Angular resolution
Encyclopedia
Angular resolution, or spatial resolution, describes the ability of any image-forming device such as an optical
or radio telescope
, a microscope
, a camera
, or an eye
, to distinguish small details of an object. It is thereby a major determinant of image 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, in general, the term "resolution" is 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.
The imaging system's resolution can be limited either by aberration or by diffraction
causing blurring
of the image. 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 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-shape 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 interplay between diffraction and aberration can be characterized by the point spread function
(PSF). The narrower the aperture of a lens the more likely the PSF is dominated by diffraction. In that case, the angular resolution of an optical system can be estimated (from the diameter
of the aperture and the wavelength
of the light) by the Rayleigh criterion invented by Lord Rayleigh:
Two point sources are regarded as just resolved when the principal diffraction maximum of one image coincides with the first minimum of the other.
If the distance is greater, the two points are well resolved and if it is smaller, they are regarded as not resolved. If one considers diffraction through a circular aperture, this translates into:
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. 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.,
), divided by π
.
The formal Rayleigh criterion is close to the empirical
resolution limit found earlier by the English astronomer W. R. Dawes who tested human observers on close binary stars of equal brightness. The result, θ = 4.56/D, with D in inches and θ in arcseconds is slightly narrower than calculated with the Rayleigh criterion: A calculation using Airy discs as point spread function shows that at Dawes' limit there is a 5% dip between the two maxima, whereas at Rayleigh's limit there is a 20% dip. Modern image processing
techniques including deconvolution
of the point spread function allow resolution of even narrower binaries.
The angular resolution may be converted into a spatial resolution, Δl, by multiplication of the angle (in radians) with the distance to the object. For a microscope, that distance is close to the focal length
f of the objective
. For this case, the Rayleigh criterion reads:
.
This is the size, in the imaging plane, 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. 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. 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.
A similar result holds for a small sensor imaging a subject at infinity: The angular resolution can be converted to a spatial resolution on the sensor by using f as the distance to the image sensor; this relates the spatial resolution of the image to the f-number
, #:
.
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
where
Resulting R is in radian
s. 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.
s: 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
where
The resulting R is in radian
s. 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 that is 120 m × 120 m.
:
where 
.
Here NA is the numerical aperture
,
is half the included angle 
of the lens, which depends on the diameter of the lens and its focal length, 
is the refractive index
of the medium between the lens and the specimen, and
is the wavelength of light illuminating or emanating from (in the case of fluorescence microscopy) the sample.
It follows that the NAs of both the objective and the condenser should be as high as possible for maximum resolution. In the case that both NAs are the same, the equation may be reduced to:
The practical limit for
is about 70°. In an air objective or condenser, this gives a maximum NA of 0.95. In a high-resolution oil immersion lens, the maximum NA is typically 1.45, when using immersion oil with a refractive index of 1.52. Due to these limitations, the resolution limit of a light microscope using visible light is about 200 nm. Given that the shortest wavelength of visible light is violet
(
≈ 400 nm),
which is near 200 nm.
Oil immersion objectives can have practical difficulties due to their shallow depth of field and extremely short working distance, which calls for the use of very thin (0.17mm) cover slips, or, in an inverted microscope, thin glass-bottomed Petri dish
es.
However, resolution below this theoretical limit can be achieved using a diffraction technique called 4Pi STED microscopy
. By destructively focusing two beams on a point source, objects as small as 30 nm have been resolved.
Optical telescope
An optical telescope is a telescope which is used to gather and focus light mainly from the visible part of the electromagnetic spectrum for directly viewing a magnified image for making a photograph, or collecting data through electronic image sensors....
or radio telescope
Radio telescope
A radio telescope is a form of directional radio antenna used in radio astronomy. The same types of antennas are also used in tracking and collecting data from satellites and space probes...
, a microscope
Microscope
A microscope is an instrument used to see objects that are too small for the naked eye. The science of investigating small objects using such an instrument is called microscopy...
, a camera
Camera
A camera is a device that records and stores images. These images may be still photographs or moving images such as videos or movies. The term camera comes from the camera obscura , an early mechanism for projecting images...
, or an eye
Human eye
The human eye is an organ which reacts to light for several purposes. As a conscious sense organ, the eye allows vision. Rod and cone cells in the retina allow conscious light perception and vision including color differentiation and the perception of depth...
, to distinguish small details of an object. It is thereby a major determinant of image resolution
Image resolution
Image resolution is an umbrella term that describes the detail an image holds. The term applies to raster digital images, film images, and other types of images. Higher resolution means more image detail....
.
