Lens Resolution

The intrinsic metrics for characterizing a lens’s imaging quality are its optical transfer function (OTF) and distortion. For users, however, only the spatial resolution of the lens needs to be understood. It is measured in line pairs per millimeter (lp/mm). The calculation formula is:Lens Resolution N = Total lines of field of view ÷ 2 ÷ frame format (L/W)

For example, for a 5-megapixel lens paired with a 2/3” sensor: 2448 ÷ 2 ÷ 8.8 = 139 lp/mm.

Resolution is a quantitative measure of the ability of an optical instrument to form discrete images. Due to diffraction effects arising from the lens aperture restricting light beams, light waves emitted by an object point cannot converge to a perfect point on the image plane. Instead, they spread into a certain intensity distribution centered at the image point. The central spot is the zero-order spot of Fraunhofer diffraction, also known as the Airy disk. This means that even if all geometric aberrations are ignored, an imaging optical instrument cannot achieve the ideal scenario of a point object forming a point image.

Therefore, two closely spaced object points on the object plane may form two overlapping diffraction disks on the image plane. In extreme cases, the disks overlap so heavily that they blur together, making it impossible for an observer to distinguish the two original object points. In short, an object-plane image is a collection of countless object points, while the intensity distribution transformed onto the image plane is a collection of diffraction disks, which cannot faithfully reproduce all fine details of the object plane.

To establish a universal standard for evaluating the capability of optical instruments to resolve fine details, the Rayleigh criterion is commonly adopted. The Rayleigh criterion states that two image spots are just resolvable when the center of one Airy disk coincides with the first dark ring of the second disk (see Figure b). Calculations show that for two such incoherently superimposed image spots meeting the Rayleigh criterion, the resulting light intensity fluctuation is approximately 20%, which the normal human eye can detect.

For objective light receivers such as photographic emulsion, phototubes and other sensors, a 20% intensity fluctuation may not be the mandatory threshold for distinguishability. Even so, the Rayleigh criterion remains a convenient reference standard for estimating and comparing the resolving power of optical instruments.

Resolving Power

The pupil diameter Dₑ of the human eye can adjust within a range of 2–8 mm. From the Rayleigh criterion and the formula for the half-angle width of the Airy disk, the formula for the minimum angular resolution of the human eye is derived:δθₑ = 1.22λ / Dₑ

Taking Dₑ = 2 mm and light wavelength λ = 0.55 μm for estimation: The minimum angular resolution of the human eye is 3.355 × 10⁻⁴ radian (radian = degree ÷ 180° × π). δθₑ ≈ 1 arcmin = 0.075 mm / 25 cm = 3 mm / 10 m.

In other words, a normal human eye can distinguish two engraved lines separated by 0.075 mm at a near vision distance of 25 cm, or two lines 3 mm apart at a distance of 10 meters. This physiological optics parameter is fundamental to the design of visual aids, television sets, and image recognition systems.

Telescope

Telescopes observe distant objects, which generally have large physical dimensions. Their resolving power is therefore directly characterized by the minimum angular resolution δθₘ. The minimum angular resolution formula for a telescope:δθₘ ≈ 1.22λ / D (radian)

Where λ is the light wavelength in the medium and D is the diameter of the objective aperture. With D = 2000 mm and λ = 0.55 μm, δθₘ ≈ 0.06 arcseconds. To reduce δθₘ and improve resolving power, the objective aperture diameter must be increased.

However, atmospheric turbulence during long-distance light propagation degrades performance, so the actual resolving power of astronomical telescopes is lower than this theoretical value. For this reason, large astronomical telescopes worldwide are preferably built on mountain tops. The Yunnan Observatory of China sits on a peak at an altitude of 2300 meters. An infrared telescope completed in Hawaii, USA in 1981 has a diameter of 3357 mm and is installed at 4200 meters above sea level. It can observe celestial bodies billions of light-years away and study molecular structures and nascent stellar envelopes that are difficult for ordinary optical telescopes to capture.

Microscope

Microscopes observe tiny, close-range specimens, so their resolving power is quantified by the minimum resolvable distance δyₘ. Based on the Rayleigh criterion, the half-angle width formula of the Airy disk, and the aplanatic-point operating condition of microscopes, the formula for the minimum resolvable distance of a microscope can be derived:

Where n = refractive index of the object medium, u₀ = aperture angle of the object light beam, λ₀ = vacuum wavelength. The product n·sinu₀ is defined as the numerical aperture, abbreviated as N.A.

For order-of-magnitude estimation, the maximum numerical aperture is approximately N.A. ≈ n ≈ 1.5 (for oil-immersion objectives). Thus δyₘ has a lower limit: δyₘ ≥ 0.4λ₀. Within the visible light spectrum, δyₘ ≥ 0.2 μm.

To fully utilize the microscope’s resolving power, δyₘ must be magnified to a distance δyₑ large enough for human eyes to resolve: δyₑ ≈ δθₑ × 25 cm ≈ 0.075 mm. From this, the transverse linear magnification of an optical microscope is estimated as:v ≈ δyₑ / δyₘ ≈ 400×

Magnification higher than this brings no benefit, since the instrument still cannot resolve features smaller than δyₘ. This magnification matched to resolving power is called the normal magnification or effective magnification of the microscope. In design practice, magnification is usually selected slightly above the normal magnification, and optical microscopes generally do not exceed 1000×.

The only way to further improve microscope resolving power is to shorten the wavelength. Modern electron microscopes use the wave nature of electron beams and magnetic lenses for imaging. Electron beams have extremely short wavelengths (determined by acceleration voltage), reaching the angstrom level. Although the aperture angle of electron beams is small (less than 10°), electron microscopes achieve resolving power several orders of magnitude higher than optical microscopes, with corresponding magnification ranging from tens of thousands to millions of times, capable of visualizing protein molecular structures.

Photographic Systems

Photographic systems (such as cameras and video cameras) typically operate in the scenario of distant objects and short focal lengths. Unlike visual aids (telescopes and microscopes), they form a reduced real image of the object through the photographic lens, which is directly recorded by light-sensitive media.

Therefore, when analyzing the resolving power of the entire system, both the diffraction effect of the lens aperture and the spatial resolution N of the recording medium itself — the number of resolvable lines per unit length on the photographic emulsion — must be considered.

The formula for the minimum object angular resolution limited by lens diffraction remains δθₑ = 1.22λ / D. The corresponding minimum resolvable linear size on the image plane of a photographic system is:δy’ₘ ≈ 1.22λf / Dwhere f = lens focal length, and the ratio D/f is known as the relative aperture. A larger relative aperture yields higher lens resolving power.

Taking a relative aperture of 1:3.5 for estimation, δy’ₘ ≈ 2.35 μm. To fully exploit the lens resolving power, the resolution of the recording medium must satisfy N ≥ 1/δy’ₘ ≈ 425 lp/mm. In other words, photographic emulsion capable of resolving more than 425 line pairs per millimeter is required.

The formulas for resolving power of optical imaging instruments given above are theoretical results that only account for diffraction effects. In practice, imaging instruments suffer from various geometric aberrations, especially photographic systems. As a result, the actual resolving power of equipment is lower than the theoretical value, sometimes by an order of magnitude.

Comprehensive evaluation of imaging quality by integrating all factors that prevent point-to-point imaging began with the concept of the optical transfer function emerging in the 1950s.