Diffraction
Diffraction refers to various phenomena that occur when a wave encounters an obstacle or a slit. It is defined as the bending of light 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. 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
Diffraction has an impact on the acoustic space. Diffraction
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 (at the atomic level), diffraction also occurs with matter and can be studied according to the principles of quantum mechanics. Italian scientist Francesco Maria Grimaldi
Francesco Maria Grimaldi coined the word "diffraction" and was the first to record accurate observations of the phenomenon in 1660.[1][2] 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. Contents 1 Examples
2 History
3 Mechanism
4
4.1 Single-slit diffraction
4.2
5 Patterns 6 Particle diffraction 7 Bragg diffraction 8 Coherence 9 See also 10 References 11 External links Examples[edit] 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.
Thomas Young's sketch of two-slit diffraction, which he presented to
the
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.[5][6][7]
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-
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. 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 d sin ( θ ) 2 displaystyle frac dsin(theta ) 2 so that the minimum intensity occurs at an angle θmin given by d sin θ min = λ displaystyle d,sin theta _ text min =lambda where d is the width of the slit, θ min displaystyle theta _ text min is the angle of incidence at which the minimum intensity occurs, and λ displaystyle lambda 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 d sin θ n = n λ displaystyle d,sin theta _ n =nlambda 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
I ( θ ) = I 0 sinc 2 ( d π λ sin θ ) displaystyle I(theta )=I_ 0 ,operatorname sinc ^ 2 left( frac dpi lambda sin theta right) where I ( θ ) displaystyle I(theta ) is the intensity at a given angle, I 0 displaystyle I_ 0 is the original intensity, and the unnormalized sinc function above is given by sinc ( x ) = sin x x displaystyle operatorname sinc (x)= frac sin x x if x ≠ 0 displaystyle xneq 0 , and sinc ( 0 ) = 1 displaystyle operatorname sinc (0)=1 This analysis applies only to the far field, that is, at a distance much larger than the width of the slit. 2-slit (top) and 5-slit diffraction of red laser light
A diffraction pattern of a 633 nm laser through a grid of 150 slits
d ( sin θ m + sin θ i ) = m λ . displaystyle dleft(sin theta _ m +sin theta _ i right)=mlambda . 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. A computer-generated image of an Airy disk. 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[edit] Main article: Airy disk 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 I ( θ ) = I 0 ( 2 J 1 ( k a sin θ ) k a sin θ ) 2 displaystyle I(theta )=I_ 0 left( frac 2J_ 1 (kasin theta ) kasin theta right)^ 2 , 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[edit] The wave that emerges from a point source has amplitude ψ displaystyle psi at location r that is given by the solution of the frequency domain wave equation for a point source (The Helmholtz Equation), ∇ 2 ψ + k 2 ψ = δ ( r ) displaystyle nabla ^ 2 psi +k^ 2 psi =delta ( mathbf r ) where δ ( r ) displaystyle delta ( mathbf r ) is the 3-dimensional delta function. The delta function has only
radial dependence, so the
∇ 2 ψ = 1 r ∂ 2 ∂ r 2 ( r ψ ) displaystyle nabla ^ 2 psi = frac 1 r frac partial ^ 2 partial r^ 2 (rpsi ) 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 e − i ω t displaystyle e^ -iomega t ) is: ψ ( r ) = e i k r 4 π r displaystyle psi (r)= frac e^ ikr 4pi r 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 r ′ displaystyle mathbf r ' and the field point is located at the point r displaystyle mathbf r , then we may represent the scalar
ψ ( r
r ′ ) = e i k
r − r ′
4 π
r − r ′
displaystyle psi ( mathbf r mathbf r ')= frac e^ ik mathbf r - mathbf r ' 4pi mathbf r - mathbf r ' 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: Ψ ( r ) ∝ ∬ a p e r t u r e E i n c ( x ′ , y ′ ) e i k
r − r ′
4 π
r − r ′
