84 Limits of Resolution: The Rayleigh Criterion

Diffraction and Image Resolution

Diffraction is often useful. For example, diffraction gratings separate light into its component wavelengths and are widely used in spectroscopy to identify chemical elements and biological molecules. However, diffraction also places a fundamental limit on how much detail any optical instrument can reveal.

Whenever light passes through an opening or is focused by a lens or mirror, it spreads slightly because of its wave nature. Instead of forming an infinitely small point, the light forms a bright central spot surrounded by much weaker rings. This pattern is called an Airy pattern, and the bright central region is known as the Airy disk.

The smaller the aperture, the more pronounced the diffraction and the larger the Airy disk becomes. As a result, details that are very close together begin to blur into one another.

Figure 84.1 illustrates this limitation. A single point source produces an Airy disk rather than a perfectly sharp image. When two point sources are observed, each produces its own diffraction pattern. If the two Airy disks overlap only slightly, the sources can still be distinguished. If they overlap too much, they appear as a single blurred object.

This limitation is unavoidable because it arises from the wave nature of light rather than imperfections in the optical system.

Diffraction Limits Many Optical Systems

Diffraction affects virtually every instrument that forms an image. The human eye, cameras, microscopes, telescopes, and medical imaging devices all rely on lenses or mirrors with finite diameters. Because these optical elements have limited sizes, diffraction always causes some spreading of the light.

For example, the pupil of the human eye acts as a circular aperture. Even a perfectly healthy eye cannot distinguish details smaller than the diffraction limit imposed by the pupil. Similarly, a microscope objective cannot resolve structures smaller than its diffraction limit, regardless of how much magnification is used.

Increasing the diameter of the lens or mirror reduces diffraction and improves resolution. This is one reason why astronomical telescopes have very large primary mirrors and why high-performance microscopes use objective lenses with large numerical apertures.

The Rayleigh Criterion

What determines the smallest angular separation at which two objects can still be distinguished?

For a circular aperture, the diffraction pattern consists of a bright central Airy disk surrounded by faint rings. The first dark ring occurs at an angle

where [latex]\lambda[/latex] is the wavelength of the light and [latex]D[/latex] is the diameter of the aperture.

In the nineteenth century, Lord Rayleigh proposed a practical definition of when two point sources are considered just resolved. According to the Rayleigh criterion, two objects are just distinguishable when the center of the Airy disk produced by one object falls on the first minimum of the diffraction pattern produced by the other.

The minimum angular separation that can be resolved is therefore

where

• [latex]\theta_{\min}[/latex] is the smallest resolvable angular separation (in radians),
• [latex]\lambda[/latex] is the wavelength of the light, and
• [latex]D[/latex] is the diameter of the aperture, lens, or mirror.

This equation shows two important trends:

• Using shorter wavelengths improves resolution.
• Using a larger aperture improves resolution.

Example 84.1: Calculating the Diffraction Limit of the Hubble Space Telescope

The primary mirror of the Hubble Space Telescope has a diameter of 2.40 m. Because Hubble operates above Earth’s atmosphere, its images are not blurred by atmospheric turbulence in the same way as images from ground-based telescopes.

• 1. What is the minimum angular separation between two point sources, such as two stars, that Hubble can just resolve? Assume an average wavelength of 550 nm.
  2. If the stars are located in the Andromeda galaxy, approximately [latex]2.0\times10^6[/latex] light-years away, what is the minimum physical separation between them for Hubble to resolve them as two distinct objects?

Strategy

The Rayleigh criterion gives the smallest angular separation that can be resolved by a circular aperture:

[latex]\theta_{\min}=1.22\frac{\lambda}{D}.[/latex]

In part (a), we use the wavelength and mirror diameter to calculate the angular resolution.

In part (b), we convert that angular separation into a physical separation using the small-angle relationship

[latex]s=r\theta,[/latex]

where [latex]r[/latex] is the distance to the stars and [latex]s[/latex] is their separation.

