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The resolution of an optical imaging system a microscope,
telescope A telescope is a device used to observe distant objects by their emission, absorption, or reflection of electromagnetic radiation. Originally meaning only an optical instrument using lenses, curved mirrors, or a combination of both to observ ...
, or camera can be limited by
factors Factor, a Latin word meaning "who/which acts", may refer to: Commerce * Factor (agent), a person who acts for, notably a mercantile and colonial agent * Factor (Scotland), a person or firm managing a Scottish estate * Factors of production, su ...
such as imperfections in the lenses or misalignment. However, there is a principal limit to the resolution of any optical system, due to the
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which relat ...
of
diffraction Diffraction is defined as the interference or bending of waves around the corners of an obstacle or through an aperture into the region of geometrical shadow of the obstacle/aperture. The diffracting object or aperture effectively becomes a s ...
. An optical system with resolution performance at the instrument's theoretical limit is said to be diffraction-limited. The diffraction-limited
angular resolution Angular 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, thereby making it a major determinant of image resolution ...
of a telescopic instrument is inversely proportional to the
wavelength In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tro ...
of the light being observed, and proportional to the diameter of its
objective Objective may refer to: * Objective (optics), an element in a camera or microscope * ''The Objective'', a 2008 science fiction horror film * Objective pronoun, a personal pronoun that is used as a grammatical object * Objective Productions, a Brit ...
's entrance aperture. For telescopes with circular apertures, the size of the smallest feature in an image that is diffraction limited is the size of the Airy disk. As one decreases the size of the aperture of a telescopic
lens A lens is a transmissive optical device which focuses or disperses a light beam by means of refraction. A simple lens consists of a single piece of transparent material, while a compound lens consists of several simple lenses (''elements''), ...
, diffraction proportionately increases. At small apertures, such as f/22, most modern lenses are limited only by diffraction and not by aberrations or other imperfections in the construction. For microscopic instruments, the diffraction-limited
spatial resolution In physics and geosciences, the term spatial resolution refers to distance between independent measurements, or the physical dimension that represents a pixel of the image. While in some instruments, like cameras and telescopes, spatial resoluti ...
is proportional to the light wavelength, and to the
numerical aperture In optics, the numerical aperture (NA) 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 propert ...
of either the objective or the object illumination source, whichever is smaller. In
astronomy Astronomy () is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution. Objects of interest include planets, moons, stars, nebulae, galax ...
, a diffraction-limited observation is one that achieves the resolution of a theoretically ideal objective in the size of instrument used. However, most observations from Earth are seeing-limited due to
atmospheric An atmosphere () is a layer of gas or layers of gases that envelop a planet, and is held in place by the gravity of the planetary body. A planet retains an atmosphere when the gravity is great and the temperature of the atmosphere is low. A s ...
effects. Optical telescopes on the
Earth Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's surface ...
work at a much lower resolution than the diffraction limit because of the distortion introduced by the passage of light through several kilometres of
turbulent In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between ...
atmosphere. Advanced observatories have started using
adaptive optics Adaptive optics (AO) is a technology used to improve the performance of optical systems by reducing the effect of incoming wavefront distortions by deforming a mirror in order to compensate for the distortion. It is used in astronomical tele ...
technology, resulting in greater image resolution for faint targets, but it is still difficult to reach the diffraction limit using adaptive optics.
Radio telescope A radio telescope is a specialized antenna and radio receiver used to detect radio waves from astronomical radio sources in the sky. Radio telescopes are the main observing instrument used in radio astronomy, which studies the radio frequency ...
s are frequently diffraction-limited, because the wavelengths they use (from millimeters to meters) are so long that the atmospheric distortion is negligible. Space-based telescopes (such as
Hubble The Hubble Space Telescope (often referred to as HST or Hubble) is a space telescope that was launched into low Earth orbit in 1990 and remains in operation. It was not the first space telescope, but it is one of the largest and most ver ...
, or a number of non-optical telescopes) always work at their diffraction limit, if their design is free of
optical aberration In optics, aberration is a property of optical systems, such as lenses, that causes light to be spread out over some region of space rather than focused to a point. Aberrations cause the image formed by a lens to be blurred or distorted, with th ...
. The beam from a
laser A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. The word "laser" is an acronym for "light amplification by stimulated emission of radiation". The firs ...
with near-ideal beam propagation properties may be described as being diffraction-limited. A diffraction-limited laser beam, passed through diffraction-limited optics, will remain diffraction-limited, and will have a spatial or angular extent essentially equal to the resolution of the optics at the wavelength of the laser.