Definition of terms
Resolving power is the ability of an imaging device to separate (i.e., to see as distinct) points of an object that are located at a small angular distance. The term resolutionOptical resolution
Optical resolution describes the ability of an imaging system to resolve detail in the object that is being imaged.An imaging system may have many individual components including a lens and recording and display components...
or minimum resolvable distance is the minimum distance between distinguishable objects
Object (image processing)
An object in image processing An object in image processing is an identifiable portion of an image that can be interpreted as a single unit or is an identifiable portion of an image that can be interpreted as a single unit....
in an image, although the term is loosely used by many users of microscopes and telescopes to describe resolving power. In scientific analysis, in general, the term "resolution" is used to describe the precision
Accuracy and precision
In the fields of science, engineering, industry and statistics, the accuracy of a measurement system is the degree of closeness of measurements of a quantity to that quantity's actual value. The precision of a measurement system, also called reproducibility or repeatability, is the degree to which...
with which any instrument measures and records (in an image or spectrum) any variable in the specimen or sample under study.
Explanation
Diffraction
Diffraction refers to various phenomena which occur when a wave encounters an obstacle. Italian scientist Francesco Maria Grimaldi coined the word "diffraction" and was the first to record accurate observations of the phenomenon in 1665...
causing blurring
Focus (optics)
In geometrical optics, a focus, also called an image point, is the point where light rays originating from a point on the object converge. Although the focus is conceptually a point, physically the focus has a spatial extent, called the blur circle. This non-ideal focusing may be caused by...
of the image. 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 lens
Lens (optics)
A lens is an optical device with perfect or approximate axial symmetry which transmits and refracts light, converging or diverging the beam. A simple lens consists of a single optical element...
' circular aperture
Aperture
In optics, an aperture is a hole or an opening through which light travels. More specifically, the aperture of an optical system is the opening that determines the cone angle of a bundle of rays that come to a focus in the image plane. The aperture determines how collimated the admitted rays are,...
is analogous to a two-dimensional version of the single-slit experiment. Light
Light
Light or visible light is electromagnetic radiation that is visible to the human eye, and is responsible for the sense of sight. Visible light has wavelength in a range from about 380 nanometres to about 740 nm, with a frequency range of about 405 THz to 790 THz...
passing through the lens interferes with itself creating a ring-shape diffraction pattern, known as the Airy pattern, if the wavefront
Wavefront
In physics, a wavefront is the locus of points having the same phase. Since infrared, optical, x-ray and gamma-ray frequencies are so high, the temporal component of electromagnetic waves is usually ignored at these wavelengths, and it is only the phase of the spatial oscillation that is described...
of the transmitted light is taken to be spherical or plane over the exit aperture.
The interplay between diffraction and aberration can be characterized by the point spread function
Point spread function
The point spread function describes the response of an imaging system to a point source or point object. A more general term for the PSF is a system's impulse response, the PSF being the impulse response of a focused optical system. The PSF in many contexts can be thought of as the extended blob...
(PSF). The narrower the aperture of a lens the more likely the PSF is dominated by diffraction. In that case, the angular resolution of an optical system can be estimated (from the diameter
Diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle. The diameters are the longest chords of the circle...
of the aperture and the wavelength
Wavelength
In physics, the wavelength of a sinusoidal wave is the spatial period of the wave—the distance over which the wave's shape repeats.It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a...
of the light) by the Rayleigh criterion invented by Lord Rayleigh:
Two point sources are regarded as just resolved when the principal diffraction maximum of one image coincides with the first minimum of the other.
If the distance is greater, the two points are well resolved and if it is smaller, they are regarded as not resolved. If one considers diffraction through a circular aperture, this translates into:
 |
where
|
The factor 1.220 is derived from a calculation of the position of the first dark ring surrounding the central Airy disc
Airy disc
In optics, the Airy disk and Airy pattern are descriptions of the best focused spot of light that a perfect lens with a circular aperture can make, limited by the diffraction of light....
of the diffraction
Diffraction
Diffraction refers to various phenomena which occur when a wave encounters an obstacle. Italian scientist Francesco Maria Grimaldi coined the word "diffraction" and was the first to record accurate observations of the phenomenon in 1665...
pattern. The calculation involves a Bessel function
Bessel function
In mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y of Bessel's differential equation:...
—1.220 is approximately the first zero of the Bessel function of the first kind, of order one (i.e.,
), divided by πPi
' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...
.
The formal Rayleigh criterion is close to the empirical
Empirical
The word empirical denotes information gained by means of observation or experimentation. Empirical data are data produced by an experiment or observation....
resolution limit found earlier by the English astronomer W. R. Dawes who tested human observers on close binary stars of equal brightness. The result, θ = 4.56/D, with D in inches and θ in arcseconds is slightly narrower than calculated with the Rayleigh criterion: A calculation using Airy discs as point spread function shows that at Dawes' limit there is a 5% dip between the two maxima, whereas at Rayleigh's limit there is a 20% dip. Modern image processing
Image processing
In electrical engineering and computer science, image processing is any form of signal processing for which the input is an image, such as a photograph or video frame; the output of image processing may be either an image or, a set of characteristics or parameters related to the image...
techniques including deconvolution
Deconvolution
In mathematics, deconvolution is an algorithm-based process used to reverse the effects of convolution on recorded data. The concept of deconvolution is widely used in the techniques of signal processing and image processing...
of the point spread function allow resolution of even narrower binaries.
