3. Resolution and Optical Instruments

Resolution and Optical Instruments

What Is Resolution?

Resolution is the ability of an optical instrument to distinguish between two objects or details that are very close together.

An instrument with high resolution can show small details clearly and can distinguish two nearby objects as separate. An instrument with low resolution may cause nearby objects to blur together.

For example, imagine looking at two stars that appear very close together:

• With poor resolution, they may appear as one point of light.
• With good resolution, they can be identified as two separate stars.

Resolution is therefore an important measure of the detail that an optical instrument can reveal.

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Resolution vs Magnification

Resolution and magnification are related, but they are not the same thing.

Magnification tells us how much larger an image appears.

Resolution tells us how much detail can actually be distinguished.

An image can be highly magnified without revealing additional information.

For example, enlarging a blurry photograph makes the photograph bigger, but it does not reveal details that were not captured in the original image.

This is sometimes called empty magnification.

Example

Suppose two microscope images are both enlarged 1000 times.

• Microscope A produces a blurry image in which two nearby cells merge together.
• Microscope B clearly shows the boundaries between the two cells.

Both have the same magnification, but Microscope B has better resolution.

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Resolving Power

Resolving power describes how well an optical instrument can distinguish between objects that are very close together.

Greater resolving power means that smaller separations can be detected.

A telescope with high resolving power might distinguish two nearby stars that appear as a single star through a poorer telescope.

A microscope with high resolving power can distinguish structures that are extremely close together.

The terms resolution and resolving power are sometimes used slightly differently depending on the field, but both describe an instrument's ability to separate fine details.

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The Resolution Limit

Every optical instrument has a limit to the amount of detail it can resolve.

This occurs partly because light behaves as a wave.

When light passes through an opening such as a telescope aperture or microscope objective, it undergoes diffraction. Instead of producing a perfectly sharp point, a point source produces a small diffraction pattern.

If two diffraction patterns overlap too strongly, the two objects become difficult or impossible to distinguish.

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The Rayleigh Criterion

A common method for deciding whether two point sources can be resolved is the Rayleigh criterion.

According to this criterion, two objects are just resolved when the central maximum of one diffraction pattern falls approximately on the first minimum of the other.

For a circular aperture such as a telescope objective:

θ = 1.22λ/D

where:

• θ = minimum angular separation that can be resolved, in radians
• λ = wavelength of the light
• D = diameter of the aperture

A smaller value of θ means better resolution because the instrument can distinguish objects separated by a smaller angle.

Example

Consider two telescopes observing the same wavelength of light.

Telescope A:

D = 0.10 m

Telescope B:

D = 1.0 m

Because Telescope B has an aperture ten times larger, its minimum resolvable angle is approximately ten times smaller.

Therefore, Telescope B has much better resolving power.

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How Wavelength Affects Resolution

The equation

θ = 1.22λ/D

shows that resolution depends strongly on wavelength.

If λ decreases, θ decreases.

Therefore:

shorter wavelengths generally produce better resolution.

For visible light:

• violet/blue light has a shorter wavelength
• red light has a longer wavelength

An optical system using shorter-wavelength light can therefore potentially distinguish finer details.

This principle extends beyond visible light. Instruments that use very short wavelengths can reveal structures that ordinary visible-light instruments cannot.

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How Aperture Size Affects Resolution

The diameter of the aperture also affects resolution.

From:

θ = 1.22λ/D

increasing D decreases the minimum resolvable angle.

Therefore:

larger aperture → better resolution

This is one reason astronomical telescopes often have very large mirrors.

A large mirror does more than collect additional light. It also improves the telescope's theoretical ability to distinguish closely spaced objects.

For example, two stars that blur together through a small telescope may be distinguishable using a larger telescope.

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Resolution in Microscopes

For microscopes, resolution depends on factors including:

• wavelength of the light
• properties of the objective lens
• numerical aperture
• refractive index of the material between the specimen and lens

A simplified relationship is:

d ≈ λ/(2NA)

where:

• d = minimum distance between two resolvable points
• λ = wavelength
• NA = numerical aperture

For this equation, smaller d means better resolution.

Therefore:

• shorter wavelength → better resolution
• larger numerical aperture → better resolution

A typical light microscope has a resolution of roughly 200 nm (0.2 μm) under good conditions.

Objects much smaller than this cannot normally be distinguished clearly using an ordinary visible-light microscope.

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Light Microscopes and Electron Microscopes

Electron microscopes can achieve much greater resolution than ordinary light microscopes.

This is because electrons can behave as waves, and their effective wavelength can be far shorter than the wavelength of visible light.

Therefore, electron microscopes can reveal extremely small structures that light microscopes cannot resolve.

For example:

The exact resolution of an electron microscope depends on its design and operating conditions.

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Resolution in Telescopes

Telescopes must distinguish objects separated by very small angles.

Important factors include:

Aperture size — larger apertures provide better theoretical resolution.

Wavelength — shorter wavelengths generally provide better diffraction-limited resolution.

Atmospheric conditions — turbulence in Earth's atmosphere can distort incoming light and reduce image quality.

