Diffraction - Wikipedia, The Free Encyclopedia

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Computer generated intensity pattern

formed on a screen by diffraction

from a square aperture.

Generation of an interference pattern

from two-slit diffraction.

Computational model of an

interference pattern from two-slit

diffraction.

DiffractionFrom Wikipedia, the free encyclopedia

Diffraction refers to various phenomena which occur when a wave

encounters an obstacle. In classical physics, the diffraction

phenomenon is described as the apparent bending of waves around

small obstacles and the spreading out of waves past small openings.

Similar effects occur when a light wave travels through a medium with

a varying refractive index, or a sound wave travels through one with

varying acoustic impedance. Diffraction occurs with all waves,

including sound waves, water waves, and electromagnetic waves such

as visible light, X-rays and radio waves. As 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 coined the

word "diffraction" and was the first to record accurate observations of

the phenomenon in 1665.[2][3]

Richard Feynman[4] said that

"no-one has ever been able to define the difference between

interference and diffraction satisfactorily. It is just a question of

usage, and there is no specific, important physical difference

between them."

He suggested that when there are only a few sources, say two, we

call it interference, as in Young's slits, but with a large number ofsources, the process be labelled diffraction.

While diffraction occurs whenever propagating waves encountersuch

changes, its effects are generally most pronounced for waves whose

wavelength is roughly similar to the dimensions of the diffracting

objects. If the obstructing object provides multiple, closely spaced

openings, a complex pattern of varying intensity can result. This is due

to the superposition, or interference, of different parts of a wave that

travels to the observer by different paths (see diffraction grating).

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 analysed using

diffraction equations.

Contents

1 Examples

2 History

3 Mechanism

4 Diffraction of light

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Colors seen in a spider web are

partially due to diffraction, according

to some analyses.[1]

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.

4.1 Single-slit diffraction

4.2 Diffraction grating

4.3 Circular aperture

4.4 General aperture

4.5 Propagation of a laser beam

4.6 Diffraction-limited imaging

4.7 Speckle patterns

5 Patterns6 Particle diffraction

7 Bragg diffraction

8 Coherence

9 See also

10 References

11 External links

Examples

The effects of diffraction are often seen in everyday life. The most

striking examples of diffraction are those involving 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 disk. 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 lightsource 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. 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 sometechnical applications; it sets a fundamental limit to the resolution of a

camera, telescope, or microscope.

History

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(16381675) 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

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Thomas Young's sketch of two-slit

diffraction, which he presented to the

Royal Society in 1803.

Photograph of single-slit diffraction in

a circular ripple tank

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

Diffraction arises because

of the way in which waves

propagate; this is described

by the HuygensFresnel principle and the principle of superposition of

waves. The propagation of a wave can be visualized by considering

every point 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 ofthe 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.

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

Main article: Diffraction formalism

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

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Numerical approximation of

diffraction pattern from a slit of width

equal to wavelength of an incident

plane wave in 3D spectrum

visualization

Numerical approximation ofdiffraction pattern from a slit of width

equal to five times the wavelength of

an incident plane wave in 3D

spectrum visualization

Diffraction of red laser beam on

the hole

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 monochromatic, 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 2 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/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 given byso that the minimum intensity occurs at an angle min given

by

where

dis the width of the slit,

is the angle of incidenceat 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 n 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

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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.

2-slit (top) and 5-slit diffraction of

red laser lightDiffraction of a red laser using a

diffraction grating.

is the intensity at a given angle,

is the original intensity, and

the sinc function is given by sinc(x) = sin(x)/(x) ifx 0, and sinc(0) = 1

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

Diffraction grating

Main article: 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 m which are given by the grating

equation

where

i is the angle at which the light is incident,

dis 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.

Circular aperture

Main article: Airy disk

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

angle is given by

,

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A diffraction pattern of a 633 nm

laser through a grid of 150 slits

A computer-generated image of an

Airy disk.