Calculation of diffraction limit


The Abbe diffraction limit for a microscope

The observation of sub-wavelength structures with microscopes is difficult because of the Abbe diffraction limit.
Ernst Abbe Ernst Karl Abbe HonFRMS (23 January 1840 – 14 January 1905) was a German physicist, optical scientist, entrepreneur, and social reformer. Together with Otto Schott and Carl Zeiss, he developed numerous optical instruments. He was also a co- ...
found in 1873 that light with wavelength \lambda, traveling in a medium with refractive index n and converging to a spot with half-angle \theta will have a minimum resolvable distance of :d=\frac = \frac The portion of the denominator n\sin \theta is called the
numerical aperture In optics, the numerical aperture (NA) 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 propert ...
(NA) and can reach about 1.4–1.6 in modern optics, hence the Abbe limit is d=\frac. Considering green light around 500 nm and a NA of 1, the Abbe limit is roughly d=\frac=250 \text (0.25 μm), which is small compared to most biological cells (1 μm to 100 μm), but large compared to viruses (100 nm), proteins (10 nm) and less complex molecules (1 nm). To increase the resolution, shorter wavelengths can be used such as UV and X-ray microscopes. These techniques offer better resolution but are expensive, suffer from lack of contrast in biological samples and may damage the sample.


Digital photography

In a digital camera, diffraction effects interact with the effects of the regular pixel grid. The combined effect of the different parts of an optical system is determined by the
convolution In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' ...
of the point spread functions (PSF). The point spread function of a diffraction limited lens is simply the Airy disk. The point spread function of the camera, otherwise called the instrument response function (IRF) can be approximated by a rectangle function, with a width equivalent to the pixel pitch. A more complete derivation of the modulation transfer function (derived from the PSF) of image sensors is given by Fliegel. Whatever the exact instrument response function, it is largely independent of the f-number of the lens. Thus at different f-numbers a camera may operate in three different regimes, as follows: # In the case where the spread of the IRF is small with respect to the spread of the diffraction PSF, in which case the system may be said to be essentially diffraction limited (so long as the lens itself is diffraction limited). # In the case where the spread of the diffraction PSF is small with respect to the IRF, in which case the system is instrument limited. # In the case where the spread of the PSF and IRF are similar, in which case both impact the available resolution of the system. The spread of the diffraction-limited PSF is approximated by the diameter of the first null of the Airy disk, : d/2 = 1.22 \lambda N,\, where λ is the wavelength of the light and ''N'' is the
f-number In optics, the f-number of an optical system such as a camera lens is the ratio of the system's focal length to the diameter of the entrance pupil ("clear aperture").Smith, Warren ''Modern Optical Engineering'', 4th Ed., 2007 McGraw-Hill Pro ...
of the imaging optics. For f/8 and green (0.5 μm wavelength) light, d = 9.76 μm. This is similar to the pixel size for the majority of commercially available 'full frame' (43mm sensor diagonal) cameras and so these will operate in regime 3 for f-numbers around 8 (few lenses are close to diffraction limited at f-numbers smaller than 8). Cameras with smaller sensors will tend to have smaller pixels, but their lenses will be designed for use at smaller f-numbers and it is likely that they will also operate in regime 3 for those f-numbers for which their lenses are diffraction limited.


Obtaining higher resolution

There are techniques for producing images that appear to have higher resolution than allowed by simple use of diffraction-limited optics. Although these techniques improve some aspect of resolution, they generally come at an enormous increase in cost and complexity. Usually the technique is only appropriate for a small subset of imaging problems, with several general approaches outlined below.


