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optics Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of optical instruments, instruments that use or Photodetector, detect it. Optics usually describes t ...
, any optical instrument or systema
microscope A microscope () is a laboratory equipment, laboratory instrument used to examine objects that are too small to be seen by the naked eye. Microscopy is the science of investigating small objects and structures using a microscope. Microscopic ...
,
telescope A telescope is a device used to observe distant objects by their emission, Absorption (electromagnetic radiation), absorption, or Reflection (physics), reflection of electromagnetic radiation. Originally, it was an optical instrument using len ...
, or
camera A camera is an instrument used to capture and store images and videos, either digitally via an electronic image sensor, or chemically via a light-sensitive material such as photographic film. As a pivotal technology in the fields of photograp ...
has a principal limit to its resolution due to the
physics Physics is the scientific study of matter, its Elementary particle, 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 whi ...
of
diffraction Diffraction is the deviation of waves from straight-line propagation without any change in their energy due to an obstacle or through an aperture. The diffracting object or aperture effectively becomes a secondary source of the Wave propagation ...
. An optical instrument is said to be diffraction-limited if it has reached this limit of resolution performance. Other factors may affect an optical system's performance, such as lens imperfections or aberrations, but these are caused by errors in the manufacture or
calculation A calculation is a deliberate mathematical process that transforms a plurality of inputs into a singular or plurality of outputs, known also as a result or results. The term is used in a variety of senses, from the very definite arithmetical ...
of a lens, whereas the diffraction limit is the maximum resolution possible for a theoretically perfect, or ideal, optical system. The diffraction-limited
angular resolution Angular resolution describes the ability of any image-forming device such as an Optical telescope, optical or radio telescope, a microscope, a camera, or an Human eye, eye, to distinguish small details of an object, thereby making it a major det ...
, in radians, of an instrument is proportional to the
wavelength In physics and mathematics, wavelength or spatial period of a wave or periodic function is the distance over which the wave's shape repeats. In other words, it is the distance between consecutive corresponding points of the same ''phase (waves ...
of the light being observed, and inversely proportional to the diameter of its objective'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 In optics, the Airy disk (or Airy disc) and Airy pattern are descriptions of the best-focus (optics), focused Point source#Light, spot of light that a perfect lens (optics), lens with a circular aperture can make, limited by the diffraction of ...
. As one decreases the size of the aperture of a telescopic
lens A lens is a transmissive optical device that 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, has the property ...
of either the objective or the object illumination source, whichever is smaller. In
astronomy Astronomy is a natural science that studies celestial objects and the phenomena that occur in the cosmos. It uses mathematics, physics, and chemistry in order to explain their origin and their overall evolution. Objects of interest includ ...
, 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 gases that envelop an astronomical object, held in place by the gravity of the object. A planet retains an atmosphere when the gravity is great and the temperature of the atmosphere is low. A stellar atmosphere ...
effects. Optical telescopes on the
Earth Earth is the third planet from the Sun and the only astronomical object known to Planetary habitability, harbor life. This is enabled by Earth being an ocean world, the only one in the Solar System sustaining liquid surface water. Almost all ...
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 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 technique of precisely deforming a mirror in order to compensate for light distortion. It is used in Astronomy, astronomical telescopes and laser communication systems to remove the effects of Astronomical seeing, atmo ...
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 (radio), 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 r ...
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, 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 Lens (optics), lenses and mirrors, that causes the ''image'' created by the optical system to not be a faithful reproduction of the ''object'' being observed. Aberrations cause the i ...
. 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'' originated as an acronym for light amplification by stimulated emission of radi ...
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 (23 January 1840 – 14 January 1905) was a German businessman, optical engineer, physicist, and social reformer. Together with Otto Schott and Carl Zeiss, he developed numerous optical instruments. He was also a co-owner of Ca ...
first mentioned the diffraction limit in his 1873 paper, page 466: „ ��die physikalische Unterscheidungsgrenze ��hängt allein vom Oeffnungswinkel ab und ist dem Sinus seines halben Betrages proportional“, or " ��the physical limit of resolution ��depends solely on the aperture angle and is proportional to the sine of half its magnitude". Abbe wrote it in form of a formula in his 1882 paper, page 461: "The smallest dimensions which are within the reach of a given aperture are indicated with sufficient accuracy by taking the limit of the resolving or separating power of that aperture for periodic or regular structures, i.e. the minimum distance apart at which given elements can be delineated separately with the aperture in question. The numerical expression of that minimum distance is" :d=\frac = \frac, where \lambda is the wavelength, n is the refractive index of the medium, and \theta is the semi-angle of the light focused by the optical system. The same formula had been proven by Hermann von Helmholtz in 1874. 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, has the property ...
(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 operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...
of the
point spread function The point spread function (PSF) describes the response of a focused optical imaging system to a point source or point object. A more general term for the PSF is the system's impulse response; the PSF is the impulse response or impulse response ...
s (PSF). The point spread function of a diffraction limited circular-aperture lens is simply the
Airy disk In optics, the Airy disk (or Airy disc) and Airy pattern are descriptions of the best-focus (optics), focused Point source#Light, spot of light that a perfect lens (optics), lens with a circular aperture can make, limited by the diffraction of ...
. 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 In optics, the Airy disk (or Airy disc) and Airy pattern are descriptions of the best-focus (optics), focused Point source#Light, spot of light that a perfect lens (optics), lens with a circular aperture can make, limited by the diffraction of ...
, : d/2 = 1.22 \lambda N,\, where \lambda is the wavelength of the light and N is the
f-number An f-number is a measure of the light-gathering ability of an optical system such as a camera lens. It is calculated by dividing the system's focal length by the diameter of the entrance pupil ("clear aperture").Smith, Warren ''Modern Optical ...
of the imaging optics, i.e., 2 NA \rightarrow (2.44N)^ in the Abbe diffraction limit formula. For instance, for an f/8 lens (N=8 and NA\approx2.5% ) and for green light (\lambda_g= 0.5 Î¼m wavelength) light, the focusing spot diameter will be d = 9.76 Î¼m or 19.5\lambda_g. 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. Given the same field of view, pixel count,
shutter speed In photography, shutter speed or exposure time is the length of time that the film or digital sensor inside the camera is exposed to light (that is, when the camera's shutter (photography), shutter is open) when taking a photograph. The am ...
and
shot noise Shot noise or Poisson noise is a type of noise which can be modeled by a Poisson process. In electronics shot noise originates from the discrete nature of electric charge. Shot noise also occurs in photon counting in optical devices, where s ...
SNR (i.e. the same amount of light collected per pixel), a small sensor and a large sensor of equivalent quality will produce the same digital image, with the same amount of blur due to both diffraction and
depth of field The depth of field (DOF) is the distance between the nearest and the farthest objects that are in acceptably sharp focus (optics), focus in an image captured with a camera. See also the closely related depth of focus. Factors affecting depth ...
. The larger sensors do have an advantage in that the lenses for them tend to have larger maximum entrance pupils, which allows, to a larger extent, trading depth of field for less diffraction blur and either more SNR or higher shutter speeds.


