formulasearchengine - User contributions [en]

Dividend discount model

2014-12-24T09:08:40Z

66.27.64.119: /* Derivation of equation */ Three lines have been add following the first paragraph, and the subsequent text has been modified to clarify the derivation.

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List of mathematical functions

2014-03-05T06:04:50Z

66.27.71.59: /* Number theoretic functions */

Hello buddy. Let me introduce myself. I am Ron but I don't like when individuals use my complete title. Bookkeeping is what I do for a living. Her family members lives in Delaware but she needs to transfer because of her family members. One of the things I adore most is climbing and now I have time to consider on new issues. My blog extended car warranty - [http://Frivergames.com/profile/boashcroft my website] -

Shear modulus

2013-12-15T19:44:42Z

66.27.101.243:

{{refimprove|date=June 2013}}
[[Image:Roessler attractor.png|right|thumb|The Rössler attractor]]
[[Image:RosslerStereo.png|thumb|right|Rössler attractor as a [[stereogram]] with <math>a=0.2</math>, <math>b=0.2</math>, <math>c=14</math>]]
The '''Rössler attractor''' {{IPAc-en|ˈ|r|ɒ|s|l|ər}} is the [[attractor]] for the '''Rössler system''', a system of three [[non-linear dynamics|non-linear]] [[ordinary differential equation]]s originally studied by [[Otto Rössler]].<ref name="r76">{{citation
| last = Rössler | first = O. E. | author-link = Otto Rössler
| issue = 5
| journal = [[Physics Letters]]
| pages = 397–398
| title = An Equation for Continuous Chaos
| volume = 57A
| year = 1976}}.</ref><ref name="r79">{{citation
| last = Rössler | first = O. E. | author-link = Otto Rössler
| issue = 2,3
| journal = [[Physics Letters]]
| pages = 155–157
| title = An Equation for Hyperchaos
| volume = 71A
| year = 1979}}.</ref> These differential equations define a [[continuous-time dynamical system]] that exhibits [[Chaos theory|chaotic]] dynamics associated with the [[fractal]] properties of the attractor.<ref name="cfnf">{{citation
| last1 = Peitgen | first1 = Heinz-Otto | author1-link = Heinz-Otto Peitgen
| last2 = Jürgens | first2 = Hartmut | author2-link = Hartmut Jürgens
| last3 = Saupe | first3 = Dietmar | author3-link = Dietmar Saupe
| contribution = 12.3 The Rössler Attractor
| pages = 636–646
| publisher = Springer
| title = Chaos and Fractals: New Frontiers of Science
| year = 2004}}.</ref>

Some properties of the Rössler system can be deduced via linear methods such as [[eigenvector]]s, but the main features of the system require non-linear methods such as [[Poincaré map]]s and [[bifurcation diagram]]s. The original Rössler paper states the Rössler attractor was intended to behave similarly to the [[Lorenz attractor]], but also be easier to analyze qualitatively.<ref name="r76"/> An orbit within the attractor follows an outward spiral close to the <math>x, y</math> plane around an unstable fixed point. Once the graph spirals out enough, a second fixed point influences the graph, causing a rise and twist in the <math>z</math>-dimension. In the time domain, it becomes apparent that although each variable is oscillating within a fixed range of values, the oscillations are chaotic. This attractor has some similarities to the Lorenz attractor, but is simpler and has only one [[manifold]]. [[Otto Rössler]] designed the Rössler attractor in 1976,<ref name="r76"/> but the originally theoretical equations were later found to be useful in modeling equilibrium in chemical reactions.

==Definition==
The defining equations of the Rössler system are:<ref name="cfnf"/>

<math> \begin{cases} \frac{dx}{dt} = -y - z \\ \frac{dy}{dt} = x + ay \\ \frac{dz}{dt} = b + z(x-c) \end{cases} </math>

Otto E. Rössler studied the [[chaotic attractor]] with <math>a = 0.2</math>, <math>b = 0.2</math>, and <math>c = 5.7</math>, though properties of <math>a = 0.1</math>, <math>b = 0.1</math>, and <math>c = 14</math> have been more commonly used since. Another line of the parameter space was investigated using the topological analysis. It corresponds to <math>b = 2</math>, <math>c = 4</math>, and <math>a</math> was chosen as the bifurcation parameter.<ref>{{cite journal|last=Letellier|first=C.|coauthors=P. Dutertre & B. Maheu|title=Unstable periodic orbits and templates of the Rössler system: toward a systematic topological characterization|journal=Chaos|year=1995|volume=5|issue=1|pages=272–281|url=http://chaos.aip.org/resource/1/chaoeh/v5/i1/p271_s1?isAuthorized=no}}</ref> How Rössler discovered this set of equations was investigated in.<ref>{{cite journal|last=Letellier|first=C.|coauthors=V. Messager|title=Influences on Otto E. Rössler’s earliest paper on chaos|journal=International Journal of Bifurcation & Chaos|year=2010|volume=20|issue=11|pages=3585–3616|url=http://dblp.uni-trier.de/db/journals/ijbc/ijbc20.html#LetellierM10}}</ref>

== An analysis ==
[[Image:RosslerstdXY.png|framed|<math>x,y</math> plane of Rössler attractor with <math>a=0.2</math>, <math>b=0.2</math>, <math>c=5.7</math>]]
Some of the Rössler attractor's elegance is due to two of its equations being linear; setting <math>z = 0</math>, allows examination of the behavior on the <math>x, y</math> plane

<math> \begin{cases} \frac{dx}{dt} = -y \\ \frac{dy}{dt} = x + ay \end{cases} </math>

