Editing Diffraction (section)

== Examples ==
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.
{{Multiple image
| image1            = Screendiffraction.jpg
| image2            = Sunlight diffraction off of cd rom.jpg
| caption1          = Pixels on smart phone screen acting as diffraction grating
| caption2          = Data is written on CDs as pits and lands; the pits on the surface act as diffracting elements.
| direction         = horizontal
| align             = center
| perrow            = 
}}
This principle can be extended to engineer a grating with a structure such that it will produce any diffraction pattern desired; the [[holography|hologram]] on a credit card is an example.

[[Atmospheric diffraction|Diffraction in the atmosphere]] by small particles can cause a [[corona (optical phenomenon)|corona]] - a bright disc and rings around a bright light source like the sun or the moon. At the opposite point one may also observe [[glory (optical phenomenon)|glory]] - bright rings around the shadow of the observer. In contrast to the corona, glory requires the particles to be transparent spheres (like fog droplets), since the [[backscatter]]ing of the light that forms the glory involves [[refraction]] and internal reflection within the droplet.
{{Multiple image
| image1            = Lunar Halo .jpg
| image2            = IMG 7474 solar glory.JPG
| caption1          = Lunar [[corona (optical phenomenon)|corona]].
| caption2          = A solar [[glory (optical phenomenon)|glory]], as seen from a plane on the underlying clouds.
| direction         = horizontal
| align             = center
| perrow            = 
}}

A shadow of a solid object, using light from a compact source, shows small fringes near its edges. 
[[File:Poissonspot simulation d4mm.jpg|thumb|center|The bright spot ([[Arago spot]]) seen in the center of the shadow of a circular obstacle is due to diffraction]]

[[Diffraction spikes]] are diffraction patterns caused due to non-circular [[aperture]] in camera or support struts in telescope; In normal vision, diffraction through eyelashes may produce such spikes.
[[File:Night London Panorama with Full Moon.jpg|thumb|center|View from the end of Millennium Bridge; Moon rising above the Southwark Bridge. Street lights are reflecting in the Thames.]]
[[File:FOFC8ZPX0AIB-Ho.png|thumb|center|Simulated diffraction spikes in hexagonal telescope mirrors]]

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.<ref>{{cite web | last=Arumugam|first=Nadia | title=Food Explainer: Why Is Some Deli Meat Iridescent? | url=http://www.slate.com/blogs/browbeat/2013/09/09/iridescent_deli_meat_why_some_sliced_ham_and_beef_shine_with_rainbow_colors.html| work=Slate|date=9 September 2013 | publisher=[[The Slate Group]] | access-date=9 September 2013 | url-status=live | archive-url=https://web.archive.org/web/20130910021203/http://www.slate.com/blogs/browbeat/2013/09/09/iridescent_deli_meat_why_some_sliced_ham_and_beef_shine_with_rainbow_colors.html | archive-date=10 September 2013}}</ref> 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 [[jetty|jetties]] and other obstacles. [[File:Wave diffraction at the Blue Lagoon, Abereiddy.jpg|thumb|center|Circular waves generated by diffraction from the narrow entrance of a flooded coastal quarry]] Sound waves can diffract around objects, which is why one can still hear someone calling even when hiding behind a tree.<ref>
{{cite book
 |title = Dynamic fields and waves of physics
 |author = Andrew Norton
 |publisher = CRC Press
 |year = 2000
 |isbn = 978-0-7503-0719-2
 |page = 102
 |url = https://books.google.com/books?id=XRRMxjr24pwC&pg=PA102
}}</ref>
Diffraction can also be a concern in some technical applications; it sets a [[Diffraction-limited system|fundamental limit]] to the resolution of a camera, telescope, or microscope.

Other examples of diffraction are considered below.

=== Single-slit diffraction ===
{{Main|Diffraction formalism}}

[[File:DiffractionSingleSlit Anim.gif|thumb|2D Single-slit diffraction with width changing animation]] 
[[Image:Wave Diffraction 4Lambda Slit.png|right|thumb|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.]]
[[Image:Single Slit Diffraction (english).svg|right|thumb|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, in accordance with the [[Huygens–Fresnel principle]].