d x ′ d y ′ , displaystyle Psi (r)propto iint limits _ mathrm aperture E_ mathrm inc (x',y')~ frac e^ ik mathbf r - mathbf r ' 4pi mathbf r - mathbf r ' ,dx',dy', On the calculation of Fraunhofer region fields where the source point in the aperture is given by the vector r ′ = x ′ x ^ + y ′ y ^ displaystyle mathbf r '=x' mathbf hat x +y' mathbf hat y In the far field, wherein the parallel rays approximation can be employed, the Green's function, ψ ( r
r ′ ) = e i k
r − r ′
4 π
r − r ′
displaystyle psi ( mathbf r mathbf r ')= frac e^ ik mathbf r - mathbf r ' 4pi mathbf r - mathbf r ' simplifies to ψ ( r
r ′ ) = e i k r 4 π r e − i k ( r ′ ⋅ r ^ ) displaystyle psi ( mathbf r mathbf r ')= frac e^ ikr 4pi r e^ -ik( mathbf r 'cdot mathbf hat r ) as can be seen in the figure to the right (click to enlarge). The expression for the far-zone (Fraunhofer region) field becomes Ψ ( r ) ∝ e i k r 4 π r ∬ a p e r t u r e E i n c ( x ′ , y ′ ) e − i k ( r ′ ⋅ r ^ ) d x ′ d y ′ , displaystyle Psi (r)propto frac e^ ikr 4pi r iint limits _ mathrm aperture E_ mathrm inc (x',y')e^ -ik( mathbf r 'cdot mathbf hat r ) ,dx',dy', Now, since r ′ = x ′ x ^ + y ′ y ^ displaystyle mathbf r '=x' mathbf hat x +y' mathbf hat y and r ^ = sin θ cos ϕ x ^ + sin θ sin ϕ y ^ + cos θ z ^ displaystyle mathbf hat r =sin theta cos phi mathbf hat x +sin theta ~sin phi ~ mathbf hat y +cos theta mathbf hat z the expression for the Fraunhofer region field from a planar aperture now becomes, Ψ ( r ) ∝ e i k r 4 π r ∬ a p e r t u r e E i n c ( x ′ , y ′ ) e − i k sin θ ( cos ϕ x ′ + sin ϕ y ′ ) d x ′ d y ′ displaystyle Psi (r)propto frac e^ ikr 4pi r iint limits _ mathrm aperture E_ mathrm inc (x',y')e^ -iksin theta (cos phi x'+sin phi y') ,dx',dy' Letting, k x = k sin θ cos ϕ displaystyle k_ x =ksin theta cos phi ,! and k y = k sin θ sin ϕ displaystyle k_ y =ksin theta sin phi ,! the Fraunhofer region field of the planar aperture assumes the form of a Fourier transform Ψ ( r ) ∝ e i k r 4 π r ∬ a p e r t u r e E i n c ( x ′ , y ′ ) e − i ( k x x ′ + k y y ′ ) d x ′ d y ′ , displaystyle Psi (r)propto frac e^ ikr 4pi r iint limits _ mathrm aperture E_ mathrm inc (x',y')e^ -i(k_ x x'+k_ y y') ,dx',dy', 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
The
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
d = 1.22 λ N , displaystyle d=1.22lambda N,, 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 sin θ = 1.22 λ D , displaystyle sin theta =1.22 frac lambda D ,, 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
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[edit]
See also: neutron diffraction and electron diffraction
Quantum theory tells us that every particle exhibits wave properties.
In particular, massive particles can interfere and therefore diffract.
λ = h p displaystyle lambda = frac h p , where h is
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. Further information: Bragg diffraction
m λ = 2 d sin θ displaystyle mlambda =2dsin theta , 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.
Angle-sensitive pixel
Atmospheric diffraction
Bragg diffraction
Brocken spectre
Cloud iridescence
References[edit] ^ 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. ^ 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
^ 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.
^ Andrew Norton (2000). Dynamic fields and waves of physics. CRC
Press. p. 102. ISBN 978-0-7503-0719-2.
^ 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].)
^ 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.
^ 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.
^ 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..
^ Thomas Young (1804-01-01). "The Bakerian Lecture: Experiments and
calculations relative to physical optics". Philosophical Transactions
of the
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:
^ 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.)
^ 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.
^ 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.
^ John M. Cowley (1975)
External links[edit] This article's use of external links may not follow's policies or guidelines. Please improve this article by removing excessive or inappropriate external links, and converting useful links where appropriate into footnote references. (May 2015) (Learn how and when to remove this template message) Wikimedia Commons has media related to Diffraction. The Wikibook Nanotechnology has a page on the topic of: Nano-optics
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