Computer generated light diffraction

pattern from a c ircular 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.

where a is the radius of the circular aperture, kis equal to 2/ 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 (aka 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 theorigin. 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:

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

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On the calculation of Fraunhofer region fields

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).

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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 Botis.

Propagation of a laser beam

The way in which the profile of a laser beam changes as it propagates is determined by diffraction. The output

mirror of the laser is an aperture, and the subsequent beam shape is determined by that aperture. Hence, the

smaller the output beam, the quicker it diverges.

Paradoxically, 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. Theresulting beam has a larger aperture, and hence a lower divergence.

Diffraction-limited imaging

Main article: Diffraction-limited system

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 planewith radius to first null of

where is the wavelength of the light andNis the f-number (focal

length divided by diameter) of the imaging optics. In object space,

the corresponding angular resolution is

whereD 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

Main article: speckle pattern

The speckle pattern which is seen when using a laser pointer is another diffraction phenomenon. It is a result of

the superpostion of many waves with different phases, which are produced when a laser beam illuminates arough surface. They add together to give a resultant wave whose amplitude, and therefore intensity varies

randomly.

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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.

Patterns

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 viceversa. (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 oneslit and the next.

Particle diffraction

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.

Diffraction of electrons and neutrons stood as one of the powerfularguments in favor of quantum mechanics. The wavelength associated

with a particle is the de Broglie wavelength

where h is Planck's constant andp 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 picometers.

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.[14]

Bragg diffraction

For more details on this topic, see Bragg diffraction.

Diffraction from a three dimensional periodic structure such as atoms in a crystal is called Bragg diffraction. It is

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Following Bragg's law, each dot (or

reflection), in this diffraction pattern

forms from the constructiveinterference of X-rays passing

through a crystal. The data can be

used to determine the crystal's atomic

structure.

similar to what occurs when waves are scattered from a diffraction grating. Bragg diffraction is a consequence o

interference between waves reflecting from different crystal planes. The condition of constructive interference is

given byBragg's law:

where

is the wavelength,dis the distance between crystal planes,

is the angle of the diffracted wave.

and m is an integer known as the orderof 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.[15] The pattern produced gives information of the

separations of crystallographic planes d, allowing one to deduce thecrystal 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

Main article: Coherence (physics)

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 tookdifferent 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.

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.

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.

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.

See also

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Atmospheric diffraction

Bragg diffraction

Brocken spectre

Cloud iridescence

Diffraction formalism

Diffraction grating

Diffraction limit

DiffractometerDynamical theory of diffraction

Electron diffraction

Fraunhofer diffraction

Fresnel diffraction

Fresnel imager

Fresnel number

Fresnel zone

Neutron diffraction

PrismPowder diffraction

Refraction

SchaeferBergmann diffraction

Thinned array curse

X-ray scattering techniques

References

1. ^ Dietrich Zawischa. "Optical effects on spider webs" (http://www.itp.uni-

hannover.de/%7Ezawischa/ITP/spiderweb.html) . http://www.itp.uni-

hannover.de/%7Ezawischa/ITP/spiderweb.html. Retrieved 2007-09-21.

2. ^ Francesco Maria Grimaldi,Physico mathesis de lumine, coloribus, et iride, aliisque annexis libri duo

(Bologna ("Bonomia"), Italy: Vittorio Bonati, 1665), page 2 (http://books.google.com/books?

id=FzYVAAAAQAAJ&pg=PA2#v=onepage&q&f=false) :

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." (http://books.google.com/books?id=KZ4C-

1CRtYQC&ots=c_YpkkbTpT&dq=Florian%20Cajori%20history%20of%20physics&pg=PA88) MacMillan

Company, New York 1899

4. ^ R. Feynman, Lectures in Physics, Vol, 1, 1963, pg. 30-1, Addison Wesley Publishing Company Reading,

Mass

5. ^ Andrew Norton (2000). Dynamic f ields and waves of physics (http://books.google.com/?

id=XRRMxjr24pwC&pg=PA102) . CRC Press. p. 102. ISBN 978-0-7503-0719-2. http://books.google.com/?

id=XRRMxjr24pwC&pg=PA102.

