formulasearchengine - User contributions [en]

Pound–Drever–Hall technique

2014-10-13T21:24:27Z

72.33.79.226: /* Laser stabilization */

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Oxaloacetic acid

2014-03-03T16:26:08Z

72.33.207.135:

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Template:Campaignbox Second Sino–Japanese War

2014-02-20T21:58:18Z

72.33.228.161:

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Canonical correlation

2013-10-08T21:25:33Z

72.33.217.73: /* Derivation */ included parentheses in C-S inequality for clarity

{{about|the mathematical algorithm|other meanings such as the branch of [[motorsport]]|Hillclimbing (disambiguation)}}
{{Tree search algorithm}}
In [[computer science]], '''hill climbing''' is a [[Optimization_(mathematics)|mathematical optimization]] technique which belongs to the family of [[Local search (optimization)|local search]]. It is an iterative algorithm that starts with an arbitrary solution to a problem, then attempts to find a better solution by [[incremental heuristic search|incrementally]] changing a single element of the solution. If the change produces a better solution, an incremental change is made to the new solution, repeating until no further improvements can be found.

For example, hill climbing can be applied to the [[travelling salesman problem]]. It is easy to find an initial solution that visits all the cities but will be very poor compared to the optimal solution. The algorithm starts with such a solution and makes small improvements to it, such as switching the order in which two cities are visited. Eventually, a much shorter route is likely to be obtained.

Hill climbing is good for finding a [[local optimum]] (a solution that cannot be improved by considering a neighbouring configuration) but it is not guaranteed to find the best possible solution (the [[global optimum]]) out of all possible solutions (the [[Candidate solution|search space]]).
The characteristic that only local optima are guaranteed can be cured by using restarts (repeated local search), or more complex schemes based
on iterations, like [[iterated local search]], on memory,
like [[reactive search optimization]] and [[tabu search]],
or memory-less stochastic modifications, like [[simulated annealing]].

The relative simplicity of the algorithm makes it a popular first choice amongst optimizing algorithms. It is used widely in [[artificial intelligence]], for reaching a goal state from a starting node. Choice of next node and starting node can be varied to give a list of related algorithms. Although more advanced algorithms such as [[simulated annealing]] or [[tabu search]] may give better results, in some situations hill climbing works just as well. Hill climbing can often produce a better result than other algorithms when the amount of time available to perform a search is limited, such as with real-time systems.
It is an [[anytime algorithm]]:
it can return a valid solution even if it's interrupted at any time before it ends.

==Mathematical description==

Hill climbing attempts to maximize (or minimize) a target [[function (mathematics)|function]] <math>f(\mathbf{x})</math>, where <math>\mathbf{x}</math> is a vector of continuous and/or discrete values. At each iteration, hill climbing will adjust a single element in <math>\mathbf{x}</math> and determine whether the change improves the value of <math>f(\mathbf{x})</math>. (Note that this differs from [[gradient descent]] methods, which adjust all of the values in <math>\mathbf{x}</math> at each iteration according to the gradient of the hill.) With hill climbing, any change that improves <math>f(\mathbf{x})</math> is accepted, and the process continues until no change can be found to improve the value of <math>f(\mathbf{x})</math>. <math>\mathbf{x}</math> is then said to be "locally optimal".

In discrete vector spaces, each possible value for <math>\mathbf{x}</math> may be visualized as a [[vertex (graph theory)|vertex]] in a [[graph (mathematics)|graph]]. Hill climbing will follow the graph from vertex to vertex, always locally increasing (or decreasing) the value of <math>f(\mathbf{x})</math>, until a [[local maximum]] (or [[local minimum]]) <math>x_m</math> is reached.

[[Image:hill climb.png|thumb|260px|''A convex surface. Hill-climbers are well-suited for optimizing over such surfaces, and will converge to the global maximum.'']]

==Variants==

In '''simple hill climbing''', the first closer node is chosen, whereas in '''steepest ascent hill climbing''' all successors are compared and the closest to the solution is chosen. Both forms fail if there is no closer node, which may happen if there are local maxima in the search space which are not solutions. Steepest ascent hill climbing is similar to [[best-first search]], which tries all possible extensions of the current path instead of only one.

'''[[Stochastic hill climbing]]''' does not examine all neighbors before deciding how to move. Rather, it selects a neighbor at random, and decides (based on the amount of improvement in that neighbor) whether to move to that neighbor or to examine another.

[[Coordinate descent]] does a [[line search]] along one coordinate direction at the current point in each iteration. Some versions of coordinate descent randomly pick a different coordinate direction each iteration.

'''Random-restart hill climbing''' is a [[meta-algorithm]] built on top of the hill climbing algorithm. It is also known as '''Shotgun hill climbing'''. It iteratively does hill-climbing, each time with a random initial condition <math>x_0</math>. The best <math>x_m</math> is kept: if a new run of hill climbing produces a better <math>x_m</math> than the stored state, it replaces the stored state.

