Diffraction

Diffraction refers to various phenomena that occur when a wave encounters an obstacle or a slit. It is defined as the bending of waves around the corners of an obstacle or aperture into the region of geometrical shadow of the obstacle. In classical physics, the diffraction phenomenon is described as the interference of waves according to the Huygens–Fresnel principle that treats each point in the wave-front as a collection of individual spherical wavelets.[1] These characteristic behaviors are exhibited when a wave encounters an obstacle or a slit that is comparable in size to its wavelength. Similar effects occur when a light wave travels through a medium with a varying refractive index, or when a sound wave travels through a medium with varying acoustic impedance. Diffraction has an impact on the acoustic space. Diffraction occurs with all waves, including sound waves, water waves, and electromagnetic waves such as visible light, X-rays and radio waves.

Since physical objects have wave-like properties (significantly at the atomic level, invisibly at macro level), diffraction also occurs with matter and can be studied according to the principles of quantum mechanics. Italian scientist Francesco Maria Grimaldi coined the word "diffraction" and was the first to record accurate observations of the phenomenon in 1660.[2][3]

While diffraction occurs whenever propagating waves encounter such changes, its effects are generally most pronounced for waves whose wavelength is roughly comparable to the dimensions of the diffracting object or slit. If the obstructing object provides multiple, closely spaced openings, a complex pattern of varying intensity can result. This is due to the addition, or interference, of different parts of a wave that travel to the observer by different paths, where different path lengths result in different phases (see diffraction grating and wave superposition). The formalism of diffraction can also describe the way in which waves of finite extent propagate in free space. For example, the expanding profile of a laser beam, the beam shape of a radar antenna and the field of view of an ultrasonic transducer can all be analyzed using diffraction equations.

Laser Interference
Diffraction pattern of red laser beam made on a plate after passing through a small circular aperture in another plate

Examples

Wave diffraction at the Blue Lagoon, Abereiddy
Circular waves generated by diffraction from the narrow entrance of a flooded coastal quarry
Solar glory at the steam from hot spring
Solar glory at the steam from hot springs. A glory is an optical phenomenon produced by light backscattered (a combination of diffraction, reflection and refraction) towards its source by a cloud of uniformly sized water droplets.

The effects of diffraction are often seen in everyday life. The most striking examples of diffraction are those that involve light; for example, the closely spaced tracks on a CD or DVD act as a diffraction grating to form the familiar rainbow pattern seen when looking at a disc. This principle can be extended to engineer a grating with a structure such that it will produce any diffraction pattern desired; the hologram on a credit card is an example. Diffraction in the atmosphere by small particles can cause a bright ring to be visible around a bright light source like the sun or the moon. A shadow of a solid object, using light from a compact source, shows small fringes near its edges. The speckle pattern which is observed when laser light falls on an optically rough surface is also a diffraction phenomenon. When deli meat appears to be iridescent, that is diffraction off the meat fibers.[4] All these effects are a consequence of the fact that light propagates as a wave.

Diffraction can occur with any kind of wave. Ocean waves diffract around jetties and other obstacles. Sound waves can diffract around objects, which is why one can still hear someone calling even when hiding behind a tree.[5] Diffraction can also be a concern in some technical applications; it sets a fundamental limit to the resolution of a camera, telescope, or microscope.

History

Young Diffraction
Thomas Young's sketch of two-slit diffraction, which he presented to the Royal Society in 1803.

The effects of diffraction of light were first carefully observed and characterized by Francesco Maria Grimaldi, who also coined the term diffraction, from the Latin diffringere, 'to break into pieces', referring to light breaking up into different directions. The results of Grimaldi's observations were published posthumously in 1665.[6][7][8] Isaac Newton studied these effects and attributed them to inflexion of light rays. James Gregory (1638–1675) observed the diffraction patterns caused by a bird feather, which was effectively the first diffraction grating to be discovered.[9] Thomas Young performed a celebrated experiment in 1803 demonstrating interference from two closely spaced slits.[10] Explaining his results by interference of the waves emanating from the two different slits, he deduced that light must propagate as waves. Augustin-Jean Fresnel did more definitive studies and calculations of diffraction, made public in 1815[11] and 1818,[12] and thereby gave great support to the wave theory of light that had been advanced by Christiaan Huygens[13] and reinvigorated by Young, against Newton's particle theory.

Mechanism

Single-slit-diffraction-ripple-tank
Photograph of single-slit diffraction in a circular ripple tank

In traditional classical physics diffraction arises because of the way in which waves propagate; this is described by the Huygens–Fresnel principle and the principle of superposition of waves. The propagation of a wave can be visualized by considering every particle of the transmitted medium on a wavefront as a point source for a secondary spherical wave. The wave displacement at any subsequent point is the sum of these secondary waves. When waves are added together, their sum is determined by the relative phases as well as the amplitudes of the individual waves so that the summed amplitude of the waves can have any value between zero and the sum of the individual amplitudes. Hence, diffraction patterns usually have a series of maxima and minima.

In the modern quantum mechanical understanding of light propagation through a slit (or slits) every photon has what is known as a wavefunction which describes its path from the emitter through the slit to the screen. The wavefunction (the path the photon will take) is determined by the physical surroundings such as slit geometry, screen distance and initial conditions when the photon is created. In important experiments (A low-intensity double-slit experiment was first performed by G. I. Taylor in 1909, see double-slit experiment) the existence of the photon's wavefunction was demonstrated. In the quantum approach the diffraction pattern is created by the distribution of paths, the observation of light and dark bands is the presence or absence of photons in these areas (no interference!). The quantum approach has some striking similarities to the Huygens-Fresnel principle, in that principle the light becomes a series of individually distributed light sources across the slit which is similar to the limited number of paths (or wave functions) available for the photons to travel through the slit.

There are various analytical models which allow the diffracted field to be calculated, including the Kirchhoff-Fresnel diffraction equation which is derived from wave equation, the Fraunhofer diffraction approximation of the Kirchhoff equation which applies to the far field and the Fresnel diffraction approximation which applies to the near field. Most configurations cannot be solved analytically, but can yield numerical solutions through finite element and boundary element methods.

