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.

Related Topics:
Microscope - Telescope - Microscopy

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

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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:

Related Topics:
Lens - Diffraction - Single-slit experiment - Rayleigh

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: sin heta = 1.22 rac{lambda}{D}

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

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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 the first positive zero of the Bessel function of the first kind, of order zero. 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.

Related Topics:
Airy disc - Bessel function - Eye - Image processing

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

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: Delta l = 1.22 rac{ f lambda}{D}.

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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.

Related Topics:
Collimated - Blue - Red - Laser

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