Solution for Part (a)

The known quantities are

[latex]\lambda=550\ \text{nm}=5.50\times10^{-7}\ \text{m}[/latex]

and

[latex]D=2.40\ \text{m}.[/latex]

Applying the Rayleigh criterion,

[latex]\theta_{\min}=1.22\frac{5.50\times10^{-7}\ \text{m}}{2.40\ \text{m}}.[/latex]

[latex]\theta_{\min}=2.80\times10^{-7}\ \text{rad}.[/latex]

Therefore, the diffraction-limited angular resolution is

[latex]\boxed{\theta_{\min}=2.80\times10^{-7}\ \text{rad}}.[/latex]

Solution for Part (b)

For small angles, the physical separation between the stars is

[latex]s=r\theta_{\min}.[/latex]

Using

[latex]r=2.0\times10^6\ \text{ly},[/latex]

we obtain

[latex]s=(2.0\times10^6\ \text{ly})(2.80\times10^{-7}).[/latex]

[latex]s=0.56\ \text{ly}.[/latex]

Thus, under ideal diffraction-limited conditions, the two stars must be separated by at least

[latex]\boxed{s=0.56\ \text{ly}}[/latex]

to be resolved as distinct objects.

Discussion

The angular resolution is extremely small because the telescope’s mirror is enormous compared with the wavelength of visible light. Even so, diffraction still places an unavoidable limit on the smallest details that can be distinguished.

The calculated value represents an ideal diffraction limit. Real instruments may perform less well because of mirror imperfections, alignment errors, detector limitations, and other optical effects. Ground-based telescopes also experience atmospheric turbulence, which causes stars to shimmer and blurs fine detail unless adaptive optics or other correction methods are used.

A separation of 0.56 light-year is much smaller than the typical spacing between many stars in a galaxy. This means that a telescope with Hubble’s resolution can distinguish many individual stars in nearby galaxies rather than seeing only an unresolved glow.

The result from Example 84.1 shows that Hubble can distinguish two stars separated by only about half a light-year in the Andromeda Galaxy. This is remarkable because Andromeda is approximately 2 million light-years away, meaning the light reaching the telescope today began its journey long before modern humans existed.

The average separation between stars is roughly 5 light-years in the outer regions of a galaxy and about 1 light-year near its center. Consequently, Hubble is capable of resolving many individual stars in Andromeda rather than recording only an unresolved glow.

Although Hubble has a much smaller mirror than many radio telescopes, its use of visible light—which has a much shorter wavelength—gives it an exceptionally high angular resolution.

Diffraction of Light Beams

Diffraction is not limited to imaging systems. Every beam of electromagnetic radiation with a finite diameter naturally spreads as it travels, even if it is produced by an ideal optical system.

The minimum angular spread of a beam is determined by the same expression used for the Rayleigh criterion:

where [latex]D[/latex] is the diameter of the beam and [latex]\lambda[/latex] is its wavelength.

This means that a perfectly parallel beam cannot exist. Even the highest-quality laser beam slowly diverges because diffraction causes light from different parts of the beam to interfere with one another.

For an ordinary flashlight this effect is negligible because imperfections in the reflector and lens produce much greater spreading than diffraction. In contrast, lasers and microwave communication systems can travel enormous distances with very little initial divergence, making diffraction the dominant factor that limits how narrow the beam remains.

One way to reduce diffraction spreading is to increase the beam diameter before transmission. For example, lasers used to measure the Earth–Moon distance are expanded by large telescopes before leaving the Earth. Increasing [latex]D[/latex] decreases the diffraction angle, allowing much more of the laser light to reach the lunar retroreflectors.

Resolution in Microscopy

 

Diffraction-limited resolution is especially important in biology and medicine because microscopes are used to distinguish extremely small structures such as bacteria, cell membranes, chromosomes, and intracellular organelles.

 

Resolution is the ability of an optical system to distinguish two nearby objects as separate. The smaller the separation that can still be distinguished, the higher the resolution of the microscope.

 

Suppose two points in a specimen are separated by a distance [latex]x[/latex] and lie a distance [latex]d[/latex] from the objective lens. According to the Rayleigh criterion, they are just resolved when the angular separation equals

 

[latex]\theta=1.22\frac{\lambda}{D}.[/latex]

 

Using the small-angle approximation,

 

[latex]\theta\approx\frac{x}{d},[/latex]

 

which gives

 

[latex]1.22\frac{\lambda}{D}=\frac{x}{d}.[/latex]

 

Solving for the smallest resolvable distance yields

 

[latex]\boxed{x=1.22\frac{\lambda d}{D}}.[/latex]

 

This equation shows that microscope resolution improves when:

 

• • the wavelength is shorter,

 

• • the objective lens has a larger diameter, or

 

• • the specimen is closer to the objective lens.