The stability in the <math>x, y</math> plane can then be found by calculating the [[eigenvalues]] of the [[Jacobian]] <math>\begin{pmatrix}0 & -1 \\ 1 & a\\\end{pmatrix}</math>, which are <math>(a \pm \sqrt{a^2 - 4})/2</math>. From this, we can see that when <math>0 < a < 2</math>, the eigenvalues are complex and both have a positive real component, making the origin unstable with an outwards spiral on the <math>x, y</math> plane. Now consider the <math>z</math> plane behavior within the context of this range for <math>a</math>. So long as <math>x</math> is smaller than <math>c</math>, the <math>c</math> term will keep the orbit close to the <math>x, y</math> plane. As the orbit approaches <math>x</math> greater than <math>c</math>, the <math>z</math>-values begin to climb. As <math>z</math> climbs, though, the <math>-z</math> in the equation for <math>dx/dt</math> stops the growth in <math>x</math>.

=== Fixed points ===
In order to find the fixed points, the three Rössler equations are set to zero and the (<math>x</math>,<math>y</math>,<math>z</math>) coordinates of each fixed point were determined by solving the resulting equations. This yields the general equations of each of the fixed point coordinates:

<math> \begin{cases} x = \frac{c\pm\sqrt{c^2-4ab}}{2} \\ y=-\left(\frac{c\pm\sqrt{c^2-4ab}}{2a}\right) \\ z=\frac{c\pm\sqrt{c^2-4ab}}{2a} \end{cases} </math>

Which in turn can be used to show the actual fixed points for a given set of parameter values:
: <math>\left(\frac{c+\sqrt{c^2-4ab}}{2}, \frac{-c-\sqrt{c^2-4ab}}{2a}, \frac{c+\sqrt{c^2-4ab}}{2a}\right)</math>
: <math>\left(\frac{c-\sqrt{c^2-4ab}}{2}, \frac{-c+\sqrt{c^2-4ab}}{2a}, \frac{c-\sqrt{c^2-4ab}}{2a}\right)</math>

As shown in the general plots of the Rössler Attractor above, one of these fixed points resides in the center of the attractor loop and the other lies comparatively removed from the attractor.

=== Eigenvalues and eigenvectors ===
The stability of each of these fixed points can be analyzed by determining their respective eigenvalues and eigenvectors. Beginning with the Jacobian:

<math>\begin{pmatrix}0 & -1 & -1 \\ 1 & a & 0 \\ z & 0 & x-c\\\end{pmatrix}</math>

the eigenvalues can be determined by solving the following cubic:

<math>-\lambda^3+\lambda^2(a+x-c) + \lambda(ac-ax-1-z)+x-c+az =0\,</math>

For the centrally located fixed point, Rössler’s original parameter values of a=0.2, b=0.2, and c=5.7 yield eigenvalues of:
: <math>\lambda_{1}= 0.0971028 + 0.995786i \,</math>
: <math>\lambda_{2}= 0.0971028 - 0.995786i \,</math>
: <math>\lambda_{3}= -5.68718 \,</math>
(Using Mathematica 7)

The magnitude of a negative eigenvalue characterizes the level of attraction along the corresponding eigenvector. Similarly the magnitude of a positive eigenvalue characterizes the level of repulsion along the corresponding eigenvector.

The eigenvectors corresponding to these eigenvalues are:
: <math>v_{1}= \begin{pmatrix} 0.7073 \\ -0.07278 - 0.7032i \\ 0.0042 - 0.0007i \\\end{pmatrix}</math>
: <math>v_{2}= \begin{pmatrix}0.7073 \\ 0.07278 + 0.7032i \\ 0.0042 + 0.0007i \\\end{pmatrix}</math>
: <math>v_{3}= \begin{pmatrix}0.1682 \\ -0.0286 \\ 0.9853 \\\end{pmatrix}</math>
[[Image:Eigenvectors.png|thumb|right|Examination of central fixed point eigenvectors: The blue line corresponds to the standard Rössler attractor generated with <math>a=0.2</math>, <math>b=0.2</math>, and <math>c=5.7</math>.]]
[[Image:Rosslerstd3D.png|framed|Rössler attractor with <math>a=0.2</math>, <math>b=0.2</math>, <math>c=5.7</math>]]
These eigenvectors have several interesting implications. First, the two eigenvalue/eigenvector pairs (<math>v_{1}</math> and <math>v_{2}</math>) are responsible for the steady outward slide that occurs in the main disk of the attractor. The last eigenvalue/eigenvector pair is attracting along an axis that runs through the center of the manifold and accounts for the z motion that occurs within the attractor. This effect is roughly demonstrated with the figure below.

The figure examines the central fixed point eigenvectors. The blue line corresponds to the standard Rössler attractor generated with <math>a=0.2</math>, <math>b=0.2</math>, and <math>c=5.7</math>. The red dot in the center of this attractor is <math>FP_{1}</math>. The red line intersecting that fixed point is an illustration of the repulsing plane generated by <math>v_{1}</math> and <math>v_{2}</math>. The green line is an illustration of the attracting <math>v_{3}</math>. The magenta line is generated by stepping backwards through time from a point on the attracting eigenvector which is slightly above <math>FP_{1}</math> – it illustrates the behavior of points that become completely dominated by that vector. Note that the magenta line nearly touches the plane of the attractor before being pulled upwards into the fixed point; this suggests that the general appearance and behavior of the Rössler attractor is largely a product of the interaction between the attracting <math>v_{3}</math> and the repelling <math>v_{1}</math> and <math>v_{2}</math> plane. Specifically it implies that a sequence generated from the Rössler equations will begin to loop around <math>FP_{1}</math>, start being pulled upwards into the <math>v_{3}</math> vector, creating the upward arm of a curve that bends slightly inward toward the vector before being pushed outward again as it is pulled back towards the repelling plane.