It is possible to obtain a qualitative understanding of many diffraction phenomena by considering how the relative phases of the individual secondary wave sources vary, and in particular, the conditions in which the phase difference equals half a cycle in which case waves will cancel one another out.

The simplest descriptions of diffraction are those in which the situation can be reduced to a two-dimensional problem. For water waves, this is already the case; water waves propagate only on the surface of the water. For light, we can often neglect one direction if the diffracting object extends in that direction over a distance far greater than the wavelength. In the case of light shining through small circular holes we will have to take into account the full three-dimensional nature of the problem.

Diffraction of light

Some examples of diffraction of light are considered below.

Single-slit diffraction

One wave slit diffraction dirichlet bw
Diffraction of a scalar wave passing through a 1-wavelength-wide slit
Four waves slit diffraction dirichlet bw
Diffraction of a scalar wave passing through a 4-wavelength-wide slit
Wave Diffraction 4Lambda Slit
Numerical approximation of diffraction pattern from a slit of width four wavelengths with an incident plane wave. The main central beam, nulls, and phase reversals are apparent.
Single Slit Diffraction (english)
Graph and image of single-slit diffraction.

A long slit of infinitesimal width which is illuminated by light diffracts the light into a series of circular waves and the wavefront which emerges from the slit is a cylindrical wave of uniform intensity.

A slit which is wider than a wavelength produces interference effects in the space downstream of the slit. These can be explained by assuming that the slit behaves as though it has a large number of point sources spaced evenly across the width of the slit. The analysis of this system is simplified if we consider light of a single wavelength. If the incident light is coherent, these sources all have the same phase. Light incident at a given point in the space downstream of the slit is made up of contributions from each of these point sources and if the relative phases of these contributions vary by 2π or more, we may expect to find minima and maxima in the diffracted light. Such phase differences are caused by differences in the path lengths over which contributing rays reach the point from the slit.

We can find the angle at which a first minimum is obtained in the diffracted light by the following reasoning. The light from a source located at the top edge of the slit interferes destructively with a source located at the middle of the slit, when the path difference between them is equal to λ/2. Similarly, the source just below the top of the slit will interfere destructively with the source located just below the middle of the slit at the same angle. We can continue this reasoning along the entire height of the slit to conclude that the condition for destructive interference for the entire slit is the same as the condition for destructive interference between two narrow slits a distance apart that is half the width of the slit. The path difference is approximately so that the minimum intensity occurs at an angle θmin given by

where

  • d is the width of the slit,
  • is the angle of incidence at which the minimum intensity occurs, and
  • is the wavelength of the light

A similar argument can be used to show that if we imagine the slit to be divided into four, six, eight parts, etc., minima are obtained at angles θn given by

where

  • n is an integer other than zero.

There is no such simple argument to enable us to find the maxima of the diffraction pattern. The intensity profile can be calculated using the Fraunhofer diffraction equation as

where

  • is the intensity at a given angle,
  • is the original intensity, and
  • the unnormalized sinc function above is given by if , and

This analysis applies only to the far field, that is, at a distance much larger than the width of the slit.

Diffraction2vs5
2-slit (top) and 5-slit diffraction of red laser light
Diffraction-red laser-diffraction grating PNr°0126
Diffraction of a red laser using a diffraction grating.
Diffraction 150 slits
A diffraction pattern of a 633 nm laser through a grid of 150 slits

Diffraction grating

A diffraction grating is an optical component with a regular pattern. The form of the light diffracted by a grating depends on the structure of the elements and the number of elements present, but all gratings have intensity maxima at angles θm which are given by the grating equation

where

  • θi is the angle at which the light is incident,
  • d is the separation of grating elements, and
  • m is an integer which can be positive or negative.

The light diffracted by a grating is found by summing the light diffracted from each of the elements, and is essentially a convolution of diffraction and interference patterns.

The figure shows the light diffracted by 2-element and 5-element gratings where the grating spacings are the same; it can be seen that the maxima are in the same position, but the detailed structures of the intensities are different.

Airy-pattern
A computer-generated image of an Airy disk.
Airy2
Computer generated light diffraction pattern from a circular aperture of diameter 0.5 micrometre at a wavelength of 0.6 micrometre (red-light) at distances of 0.1 cm – 1 cm in steps of 0.1 cm. One can see the image moving from the Fresnel region into the Fraunhofer region where the Airy pattern is seen.

Circular aperture

The far-field diffraction of a plane wave incident on a circular aperture is often referred to as the Airy Disk. The variation in intensity with angle is given by

,

where a is the radius of the circular aperture, k is equal to 2π/λ and J1 is a Bessel function. The smaller the aperture, the larger the spot size at a given distance, and the greater the divergence of the diffracted beams.

General aperture

The wave that emerges from a point source has amplitude at location r that is given by the solution of the frequency domain wave equation for a point source (The Helmholtz Equation),

where is the 3-dimensional delta function. The delta function has only radial dependence, so the Laplace operator (a.k.a. scalar Laplacian) in the spherical coordinate system simplifies to (see del in cylindrical and spherical coordinates)

By direct substitution, the solution to this equation can be readily shown to be the scalar Green's function, which in the spherical coordinate system (and using the physics time convention ) is:

This solution assumes that the delta function source is located at the origin. If the source is located at an arbitrary source point, denoted by the vector and the field point is located at the point , then we may represent the scalar Green's function (for arbitrary source location) as:

Therefore, if an electric field, Einc(x,y) is incident on the aperture, the field produced by this aperture distribution is given by the surface integral:

Fraunhofer
On the calculation of Fraunhofer region fields

where the source point in the aperture is given by the vector

In the far field, wherein the parallel rays approximation can be employed, the Green's function,

simplifies to

as can be seen in the figure to the right (click to enlarge).