 

 

Numerical Aperture and Resolving Power

 

Microscope objectives are usually characterized by their numerical aperture (NA), which measures how effectively the lens collects light from the specimen.

 

If the lens subtends a half-angle [latex]\alpha[/latex], geometry gives

 

[latex]\sin\alpha=\frac{D}{2d}.[/latex]

 

The numerical aperture is defined as

 

[latex]\boxed{\mathrm{NA}=n\sin\alpha},[/latex]

 

where [latex]n[/latex] is the refractive index of the medium between the specimen and the objective lens.

 

Substituting this definition into the diffraction-limited resolution equation gives

 

[latex]x=1.22\frac{\lambda d}{D}=1.22\frac{\lambda}{2\sin\alpha}=0.61\frac{\lambda n}{\mathrm{NA}}.[/latex]

 

The diffraction-limited resolution of a microscope is therefore

 

[latex]\boxed{x=0.61\frac{\lambda n}{\mathrm{NA}}}.[/latex]

 

 

Why Numerical Aperture Matters

 

High-quality microscope objectives are designed to maximize their numerical aperture.

 

A larger numerical aperture:

 

• • collects more light from the specimen,

 

• • produces brighter images,

 

• • captures more diffraction information, and

 

• • allows finer details to be distinguished.

 

 

This is why oil-immersion objectives, which increase the refractive index [latex]n[/latex], can resolve structures significantly smaller than comparable objectives operating in air.

 

 

 

Healthcare Connection

 

 

 

Modern pathology, microbiology, hematology, and cell biology all depend on high-resolution microscopes. Increasing the numerical aperture allows clinicians to distinguish neighboring bacteria, identify subtle cellular abnormalities, and observe intracellular structures that would otherwise blur together because of diffraction. Advanced techniques such as confocal microscopy and fluorescence microscopy are designed to maximize the information collected within these diffraction limits.

 

 

 

 

Diffraction and the Focal Spot

Diffraction also affects the way a lens focuses light. In geometric optics, parallel rays are represented as converging to an exact focal point. This model is useful for locating images, but it does not describe the complete behavior of real waves.
Because light diffracts, a lens cannot concentrate all of the incoming light into an infinitely small point. Instead, the light is distributed over a small three-dimensional region known as the focal spot or focal region.
The size of this focal spot depends strongly on the numerical aperture of the lens. A larger numerical aperture produces a narrower focal spot because the lens captures light over a wider range of angles. A smaller focal spot improves the ability to illuminate or image fine structures.

Figure 84.7: (a) In the geometric-optics model, rays converge to a single focal point. (b) In wave optics, diffraction prevents the light from forming a perfect point. The energy is distributed across a finite focal region whose size decreases as the numerical aperture increases.

A smaller focal spot also concentrates the light into a smaller area, which increases the intensity. This can be useful in techniques such as confocal microscopy, laser surgery, optical trapping, and fluorescence imaging.

However, higher intensity can also damage biological specimens. Fluorescent molecules may permanently lose their ability to emit light, a process known as photobleaching. Living cells may also be harmed through phototoxicity, in which absorbed light initiates damaging chemical reactions.

Microscope design therefore involves a trade-off. A high numerical aperture improves resolution and produces a smaller focal spot, but the resulting increase in light intensity may damage delicate samples.

Section Summary

• Diffraction places a fundamental limit on the resolving power of all optical systems, including the human eye, microscopes, cameras, telescopes, and laser systems.
• For a circular aperture, the Rayleigh criterion states that two point objects are just resolvable when the center of one Airy disk coincides with the first diffraction minimum of the other.
• The minimum angular separation that can be resolved is

• where [latex]\lambda[/latex] is the wavelength of the radiation and [latex]D[/latex] is the diameter of the aperture, lens, or mirror. The angle [latex]\theta_{\min}[/latex] is measured in radians.
• Resolution improves when shorter wavelengths are used or when the diameter of the optical system is increased.
• The same relationship also describes the minimum divergence of a beam of light with diameter [latex]D[/latex]. Even an ideal laser beam spreads because of diffraction.
• For microscopes, the diffraction-limited spatial resolution can be written as

• showing that higher numerical aperture and shorter wavelengths allow finer details to be resolved.