The angular resolution may be converted into a spatial resolution, Δl, by multiplication of the angle (in radians) with the distance to the object. For a microscope, that distance is close to the focal length
Focal length
The focal length of an optical system is a measure of how strongly the system converges or diverges light. For an optical system in air, it is the distance over which initially collimated rays are brought to a focus...
f of the objective
Objective (optics)
In an optical instrument, the objective is the optical element that gathers light from the object being observed and focuses the light rays to produce a real image. Objectives can be single lenses or mirrors, or combinations of several optical elements. They are used in microscopes, telescopes,...
. For this case, the Rayleigh criterion reads:
.This is the size, in the imaging plane, of smallest object that the lens can resolve, and also the radius
Radius
In classical geometry, a radius of a circle or sphere is any line segment from its center to its perimeter. By extension, the radius of a circle or sphere is the length of any such segment, which is half the diameter. If the object does not have an obvious center, the term may refer to its...
of the smallest spot to which a collimated beam of light
Light
Light or visible light is electromagnetic radiation that is visible to the human eye, and is responsible for the sense of sight. Visible light has wavelength in a range from about 380 nanometres to about 740 nm, with a frequency range of about 405 THz to 790 THz...
can be focused. The size is proportional to wavelength, λ, and thus, for example, blue
Blue
Blue is a colour, the perception of which is evoked by light having a spectrum dominated by energy with a wavelength of roughly 440–490 nm. It is considered one of the additive primary colours. On the HSV Colour Wheel, the complement of blue is yellow; that is, a colour corresponding to an equal...
light can be focused to a smaller spot than red
Red
Red is any of a number of similar colors evoked by light consisting predominantly of the longest wavelengths of light discernible by the human eye, in the wavelength range of roughly 630–740 nm. Longer wavelengths than this are called infrared , and cannot be seen by the naked eye...
light. If the lens is focusing a beam of light
Light
Light or visible light is electromagnetic radiation that is visible to the human eye, and is responsible for the sense of sight. Visible light has wavelength in a range from about 380 nanometres to about 740 nm, with a frequency range of about 405 THz to 790 THz...
with a finite extent (e.g., a laser
Laser
A laser is a device that emits light through a process of optical amplification based on the stimulated emission of photons. The term "laser" originated as an acronym for Light Amplification by Stimulated Emission of Radiation...
beam), the value of D corresponds to the diameter
Diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle. The diameters are the longest chords of the circle...
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 focused to a smaller spot than a narrow one. This result is related to the Fourier properties of a lens.
A similar result holds for a small sensor imaging a subject at infinity: The angular resolution can be converted to a spatial resolution on the sensor by using f as the distance to the image sensor; this relates the spatial resolution of the image to the f-number
F-number
In optics, the f-number of an optical system expresses the diameter of the entrance pupil in terms of the focal length of the lens; in simpler terms, the f-number is the focal length divided by the "effective" aperture diameter...
, #:
.Single telescope
Point-like sources separated by an angleAngle
In geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...
smaller than the angular resolution cannot be resolved. A single optical telescope may have an angular resolution less than one arcsecond, but astronomical seeing
Astronomical seeing
Astronomical seeing refers to the blurring and twinkling of astronomical objects such as stars caused by turbulent mixing in the Earth's atmosphere varying the optical refractive index...
and other atmospheric effects make attaining this very hard.
The angular resolution R of a telescope can usually be approximated by
where
- λ is the wavelengthWavelengthIn physics, the wavelength of a sinusoidal wave is the spatial period of the wave—the distance over which the wave's shape repeats.It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a...
of the observed radiation - and D is the diameter of the telescope's objectiveObjective (optics)In an optical instrument, the objective is the optical element that gathers light from the object being observed and focuses the light rays to produce a real image. Objectives can be single lenses or mirrors, or combinations of several optical elements. They are used in microscopes, telescopes,...
.
Resulting R is in radian
Radian
Radian is the ratio between the length of an arc and its radius. The radian is the standard unit of angular measure, used in many areas of mathematics. The unit was formerly a SI supplementary unit, but this category was abolished in 1995 and the radian is now considered a SI derived unit...
s. 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 interferometerAstronomical interferometer
An astronomical interferometer is an array of telescopes or mirror segments acting together to probe structures with higher resolution by means of interferometry....
s: 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
Aperture synthesis
Aperture synthesis or synthesis imaging is a type of interferometry that mixes signals from a collection of telescopes to produce images having the same angular resolution as an instrument the size of the entire collection...