6. ^ Francesco Maria Grimaldi,Physico-mathesis de lumine, coloribus, et iride, aliisque adnexis... [The physical

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mathematics of light, color, and the rainbow, and other things appended...] (Bologna ("Bonomia"), Italy: Vittorio

Bonati, 1665), pp. 111 (http://books.google.com/books?

id=FzYVAAAAQAAJ&pg=PA1#v=onepage&q&f=false) : "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.)

7. ^ Jean Louis Aubert (1760).Memoires pour l'histoire des sciences et des beaux arts (http://books.google.com/?

id=3OgDAAAAMAAJ&pg=PP151&lpg=PP151&dq=grimaldi+diffraction+date:0-1800) . Paris: Impr. de S. A.

S.; Chez E. Ganeau. pp. 149. http://books.google.com/?id=3OgDAAAAMAAJ&pg=PP151&lpg=PP151&dq=grimaldi+diffraction+date:0-1800.

8. ^ Sir David Brewster (1831). A Treatise on Optics (http://books.google.com/?id=opYAAAAAMAAJ&pg=RA1-

PA95&lpg=RA1-PA95&dq=grimaldi+diffraction+date:0-1840) . London: Longman, Rees, Orme, Brown &

Green and John Taylor. pp. 95. http://books.google.com/?id=opYAAAAAMAAJ&pg=RA1-PA95&lpg=RA1-

PA95&dq=grimaldi+diffraction+date:0-1840.

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. 251255, especially p. 254 (http://books.google.com/books?id=0h45L_66bcYC&pg=PA254)

.

10. ^ Young, Thomas (1804-01-01). "The Bakerian Lecture: Experiments and calculations relative to physical

optics" (http://books.google.com/?id=7AZGAAAAMAAJ&pg=PA1) .Philosophical Transactions of the RoyalSociety of London (Royal Society of London.) 94: 116. doi:10.1098/rstl.1804.0001

(http://dx.doi.org/10.1098%2Frstl.1804.0001) . http://books.google.com/?id=7AZGAAAAMAAJ&pg=PA1.

(Note: This lecture was presented before the Royal Society on 24 November 1803.)

11. ^ Augustin-Jean Fresnel (1816) "Mmoire sur la diffraction de la lumire ,"Annales de la Chemie et de

Physique, 2nd series, vol. 1, pages 239281. (Presented before l'Acadmie des sciences on 15 October 1815.)

Available on-line at: Bibnum.education.fr (http://www.bibnum.education.fr/physique/optique/premier-memoire-

sur-la-diffraction-de-la-lumiere) (French)

12. ^ Augustin-Jean Fresnel (1826) "Mmoire sur la diffraction de la lumire,"Mmoires de l'Acadmie des

Sciences (Paris), vol. 5, pages 33475. (Summitted to l'Acadmie des sciences of Paris on 20 April 1818.)

13. ^ Christiaan Huygens, Trait de la lumiere... (http://books.google.com/books?id=X9PKaZlChggC&pg=PP5)

(Leiden, Netherlands: Pieter van der Aa, 1690), Chapter 1. From p. 15 (http://books.google.com/books?

id=X9PKaZlChggC&pg=PA15) : "J'ay donc monstr de quelle faon 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 Trait 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. ^ Brezger, B.; Hackermller, L.; Uttenthaler, S.; Petschinka, J.; Arndt, M.; Zeilinger, A. (February 2002).

