Rayleigh criterion: Wikis

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Angular resolution or 'spatial resolution' describes the resolving power of any image-forming device such as an optical or radio telescope, a microscope, a camera, or an eye.

Definition of terms

Resolving power is the ability of the components of an imaging device to measure the angular separation of the points in an object. The term resolution or minimum resolvable distance is the minimum distance between distinguishable objects in an image, although the term is loosely used by many users of microscopes and telescopes to describe resolving power. In scientific analysis the term "resolution" is generally used to describe the precision with which any instrument measures and records (in an image or spectrum) any variable in the specimen or sample under study.

Explanation

Airy diffraction pattern generated by a plane wave falling on a circular aperture, such as

The imaging system's resolution can be limited either by aberration or by diffraction. These two phenomena have different origins and are unrelated. Aberrations can be explained by geometrical optics and can in principle be solved by increasing the optical quality—and cost—of the system. On the other hand, diffraction comes from the wave nature of light and is determined by the finite aperture of the optical elements.

The interplay between diffraction and aberration can be characterized by the point spread function (PSF). The PSF of a lens is ultimately limited by diffraction. The lens' circular aperture is analogous to a two-dimensional version of the single-slit experiment. Light passing through the lens interferes with itself creating a ring-shaped diffraction pattern, known as the Airy pattern, if the wavefront of the transmitted light is taken to be spherical or plane over the exit aperture. The result is a blurring of the image. An empirical diffraction limit is given by the Rayleigh criterion invented by Lord Rayleigh:

The images of two different points are regarded as just resolved when the principal diffraction maximum of one image coincides with the first minimum of the other.[1] If the distance is greater, the two points are well resolved and if it is smaller, they are not resolved. Mathematically, this translates into:

 $\sin \theta = 1.220 \frac{\lambda}{D}$ where θ is the angular resolution in radians, λ is the wavelength of light, and D is the diameter of the lens' aperture.

The factor 1.220 is derived from a calculation of the position of the first dark ring surrounding the central Airy disc of the diffraction pattern. If one considers diffraction through a circular aperture, then the calculation involves a Bessel function -- 1.220 is approximately the first zero of the Bessel function of the first kind, of order one (i.e. J1), divided by π. 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: the minimum of one point source is located at the maximum of the other. 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:

$\Delta l = 1.220 \frac{ f \lambda}{D}$.

This is the size of smallest object that the lens can resolve, and also the radius of the smallest spot to which a collimated beam of light can be focused.[2] The size is proportional to wavelength, λ, and thus, for example, blue light can be focused to a smaller spot than red light. If the lens is focusing 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.Note Since the spatial resolution is inversely proportional to D, this leads to the slightly surprising result that a wide beam of light may be focused to a smaller spot than a narrow one. This result is related to the Fourier properties of a lens.

Specific cases

Single telescope

Point-like sources separated by an angle smaller than the angular resolution cannot be resolved. A single optical telescope may have an angular resolution less than one arcsecond, but astronomical seeing and other atmospheric effects make attaining this very hard.

The angular resolution R of a telescope can usually be approximated by

$R = \frac {\lambda}{D}$

where

λ is the wavelength of the observed radiation
and D is the diameter of the telescope's objective.

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 0.1 arc second, we need D = 1.2 m.

This formula, for light with a wavelength of ca 562 nm, is also called the Dawes' limit.

Telescope array

The highest angular resolutions can be achieved by arrays of telescopes called astronomical interferometers: these instruments can achieve angular resolutions of 0.001 arcsecond at optical wavelengths, and much higher resolutions at radio wavelengths. In order to perform aperture synthesis imaging, a large number of telescopes are required laid out in a 2-dimensional arrangement.

The angular resolution R of an interferometer array can usually be approximated by

$R = \frac {\lambda}{B}$

where

λ is the wavelength of the observed radiation
and B is the length of the maximum physical separation of the telescopes in the array, called the baseline.

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 order to form an image in yellow light with a wavelength of 580 nm, for a resolution of 1 milli-arcsecond, we need telescopes laid out in an array which is 120 m × 120 m.

Microscope

The resolution R (here measured as a distance, not to be confused with the angular resolution of a previous subsection) depends on the angular aperture α:

$R=\frac{1.22\lambda}{2\times N.A.}=\frac{1.22\lambda}{2n\sin\theta}$.[3]

Here θ is the collecting angle of the lens, which depends on the width of objective lens and its focal 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. The quantity n × sin θ is also known as the numerical aperture or N.A.

Due to the limitations of the values θ, 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° (sin α = 0.94), the shortest wavelength of visible light is blue (λ = 450 nm), and the typical high resolution lenses are oil immersion lenses (n = 1.56):

$R=\frac{1.22 \times 450\,\mbox{nm}}{2 \times 1.56 \times 0.94} = 187\,\mbox{nm}$

However, resolution below this theoretical limit can achieved using a diffraction technique called 4Pi_STED_microscopy. By destructively focusing two beams on a point source, objects as small as 30nm have been resolved.[4]

Notes

1.^  In the case of laser beams, a Gaussian Optics analysis is more appropriate than the Raleigh criterion, and may reveal a smaller diffraction-limited spot size than that indicated by the formula above.

References

1. ^ Born, Max; Wolf, Emil (October 1999). Principle of Optics. Cambridge: Cambridge University Press. pp. 461. ISBN 0521642221.
2. ^ "Diffraction: Fraunhofer Diffraction at a Circular Aperture". Melles Griot Optics Guide. Melles Griot. 2002. Retrieved 2009-02-16.
3. ^ Nikon MicroscopyU: Concepts and Formulas in Microscopy: Resolution
4. ^ http://www.mpibpc.mpg.de/groups/hell/4Pi-STED.htm