An illuminated slit that is wider than a wavelength produces interference effects in the space downstream of the slit.  Assuming that the slit behaves as though it has a large number of point sources spaced evenly across the width of the slit interference effects can be calculated. The analysis of this system is simplified if we consider light of a single wavelength. If the incident light is [[Coherence (physics)#Examples|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 <math>2\pi</math> 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 {{nowrap|<math>\lambda/2</math>.}} 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 <math>\frac{d \sin(\theta)}{2}</math> so that the minimum intensity occurs at an angle <math>\theta_\text{min}</math> given by
<math display="block">d\,\sin\theta_\text{min} = \lambda,</math>
where <math>d</math> is the width of the slit, <math>\theta_\text{min}</math> is the [[Angle of incidence (optics)|angle of incidence]] at which the minimum intensity occurs, and <math>\lambda</math> 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 <math>\theta_{n}</math> given by
<math display="block">d\,\sin\theta_{n} = n \lambda,</math>
where <math>n</math> is an integer other than zero.

There is no such simple argument to enable us to find the maxima of the diffraction pattern. The [[diffraction formalism#Quantitative analysis of single-slit diffraction|intensity profile]] can be calculated using the [[Fraunhofer diffraction]] equation as
<math display="block">I(\theta) = I_0 \, \operatorname{sinc}^2 \left( \frac{d \pi}{\lambda} \sin\theta \right),</math>
where <math>I(\theta)</math> is the intensity at a given angle, <math>I_0</math> is the intensity at the central maximum {{nowrap|(<math>\theta = 0</math>),}} which is also a normalization factor of the intensity profile that can be determined by an integration from <math display="inline">\theta = -\frac{\pi}{2}</math> to <math display="inline">\theta = \frac{\pi}{2}</math> and conservation of energy, and {{nowrap|<math>\operatorname{sinc} x = \frac{\sin x}{x}</math>,}} which is the [[unnormalized sinc function]].

This analysis applies only to the [[far field]] ([[Fraunhofer diffraction]]), that is, at a distance much larger than the width of the slit.

From the [[diffraction formalism#Quantitative analysis of single-slit diffraction|intensity profile]] above, if {{nowrap|<math>d \ll \lambda</math>,}} the intensity will have little dependency on {{nowrap|<math>\theta</math>,}} hence the wavefront emerging from the slit would resemble a cylindrical wave with azimuthal symmetry; If {{nowrap|<math>d \gg \lambda</math>,}} only <math>\theta \approx 0</math> would have appreciable intensity, hence the wavefront emerging from the slit would resemble that of [[geometrical optics]].

When the incident angle <math>\theta_\text{i}</math> of the light onto the slit is non-zero (which causes a change in the [[Optical path length|path length]]), the intensity profile in the Fraunhofer regime (i.e. far field) becomes:
<math display="block">I(\theta) = I_0 \, \operatorname{sinc}^2 \left[ \frac{d \pi}{\lambda} (\sin\theta \pm \sin\theta_\text{i})\right]</math>
The choice of plus/minus sign depends on the definition of the incident angle {{nowrap|<math>\theta_\text{i}</math>.}}[[File:Diffraction2vs5.jpg|right|thumb|2-slit (top) and 5-slit diffraction of red laser light]]
[[File:Diffraction-red laser-diffraction grating PNr°0126.jpg|thumb|left|Diffraction of a red laser using a diffraction grating]]
[[File:Diffraction 150 slits.jpg|right|thumb|A diffraction pattern of a 633 nm laser through a grid of 150 slits]]

=== Diffraction grating ===
{{Main|Diffraction grating}}

[[File:Diffraction grating demo.webm|thumb|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 ''θ''<sub>''m''</sub> which are given by the grating equation
<math display="block"> d \left( \sin{\theta_m} \pm \sin{\theta_i} \right) = m \lambda,</math>
where <math>\theta_{i}</math> is the angle at which the light is incident, <math>d</math> is the separation of grating elements, and <math>m</math> 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|Airy disk}}

[[Image:Airy-pattern.svg|thumb|A computer-generated image of an '''Airy disk''']]

[[Image:Fresnel to Fraunhofer transition.gif|thumb|Diffraction pattern from a circular aperture at various distances]]