The expression for the far-zone (Fraunhofer region) field becomes

Now, since

and

the expression for the Fraunhofer region field from a planar aperture now becomes,

Letting,

and

the Fraunhofer region field of the planar aperture assumes the form of a Fourier transform

In the far-field / Fraunhofer region, this becomes the spatial Fourier transform of the aperture distribution. Huygens' principle when applied to an aperture simply says that the far-field diffraction pattern is the spatial Fourier transform of the aperture shape, and this is a direct by-product of using the parallel-rays approximation, which is identical to doing a plane wave decomposition of the aperture plane fields (see Fourier optics).

Propagation of a laser beam

The way in which the beam profile of a laser beam changes as it propagates is determined by diffraction. When the entire emitted beam has a planar, spatially coherent wave front, it approximates Gaussian beam profile and has the lowest divergence for a given diameter. The smaller the output beam, the quicker it diverges. It is possible to reduce the divergence of a laser beam by first expanding it with one convex lens, and then collimating it with a second convex lens whose focal point is coincident with that of the first lens. The resulting beam has a larger diameter, and hence a lower divergence. Divergence of a laser beam may be reduced below the diffraction of a Gaussian beam or even reversed to convergence if the refractive index of the propagation media increases with the light intensity.[14] This may result in a self-focusing effect.

When the wave front of the emitted beam has perturbations, only the transverse coherence length (where the wave front perturbation is less than 1/4 of the wavelength) should be considered as a Gaussian beam diameter when determining the divergence of the laser beam. If the transverse coherence length in the vertical direction is higher than in horizontal, the laser beam divergence will be lower in the vertical direction than in the horizontal.

Diffraction-limited imaging

Zboo lucky image 1pc
The Airy disk around each of the stars from the 2.56 m telescope aperture can be seen in this lucky image of the binary star zeta Boötis.

The ability of an imaging system to resolve detail is ultimately limited by diffraction. This is because a plane wave incident on a circular lens or mirror is diffracted as described above. The light is not focused to a point but forms an Airy disk having a central spot in the focal plane with radius to first null of

where λ is the wavelength of the light and N is the f-number (focal length divided by diameter) of the imaging optics. In object space, the corresponding angular resolution is

where D is the diameter of the entrance pupil of the imaging lens (e.g., of a telescope's main mirror).

Two point sources will each produce an Airy pattern – see the photo of a binary star. As the point sources move closer together, the patterns will start to overlap, and ultimately they will merge to form a single pattern, in which case the two point sources cannot be resolved in the image. The Rayleigh criterion specifies that two point sources can be considered to be resolvable if the separation of the two images is at least the radius of the Airy disk, i.e. if the first minimum of one coincides with the maximum of the other.

Thus, the larger the aperture of the lens, and the smaller the wavelength, the finer the resolution of an imaging system. This is why telescopes have very large lenses or mirrors, and why optical microscopes are limited in the detail which they can see.

Speckle patterns

The speckle pattern which is seen when using a laser pointer is another diffraction phenomenon. It is a result of the superposition of many waves with different phases, which are produced when a laser beam illuminates a rough surface. They add together to give a resultant wave whose amplitude, and therefore intensity, varies randomly.

Babinet's Principle

Babinet's Principle is a useful theorem stating that the diffraction pattern from an opaque body is identical to that from a hole of the same size and shape, but with differing intensities. This means that the interference conditions of a single obstruction would be the same as that of a single slit.

Patterns

Diffraction on elliptic aperture with fft
The upper half of this image shows a diffraction pattern of He-Ne laser beam on an elliptic aperture. The lower half is its 2D Fourier transform approximately reconstructing the shape of the aperture.

Several qualitative observations can be made of diffraction in general:

  • The angular spacing of the features in the diffraction pattern is inversely proportional to the dimensions of the object causing the diffraction. In other words: The smaller the diffracting object, the 'wider' the resulting diffraction pattern, and vice versa. (More precisely, this is true of the sines of the angles.)
  • The diffraction angles are invariant under scaling; that is, they depend only on the ratio of the wavelength to the size of the diffracting object.
  • When the diffracting object has a periodic structure, for example in a diffraction grating, the features generally become sharper. The third figure, for example, shows a comparison of a double-slit pattern with a pattern formed by five slits, both sets of slits having the same spacing, between the center of one slit and the next.

Particle diffraction

Quantum theory tells us that every particle exhibits wave properties. In particular, massive particles can interfere and therefore diffract. Diffraction of electrons and neutrons stood as one of the powerful arguments in favor of quantum mechanics. The wavelength associated with a particle is the de Broglie wavelength

where h is Planck's constant and p is the momentum of the particle (mass × velocity for slow-moving particles).

For most macroscopic objects, this wavelength is so short that it is not meaningful to assign a wavelength to them. A sodium atom traveling at about 30,000 m/s would have a De Broglie wavelength of about 50 pico meters.

Because the wavelength for even the smallest of macroscopic objects is extremely small, diffraction of matter waves is only visible for small particles, like electrons, neutrons, atoms and small molecules. The short wavelength of these matter waves makes them ideally suited to study the atomic crystal structure of solids and large molecules like proteins.

Relatively larger molecules like buckyballs were also shown to diffract.[15]

Bragg diffraction

X-ray diffraction pattern 3clpro
Following Bragg's law, each dot (or reflection) in this diffraction pattern forms from the constructive interference of X-rays passing through a crystal. The data can be used to determine the crystal's atomic structure.

Diffraction from a three-dimensional periodic structure such as atoms in a crystal is called Bragg diffraction. It is similar to what occurs when waves are scattered from a diffraction grating. Bragg diffraction is a consequence of interference between waves reflecting from different crystal planes. The condition of constructive interference is given by Bragg's law:

where

λ is the wavelength,
d is the distance between crystal planes,
θ is the angle of the diffracted wave.
and m is an integer known as the order of the diffracted beam.

