# Dictionary Definition

diffract v : undergo diffraction; "laser light
diffracts electrons"

# User Contributed Dictionary

## English

### Verb

- To cause diffraction
- To undergo diffraction

#### Translations

transitive

- Italian: diffrangere

intransitive

- Italian: diffrangersi

# Extensive Definition

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. 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. In 1803 Thomas
Young did his famous experiment observing interference from two
closely spaced slits. 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, published in 1815 and 1818, and thereby gave great
support to the wave theory of light that had been advanced by
Christiaan
Huygens and reinvigorated by Young, against Newton's particle
theory.

## The mechanism of diffraction

Diffraction arises because of the way in which waves propagate; this is described by the Huygens–Fresnel principle. The propagation of a wave can be visualized by considering every point on a wavefront as a point source for a secondary radial wave. The subsequent propagation and addition of all these radial waves form the new wavefront. When waves are added together, their sum is determined by the relative phases as well as the amplitudes of the individual waves, an effect which is often known as wave interference. 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.To determine the form of a diffraction pattern,
we must determine the phase and amplitude of each of the Huygens
wavelets at each point in space and then find the sum of these
waves. There are various analytical models which can be used to do
this including the Fraunhoffer
diffraction equation for the far field and the Fresnel
Diffraction equation for the near-field. Most configurations
cannot be solved analytically; solutions can be found using various
numerical analytical methods including Finite
element and boundary
element methods

## Diffraction systems

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.

Some of the simpler cases of diffraction are
considered below.

### Single-slit diffraction

A long slit of infinitesmal 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 has a
large number of point sources spaced evenly across the width of the
slit. The light at a given angle is made up contributions from each
of these point sources and if the relative phases of these
contributions vary by more than 2π, we expect to find minima and
maxima in the diffracted light.

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 interferes destructively
with the source located just to 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 by
(d sinθ)/2 so that the minimum intensity occurs at an angle θmin
given by

d \sin \theta_ = \lambda \,

where d is the width of the slit.

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 \theta_ = n\lambda \,

where n is an integer greater than zero.

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

where the sinc
function is given by sinc(x)=sin(x)/x.

It should be noted that this analysis applies
only to the far field, i.e
a significant distance from the diffracting slit.

### 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- d \left( \sin + \sin \right) = m \lambda.

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.

### Diffraction by a 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- I(\theta) = I_0 \left ( \frac \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.

### Diffraction of a plane wave reflected by a mirror

Another conceptually simple example is
diffraction of a plane wave on a large (compared to the wavelength)
plane mirror. The only direction at which all electrons oscillating
in the mirror are seen oscillating in phase with each other is the
specular
(mirror) direction – thus a typical mirror reflects at the angle
which is equal to the angle of incidence of the wave. This result
is called the law of
reflection. Smaller and smaller mirrors diffract light over a
progressively larger and larger range of angles.

### 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. Diode lasers have much greater divergence than He-Ne lasers for this reason.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 co-incident with that of the first lens. The resulting beam has
a larger aperture, and hence a lower divergence.

### Diffraction limited imaging

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 is diffracted as described above. The light is not focused to a point but to a pattern of rings with a central spot of diameter- d = 1.22 \lambda \frac,\,

where λ is the wavelength of the light, f is the
focal length of the lens, and a is the diameter of the beam of
light, or (if the beam is filling the lens) the diameter of the
lens.

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 superpostion 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.## Common features of diffraction 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 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- \lambda=\frac \,

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 recently, larger molecules like
buckyballs, have been
shown to diffract. Currently, research is underway into the
diffraction of viruses,
which, being huge relative to electrons and other more
commonly diffracted particles, have tiny wavelengths so must be
made to travel very slowly through an extremely narrow slit in
order to diffract.

## Bragg diffraction

details Bragg diffraction 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:- m \lambda = 2 d \sin \theta \,

- 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. 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.

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
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 spacial extent of
the wave function that describes the particle.

## References

## See also

- Atmospheric diffraction
- Bragg diffraction
- Diffraction formalism
- Diffractometer
- Dynamical theory of diffraction
- Diffraction grating
- Electron diffraction
- Fraunhofer diffraction
- Fresnel diffraction
- Fresnel number
- Fresnel zone
- Iridescent Cloud
- Neutron diffraction
- Prism
- Powder diffraction
- Refraction
- Schaefer-Bergmann diffraction
- Thinned array curse
- X-ray diffraction

## External links

- How to build a diffraction spectrometer
- Diffraction and acoustics.
- On Diffraction at MathPages.
- Wave Optics - A chapter of an online textbook.
- 2-D wave Java applet - Displays diffraction patterns of various slit configurations.
- Diffraction Java applet - Displays diffraction patterns of various 2-D apertures.
- Diffraction approximations illustrated - MIT site that illustrates the various approximations in diffraction and intuitively explains the Fraunhofer regime from the perspective of linear system theory.
- Gap Obstacle Corner - Java simulation of diffraction of water wave.
- Google Maps - Satellite image of Panama Canal entry ocean wave diffraction.

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