The far-field diffraction of a plane wave incident on a circular aperture is often referred to as the [[Airy disk]].  The [[Airy disk#Mathematical details|variation]] in intensity with angle is given by
<math display="block">I(\theta) = I_0 \left ( \frac{2 J_1(ka \sin \theta)}{ka \sin \theta} \right )^2 ,</math>
where <math>a</math> is the radius of the circular aperture, <math>k</math> is equal to <math>2\pi/\lambda</math> and <math>J_1</math> 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 <math>\psi</math> at location <math>\mathbf r</math> that is given by the solution of the [[frequency domain]] [[wave equation]] for a point source (the [[Helmholtz equation]]),
<math display="block">\nabla^2 \psi + k^2 \psi = \delta(\mathbf r),</math>
where <math> \delta(\mathbf r)</math> 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
<math display="block">\nabla ^2\psi = \frac{1}{r} \frac {\partial ^2}{\partial r^2} (r \psi) .</math>

(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 <math>e^{-i \omega t}</math>) is
<math display="block">\psi(r) = \frac{e^{ikr}}{4 \pi r}.</math>

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 <math>\mathbf r'</math> and the field point is located at the point <math>\mathbf r</math>, then we may represent the scalar [[Green's function]] (for arbitrary source location) as
<math display="block">\psi(\mathbf r | \mathbf r') = \frac{e^{ik | \mathbf r - \mathbf r' | }}{4 \pi | \mathbf r - \mathbf r' |}.</math>

Therefore, if an electric field <math>E_\mathrm{inc}(x, y)</math> is incident on the aperture, the field produced by this aperture distribution is given by the [[surface integral]]
<math display="block">\Psi(r)\propto \iint\limits_\mathrm{aperture} \!\! E_\mathrm{inc}(x',y') ~ \frac{e^{ik | \mathbf r - \mathbf r'|}}{4 \pi | \mathbf r - \mathbf r' |} \,dx'\, dy',</math>

[[Image:Fraunhofer.svg|upright=1.4|thumb|On the calculation of Fraunhofer region fields]]
where the source point in the aperture is given by the vector
<math display="block">\mathbf{r}' = x' \mathbf{\hat{x}} + y' \mathbf{\hat{y}}.</math>

In the far field, wherein the parallel rays approximation can be employed, the Green's function,
<math display="block">\psi(\mathbf r | \mathbf r') = \frac{e^{ik | \mathbf r - \mathbf r' |} }{4 \pi | \mathbf r - \mathbf r' |},</math>
simplifies to
<math display="block"> \psi(\mathbf{r} | \mathbf{r}') = \frac{e^{ik r}}{4 \pi r} e^{-ik ( \mathbf{r}' \cdot \mathbf{\hat{r}})}</math>
as can be seen in the adjacent <!-- "adjacent" is not the best description but is better than "to the right". The latter is affected by screen size (esp smartphones), aspect ratio and font size. -->figure<!-- to the right -->.

The expression for the far-zone (Fraunhofer region) field becomes
<math display="block">\Psi(r)\propto \frac{e^{ik r}}{4 \pi r} \iint\limits_\mathrm{aperture} \!\! E_\mathrm{inc}(x',y') e^{-ik  ( \mathbf{r}' \cdot \mathbf{\hat{r}} ) } \, dx' \,dy'.</math>

Now, since
<math display="block">\mathbf{r}' = x' \mathbf{\hat{x}} + y' \mathbf{\hat{y}}</math>
and
<math display="block">\mathbf{\hat{r}} =  \sin \theta \cos \phi \mathbf{\hat{x}} + \sin \theta ~ \sin \phi ~ \mathbf{\hat{y}} +  \cos \theta \mathbf{\hat{z}},</math>
the expression for the Fraunhofer region field from a planar aperture now becomes
<math display="block">\Psi(r) \propto \frac{e^{ik r}}{4 \pi r} \iint\limits_\mathrm{aperture} \!\! E_\mathrm{inc}(x',y') e^{-ik \sin \theta (\cos \phi x' + \sin \phi y')} \, dx' \, dy'.</math>

Letting
<math display="block">k_x = k \sin \theta \cos \phi </math>
and
<math display="block">k_y = k \sin \theta \sin \phi \,,</math>
the Fraunhofer region field of the planar aperture assumes the form of a [[Fourier transform]]
<math display="block">\Psi(r)\propto \frac{e^{ik r}}{4 \pi r} \iint\limits_\mathrm{aperture} \!\! E_\mathrm{inc}(x',y') e^{-i (k_x x' + k_y y') } \, dx' \, dy' ,</math>