Extending numerical aperture

The effective resolution of a microscope can be improved by illuminating from the side. In conventional microscopes such as bright-field or
differential interference contrast Differential interference contrast (DIC) microscopy, also known as Nomarski interference contrast (NIC) or Nomarski microscopy, is an optical microscopy technique used to enhance the contrast in unstained, transparent samples. DIC works on the p ...
, this is achieved by using a condenser. Under spatially incoherent conditions, the image is understood as a composite of images illuminated from each point on the condenser, each of which covers a different portion of the object's spatial frequencies. This effectively improves the resolution by, at most, a factor of two. Simultaneously illuminating from all angles (fully open condenser) drives down interferometric contrast. In conventional microscopes, the maximum resolution (fully open condenser, at N = 1) is rarely used. Further, under partially coherent conditions, the recorded image is often non-linear with object's scattering potential—especially when looking at non-self-luminous (non-fluorescent) objects. To boost contrast, and sometimes to linearize the system, unconventional microscopes (with
structured illumination A structured light pattern designed for surface inspection An Automatix Seamtracker arc welding robot equipped with a camera and structured laser light source, enabling the robot to follow a welding seam automatically Structured light is the p ...
) synthesize the condenser illumination by acquiring a sequence of images with known illumination parameters. Typically, these images are composited to form a single image with data covering a larger portion of the object's spatial frequencies when compared to using a fully closed condenser (which is also rarely used). Another technique, 4Pi microscopy, uses two opposing objectives to double the effective numerical aperture, effectively halving the diffraction limit, by collecting the forward and backward scattered light. When imaging a transparent sample, with a combination of incoherent or structured illumination, as well as collecting both forward, and backward scattered light it is possible to image the complete scattering sphere. Unlike methods relying on localization, such systems are still limited by the diffraction limit of the illumination (condenser) and collection optics (objective), although in practice they can provide substantial resolution improvements compared to conventional methods.


Near-field techniques

The diffraction limit is only valid in the far field as it assumes that no
evanescent field In electromagnetics, an evanescent field, or evanescent wave, is an oscillating electric and/or magnetic field that does not propagate as an electromagnetic wave but whose energy is spatially concentrated in the vicinity of the source (oscillati ...
s reach the detector. Various near-field techniques that operate less than ≈1 wavelength of light away from the image plane can obtain substantially higher resolution. These techniques exploit the fact that the evanescent field contains information beyond the diffraction limit which can be used to construct very high resolution images, in principle beating the diffraction limit by a factor proportional to how well a specific imaging system can detect the near-field signal. For scattered light imaging, instruments such as
near-field scanning optical microscope Near-field scanning optical microscopy (NSOM) or scanning near-field optical microscopy (SNOM) is a microscopy technique for nanostructure investigation that breaks the far field resolution limit by exploiting the properties of evanescent waves ...
s and
nano-FTIR Nano-FTIR (nanoscale Fourier transform infrared spectroscopy) is a scanning probe technique that utilizes as a combination of two techniques: Fourier transform infrared spectroscopy (FTIR) and scattering-type scanning near-field optical microsco ...
, which are built atop
atomic force microscope Atomic force microscopy (AFM) or scanning force microscopy (SFM) is a very-high-resolution type of scanning probe microscopy (SPM), with demonstrated resolution on the order of fractions of a nanometer, more than 1000 times better than the opti ...
systems, can be used to achieve up to 10-50 nm resolution. The data recorded by such instruments often requires substantial processing, essentially solving an optical inverse problem for each image.
Metamaterial A metamaterial (from the Greek word μετά ''meta'', meaning "beyond" or "after", and the Latin word ''materia'', meaning "matter" or "material") is any material engineered to have a property that is not found in naturally occurring materials ...
-based
superlens A superlens, or super lens, is a lens which uses metamaterials to go beyond the diffraction limit. For example, in 1995, Guerra combined a transparent grating having 50nm lines and spaces (the "metamaterial") with a conventional microscope immersio ...
es can image with a resolution better than the diffraction limit by locating the
objective lens In optical engineering, 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 a single lens or mirror, or combinations of several optical elem ...
extremely close (typically hundreds of nanometers) to the object. In fluorescence microscopy the excitation and emission are typically on different wavelengths. In
total internal reflection fluorescence microscopy A total internal reflection fluorescence microscope (TIRFM) is a type of microscope with which a thin region of a specimen, usually less than 200 nanometers can be observed. TIRFM is an imaging modality which uses the excitation of fluorescent cel ...
a thin portion of the sample located immediately on the cover glass is excited with an evanescent field, and recorded with a conventional diffraction-limited objective, improving the axial resolution. However, because these techniques cannot image beyond 1 wavelength, they cannot be used to image into objects thicker than 1 wavelength which limits their applicability.


