Angular resolution

From Wikinfo

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

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. The diffraction limit is given by the Rayleigh criterion:

<math> \sin \theta = 1.22 \frac{\lambda}{D}</math>

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.

For a ideal lens of focal length f, this results in a minimum spatial resolution, Δl:

<math> \Delta l = 1.22 \frac{ f \lambda}{D}</math>.

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 can not be resolved. A single optical telescope has an angular resolution less than one arcsecond, but seeing and other atmospheric effects make attaining this very hard. The highest angular resolutions can be achieved by interferometry.

The angular resolution of a telecope 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.

Microscope case

The resolution D depends on the angular aperture α:

<math>D=\frac{(0.61 \lambda )}{(N \times \sin \alpha )}</math>.

Here α depends on the width of objective lens and its distance from the specimen; and N is a measure of the number of degrees to which a medium bends a light ray which passes through it. The shorter λ, the lower the value of D, the higher the resolution.

Due to the limitations of the values α, λ, and N, the limit of a light microscope using visible light is 200nm. This is because: α for the best lens is 70� (sinα=0.94), shortest wavelength of visible light is blue (λ=450nm) and

<math>D=\frac{0.61 \times 450}{1.5 \times 0.94} = 194\,\mbox{nm}</math>

To improve the resolving power of microscopes oil immersion lenses can be used; these use a layer of high-density transparent oil to reduce the effective index of refraction of the lens.

References

• Adapted from the Wikipedia article, "Angular_resolution" http://en.wikipedia.org/wiki/Angular_resolution, used under the GNU Free Documentation License