For the outlier fixed point, Rössler’s original parameter values of <math>a=0.2</math>, <math>b=0.2</math>, and <math>c=5.7</math> yield eigenvalues of:
: <math>\lambda_{1}= -0.0000046 + 5.4280259i</math>
: <math>\lambda_{2}= -0.0000046 - 5.4280259i </math>
: <math>\lambda_{3}= 0.1929830</math>

The eigenvectors corresponding to these eigenvalues are:
: <math>v_{1}= \begin{pmatrix}0.0002422 + 0.1872055i \\ 0.0344403 - 0.0013136i \\ 0.9817159 \\\end{pmatrix}</math>
: <math>v_{2}= \begin{pmatrix}0.0002422 - 0.1872055i \\ 0.0344403 + 0.0013136i \\ 0.9817159 \\\end{pmatrix}</math>
: <math>v_{3}= \begin{pmatrix}0.0049651 \\ -0.7075770 \\ 0.7066188 \\\end{pmatrix}</math>

Although these eigenvalues and eigenvectors exist in the Rössler attractor, their influence is confined to iterations of the Rössler system whose initial conditions are in the general vicinity of this outlier fixed point. Except in those cases where the initial conditions lie on the attracting plane generated by <math>\lambda_{1}</math> and <math>\lambda_{2}</math>, this influence effectively involves pushing the resulting system towards the general Rössler attractor. As the resulting sequence approaches the central fixed point and the attractor itself, the influence of this distant fixed point (and its eigenvectors) will wane.

=== Poincaré map ===
[[Image:Poincare2.png|framed|Poincaré map for Rössler attractor with <math>a=0.1</math>, <math>b=0.1</math>, <math>c=14</math>]]
The [[Poincaré map]] is constructed by plotting the value of the function every time it passes through a set plane in a specific direction. An example would be plotting the <math>y, z</math> value every time it passes through the <math>x = 0</math> plane where <math>x</math> is changing from negative to positive, commonly done when studying the Lorenz attractor. In the case of the Rössler attractor, the <math>x = 0</math> plane is uninteresting, as the map always crosses the <math>x = 0 </math> plane at <math>z = 0</math> due to the nature of the Rössler equations. In the <math>x=0.1</math> plane for <math>a=0.1</math>, <math>b=0.1</math>, <math>c=14</math>, the Poincaré map shows the upswing in <math>z</math> values as <math>x</math> increases, as is to be expected due to the upswing and twist section of the Rössler plot. The number of points in this specific Poincaré plot is infinite, but when a different <math>c</math> value is used, the number of points can vary. For example, with a <math>c</math> value of 4, there is only one point on the Poincaré map, because the function yields a periodic orbit of period one, or if the <math>c</math> value is set to 12.8, there would be six points corresponding to a period six orbit.

=== Mapping local maxima ===
[[Image:Tentmap.png|framed|<math>Z_n</math> vs. <math>Z_{n+1}</math>]]
In the original paper on the Lorenz Attractor,<ref>{{citation
| last = Lorenz | first = E. N. | author-link = Edward Norton Lorenz
| bibcode=1963JAtS...20..130L
| doi = 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2
| issue = 2
| journal = [[J. Atmos. Sci.]]
| pages = 130–141
| title = Deterministic nonperiodic flow
| volume = 20
| year = 1963}}.</ref> [[Edward Lorenz]] analyzed the local maxima of <math>z</math> against the immediately preceding local maxima. When visualized, the plot resembled the tent map, implying that similar analysis can be used between the map and attractor. For the Rössler attractor, when the <math>z_n</math> local maximum is plotted against the next local <math>z</math> maximum, <math>z_{n+1}</math>, the resulting plot
(shown here for <math>a=0.2</math>, <math>b=0.2</math>, <math>c=5.7</math>) is unimodal, resembling a skewed [[Hénon map]]. Knowing that the Rössler attractor can be used to create a pseudo 1-d map, it then follows to use similar analysis methods. The bifurcation diagram is specifically a useful analysis method.

===Variation of parameters===
Rössler attractor's behavior is largely a factor of the values of its constant parameters <math>a</math>, <math>b</math>, and <math>c</math>. In general, varying each parameter has a comparable effect by causing the system to converge toward a periodic orbit, fixed point, or escape towards infinity, however the specific ranges and behaviors induced vary substantially for each parameter. Periodic orbits, or "unit cycles," of the Rössler system are defined by the number of loops around the central point that occur before the loops series begins to repeat itself.

[[Bifurcation diagram]]s are a common tool for analyzing the behavior of [[dynamical system]]s, of which the Rössler attractor is one. They are created by running the equations of the system, holding all but one of the variables constant and varying the last one. Then, a graph.is plotted of the points that a particular value for the changed variable visits after transient factors have been neutralised. Chaotic regions are indicated by filled-in regions of the plot.

====Varying a====
Here, <math>b</math> is fixed at 0.2, <math>c</math> is fixed at 5.7 and <math>a</math> changes. Numerical examination of the attractor's behavior over changing <math>a</math> suggests it has a disproportional influence over the attractor's behavior. The results of the analysis are:
* <math>a \leq 0</math>: Converges to the centrally located fixed point
* <math>a = 0.1 </math>: Unit cycle of period 1
* <math>a = 0.2 </math>: Standard parameter value selected by Rössler, chaotic
* <math>a = 0.3</math>: Chaotic attractor, significantly more [[Möbius strip]]-like (folding over itself).
* <math>a = 0.35</math>: Similar to .3, but increasingly chaotic
* <math>a = 0.38</math>: Similar to .35, but increasingly chaotic.