Far-field techniques

Far-field imaging techniques are most desirable for imaging objects that are large compared to the illumination wavelength but that contain fine structure. This includes nearly all biological applications in which cells span multiple wavelengths but contain structure down to molecular scales. In recent years several techniques have shown that sub-diffraction limited imaging is possible over macroscopic distances. These techniques usually exploit optical
nonlinearity In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...
in a material's reflected light to generate resolution beyond the diffraction limit. Among these techniques, the STED microscope has been one of the most successful. In STED, multiple laser beams are used to first excite, and then quench
fluorescent Fluorescence is the emission of light by a substance that has absorbed light or other electromagnetic radiation. It is a form of luminescence. In most cases, the emitted light has a longer wavelength, and therefore a lower photon energy, tha ...
dyes. The nonlinear response to illumination caused by the quenching process in which adding more light causes the image to become less bright generates sub-diffraction limited information about the location of dye molecules, allowing resolution far beyond the diffraction limit provided high illumination intensities are used.


Laser beams

The limits on focusing or collimating a laser beam are very similar to the limits on imaging with a microscope or telescope. The only difference is that laser beams are typically soft-edged beams. This non-uniformity in light distribution leads to a coefficient slightly different from the 1.22 value familiar in imaging. However, the scaling with wavelength and aperture is exactly the same. The beam quality of a laser beam is characterized by how well its propagation matches an ideal
Gaussian beam In optics, a Gaussian beam is a beam of electromagnetic radiation with high monochromaticity whose amplitude envelope in the transverse plane is given by a Gaussian function; this also implies a Gaussian intensity (irradiance) profile. Th ...
at the same wavelength. The beam quality factor
M squared In laser science, the parameter M2, also known as the beam propagation ratio or beam quality factor is a measure of laser beam quality. It represents the degree of variation of a beam from an ideal Gaussian beam. It is calculated from the ratio of ...
(M2) is found by measuring the size of the beam at its waist, and its divergence far from the waist, and taking the product of the two, known as the
beam parameter product In laser science, the beam parameter product (BPP) is the product of a laser beam's divergence angle (half-angle) and the radius of the beam at its narrowest point (the beam waist). The BPP quantifies the quality of a laser beam, and how well it ca ...
. The ratio of this measured beam parameter product to that of the ideal is defined as M2, so that M2=1 describes an ideal beam. The M2 value of a beam is conserved when it is transformed by diffraction-limited optics. The outputs of many low and moderately powered lasers have M2 values of 1.2 or less, and are essentially diffraction-limited.


Other waves

The same equations apply to other wave-based sensors, such as radar and the human ear. As opposed to light waves (i.e., photons), massive particles have a different relationship between their quantum mechanical wavelength and their energy. This relationship indicates that the effective "de Broglie" wavelength is inversely proportional to the momentum of the particle. For example, an electron at an energy of 10 keV has a wavelength of 0.01 nm, allowing the electron microscope (
SEM SEM can refer to: Computing * Search engine marketing, promoting websites by increasing their visibility in search engine results * Security event manager, a security log tool used on data networks Economics and management * Stock Exchang ...
or TEM) to achieve high resolution images. Other massive particles such as helium, neon, and gallium ions have been used to produce images at resolutions beyond what can be attained with visible light. Such instruments provide nanometer scale imaging, analysis and fabrication capabilities at the expense of system complexity.


See also

*
Rayleigh criterion Angular 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, thereby making it a major determinant of image resolution ...


References


External links

* Describes the Leica APO-Telyt-R 280mm f/4, a diffraction-limited photographic lens. {{Optical microscopy Diffraction Telescopes Microscopes