous distance but appear extremely close together because they are so far from Earth.

A telescope must distinguish their tiny angular separation.

The theoretical resolution depends primarily on:

• aperture diameter
• wavelength

However, real ground-based telescopes are also affected by:

• atmospheric turbulence
• mirror imperfections
• focusing errors
• vibration
• detector limitations

Therefore, a telescope may perform worse than its theoretical diffraction limit.

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Diffraction Limits in Microscopes

Microscopes face the same fundamental problem, but the situation is described somewhat differently.

Instead of asking for the smallest angular separation, we usually want to know the smallest distance between two points that can be distinguished.

A commonly used approximate relationship is:

d ≈ λ/(2NA)

where:

• d = minimum resolvable distance
• λ = wavelength
• NA = numerical aperture

The numerical aperture describes the ability of the microscope objective to collect light.

Again:

smaller d = better resolution

Therefore:

• shorter wavelength improves resolution
• larger numerical aperture improves resolution

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Numerical Aperture

For a microscope:

NA = n sin θ

where:

• NA = numerical aperture
• n = refractive index of the material between the specimen and objective
• θ = half-angle of the cone of light entering the objective

A microscope can therefore improve its resolving ability by increasing its numerical aperture.

This can be achieved through:

• improved objective design
• collecting light over a wider angle
• using materials with a higher refractive index

Some high-power microscopes use immersion oil between the specimen and objective lens. The oil has a higher refractive index than air, helping increase the numerical aperture and improve resolution.

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Worked Example: Microscope Resolution

A microscope uses light of wavelength:

λ = 500 nm

Its objective has:

NA = 1.25

Using:

d ≈ λ/(2NA)

we obtain:

d ≈ (500 × 10⁻⁹)/(2 × 1.25)

d ≈ 2.0 × 10⁻⁷ m

Therefore:

d ≈ 200 nm

Objects separated by much less than about 200 nm would not normally be resolved by this optical system.

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Microscopes vs Telescopes

Both microscopes and telescopes are limited by diffraction, but their resolution is often described differently.

The underlying principle is the same:

Diffraction spreads light and prevents infinitely fine detail from being resolved.

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Why Electron Microscopes Have Better Resolution

Visible-light microscopes eventually reach a diffraction limit because visible light has wavelengths of roughly hundreds of nanometres.

Electrons also have wave properties.

Under suitable conditions, their associated wavelength can be far shorter than visible-light wavelengths.

Therefore, electron microscopes can achieve much better resolution than conventional light microscopes.

This illustrates a fundamental principle:

To resolve smaller structures, use a shorter wavelength.

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Diffraction Limit vs Other Limitations

It is important to distinguish between a diffraction limit and poor image quality caused by equipment.

Suppose an image is blurry because the lens is incorrectly focused. Correcting the focus may improve the image.

Suppose instead that two objects are closer together than the diffraction limit. Simply improving the focus cannot make an ordinary diffraction-limited optical system resolve them.

The diffraction limit exists because of the physics of waves, not because the instrument was badly manufactured.

Real instruments may therefore be limited by both:

Fundamental limitations

• diffraction
• wavelength

Practical limitations

• aberrations
• poor focusing
• vibration
• atmospheric turbulence
• detector resolution
• optical imperfections

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Evaluating Changes to an Optical Instrument

The equation:

θ = 1.22λ/D

allows us to predict how changes affect a telescope's diffraction limit.

Double the aperture

If D doubles:

θ becomes half as large.

Resolution improves by a factor of 2.

Halve the wavelength

If λ is halved:

θ becomes half as large.

Resolution improves by a factor of 2.

Double both wavelength and aperture

If λ doubles and D doubles:

λ/D remains unchanged.

Therefore:

the diffraction-limited resolution remains unchanged.

This type of proportional reasoning is often faster than performing a complete calculation.

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Common Mistakes

Mistake: Larger θ means better resolution.
For minimum angular resolution, a smaller θ means better resolution.

Mistake: A smaller aperture gives a sharper image because it blocks unwanted light.
Reducing the aperture increases diffraction and eventually reduces resolving power.

Mistake: Diffraction only occurs with small apertures.
Diffraction occurs at every aperture. It simply becomes more significant when the aperture is small relative to the wavelength.

Mistake: Increasing magnification overcomes the diffraction limit.
Magnifying an unresolved image only produces a larger unresolved image.

Mistake: Forgetting to convert nanometres.
Remember:

1 nm = 10⁻⁹ m

Mistake: Comparing microscope and telescope resolution values directly.
Microscope resolution is often given as a distance, while telescope resolution is commonly given as an angle.

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Key Terms

Diffraction — spreading of a wave when it passes through an opening or around an obstacle.

Diffraction limit — the fundamental limit on resolution caused by diffraction.

Resolution — the ability to distinguish closely spaced objects or details.

Resolving power — the ability of an instrument to distinguish fine detail.

Airy pattern — the diffraction pattern produced by light passing through a circular aperture.

Airy disk — the bright central region of an Airy diffraction pattern.

Rayleigh's criterion — the condition in which two point sources are considered just resolved when the central maximum of one diffraction pattern coincides approximately with the first minimum of the other.

Angular resolution — the smallest angular separation that can be distinguished.

Aperture — the opening through which light enters an optical system.

Numerical aperture — a measure of a microscope objective's ability to collect light and resolve fine detail.

Refractive index — a measure of how strongly a material affects the speed and direction of light.

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Key Takeaways

• Diffraction places a fundamental limit on optical resolution.
• A point source forms an Airy pattern rather than a perfect point.
• Overlapping diffraction patterns make nearby objects difficult to distinguish.
• Rayleigh's criterion provides a practical definition of when two point sources are just resolved.
• For a circular aperture, θ = 1.22λ/D.
• A smaller minimum angular separation means better resolution.
• Increasing aperture diameter improves telescope resolution.
• Decreasing wavelength improves resolution.
• Microscope resolution depends strongly on wavelength and numerical aperture.
• Telescopes usually describe resolution using angular separation, while microscopes often use minimum linear distance.
• Diffraction cannot simply be eliminated by perfect focusing or increased magnification.
• The diffraction limit results from the wave nature of light itself.

Check Your Understanding

1. What causes the diffraction limit of an optical instrument?
2. Explain Rayleigh's criterion in words.
3. What happens to angular resolution if a telescope's aperture diameter doubles?
4. What happens to resolution if the wavelength decreases?
5. Why does a larger telescope aperture produce better resolving power?
6. A telescope uses λ = 550 nm and has D = 0.80 m. Calculate its diffraction-limited angular resolution.
7. Two telescopes have apertures of 0.25 m and 1.00 m. If they observe the same wavelength, compare their minimum resolvable angles.
8. Why is microscope resolution usually expressed as a distance while telescope resolution is often expressed as an angle?
9. Explain why increasing magnification cannot overcome the diffraction limit.
10. A microscope uses 450 nm light instead of 600 nm light without changing its numerical aperture. Explain how its diffraction-limited resolution changes.