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|laser beam]] changes as it propagates is determined by diffraction. When the entire emitted beam has a planar, spatially [[Coherence (physics)|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.<ref>{{cite journal | last1=Chiao|first1=R. Y. | last2=Garmire|first2=E. | last3=Townes|first3=C. H. | title=Self-Trapping of Optical Beams | journal=Physical Review Letters | date=1964 | volume=13 | issue=15 | pages=479–482 | url=http://journals.aps.org/prl/pdf/10.1103/PhysRevLett.13.479 | bibcode = 1964PhRvL..13..479C | doi = 10.1103/PhysRevLett.13.479 | url-access=subscription }}</ref> 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 ===
{{Main|Diffraction-limited system}}

[[Image:zboo lucky image 1pc.png|frame|The Airy disk around each of the stars from the 2.56 m telescope aperture can be seen in this ''[[lucky imaging|lucky image]]'' of the [[binary star]] [[zeta Boötis]].]]

The ability of an imaging system to resolve detail is ultimately limited by [[diffraction-limited|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 whose radius (as measured to the first null) is
<math display="block"> \Delta x = 1.22 \lambda N ,</math>
where <math>\lambda</math> is the wavelength of the light and <math>N</math> is the [[f-number]] (focal length <math>f</math> divided by aperture diameter <math>D</math>) of the imaging optics; this is strictly accurate for <math>N \gg 1</math> ([[paraxial]] case).  In object space, the corresponding [[angular resolution]] is
<math display="block">  \theta \approx  \sin \theta = 1.22 \frac{\lambda}{D},</math>
where <math>D</math> 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 [[Angular resolution#The Rayleigh criterion|Rayleigh criterion]] specifies that two point sources are considered "resolved" 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 compared to the wavelength, the finer the resolution of an imaging system. This is one reason astronomical telescopes require large objectives, and why [[Objective (optics)#Microscope|microscope objective]]s require a large [[numerical aperture]] (large aperture diameter compared to working distance) in order to obtain the highest possible resolution.

=== Speckle patterns ===
{{Main|Speckle pattern}}

The [[speckle pattern]] 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 ===
{{Main|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.

=== "Knife edge" <span class="anchor" id="Knife edge"></span> ===
The '''knife-edge effect''' or '''knife-edge diffraction''' is a truncation of a portion of the incident [[radiation]] that strikes a sharp well-defined obstacle, such as a mountain range or the wall of a building.
The knife-edge effect is explained by the [[Huygens–Fresnel principle]], which states that a well-defined obstruction to an electromagnetic wave acts as a secondary source, and creates a new [[wavefront]]. This new wavefront propagates into the geometric shadow area of the obstacle.

Knife-edge diffraction is an outgrowth of the "[[half-plane]] problem", originally solved by [[Arnold Sommerfeld]] using a plane wave spectrum formulation.  A generalization of the half-plane problem is the "wedge problem", solvable as a boundary value problem in cylindrical coordinates. The solution in cylindrical coordinates was then extended to the optical regime by [[Joseph B. Keller]], who introduced the notion of diffraction coefficients through his [[geometrical theory of diffraction]] (GTD). In 1974, Prabhakar Pathak and [[Robert Kouyoumjian]] extended the (singular) Keller coefficients via the [[uniform theory of diffraction]] (UTD).<ref>{{cite journal |last1=Rahmat-Samii |first1=Yahya |title=GTD, UTD, UAT, and STD: A Historical Revisit and Personal Observations |journal=IEEE Antennas and Propagation Magazine |date=June 2013 |volume=55 |issue=3 |pages=29–40 |doi=10.1109/MAP.2013.6586622 |bibcode=2013IAPM...55...29R |author-link=Yahya Rahmat-Samii}}</ref><ref>{{cite journal |last1=Kouyoumjian |first1=R. G. |last2=Pathak |first2=P. H. |title=A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface |journal=[[Proceedings of the IEEE]] |date=November 1974 |volume=62 |issue=11 |pages=1448–1461 |doi=10.1109/PROC.1974.9651}}</ref>

<gallery mode="nolines" widths="300" heights="238">
File:Diffraction sharp edge.gif|Diffraction on a sharp metallic edge
File:Diffraction softest edge.gif|Diffraction on a soft aperture, with a gradient of conductivity over the image width
</gallery>