Bragg diffraction may be carried out using either light of very short wavelength like X-rays or matter waves like neutrons (and electrons) whose wavelength is on the order of (or much smaller than) the atomic spacing.[16] The pattern produced gives information of the separations of crystallographic planes d, allowing one to deduce the crystal structure. Diffraction contrast, in electron microscopes and x-topography devices in particular, is also a powerful tool for examining individual defects and local strain fields in crystals.

Coherence

The description of diffraction relies on the interference of waves emanating from the same source taking different paths to the same point on a screen. In this description, the difference in phase between waves that took different paths is only dependent on the effective path length. This does not take into account the fact that waves that arrive at the screen at the same time were emitted by the source at different times. The initial phase with which the source emits waves can change over time in an unpredictable way. This means that waves emitted by the source at times that are too far apart can no longer form a constant interference pattern since the relation between their phases is no longer time independent.[17]:919

The length over which the phase in a beam of light is correlated, is called the coherence length. In order for interference to occur, the path length difference must be smaller than the coherence length. This is sometimes referred to as spectral coherence, as it is related to the presence of different frequency components in the wave. In the case of light emitted by an atomic transition, the coherence length is related to the lifetime of the excited state from which the atom made its transition.[18]:71–74[19]:314–316

If waves are emitted from an extended source, this can lead to incoherence in the transversal direction. When looking at a cross section of a beam of light, the length over which the phase is correlated is called the transverse coherence length. In the case of Young's double slit experiment, this would mean that if the transverse coherence length is smaller than the spacing between the two slits, the resulting pattern on a screen would look like two single slit diffraction patterns.[18]:74–79

In the case of particles like electrons, neutrons and atoms, the coherence length is related to the spatial extent of the wave function that describes the particle.[20]:107

See also

References

  1. ^ Wireless Communications: Principles and Practice, Prentice Hall communications engineering and emerging technologies series, T. S. Rappaport, Prentice Hall, 2002 pg 126
  2. ^ Francesco Maria Grimaldi, Physico mathesis de lumine, coloribus, et iride, aliisque annexis libri duo (Bologna ("Bonomia"), Italy: Vittorio Bonati, 1665), page 2 Archived 2016-12-01 at the Wayback Machine:

    Original : Nobis alius quartus modus illuxit, quem nunc proponimus, vocamusque; diffractionem, quia advertimus lumen aliquando diffringi, hoc est partes eius multiplici dissectione separatas per idem tamen medium in diversa ulterius procedere, eo modo, quem mox declarabimus.

    Translation : It has illuminated for us another, fourth way, which we now make known and call "diffraction" [i.e., shattering], because we sometimes observe light break up; that is, that parts of the compound [i.e., the beam of light], separated by division, advance farther through the medium but in different [directions], as we will soon show.

  3. ^ Cajori, Florian "A History of Physics in its Elementary Branches, including the evolution of physical laboratories." Archived 2016-12-01 at the Wayback Machine MacMillan Company, New York 1899
  4. ^ Arumugam, Nadia. "Food Explainer: Why Is Some Deli Meat Iridescent?". Slate. The Slate Group. Archived from the original on 10 September 2013. Retrieved 9 September 2013.
  5. ^ Andrew Norton (2000). Dynamic fields and waves of physics. CRC Press. p. 102. ISBN 978-0-7503-0719-2.
  6. ^ Francesco Maria Grimaldi, Physico-mathesis de lumine, coloribus, et iride, aliisque adnexis … [The physical mathematics of light, color, and the rainbow, and other things appended …] (Bologna ("Bonomia"), (Italy): Vittorio Bonati, 1665), pp. 1–11 Archived 2016-12-01 at the Wayback Machine: "Propositio I. Lumen propagatur seu diffunditur non solum directe, refracte, ac reflexe, sed etiam alio quodam quarto modo, diffracte." (Proposition 1. Light propagates or spreads not only in a straight line, by refraction, and by reflection, but also by a somewhat different fourth way: by diffraction.) On p. 187, Grimaldi also discusses the interference of light from two sources: "Propositio XXII. Lumen aliquando per sui communicationem reddit obscuriorem superficiem corporis aliunde, ac prius illustratam." (Proposition 22. Sometimes light, as a result of its transmission, renders dark a body's surface, [which had been] previously illuminated by another [source].)
  7. ^ Jean Louis Aubert (1760). Memoires pour l'histoire des sciences et des beaux arts. Paris: Impr. de S. A. S.; Chez E. Ganeau. p. 149.
  8. ^ Sir David Brewster (1831). A Treatise on Optics. London: Longman, Rees, Orme, Brown & Green and John Taylor. p. 95. Archived from the original on 2016-12-01.
  9. ^ Letter from James Gregory to John Collins, dated 13 May 1673. Reprinted in: Correspondence of Scientific Men of the Seventeenth Century …, ed. Stephen Jordan Rigaud (Oxford, England: Oxford University Press, 1841), vol. 2, pp. 251–255, especially p. 254 Archived 2016-12-01 at the Wayback Machine.
  10. ^ Thomas Young (1804-01-01). "The Bakerian Lecture: Experiments and calculations relative to physical optics". Philosophical Transactions of the Royal Society of London. 94: 1–16. doi:10.1098/rstl.1804.0001.. (Note: This lecture was presented before the Royal Society on 24 November 1803.)
  11. ^ Augustin-Jean Fresnel (1816) "Mémoire sur la Diffraction de la lumière, où l'on examine particulièrement le phénomène des franges colorées que présentent les ombres des corps éclairés par un point lumineux" (Memoir on the diffraction of light, in which is examined particularly the phenomenon of colored fringes that the shadows of bodies illuminated by a point source display), Annales de la Chimie et de Physique, 2nd series, vol. 1, pages 239–281. (Presented before l'Académie des sciences on 15 October 1815.)
  12. ^ See:
    • Excerpts from Fresnel's paper on diffraction were published in 1819: A. Fresnel (1819) "Mémoire sur la diffraction de la lumière" (Memoir on the diffraction of light), Annales de chimie et de physique, 11 : 246–296 Archived 2016-12-01 at the Wayback Machine and 337–378. Archived 2016-12-01 at the Wayback Machine
    • The complete version of Fresnel's paper on diffraction was published in 1821: Augustin-Jean Fresnel (1821) "Mémoire sur la diffraction de la lumière" Archived 2014-09-07 at the Wayback Machine (Memoir on the diffraction of light), Mémoires de l'Académie des sciences de l'Institut de France, 5 : 339–475. (Summitted to l'Académie des sciences of Paris on 20 April 1818.)
  13. ^ Christiaan Huygens, Traité de la lumiere Archived 2016-06-16 at the Wayback Machine (Leiden, Netherlands: Pieter van der Aa, 1690), Chapter 1. From p. 15 Archived 2016-12-01 at the Wayback Machine: "J'ay donc monstré de quelle façon l'on peut concevoir que la lumiere s'etend successivement par des ondes spheriques, … " (I have thus shown in what manner one can imagine that light propagates successively by spherical waves, … ) (Note: Huygens published his Traité in 1690; however, in the preface to his book, Huygens states that in 1678 he first communicated his book to the French Royal Academy of Sciences.)
  14. ^ Chiao, R. Y.; Garmire, E.; Townes, C. H. (1964). "SELF-TRAPPING OF OPTICAL BEAMS". Physical Review Letters. 13 (15): 479–482. Bibcode:1964PhRvL..13..479C. doi:10.1103/PhysRevLett.13.479.
  15. ^ Brezger, B.; Hackermüller, L.; Uttenthaler, S.; Petschinka, J.; Arndt, M.; Zeilinger, A. (February 2002). "Matter–Wave Interferometer for Large Molecules" (reprint). Physical Review Letters. 88 (10): 100404. arXiv:quant-ph/0202158. Bibcode:2002PhRvL..88j0404B. doi:10.1103/PhysRevLett.88.100404. PMID 11909334. Archived (PDF) from the original on 2007-08-13. Retrieved 2007-04-30.
  16. ^ John M. Cowley (1975) Diffraction physics (North-Holland, Amsterdam) ISBN 0-444-10791-6
  17. ^ Halliday, David; Resnick, Robert; Walker, Jerl (2005), Fundamental of Physics (7th ed.), USA: John Wiley and Sons, Inc., ISBN 978-0-471-23231-5
  18. ^ a b Grant R. Fowles (1975). Introduction to Modern Optics. Courier Corporation. ISBN 978-0-486-65957-2.
  19. ^ Hecht, Eugene (2002), Optics (4th ed.), United States of America: Addison Wesley, ISBN 978-0-8053-8566-3
  20. ^ Ayahiko Ichimiya; Philip I. Cohen (13 December 2004). Reflection High-Energy Electron Diffraction. Cambridge University Press. ISBN 978-0-521-45373-8. Archived from the original on 16 July 2017.
Airy disk