Optical quality — imperfections in mirrors and lenses can reduce practical resolution.

Detector quality — the sensor must be capable of recording the detail produced by the optical system.

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Atmospheric Effects

Ground-based telescopes face an additional problem: Earth's atmosphere.

Different layers of air have slightly different temperatures and densities. As light passes through these moving layers, it is refracted in slightly different directions.

This produces atmospheric turbulence, often called astronomical seeing.

Stars may appear to:

• twinkle
• move slightly
• blur
• change shape

Modern observatories can use adaptive optics, in which mirrors are adjusted rapidly to compensate for atmospheric distortion.

Space telescopes avoid most atmospheric distortion because they operate above Earth's atmosphere.

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Other Factors That Limit Image Quality

The theoretical resolution of an instrument does not guarantee that every image will have that quality.

Practical image quality can also be reduced by:

• incorrect focusing
• lens imperfections
• mirror imperfections
• chromatic aberration
• spherical aberration
• vibrations
• atmospheric turbulence
• poor lighting
• detector pixel size
• electronic noise
• motion of the object or instrument

This means an instrument's theoretical resolution and its actual image quality may be different.

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Comparing Optical Instruments

Different optical instruments are designed for different purposes.

There is therefore no single "best" optical instrument. The appropriate instrument depends on the size, distance and nature of the object being observed.

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Evaluating Image Quality

When evaluating an image, do not simply ask:

"How large is the image?"

Instead ask:

"What details can actually be distinguished?"

A good-resolution image should allow closely spaced structures to be identified separately.

For example, if two tiny structures appear as one blurred feature, increasing the display size does not necessarily solve the problem. The original optical system must have captured enough information to resolve them.

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Worked Example

A telescope has an aperture diameter of 0.25 m and observes light with a wavelength of 550 nm.

First convert the wavelength:

550 nm = 550 × 10⁻⁹ m

Use:

θ = 1.22λ/D

θ = (1.22 × 550 × 10⁻⁹) / 0.25

θ ≈ 2.68 × 10⁻⁶ rad

The telescope can theoretically resolve objects separated by an angle of approximately:

2.7 × 10⁻⁶ rad

If the aperture diameter were increased while the wavelength remained constant, θ would decrease and the resolution would improve.

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Common Mistakes

Mistake: "Higher magnification always means better resolution."
Magnification only makes an image appear larger. It does not necessarily reveal additional detail.

Mistake: "A larger resolution value means better resolution."
When resolution is expressed as the minimum resolvable distance or angle, a smaller value represents better resolution.

Mistake: "Only the lens determines resolution."
Wavelength, aperture, diffraction, optical quality, atmospheric conditions and detector performance can all affect the final image.

Mistake: "Diffraction only happens with poor-quality equipment."
Diffraction is a fundamental property of waves and occurs even in an ideal optical system.

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Key Terms

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

Resolving power — a measure of an instrument's ability to separate fine details.

Diffraction — the spreading of waves when they pass through an opening or around an obstacle.

Resolution limit — the smallest separation that an optical system can distinguish.

Rayleigh criterion — a commonly used criterion for deciding when two diffraction patterns are just resolvable.

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

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

Numerical aperture (NA) — a measure related to the light-gathering and resolving ability of a microscope objective.

Airy pattern — the diffraction pattern produced when light from a point source passes through a circular aperture.

Atmospheric seeing — distortion of astronomical images caused by turbulence in Earth's atmosphere.

Adaptive optics — technology that compensates for atmospheric distortion in astronomical observations.

Empty magnification — enlargement of an image without an increase in useful detail.

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Key Takeaways

• Resolution describes an instrument's ability to distinguish fine details or closely spaced objects.
• Resolution and magnification are not the same thing.
• Diffraction places a fundamental limit on the resolution of optical instruments.
• For telescopes, angular resolution can be estimated using θ = 1.22λ/D.
• Larger apertures improve resolution.
• Shorter wavelengths improve resolution.
• Microscopes can improve resolution by using shorter wavelengths and larger numerical apertures.
• Electron microscopes achieve much greater resolution than ordinary light microscopes because electrons can have extremely short effective wavelengths.
• Practical image quality can be worse than theoretical resolution because of aberrations, atmospheric turbulence, focusing, detector limitations and other factors.
• The best way to evaluate resolution is to ask how much real detail can be distinguished, rather than simply how large the image appears.

Check Your Understanding

1. What is meant by the resolution of an optical instrument?
2. Why are magnification and resolution not the same thing?
3. Two telescopes observe the same wavelength. Which should have better resolution: one with a 0.20 m aperture or one with a 2.0 m aperture? Explain.
4. How does decreasing the wavelength of light affect the diffraction-limited resolution?
5. Why can electron microscopes reveal smaller structures than light microscopes?
6. Why might a real telescope produce poorer images than predicted by θ = 1.22λ/D?
7. A photograph is enlarged five times but becomes increasingly blurry. Has its resolution improved? Explain.
8. Explain why large telescope mirrors improve both light collection and resolving power.