====Varying b====
[[Image:Bifurcation DiagramB.png|thumb|Bifurcation diagram for the Rössler attractor for varying <math>b</math>]]
Here, <math>a</math> is fixed at 0.2, <math>c</math> is fixed at 5.7 and <math>b</math> changes. As shown in the accompanying diagram, as <math>b</math> approaches 0 the attractor approaches infinity (note the upswing for very small values of <math>b</math>. Comparative to the other parameters, varying <math>b</math> generates a greater range when period-3 and period-6 orbits will occur. In contrast to <math>a</math> and <math>c</math>, higher values of <math>b</math> converge to period-1, not to a chaotic state.

====Varying c====
[[Image:Bifurcation.png|framed|Bifurcation diagram for the Rössler attractor for varying <math>c</math>]]
Here, <math>a = b = 0.1</math> and <math>c</math> changes. The [[bifurcation diagram]] reveals that low values of <math>c</math> are periodic, but quickly become chaotic as <math>c</math> increases. This pattern repeats itself as <math>c</math> increases – there are sections of periodicity interspersed with periods of chaos, and the trend is towards higher-period orbits as <math>c</math> increases. For example, the period one orbit only appears for values of <math>c</math> around 4 and is never found again in the bifurcation diagram. The same phenomenon is seen with period three; until <math>c=12</math>, period three orbits can be found, but thereafter, they do not appear.

A graphical illustration of the changing attractor over a range of <math>c</math> values illustrates the general behavior seen for all of these parameter analyses – the frequent transitions between periodicity and aperiodicity.

[[Image:VaryingC.png|center|Variations in the post-transient Rössler system as <math>c</math> is varied over a range of values.]]
The above set of images illustrates the variations in the post-transient Rössler system as <math>c</math> is varied over a range of values. These images were generated with <math>a=b=.1</math>.
*<math>c = 4</math>, period-1 orbit.
*<math>c = 6</math>, period-2 orbit.
*<math>c = 8.5</math>, period-4 orbit.
*<math>c = 8.7</math>, period-8 orbit.
*<math>c = 9</math>, sparse chaotic attractor.
*<math>c = 12</math>, period-3 orbit.
*<math>c = 12.6</math>, period-6 orbit.
* <math>c = 13</math>, sparse chaotic attractor.
*<math>c = 18</math>, filled-in chaotic attractor.

== Links to other topics ==
The banding evident in the Rössler attractor is similar to a [[Cantor set]] rotated about its midpoint. Additionally, the half-twist that occurs in the Rössler attractor only affects a part of the attractor. Rössler showed that its attractor was in fact the combination of a "normal band" and a [[Möbius strip]].<ref>{{cite journal|last=Rössler|first=Otto E.|title=Chaotic behavior in simple reaction system|journal=Zeitschrift für Naturfoschung A|year=1976|volume=31|pages=259–264|url=http://www.atomosyd.net/spip.php?article6}}</ref>

== References ==
{{reflist}}

==External links==
* [http://lagrange.physics.drexel.edu/flash/rossray Flash Animation using PovRay]
* [http://www.soe.ucsc.edu/classes/ams214/Winter09/foundingpapers/Rossler1976.pdf]
* [http://to-campos.planetaclix.pt/fractal/lorenz_eng.html Lorenz and Rössler attractors] – Java animation
* [http://amath.colorado.edu/faculty/juanga/3DAttractors.html 3D Attractors: Mac program to visualize and explore the Rössler and Lorenz attractors in 3 dimensions]
* [http://scholarpedia.org/article/Rossler_attractor Rössler attractor in Scholarpedia]

{{Chaos theory}}

{{DEFAULTSORT:Rossler Attractor}}
[[Category:Chaotic maps]]

Zwanzig projection operator

2013-06-15T02:04:15Z

66.27.100.204: /* Slow variables and scalar product */ changed "partices" to "particles"

{{multiple issues|
{{confusing|date=February 2011}}
{{more footnotes|date=February 2011}}
}}

The '''Van Cittert–Zernike theorem''' is a formula in [[coherence theory]] that states that under certain conditions the [[Fourier transform]] of the [[mutual coherence]] function of a distant, incoherent source is equal to its complex [[Interferometric visibility|visibility]]. This implies that the [[wavefront]] from an incoherent source will appear mostly coherent at large distances. Intuitively, this can be understood by considering the wavefronts created by two incoherent sources. If we measure the wavefront immediately in front of one of the sources, our measurement will be dominated by the nearby source. If we make the same measurement far from the sources, our measurement will no longer be dominated by a single source; both sources will contribute almost equally to the wavefront at large distances.

This reasoning can be easily visualized by dropping two stones in the center of a calm pond. Near the center of the pond, the disturbance created by the two stones will be very complicated. As the disturbance propagates towards the edge of pond, however, the waves will smooth out and will appear to be nearly circular.

The van Cittert–Zernike theorem has important implications for [[radio astronomy]]. With the exception of [[pulsars]] and [[masers]], all astronomical sources are spatially incoherent. Nevertheless, because they are observed at distances large enough to satisfy the van Cittert–Zernike theorem, these objects exhibit a non-zero degree of coherence at different points in the imaging plane. By measuring the [[degree of coherence]] at different points in the imaging plane (the so-called "[[interferometric visibility|visibility]] function") of an astronomical object, a radio astronomer can thereby reconstruct the source's brightness distribution and make a two-dimensional map of the source's appearance.