In optics, the Airy disk (or Airy disc) 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. The Airy disk is of importance in physics, optics, and astronomy.

The diffraction pattern resulting from a uniformly illuminated, circular aperture has a bright central region, known as the Airy disk, which together with the series of concentric rings around is called the Airy pattern. Both are named after George Biddell Airy. The disk and rings phenomenon had been known prior to Airy; John Herschel described the appearance of a bright star seen through a telescope under high magnification for an 1828 article on light for the Encyclopedia Metropolitana:

...the star is then seen (in favourable circumstances of tranquil atmosphere, uniform temperature, etc.) as a perfectly round, well-defined planetary disc, surrounded by two, three, or more alternately dark and bright rings, which, if examined attentively, are seen to be slightly coloured at their borders. They succeed each other nearly at equal intervals round the central disc....

However, Airy wrote the first full theoretical treatment explaining the phenomenon (his 1835 "On the Diffraction of an Object-glass with Circular Aperture").Mathematically, the diffraction pattern is characterized by the wavelength of light illuminating the circular aperture, and the aperture's size. The appearance of the diffraction pattern is additionally characterized by the sensitivity of the eye or other detector used to observe the pattern.

The most important application of this concept is in cameras and telescopes. Due to diffraction, the smallest point to which a lens or mirror can focus a beam of light is the size of the Airy disk. Even if one were able to make a perfect lens, there is still a limit to the resolution of an image created by such a lens. An optical system in which the resolution is no longer limited by imperfections in the lenses but only by diffraction is said to be diffraction limited.

Angular resolution

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, thereby making it a major determinant of image resolution. In physics and geosciences, the term spatial resolution refers to the precision of a measurement with respect to space.

Bragg's law

In physics, Bragg's law, or Wulff–Bragg's condition, a special case of Laue diffraction, gives the angles for coherent and incoherent scattering from a crystal lattice. When X-rays are incident on an atom, they make the electronic cloud move, as does any electromagnetic wave. The movement of these charges re-radiates waves with the same frequency, blurred slightly due to a variety of effects; this phenomenon is known as Rayleigh scattering (or elastic scattering). The scattered waves can themselves be scattered but this secondary scattering is assumed to be negligible.

A similar process occurs upon scattering neutron waves from the nuclei or by a coherent spin interaction with an unpaired electron. These re-emitted wave fields interfere with each other either constructively or destructively (overlapping waves either add up together to produce stronger peaks or are subtracted from each other to some degree), producing a diffraction pattern on a detector or film. The resulting wave interference pattern is the basis of diffraction analysis. This analysis is called Bragg diffraction.

Crystallography

Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids (see crystal structure). The word "crystallography" derives from the Greek words crystallon "cold drop, frozen drop", with its meaning extending to all solids with some degree of transparency, and graphein "to write". In July 2012, the United Nations recognised the importance of the science of crystallography by proclaiming that 2014 would be the International Year of Crystallography. X-ray crystallography is used to determine the structure of large biomolecules such as proteins.