Problems & Exercises

57. The Arecibo radio telescope had a diameter of 300 m and detected radio waves with an average wavelength of 4.00 cm.
    1. What is the minimum angular separation between two point sources that the telescope could just resolve?
    2. How close together could these sources be if they were located in the Andromeda Galaxy, [latex]2.0\times10^6[/latex] light-years away?
58. Using the angular resolution calculated for the Hubble Space Telescope in Example 84.1, determine the smallest detail that could be observed on the Moon. Use a mean Earth–Moon distance of [latex]3.84\times10^8\ \text{m}[/latex].
59. Diffraction contributes very little to the spreading of a flashlight beam compared with optical imperfections such as spherical aberration. Calculate the minimum angular spread of a flashlight beam that has an initial diameter of 5.00 cm and an average wavelength of 600 nm.
60. A helium-neon laser emits light with a wavelength of 633 nm and has an initial beam diameter of 1.00 mm.
    1. What is the minimum angular spread of the beam?
    2. If the beam is directed toward a cliff 15.0 km away, what is the diameter of the illuminated spot?
    3. What would be the diameter of the spot on the Moon, neglecting atmospheric effects and using a lunar distance of [latex]3.84\times10^8\ \text{m}[/latex]?

    Show clearly how you apply the problem-solving strategy for wave optics.

61. A telescope can be used in reverse to expand a laser beam and reduce diffraction spreading.
    1. If a telescope produces a 2.54-m-diameter beam of 633-nm light, what is the minimum angular spread of the beam?
    2. Neglecting atmospheric effects, what diameter spot would the beam form on the Moon at a distance of [latex]3.84\times10^8\ \text{m}[/latex]?
62. The diffraction limit of the pupil contributes to the resolving ability of the human eye.
    1. What is the minimum angular separation between two point sources for a pupil with a diameter of 3.00 mm? Assume a wavelength of 550 nm.
    2. If the headlights of a car are 1.30 m apart, what is the maximum distance at which the eye could resolve them under these ideal conditions?
    3. What is the minimum resolvable separation between two points held 0.800 m from the eye?
    4. How does the answer to part (c) compare with the level of detail normally observed in everyday situations?
63. What is the minimum diameter of a telescope mirror that would allow details as small as 5.00 km to be resolved on the Moon, which is approximately 384,000 km away? Assume an average wavelength of 550 nm.
64. The centers of a person’s eyes are separated by 6.5 cm. If the observer’s pupil has a diameter of 5.0 mm, at what maximum distance could the two eyes be resolved using light with a wavelength of 555 nm?
65. Pluto and its moon Charon are separated by approximately 19,600 km. They are observed from a distance of [latex]4.50\times10^9\ \text{km}[/latex] using a telescope with a mirror diameter of 5.08 m. Assume an average wavelength of 550 nm.
    1. Neglecting atmospheric effects, should the telescope be able to resolve Pluto and Charon as separate objects?
    2. In practice, they are only barely distinguishable with a ground-based telescope. Explain why the actual resolution is worse than the diffraction limit.
66. The headlights of a car are separated by 1.30 m. What is the maximum distance at which a person can resolve the headlights if the pupil diameter is 0.40 cm? Assume visible light with a wavelength of 550 nm.
67. Dots printed on a page must be sufficiently close together that the eye does not resolve them individually. Assume a pupil diameter of 3.0 mm, a viewing distance of 35 cm, and a wavelength of 550 nm.
    1. Find the approximate separation below which two dots cannot be resolved.
    2. Convert this separation into a printer resolution in dots per inch (dpi).
68. Unreasonable Results. An amateur astronomer wants to construct a telescope capable of resolving a 1.00-m object on a moon of Jupiter located [latex]7.50\times10^8\ \text{km}[/latex] from Earth. Assume an average wavelength of 600 nm.
    1. What mirror diameter would be required according to the Rayleigh criterion?
    2. What is unreasonable about the result?
    3. Identify assumptions that make the proposal unrealistic or internally inconsistent.
69. Construct Your Own Problem. Design a problem involving the diffraction limit of a circular aperture, lens, mirror, or antenna. Your problem should require the calculation of:
    1. the minimum angular separation that the device can resolve, and
    2. the minimum spatial separation or object size that can be resolved at a specified distance.

    Choose realistic values for the wavelength, aperture diameter, and observation distance, and explain the physical meaning of your result.