, 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
where
- λ is the wavelengthWavelengthIn physics, the wavelength of a sinusoidal wave is the spatial period of the wave—the distance over which the wave's shape repeats.It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a...
of the observed radiation - and B is the length of the maximum physical separation of the telescopes in the array, called the baselineBaselineA baseline is a line that is a base for measurement or for construction; see datum or point of reference .The word baseline may refer to:...
.
The resulting R is in radian
Radian
Radian is the ratio between the length of an arc and its radius. The radian is the standard unit of angular measure, used in many areas of mathematics. The unit was formerly a SI supplementary unit, but this category was abolished in 1995 and the radian is now considered a SI derived unit...
s. 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 that 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 apertureAngular aperture
The angular aperture of a lens is the apparent angle of the lens aperture as seen from the focal point:a = 2 \arctan \leftwheref is the focal lengthD is the diameter of the aperture.- Relation to numerical aperture :...
: where .Here NA is the numerical aperture
Numerical aperture
In optics, the numerical aperture of an optical system is a dimensionless number that characterizes the range of angles over which the system can accept or emit light. By incorporating index of refraction in its definition, NA has the property that it is constant for a beam as it goes from one...
,
is half the included angle of the lens, which depends on the diameter of the lens and its focal length, is the refractive indexRefractive index
In optics the refractive index or index of refraction of a substance or medium is a measure of the speed of light in that medium. It is expressed as a ratio of the speed of light in vacuum relative to that in the considered medium....
of the medium between the lens and the specimen, and
is the wavelength of light illuminating or emanating from (in the case of fluorescence microscopy) the sample.It follows that the NAs of both the objective and the condenser should be as high as possible for maximum resolution. In the case that both NAs are the same, the equation may be reduced to:
The practical limit for
is about 70°. In an air objective or condenser, this gives a maximum NA of 0.95. In a high-resolution oil immersion lens, the maximum NA is typically 1.45, when using immersion oil with a refractive index of 1.52. Due to these limitations, the resolution limit of a light microscope using visible light is about 200 nm. Given that the shortest wavelength of visible light is violetViolet (color)
As the name of a color, violet is synonymous with a bluish purple, when the word "purple" is used in the common English language sense of any color between blue and red, not including either blue or red...
(
≈ 400 nm),which is near 200 nm.
Oil immersion objectives can have practical difficulties due to their shallow depth of field and extremely short working distance, which calls for the use of very thin (0.17mm) cover slips, or, in an inverted microscope, thin glass-bottomed Petri dish
Petri dish
A Petri dish is a shallow glass or plastic cylindrical lidded dish that biologists use to culture cells or small moss plants. It was named after German bacteriologist Julius Richard Petri, who invented it when working as an assistant to Robert Koch...
es.
However, resolution below this theoretical limit can be achieved using a diffraction technique called 4Pi STED microscopy
4Pi STED microscopy
The 4Pi-STED-microscope is the result of combining the two unrelated concepts of STED- and 4Pi-microscopy. Here, the fluorescent sample is placed in the common focus of two opposing lenses, but excitation and detection are performed through a single lens...
. By destructively focusing two beams on a point source, objects as small as 30 nm have been resolved.
Common photography
The angular resolution is a major determinant of the quality of common images such as raster digital images, film images, and other types of images. As with telescopy and microscopy, higher angular resolution in general means more image detail. For digital images, the pixel count is another major measure of image quality, and is often given instead of the angular resolution, mainly because of easier measuring.See also
- Diffraction-limited system
- Angular diameterAngular diameterThe angular diameter or apparent size of an object as seen from a given position is the “visual diameter” of the object measured as an angle. In the vision sciences it is called the visual angle. The visual diameter is the diameter of the perspective projection of the object on a plane through its...
- Dawes limitDawes limitDawes' limit is a formula to express the maximum resolving power of a microscope or telescope. It is so named for its discoverer, W. R. Dawes, although it is also credited to Lord Rayleigh.The formula takes different forms depending on the units....
- Sparrow's resolution limitSparrow's resolution limitSparrow's Resolution Limit is an estimate of the angular resolution limit of an optical telescope or microscope.When a star is observed with a telescope, the light is spread into an Airy disk...
- Visual acuityVisual acuityVisual acuity is acuteness or clearness of vision, which is dependent on the sharpness of the retinal focus within the eye and the sensitivity of the interpretative faculty of the brain....
External links
- "Concepts and Formulas in Microscopy: Resolution" by Michael W. Davidson, Nikon MicroscopyU (website).
- Melles Griot Technical Guide: Fundamental Optics.
- Melles Griot Technical Guide: Gaussian Beam Optics.