Before the development of X-ray diffraction crystallography (see below), the study of crystals was based on physical measurements of their geometry. This involved measuring the angles of crystal faces relative to each other and to theoretical reference axes (crystallographic axes), and establishing the symmetry of the crystal in question. This physical measurement is carried out using a goniometer. The position in 3D space of each crystal face is plotted on a stereographic net such as a Wulff net or Lambert net. The pole to each face is plotted on the net. Each point is labelled with its Miller index. The final plot allows the symmetry of the crystal to be established.

Crystallographic methods now depend on analysis of the diffraction patterns of a sample targeted by a beam of some type. X-rays are most commonly used; other beams used include electrons or neutrons. This is facilitated by the wave properties of the particles. Crystallographers often explicitly state the type of beam used, as in the terms X-ray crystallography, neutron diffraction and electron diffraction. These three types of radiation interact with the specimen in different ways.

X-rays interact with the spatial distribution of electrons in the sample.

Electrons are charged particles and therefore interact with the total charge distribution of both the atomic nuclei and the electrons of the sample.

Neutrons are scattered by the atomic nuclei through the strong nuclear forces, but in addition, the magnetic moment of neutrons is non-zero. They are therefore also scattered by magnetic fields. When neutrons are scattered from hydrogen-containing materials, they produce diffraction patterns with high noise levels. However, the material can sometimes be treated to substitute deuterium for hydrogen.Because of these different forms of interaction, the three types of radiation are suitable for different crystallographic studies.

Depth of field

Depth of field is the distance between the nearest and the furthest objects that are in acceptably sharp focus in an image. The depth of field is determined by focal length, distance to subject, the acceptable circle of confusion size, and aperture. A particular depth of field may be chosen for technical or artistic purposes. Some post-processing methods, such as focus stacking allow extended depth of field that would be impossible with traditional techniques.

Diffraction-limited system

The resolution of an optical imaging system – a microscope, telescope, or camera – can be limited by factors 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 of diffraction. An optical system with resolution performance at the instrument's theoretical limit is said to be diffraction-limited.The diffraction-limited angular resolution of a telescopic instrument is proportional to the wavelength 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. As one decreases the size of the aperture of a telescopic lens, 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 is proportional to the light wavelength, and to the numerical aperture of either the objective or the object illumination source, whichever is smaller.

In astronomy, 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 effects. Optical telescopes on the Earth 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 atmosphere. Some advanced observatories have recently started using adaptive optics technology, resulting in greater image resolution for faint targets, but it is still difficult to reach the diffraction limit using adaptive optics.

Radiotelescopes 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.

The beam from a laser 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.

Diffraction grating

In optics, a diffraction grating is an optical component with a periodic structure that splits and diffracts light into several beams travelling in different directions. The emerging coloration is a form of structural coloration. The directions of these beams depend on the spacing of the grating and the wavelength of the light so that the grating acts as the dispersive element. Because of this, gratings are commonly used in monochromators and spectrometers.

For practical applications, gratings generally have ridges or rulings on their surface rather than dark lines. Such gratings can be either transmissive or reflective. Gratings that modulate the phase rather than the amplitude of the incident light are also produced, frequently using holography.The principles of diffraction gratings were discovered by James Gregory, about a year after Newton's prism experiments, initially with items such as bird feathers. The first man-made diffraction grating was made around 1785 by Philadelphia inventor David Rittenhouse, who strung hairs between two finely threaded screws. This was similar to notable German physicist Joseph von Fraunhofer's wire diffraction grating in 1821. In the 1860s, gratings with the lowest line-distance d were created by Friedrich Adolph Nobert (1806-1881) in Greifswald, then the two Americans Lewis Morris Rutherfurd (1816-1892) and William B. Rogers (1804-1882) took over the lead, and by the end of the 19th century, the concave gratings of Henry Augustus Rowland (1848-1901) were the best gratings available.Diffraction can create "rainbow" colors when illuminated by a wide spectrum (e.g., continuous) light source. The sparkling effects from the closely spaced narrow tracks on optical storage disks such as CDs or DVDs are an example, while the similar rainbow effects caused by thin layers of oil (or gasoline, etc.) on water are not caused by a grating, but rather by interference effects in reflections from the closely spaced transmissive layers (see Examples, below). A grating has parallel lines, while a CD has a spiral of finely-spaced data tracks. Diffraction colors also appear when one looks at a bright point source through a translucent fine-pitch umbrella-fabric covering. Decorative patterned plastic films based on reflective grating patches are very inexpensive, and are commonplace.

Double-slit experiment

In modern physics, the double-slit experiment is a demonstration that light and matter can display characteristics of both classically defined waves and particles; moreover, it displays the fundamentally probabilistic nature of quantum mechanical phenomena. The experiment was first performed with light by Thomas Young in 1801. In 1927, Davisson and Germer demonstrated that electrons show the same behavior, which was later extended to atoms and molecules.Thomas Young's experiment with light was part of classical physics well before quantum mechanics, and the concept of wave-particle duality. He believed it demonstrated that the wave theory of light was correct, and his experiment is sometimes referred to as Young's experiment or Young's slits.

The experiment belongs to a general class of "double path" experiments, in which a wave is split into two separate waves that later combine into a single wave. Changes in the path lengths of both waves result in a phase shift, creating an interference pattern. Another version is the Mach–Zehnder interferometer, which splits the beam with a mirror. In the basic version of this experiment, a coherent light source, such as a laser beam, illuminates a plate pierced by two parallel slits, and the light passing through the slits is observed on a screen behind the plate. The wave nature of light causes the light waves passing through the two slits to interfere, producing bright and dark bands on the screen — a result that would not be expected if light consisted of classical particles. However, the light is always found to be absorbed at the screen at discrete points, as individual particles (not waves), the interference pattern appearing via the varying density of these particle hits on the screen. Furthermore, versions of the experiment that include detectors at the slits find that each detected photon passes through one slit (as would a classical particle), and not through both slits (as would a wave). However, such experiments demonstrate that particles do not form the interference pattern if one detects which slit they pass through. These results demonstrate the principle of wave–particle duality.Other atomic-scale entities, such as electrons, are found to exhibit the same behavior when fired towards a double slit. Additionally, the detection of individual discrete impacts is observed to be inherently probabilistic, which is inexplicable using classical mechanics.The experiment can be done with entities much larger than electrons and photons, although it becomes more difficult as size increases. The largest entities for which the double-slit experiment has been performed were molecules that each comprised 810 atoms (whose total mass was over 10,000 atomic mass units).The double-slit experiment (and its variations) has become a classic thought experiment, for its clarity in expressing the central puzzles of quantum mechanics. Because it demonstrates the fundamental limitation of the ability of the observer to predict experimental results, Richard Feynman called it "a phenomenon which is impossible […] to explain in any classical way, and which has in it the heart of quantum mechanics. In reality, it contains the only mystery [of quantum mechanics]."