"MatterWave Interferometer for Large Molecules"

(http://homepage.univie.ac.at/Lucia.Hackermueller/unsereArtikel/Brezger2002a.pdf) (reprint).Physical Review

Letters88 (10): 100404. arXiv:quant-ph/0202158 (http://arxiv.org/abs/quant-ph/0202158) . Bibcode

2002PhRvL..88j0404B (http://adsabs.harvard.edu/abs/2002PhRvL..88j0404B) .doi:10.1103/PhysRevLett.88.100404 (http://dx.doi.org/10.1103%2FPhysRevLett.88.100404) . PMID 11909334

(//www.ncbi.nlm.nih.gov/pubmed/11909334) .

http://homepage.univie.ac.at/Lucia.Hackermueller/unsereArtikel/Brezger2002a.pdf. Retrieved 2007-04-30.

15. ^ John M. Cowley (1975) Diff raction physics (North-Holland, Amsterdam) ISBN 0-444-10791-6

External links

Diffraction (http://richannel.org/tales-from-the-prep-room-diffraction,) , Ri Channel Video, December

2011

Diffraction and Crystallography for beginners (http://www.xtal.iqfr.csic.es/Cristalografia/index-en.html)

Do Sensors Outresolve Lenses? (http://luminous-landscape.com/tutorials/resolution.shtml) ; on lens and

sensor resolution interaction.

Diffraction and acoustics. (http://www.acoustics.salford.ac.uk/feschools/waves/diffract.htm)

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Diffraction in photography. (http://www.johnsankey.ca/diffraction.html)

On Diffraction (http://www.mathpages.com/home/kmath636/kmath636.htm) at MathPages.

Diffraction pattern calculators (http://demonstrations.wolfram.com/search.html?query=diffraction) at The

Wolfram Demonstrations Project

Wave Optics (http://www.lightandmatter.com/html_books/5op/ch05/ch05.html) A chapter of an online

textbook.

2-D wave Java applet (http://www.falstad.com/wave2d/) Displays diffraction patterns of various slit

configurations.Diffraction Java applet (http://www.falstad.com/diffraction/) Displays diffraction patterns of various 2-D

apertures.

Diffraction approximations illustrated (http://www.mit.edu/~birge/diffraction/) MIT site that illustrates

the various approximations in diffraction and intuitively explains the Fraunhofer regime from the

perspective of linear system theory.

Gap (http://www.phy.hk/wiki/englishhtm/Diffraction.htm) Obstacle

(http://www.phy.hk/wiki/englishhtm/Diffraction2.htm) Corner

(http://www.phy.hk/wiki/englishhtm/Diffraction3.htm) Java simulation of diffraction of water wave.

Google Maps (http://maps.google.com/maps?q=Panama+canal&hl=en&ie=UTF8&om=1&z=16&ll=9.385048,-79.918799&spn=0.015539,0.02712

2&t=k&iwloc=addr) Satellite image of Panama Canal entry ocean wave diffraction.

Google Maps (http://maps.google.com/maps?

f=q&source=s_q&hl=en&geocode=&q=&sll=52.788632,1.609969&sspn=0.010472,0.016093&ie=U

TF8&t=h&ll=52.788217,1.606772&spn=0.010472,0.016093&z=16) and Bing Maps

(http://www.bing.com/maps/?

v=2&cp=52.788763840321245~1.6073888540267944&lvl=16&sty=h&eo=0) Aerial photo of

waves diffracting through sea barriers at Sea Palling in Norfolk, UK.

Diffraction Effects

(http://www.cvimellesgriot.com/products/Documents/TechnicalGuide/Diffraction_Effects.pdf)

An Introduction to The Wigner Distribution in Geometric Optics

(http://scripts.mit.edu/~raskar/lightfields/index.php?

title=An_Introduction_to_The_Wigner_Distribution_in_Geometric_Optics)

DoITPoMS Teaching and Learning Package Diffraction and Imaging

(http://www.doitpoms.ac.uk/tlplib/diffraction/index.php)

Animations demonstrating Diffraction (http://qed.wikina.org/interference/) by QED

FDTD Animation of single slit diffraction (http://www.youtube.com/watch?v=uPQMI2q_vPQ) on

YouTube

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