Obtaining higher resolution

Using special optical systems and
digital image processing Digital image processing is the use of a digital computer to process digital images through an algorithm. As a subcategory or field of digital signal processing, digital image processing has many advantages over analog image processing. It allo ...
, it is possible to produce images that have higher resolution (in some specific aspects of the subject, such as color or shape) than would be allowed by simple use of diffraction-limited optics. These largely computational methods offer advantages over other workarounds such as electron microscopy, and have been used to produce reasonably accurate images of individual molecules. Although these techniques improve some aspect of resolution, they generally come at an enormous increase in cost and complexity compared to using a simple light microscope. 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, 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) 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 (oscillat ...
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 microscopes and nano-FTIR, 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 diffr ...
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 a type of material engineered to have a property, typically rarely observed in naturally occu ...
-based superlenses can image with a resolution better than the diffraction limit by locating the
objective lens In optical engineering, an objective is an optical element that gathers light from an object being observed and focuses the light rays from it to produce a real image of the object. Objectives can be a single lens or mirror, or combinations of ...
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 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 (or a non-linear 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, mathe ...
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 one of two kinds of photoluminescence, the emission of light by a substance that has absorbed light or other electromagnetic radiation. When exposed to ultraviolet radiation, many substances will glow (fluoresce) with color ...
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 an idealized beam of electromagnetic radiation whose amplitude envelope in the transverse plane is given by a Gaussian function; this also implies a Gaussian intensity (irradiance) profile. This fundamental (or ...
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. 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 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


References


External links

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