Electron diffraction

Electron diffraction refers to the wave nature of electrons. However, from a technical or practical point of view, it may be regarded as a technique used to study matter by firing electrons at a sample and observing the resulting interference pattern. This phenomenon is commonly known as wave–particle duality, which states that a particle of matter (in this case the incident electron) can be described as a wave. For this reason, an electron can be regarded as a wave much like sound or water waves. This technique is similar to X-ray and neutron diffraction.

Electron diffraction is most frequently used in solid state physics and chemistry to study the crystal structure of solids. Experiments are usually performed in a transmission electron microscope (TEM), or a scanning electron microscope (SEM) as electron backscatter diffraction. In these instruments, electrons are accelerated by an electrostatic potential in order to gain the desired energy and determine their wavelength before they interact with the sample to be studied.

The periodic structure of a crystalline solid acts as a diffraction grating, scattering the electrons in a predictable manner. Working back from the observed diffraction pattern, it may be possible to deduce the structure of the crystal producing the diffraction pattern. However, the technique is limited by phase problem.

Apart from the study of crystals i.e. electron crystallography, electron diffraction is also a useful technique to study the short range order of amorphous solids, and the geometry of gaseous molecules.

Fraunhofer diffraction

In optics, the Fraunhofer diffraction equation is used to model the diffraction of waves when the diffraction pattern is viewed at a long distance from the diffracting object, and also when it is viewed at the focal plane of an imaging lens. In contrast, the diffraction pattern created near the object, in the near field region, is given by the Fresnel diffraction equation.

The equation was named in honor of Joseph von Fraunhofer although he was not actually involved in the development of the theory.This article explains where the Fraunhofer equation can be applied, and shows the form of the Fraunhofer diffraction pattern for various apertures. A detailed mathematical treatment of Fraunhofer diffraction is given in Fraunhofer diffraction equation.

Huygens–Fresnel principle

The Huygens–Fresnel principle (named after Dutch physicist Christiaan Huygens and French physicist Augustin-Jean Fresnel) is a method of analysis applied to problems of wave propagation both in the far-field limit and in near-field diffraction.

It states that every point on a wavefront is itself the source of spherical wavelets. The sum of these spherical wavelets forms the wavefront.

Neutron diffraction

Neutron diffraction or elastic neutron scattering is the application of neutron scattering to the determination of the atomic and/or magnetic structure of a material. A sample to be examined is placed in a beam of thermal or cold neutrons to obtain a diffraction pattern that provides information of the structure of the material. The technique is similar to X-ray diffraction but due to their different scattering properties, neutrons and X-rays provide complementary information: X-Rays are suited for superficial analysis, strong x-rays from synchrotron radiation are suited for shallow depths or thin specimens, while neutrons having high penetration depth are suited for bulk samples.

Powder diffraction

Powder diffraction is a scientific technique using X-ray, neutron, or electron diffraction on powder or microcrystalline samples for structural characterization of materials. An instrument dedicated to performing such powder measurements is called a powder diffractometer.

Powder diffraction stands in contrast to single crystal diffraction techniques, which work best with a single, well-ordered crystal.

Superposition principle

The superposition principle, also known as superposition property, states that, for all linear systems, the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually. So that if input A produces response X and input B produces response Y then input (A + B) produces response (X + Y).

A function that satisfies the superposition principle is called a linear function. Superposition can be defined by two simpler properties; additivity and homogeneity

 Additivity
 Homogeneity
for scalar a.

This principle has many applications in physics and engineering because many physical systems can be modeled as linear systems. For example, a beam can be modeled as a linear system where the input stimulus is the load on the beam and the output response is the deflection of the beam. The importance of linear systems is that they are easier to analyze mathematically; there is a large body of mathematical techniques, frequency domain linear transform methods such as Fourier, Laplace transforms, and linear operator theory, that are applicable. Because physical systems are generally only approximately linear, the superposition principle is only an approximation of the true physical behaviour.

The superposition principle applies to any linear system, including algebraic equations, linear differential equations, and systems of equations of those forms. The stimuli and responses could be numbers, functions, vectors, vector fields, time-varying signals, or any other object that satisfies certain axioms. Note that when vectors or vector fields are involved, a superposition is interpreted as a vector sum.

Transmission electron microscopy

Transmission electron microscopy (TEM, also sometimes conventional transmission electron microscopy or CTEM) is a microscopy technique in which a beam of electrons is transmitted through a specimen to form an image. The specimen is most often an ultrathin section less than 100 nm thick or a suspension on a grid. An image is formed from the interaction of the electrons with the sample as the beam is transmitted through the specimen. The image is then magnified and focused onto an imaging device, such as a fluorescent screen, a layer of photographic film, or a sensor such as a charge-coupled device.

Transmission electron microscopes are capable of imaging at a significantly higher resolution than light microscopes, owing to the smaller de Broglie wavelength of electrons. This enables the instrument to capture fine detail—even as small as a single column of atoms, which is thousands of times smaller than a resolvable object seen in a light microscope. Transmission electron microscopy is a major analytical method in the physical, chemical and biological sciences. TEMs find application in cancer research, virology, and materials science as well as pollution, nanotechnology and semiconductor research.