==Statement of the theorem==
If <math>\Gamma_{12}(u,v,0)</math> is the mutual coherence function between two points on a plane perpendicular to the line of sight, then

:<math>\Gamma_{12} (u,v,0) = \iint I(l,m) e^{-2\pi i(ul+vm)} \, dl \, dm</math>

where <math>l</math> and <math>m</math> are the direction cosines of a point on a distant source and <math>I</math> is the intensity of the source. This theorem was first derived by [[Pieter H. van Cittert]]<ref name="vancittert">{{cite journal | author = P.H. van Cittert | year = 1934 | title = Die Wahrscheinliche Schwingungsverteilung in Einer von Einer Lichtquelle Direkt Oder Mittels Einer Linse Beleuchteten Ebene | trans_title = | journal = Physica | volume = 1 | pages = 201–210 | doi = 10.1016/S0031-8914(34)90026-4|bibcode = 1934Phy.....1..201V }}</ref> in 1934 with a simpler proof provided by [[Frits Zernike]] in 1938.<ref name="zernike">{{cite journal | author = F. Zernike | year = 1938 | title = The concept of degree of coherence and its application to optical problems | trans_title = | journal = Physica | volume = 5 | pages = 785–795 | doi = 10.1016/S0031-8914(38)80203-2|bibcode = 1938Phy.....5..785Z }}</ref>

==The mutual coherence function==

The space-time mutual coherence function for some [[electric field]] <math>E(t)</math> measured at two points in a plane of observation (call them 1 and 2), is defined to be

:<math>\Gamma_{12} (u, v, \tau) = \lim_{T \to \infty} \frac{1}{2T} \int_{-T}^T E_1(t) E_2^*(t-\tau) dt</math>

where <math>\tau</math> is the time offset between the measurement of <math>E(t)</math> at observation points 1 and 2, while <math>u</math> and <math>v</math> give the number of [[wavelength]]s between points 1 and 2 along the ''x''- and ''y''-axes of the observation plane, respectively. A special case of the mutual coherence function when <math>\tau = 0</math> is called the visibility function and measures the equal-delay spatial coherence.<ref name="thompson">Thompson, Moran, and Swenson, ''Interferometry and Synthesis in Radio Astronomy'', pp. 595</ref>

The mutual coherence between two points may be thought of as the time-averaged cross-correlation between the electric fields at the two points separated in time by <math>\tau</math>. Thus, if we are observing two fully incoherent sources we should expect the mutual coherence function to be relatively small between the two random points in the observation plane, because the sources will interfere destructively as well as constructively. Far away from the sources, however, we should expect the mutual coherence function to be relatively large because the sum of the observed fields will be almost the same at any two points.

Normalization of the mutual coherence function to the product of the square roots of the intensities of the two electric fields yields the complex degree of second-order coherence (correlation coefficient function):

:<math>\gamma_{12} (\tau) = \frac{\Gamma_{12}(\tau)}{\sqrt{I_1} \sqrt{I_2}}</math>

==Proof of the theorem==

Consider a distant, incoherent, extended source located in a plane which is defined by two axes called the ''X''- and ''Y''-axes. This source is observed in a parallel plane defined by two axes which we shall call the ''x''- and ''y''-axes. Suppose the electric field due to some point from this source is measured at two points, <math>P_1</math> and <math>P_2</math>, in the observation plane. The position of a point in the source may be referred to by its direction cosines <math>(l, m)</math>. (Since the source is distant, its direction should be the same at <math>P_1</math> as at <math>P_2</math>.) The electric field measured at <math>P_1</math> can then be written using [[phasor]]s:

[[File:Van Cittert-Zernike theorem.jpg|thumb|The source is in the ''XY''-plane, shown at the top of the figure, and the detector is in the ''xy''-plane, shown at the bottom of the figure. Consider the electric field at two points, <math>P_1</math> and <math>P_2</math>, in the detection plane due to some point in the source whose coordinates are given by the direction cosines <math>l</math> and <math>m</math>]]

:<math>E_1(l, m, t) = A \left( l, m, t - \frac{R_1}{c} \right) \frac{e^{-i \omega \left( t - \frac{R_1}{c} \right) }}{R_1}</math>

where <math>R_1</math> is the distance from the source to <math>P_1</math>, <math>\omega</math> is the [[angular frequency]] of the [[light]], and <math>A</math> is the complex amplitude of the electric field. Similarly, the electric field measured at <math>P_2</math> can be written as

:<math>E_2(l, m, t) = A \left( l, m, t - \frac{R_2}{c} \right) \frac{ e^{-i \omega \left( t - \frac{R_2}{c} \right) }}{R_2}</math>

Let us now calculate the time-averaged cross-correlation between the electric field at <math>P_1</math> and <math>P_2</math>:

:<math>\big \langle E_1(l, m, t) E_2^*(l, m, t) \big \rangle = \Bigg \langle A \left( l, m, t - \frac{R_1}{c} \right) A^* \left( l, m, t - \frac{R_2}{c} \right) \Bigg \rangle \times \frac{e^{-i \omega \left( t - \frac{R_1}{c} \right) }}{R_1} \times \frac{e^{i \omega \left(t - \frac{R_2}{c} \right)}}{R_2}</math>

Because the quantity in the angle brackets is time-averaged an arbitrary offset to the temporal term of the amplitudes may be added as long as the same offset is added to both. Let us now add <math>\frac{R_1}{c}</math> to the temporal term of both amplitudes. The time-averaged cross-correlation of the electric field at the two points therefore simplifies to

