Angular resolution

Resolving power is the ability of a microscope or telescope to measure the angular separation of images that are close together. Angular resolution describes the resolving power of a telescope. Resolution is the minimum distance between distinguishable objects, in microscopy. These terms also apply to other angle and position measuring devices. Resolution, more generally, is the precision of any instrument to measure a continuous variable.

Resolving power is also relevant in the inverse case, where one is focusing a beam of light from an emitter onto a target.

The resolving power of a lens is ultimately limited by diffraction effects. The lens' aperture is a "hole" that is analogous to a two-dimensional version of the single-slit experiment; light passing through it interferes with itself, creating a ring-shaped diffraction pattern, known as the Airy pattern, that blurs the image. An empirical diffraction limit is given by the Rayleigh criterion:

$\displaystyle \sin \theta = 1.22 \frac{\lambda}{D}$

where θ is the angular resolution, λ is the wavelength of light, and D is the diameter of the lens.

The factor 1.22 is derived from a calculation of the position of the first dark ring surrounding the central Airy disc of the diffraction pattern. The calculation involves a Bessel function and 1.22 is approximately the first positive zero of the Bessel function of the first kind, of order one divided by pi. This factor is used to approximate the ability of the human eye to distinguish two separate point sources depending on the overlap of their Airy discs. Modern telescopes and microscopes with video sensors may be slightly better than the human eye in their ability to discern overlap of Airy discs. Thus it is worth bearing in mind that the Rayleigh criterion is an empirical estimate of resolution based on the assumption of a human observer, and may slightly underestimate the resolving power of a particular optical train. For specialized imaging, foreknowledge of some characteristics of the image can also improve on technical resolution limits through computerized image processing.

For an ideal lens of focal length f, the Rayleigh criterion yields a minimum spatial resolution, Δl:

$\displaystyle \Delta l = 1.22 \frac{ f \lambda}{D}$ .

This is the size of smallest object that the lens can resolve, and also the radius of the smallest spot that a collimated beam of light can be focussed to. The size is proportional to wavelength, λ, and thus, for example, blue light can be focussed to a smaller spot than red light. If the lens is focussing a beam of light with a finite extent (e.g., a laser beam), the value of D corresponds to the diameter of the light beam, not the lens. Since the spatial resolution is inversely proportional to D, this leads to the slightly surprising result that a wide beam of light may be focussed to a smaller spot than a narrow one.

Telescope case

Point-like sources separated by an angle smaller than the angular resolution cannot be resolved. A single optical telescope has an angular resolution less than one arcsecond, but astronomical seeing and other atmospheric effects make attaining this very hard. The highest angular resolutions can be achieved by interferometry: the VLTI is intended to achieve an effective angular resolution of 0.001 arcsecond.

The angular resolution of a telescope can usually be approximated by R = L/D where L is the wavelength of the observed radiation and D is the diameter of the telescope. The resulting R is in radians. Sources larger than the angular resolution are called extended sources or diffuse sources, and smaller sources are called point sources.

For example, in the case of yellow light with a wavelength of 580 nm, for a resolution of 1 arc second, we need D = 12 cm.

Microscope case

The resolution D depends on the angular aperture α:

$\displaystyle D=\frac{1.22\lambda}{2n\sin\alpha}$ .

Here α is the collecting angle of the lens, which depends on the width of objective lens and its distance from the specimen. n is the refractive index of the medium in which the lens operates. λ is the wavelength of light illuminating or emanating from (in the case of fluorescence microscopy) the sample.

Due to the limitations of the values α, λ, and n, the resolution limit of a light microscope using visible light is about 200 nm. This is because: α for the best lens is about 70° (sinα = 0.94), the shortest wavelength of visible light is blue (λ = 450nm), and the typical high resolution lenses are oil immersion lenses (n = 1.56):

$\displaystyle D=\frac{0.61 \times 450}{1.56 \times 0.94} = 187\,\mbox{nm}$