At lower magnifications TEM image contrast is due to differential absorption of electrons by the material due to differences in composition or thickness of the material. At higher magnifications complex wave interactions modulate the intensity of the image, requiring expert analysis of observed images. Alternate modes of use allow for the TEM to observe modulations in chemical identity, crystal orientation, electronic structure and sample induced electron phase shift as well as the regular absorption based imaging.

The first TEM was demonstrated by Max Knoll and Ernst Ruska in 1931, with this group developing the first TEM with resolution greater than that of light in 1933 and the first commercial TEM in 1939. In 1986, Ruska was awarded the Nobel Prize in physics for the development of transmission electron microscopy.

Wavelength

In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats.

It is thus the inverse of the spatial frequency. Wavelength 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 characteristic of both traveling waves and standing waves, as well as other spatial wave patterns.

Wavelength is commonly designated by the Greek letter lambda (λ).

The term wavelength is also sometimes applied to modulated waves, and to the sinusoidal envelopes of modulated waves or waves formed by interference of several sinusoids.Assuming a sinusoidal wave moving at a fixed wave speed, wavelength is inversely proportional to frequency of the wave: waves with higher frequencies have shorter wavelengths, and lower frequencies have longer wavelengths.Wavelength depends on the medium (for example, vacuum, air, or water) that a wave travels through.

Examples of wave-like phenomena are sound waves, light, water waves and periodic electrical signals in a conductor. A sound wave is a variation in air pressure, while in light and other electromagnetic radiation the strength of the electric and the magnetic field vary. Water waves are variations in the height of a body of water. In a crystal lattice vibration, atomic positions vary.

Wavelength is a measure of the distance between repetitions of a shape feature such as peaks, valleys, or zero-crossings, not a measure of how far any given particle moves. For example, in sinusoidal waves over deep water a particle near the water's surface moves in a circle of the same diameter as the wave height, unrelated to wavelength. The range of wavelengths or frequencies for wave phenomena is called a spectrum. The name originated with the visible light spectrum but now can be applied to the entire electromagnetic spectrum as well as to a sound spectrum or vibration spectrum.

X-ray crystallography

X-ray crystallography (XRC) is a technique used for determining the atomic and molecular structure of a crystal, in which the crystalline structure causes a beam of incident X-rays to diffract into many specific directions. By measuring the angles and intensities of these diffracted beams, a crystallographer can produce a three-dimensional picture of the density of electrons within the crystal. From this electron density, the mean positions of the atoms in the crystal can be determined, as well as their chemical bonds, their crystallographic disorder, and various other information.

Since many materials can form crystals—such as salts, metals, minerals, semiconductors, as well as various inorganic, organic, and biological molecules—X-ray crystallography has been fundamental in the development of many scientific fields. In its first decades of use, this method determined the size of atoms, the lengths and types of chemical bonds, and the atomic-scale differences among various materials, especially minerals and alloys. The method also revealed the structure and function of many biological molecules, including vitamins, drugs, proteins and nucleic acids such as DNA. X-ray crystallography is still the primary method for characterizing the atomic structure of new materials and in discerning materials that appear similar by other experiments. X-ray crystal structures can also account for unusual electronic or elastic properties of a material, shed light on chemical interactions and processes, or serve as the basis for designing pharmaceuticals against diseases.

In a single-crystal X-ray diffraction measurement, a crystal is mounted on a goniometer. The goniometer is used to position the crystal at selected orientations. The crystal is illuminated with a finely focused monochromatic beam of X-rays, producing a diffraction pattern of regularly spaced spots known as reflections. The two-dimensional images taken at different orientations are converted into a three-dimensional model of the density of electrons within the crystal using the mathematical method of Fourier transforms, combined with chemical data known for the sample. Poor resolution (fuzziness) or even errors may result if the crystals are too small, or not uniform enough in their internal makeup.

X-ray crystallography is related to several other methods for determining atomic structures. Similar diffraction patterns can be produced by scattering electrons or neutrons, which are likewise interpreted by Fourier transformation. If single crystals of sufficient size cannot be obtained, various other X-ray methods can be applied to obtain less detailed information; such methods include fiber diffraction, powder diffraction and (if the sample is not crystallized) small-angle X-ray scattering (SAXS).

If the material under investigation is only available in the form of nanocrystalline powders or suffers from poor crystallinity, the methods of electron crystallography can be applied for determining the atomic structure.

For all above mentioned X-ray diffraction methods, the scattering is elastic; the scattered X-rays have the same wavelength as the incoming X-ray. By contrast, inelastic X-ray scattering methods are useful in studying excitations of the sample such as plasmons, crystal-field and orbital excitations, magnons, and phonons, rather than the distribution of its atoms.

X-ray scattering techniques

X-ray scattering techniques are a family of non-destructive analytical techniques which reveal information about the crystal structure, chemical composition, and physical properties of materials and thin films. These techniques are based on observing the scattered intensity of an X-ray beam hitting a sample as a function of incident and scattered angle, polarization, and wavelength or energy.

Note that X-ray diffraction is now often considered a sub-set of X-ray scattering, where the scattering is elastic and the scattering object is crystalline, so that the resulting pattern contains sharp spots analyzed by X-ray crystallography (as in the Figure). However, both scattering and diffraction are related general phenomena and the distinction has not always existed. Thus Guinier's classic text from 1963 is titled "X-ray diffraction in Crystals, Imperfect Crystals and Amorphous Bodies" so 'diffraction' was clearly not restricted to crystals at that time.

Zone plate

A zone plate is a device used to focus light or other things exhibiting wave character. Unlike lenses or curved mirrors however, zone plates use diffraction instead of refraction or reflection. Based on analysis by Augustin-Jean Fresnel, they are sometimes called Fresnel zone plates in his honor. The zone plate's focusing ability is an extension of the Arago spot phenomenon caused by diffraction from an opaque disc.A zone plate consists of a set of radially symmetric rings, known as Fresnel zones, which alternate between opaque and transparent. Light hitting the zone plate will diffract around the opaque zones. The zones can be spaced so that the diffracted light constructively interferes at the desired focus, creating an image there.

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