:<math>\big \langle E_1(l, m, t) E_2^*(l, m, t) \big \rangle = \Bigg \langle A (l, m, t) A^* \left( l, m, t - \frac{R_2 - R_1}{c} \right) \Bigg \rangle \times \frac{ e^{i \omega \left( \frac{R_1 - R_2}{c} \right)}}{R_1 R_2}</math>

But if the source is in the [[far field]] then the difference between <math>R_1</math> and <math>R_2</math> will be small compared to the distance light travels in time <math>t</math>. (<math>t</math> is on the same order as the inverse [[Band matrix|bandwidth]].) This small correction can therefore be neglected, further simplifying our expression for the cross-correlation of the electric field at <math>P_1</math> and <math>P_2</math> to

:<math>\langle E_1(l, m, t) E_2^*(l, m, t) \rangle = \langle A(l, m, t) A^*(l, m, t) \rangle \times \frac{e^{i \omega \left( \frac{R_1 - R_2}{c} \right)}}{R_1 R_2}</math>

Now, <math>\langle A(l, m, t) A^*(l, m, t) \rangle</math> is simply the intensity of the source at a particular point, <math>I(l, m)</math>. So our expression for the cross-correlation simplifies further to

:<math>\langle E_1(l, m, t) E_2^*(l, m, t) \rangle = I(l, m) \frac{e^{i \omega \left( \frac{R_1 - R_2}{c} \right) }}{R_1 R_2}</math>

To calculate the mutual coherence function from this expression, simply integrate over the entire source.

:<math>\Gamma_{12} (u, v, 0) = \iint_{\textrm{source}} I(l, m) \frac{e^{i \omega \left( \frac{R_1 - R_2}{c} \right) }}{R_1 R_2} \, dS</math>

Note that cross terms of the form <math>\langle A_1 (l, m, t) A_2^* (l, m, t) \rangle</math> are not included due to the assumption that the source is incoherent. The time-averaged correlation between two different points from the source will therefore be zero.

Next rewrite the <math>R_2 - R_1</math> term using <math>u, v, l</math> and <math>m</math>. To do this, let <math>P_1 = (x_1, y_1)</math> and <math>P_2 = (x_2, y_2)</math>. This gives

:<math>R_1 = \sqrt{R^2 + x_1^2 + y_1^2} \, </math>

:<math>R_2 = \sqrt{R^2 + x_2^2 + y_2^2} \, </math>

where <math>R</math> is the distance between the center of the plane of observation and the center of the source. The difference between <math>R_1</math> and <math>R_2</math> thus becomes

:<math>R_2 - R_1 = R \sqrt{1 + \frac{x_2^2}{R^2} + \frac{y_2^2}{R^2}} - R \sqrt{1 + \frac{x_1^2}{R^2} + \frac{y_1^2}{R^2}}</math>

But because <math>x_1, x_2, y_1</math> and <math>y_2</math> are all much less than <math>R</math>, the square roots may be [[Taylor series|Taylor expanded]], yielding, to first order,

:<math>R_2 - R_1 = R \left( 1 + \frac{1}{2} \left( \frac{x_2^2 + y_2^2}{R^2} \right) \right) - R \left( 1 + \frac{1}{2} \left( \frac{x_1^2 + y_1^2}{R^2} \right) \right)</math>

which, after some algebraic manipulation, simplifies to

:<math>R_2 - R_1 = \frac{1}{2R} \left( (x_2 - x_1)(x_2 + x_1) + (y_2 - y_1)(y_2 + y_1) \right)</math>

Now, <math>\frac{1}{2}(x_2 + x_1)</math> is the midpoint along the <math>x</math>-axis between <math>P_1</math> and <math>P_2</math>, so <math>\frac{1}{2R}(x_2 + x_1)</math> gives us <math>l</math>, one of the direction cosines to the sources. Similarly, <math>m = \frac{1}{2R}(y_2 + y_1)</math>. Moreover, recall that <math>u</math> was defined to be the number of wavelengths along the <math>x</math>-axis between <math>P_1</math> and <math>P_2</math>. So

:<math>u = \frac{\omega}{2 \pi c} (x_1 - x_2)</math>

Similarly, <math>v</math> is the number of wavelengths between <math>P_1</math> and <math>P_2</math> along the <math>y</math>-axis, so

:<math>v = \frac{\omega}{2 \pi c} (y_1 - y_2)</math>

Hence

:<math>R_2 - R_1 = \frac{2 \pi c}{\omega}(ul + vm)</math>

Because <math>x_1, x_2, y_1,</math> and <math>y_2</math> are all much less than <math>R</math>, <math>R_1 \simeq R_2 \simeq R</math>. The differential area element, <math>dS</math>, may then be written as a differential element of [[solid angle]] of <math>R^2 \, dl \, dm</math>. Our expression for the mutual coherence function becomes

:<math>\Gamma_{12}(u, v, 0) = \iint_{\textrm{source}} I(l, m) e^{- \frac{i \omega}{c} \frac{2 \pi c}{ \omega} (ul + vm)} \, dl \, dm</math>

Which reduces to

:<math>\Gamma_{12} (u, v, 0) = \iint_{\textrm{source}} I(l, m) e^{-2 \pi i (ul + vm)} \, dl \, dm</math>

But the limits of these two integrals can be extended to cover the entire plane of the source as long as the source's intensity function is set to be zero over these regions. Hence,

:<math>\Gamma_{12}(u, v, 0) = \iint I(l, m) e^{-2 \pi i (ul + vm)} \, dl \, dm</math>

which is the two-dimensional Fourier transform of the intensity function. This completes the proof.

==Assumptions of the theorem==

The van Cittert–Zernike theorem rests on a number of assumptions, all of which are approximately true for nearly all astronomical sources. The most important assumptions of the theorem and their relevance to astronomical sources are discussed here.

===Incoherence of the source===

A spatially coherent source does not obey the van Cittert–Zernike theorem. To see why this is, suppose we observe a source consisting of two points, <math>a</math> and <math>b</math>. Let us calculate the mutual coherence function between <math>P_1</math> and <math>P_2</math> in the plane of observation. From the [[principle of superposition]], the electric field at <math>P_1</math> is

:<math>E_1 = E_{a1} + E_{b1}</math>

and at <math>P_2</math> is

:<math>E_2 = E_{a2} + E_{b2}</math>

so the mutual coherence function is

:<math>\langle E_1(t) E_2^*(t - \tau) \rangle = \langle ( E_{a1}(t) + E_{b1}(t) ) ( E_{a2}^*(t - \tau) + E_{b2}^*(t - \tau)) \rangle</math>

Which becomes

:<math>\langle E_1(t) E_2^*(t - \tau) \rangle = \langle E_{a1}(t) E_{a2}^*(t - \tau) \rangle + \langle E_{a1}(t) E_{b2}^*(t - \tau) \rangle + \langle E_{b1}(t) E_{a2}^*(t - \tau) \rangle + \langle E_{b1}(t) E_{b2}^*(t - \tau) \rangle</math>

If points <math>a</math> and <math>b</math> are coherent then the cross terms in the above equation do not vanish. In this case, when we calculate the mutual coherence function for an extended coherent source, we would not be able to simply integrate over the intensity function of the source; the presence of non-zero cross terms would give the mutual coherence function no simple form.

This assumption holds for most astronomical sources. Pulsars and masers are the only astronomical sources which exhibit coherence.

===Distance to the source===

In the proof of the theorem we assume that <math>R \gg x_1 - x_2</math> and <math>R \gg y_1 - y_2</math>. That is, we assume that the distance to the source is much greater than the size of the observation area. More precisely, the van Cittert–Zernike theorem requires that we observe the source in the so-called far field. Hence if <math>D</math> is the characteristic size of the observation area (e.g. in the case of a two-dish [[radio telescope]], the length of the baseline between the two telescopes) then

:<math>R \gg \frac{D^2}{\lambda}</math>

Using a reasonable baseline of 20 km for the [[Very Large Array]] at a wavelength of 1 cm, the far field distance is of order <math>4 \times 10^{10}</math> m. Hence any astronomical object farther away than a [[parsec]] is in the far field. Objects in the [[Solar System]] are not necessarily in the far field, however, and so the van Cittert–Zernike theorem does not apply to them.

===Angular size of the source===

In the derivation of the van Cittert–Zernike theorem we write the direction cosines <math>l</math> and <math>m</math> as <math>\frac{1}{2}(x_1+x_2)/R</math> and <math>\frac{1}{2}(y_1+y_2)/R</math>. There is, however, a third direction cosine which is neglected since <math>R \gg \frac{1}{2}(x_1 + x_2)</math> and <math>R \gg \frac{1}{2}(y_1 + y_2)</math>; under these assumptions it is very close to unity. But if the source has a large angular extent, we cannot neglect this third direction cosine and the van Cittert–Zernike theorem no longer holds.

Because most astronomical sources subtend very small angles on the sky (typically much less than a degree), this assumption of the theorem is easily fulfilled in the domain of radio astronomy.

===Quasi-monochromatic waves===

The van Cittert–Zernike theorem assumes that the source is quasi-monochromatic. That is, if the source emits light over a range of frequencies, <math>\Delta \nu</math>, with mean frequency <math>\nu</math>, then it should satisfy

:<math>\frac{\Delta \nu}{\nu} \lesssim 1</math>

Moreover, the bandwidth must be narrow enough that

:<math>\frac{\Delta \nu}{\nu} \ll \frac{1}{l u}</math>

where <math>l</math> is again the direction cosine indicating the size of the source and <math>u</math> is the number of wavelengths between one end of the aperture and the other. Without this assumption, we cannot neglect <math>(R_2 - R_1)/c</math> compared to <math>t</math>

This requirement implies that a radio astronomer must restrict signals through a [[bandpass filter]]. Because radio telescopes almost always pass the signal through a relatively narrow bandpass filter, this assumption is typically satisfied in practice.

===Two-dimensional source===

We assume that our source lies in a two-dimensional plane. In reality, astronomical sources are three-dimensional. However, because they are in the far field, their angular distribution does not change with distance. Therefore when we measure an astronomical source, its three dimensional structure becomes projected upon a two-dimensional plane. This means that the van Cittert–Zernike theorem may be applied to measurements of astronomical sources, but we cannot determine structure along the line of sight with such measurements.

===Homogeneity of the medium===

The van Cittert–Zernike theorem assumes that the medium between the source and the imaging plane is homogeneous. If the medium is not homogeneous then light from one region of the source will be differentially [[refraction|refracted]] relative to other regions of the source due to the difference in light travel time through the medium. In the case of a heterogeneous medium one must use a generalization of the van Cittert–Zernike theorem, called Hopkins's formula.

Because the wavefront does not pass through a perfectly uniform medium as it travels through the [[interstellar medium|interstellar]] (and possibly [[intergalactic medium|intergalactic]]) medium and into the [[Earth's atmosphere]], the van Cittert–Zernike theorem does not hold exactly true for astronomical sources. In practice, however, variations in the [[refractive index]] of the interstellar and intergalactic media and Earth's atmosphere are small enough that the theorem is approximately true to within any reasonable experimental error. Such variations in the refractive index of the medium result only in slight perturbations from the case of a wavefront traveling through a homogeneous medium.

==Hopkins' formula==

Suppose we have a situation identical to that considered when the van Cittert–Zernike theorem was derived, except that the medium is now heterogeneous. We therefore introduce the transmission function of the medium, <math>K(l, m, P, \nu)</math>. Following a similar derivation as before, we find that

:<math>\Gamma_{12}(l, m, 0) = \lambda^2 \iint I(l, m) K(l, m, P_1, \nu) K^*(l, m, P_2, \nu) \, dS</math>

If we define

:<math>U(l, m, P_1) \equiv i \lambda K(l, m, P_1, \nu) \sqrt{I(l, m)}</math>

then the mutual coherence function becomes

:<math>\Gamma_{12}(l, m, 0) = \iint U(l, m, P_1) U^*(l, m, P_2) \, dS</math>

which is Hopkins's generalization of the van Cittert–Zernike theorem.<ref name="born">Born and Wolf, ''Principles of Optics'', pp. 510</ref> In the special case of a homogeneous medium, the transmission function becomes

:<math>K(l, m, P, \nu) = -\frac{i e^{ikR}}{\lambda R}</math>

in which case the mutual coherence function reduces to the Fourier transform of the brightness distribution of the source. The primary advantage of Hopkins's formula is that one may calculate the mutual coherence function of a source indirectly by measuring its brightness distribution.

==Applications of the theorem==

===Aperture synthesis===

The van Cittert–Zernike theorem is crucial to the measurement of the brightness distribution of a source. With two telescopes, a radio astronomer (or an infrared or submillimeter astronomer) can measure the correlation between the electric field at the two dishes due to some point from the source. By measuring this correlation for many points on the source, the astronomer can reconstruct the visibility function of the source. By applying the van Cittert–Zernike theorem, the astronomer can then take the inverse Fourier transform of the visibility function to discover the brightness distribution of the source. This technique is known as [[aperture synthesis]] or synthesis imaging.

In practice, radio astronomers rarely recover the brightness distribution of a source by directly taking the inverse Fourier transform of a measured visibility function. Such a process would require a sufficient number of samples to satisfy the [[Nyquist sampling theorem]]; this is many more observations than are needed to approximately reconstruct the brightness distribution of the source. Astronomers therefore take advantage of physical constraints on the brightness distribution of astronomical sources to reduce the number of observations which must be made. Because the brightness distribution must be real and positive everywhere, the visibility function cannot take on arbitrary values in unsampled regions. Thus, a non-linear deconvolution algorithm like [[CLEAN (algorithm)|CLEAN]] or Maximum Entropy may be used to approximately reconstruct the brightness distribution of the source from a limited number of observations.<ref name="burke">Burke and Graham-Smith, ''Introduction to Radio Astronomy'', pp. 92</ref>

===Adaptive optics===

The van Cittert–Zernike theorem also places constraints on the sensitivity of an [[adaptive optics]] system. In an adaptive optics (AO) system, a distorted wavefront is provided and must be transformed to a distortion-free wavefront. An AO system must make a number of different corrections to remove the distortions from the wavefront. One such correction involves splitting the wavefront into two identical wavefronts and shifting one by some physical distance <math>s</math> in the plane of the wavefront. The two wavefronts are then superimposed, creating a fringe pattern. By measuring the size and separation of the fringes, the AO system can determine phase differences along the wavefront.<ref name="roddier">F. Roddier, ''Adaptive Optics in Astronomy'', pp. 95</ref> This technique is known as "shearing."

The sensitivity of this technique is limited by the van Cittert–Zernike theorem.<ref name="hardy">J. Hardy, ''Adaptive Optics for Astronomical Telescopes'', pp. 159</ref> If an extended source is imaged, the contrast between the fringes will be reduced by a factor proportional to the Fourier transform of the brightness distribution of the source.<ref name="koliopoulos">Koliopoulos, ''Appl. Opt'', '''19''', 1523 (1980)</ref> The van Cittert–Zernike theorem implies that the mutual coherence of an extended source imaged by an AO system will be the Fourier transform of its brightness distribution. An extended source will therefore change the mutual coherence of the fringes, reducing their contrast.

===Free-electron laser===
The Van Cittert–Zernike theorem can be used to calculate the partial spatial coherence of radiation from a [[free-electron laser]].

==See also==
*[[Degree of coherence]]
*[[Coherence theory]]
*[[Visibility]]
*[[Hanbury Brown and Twiss effect]]
*[[Bose-Einstein correlations]]

==References==
<References/>

==Textbooks==
* {{aut|Born, M. & Wolf, E.}}: ''Principles of optics'', Pergamon Press, Oxford, 1987, p. 510
* {{aut|Klein, Miles V. & Furtak, Thomas E.}}: ''Optics'', John Wiley & Sons, New York, 1986, 2nd edition, p. 544-545

==External links==
* [http://www.youtube.com/watch?v=3Q_IwvWBQMc Lecture on the Van Cittert-Zernike-theorem with applications. University of Berkeley, prof. David T. Attwood on YouTube] (AST 210/EE 213 Lecture 23)]

{{DEFAULTSORT:Van Cittert-Zernike theorem}}
[[Category:Optics]]
[[Category:Radio astronomy]]