Random-restart hill climbing is a surprisingly effective algorithm in many cases. It turns out that it is often better to spend CPU time exploring the space, than carefully optimizing from an initial condition. {{Or|date=September 2007}}

==Problems==
===Local maxima===
[[Image:local maximum.png|thumb|260px|''A surface with two local maxima. (Only one of them is the global maximum.) If a hill-climber begins in a poor location, it may converge to the lower maximum.'']]

A problem with hill climbing is that it will find only [[maxima and minima|local maxima]]. Unless the heuristic is convex, it may not reach a global maximum. Other local search algorithms try to overcome this problem such as [[stochastic hill climbing]], [[random walk]]s and [[simulated annealing]].

[[File:Hill Climbing with Simulated Annealing.gif|thumb|left|500px|Despite the many local maxima in this graph, the global maximum can still be found using simulated annealing. Unfortunately, the applicability of simulated annealing is problem-specific because it relies on finding ''lucky jumps'' that improve the position. In problems that involve more dimensions, the cost of finding such a jump may increase exponentially with dimensionality. Consequently, there remain many problems for which hill climbers will efficiently find good results while simulated annealing will seemingly run forever without making progress. This depiction shows an extreme case involving only one dimension.]]
{{clear}}

===Ridges and alleys===

[[Image:ridge.png|thumb|190px|''A ridge'']]

Ridges are a challenging problem for hill climbers that optimize in continuous spaces. Because hill climbers only adjust one element in the vector at a time, each step will move in an axis-aligned direction. If the target function creates a narrow ridge that ascends in a non-axis-aligned direction (or if the goal is to minimize, a narrow alley that descends in a non-axis-aligned direction), then the hill climber can only ascend the ridge (or descend the alley) by zig-zagging. If the sides of the ridge (or alley) are very steep, then the hill climber may be forced to take very tiny steps as it zig-zags toward a better position. Thus, it may take an unreasonable length of time for it to ascend the ridge (or descend the alley).

By contrast, gradient descent methods can move in any direction that the ridge or alley may ascend or descend. Hence, gradient descent or [[conjugate gradient method]] is generally preferred over hill climbing when the target function is differentiable. Hill climbers, however, have the advantage of not requiring the target function to be differentiable, so hill climbers may be preferred when the target function is complex.

===Plateau===
Another problem that sometimes occurs with hill climbing is that of a plateau. A plateau is encountered when the search space is flat, or sufficiently flat that the value returned by the target function is indistinguishable from the value returned for nearby regions due to the precision used by the machine to represent its value. In such cases, the hill climber may not be able to determine in which direction it should step, and may wander in a direction that never leads to improvement.

==Pseudocode==

<pre>
Discrete Space Hill Climbing Algorithm
currentNode = startNode;
loop do
L = NEIGHBORS(currentNode);
nextEval = -INF;
nextNode = NULL;
for all x in L
if (EVAL(x) > nextEval)
nextNode = x;
nextEval = EVAL(x);
if nextEval <= EVAL(currentNode)
//Return current node since no better neighbors exist
return currentNode;
currentNode = nextNode;
</pre>

<pre>
Continuous Space Hill Climbing Algorithm
currentPoint = initialPoint; // the zero-magnitude vector is common
stepSize = initialStepSizes; // a vector of all 1's is common
acceleration = someAcceleration; // a value such as 1.2 is common
candidate[0] = -acceleration;
candidate[1] = -1 / acceleration;
candidate[2] = 0;
candidate[3] = 1 / acceleration;
candidate[4] = acceleration;
loop do
before = EVAL(currentPoint);
for each element i in currentPoint do
best = -1;
bestScore = -INF;
for j from 0 to 4 // try each of 5 candidate locations
currentPoint[i] = currentPoint[i] + stepSize[i] * candidate[j];
temp = EVAL(currentPoint);
currentPoint[i] = currentPoint[i] - stepSize[i] * candidate[j];
if(temp > bestScore)
bestScore = temp;
best = j;
if candidate[best] is not 0
currentPoint[i] = currentPoint[i] + stepSize[i] * candidate[best];
stepSize[i] = stepSize[i] * candidate[best]; // accelerate
if (EVAL(currentPoint) - before) < epsilon
return currentPoint;
</pre>

Contrast [[genetic algorithm]]; [[random optimization]].

==See also==
* [[Gradient descent]]
* [[Greedy algorithm]]
* [[Walrasian auction|Tâtonnement]]
* [[Mean-shift]]

==References==
*{{Russell Norvig 2003| pages=111–114}}

{{FOLDOC}}

==External links==
{{Wikibooks}}
* [http://yuval.bar-or.org/index.php?item=9 Hill Climbing visualization] A visualization of a hill-climbing (greedy) solution to the N-Queens puzzle by Yuval Baror.

[[Category:Optimization algorithms and methods]]
[[Category:Search algorithms]]

{{Optimization algorithms}}

Taylor expansions for the moments of functions of random variables

2013-07-09T20:35:37Z

72.33.244.48: /* See also */

In [[backtracking]] [[algorithm]]s, '''look ahead''' is the generic term for a [[Subroutine|subprocedure]] that attempts to foresee the effects of choosing a [[branch (computer science)|branching]] [[Variable (programming)|variable]] to evaluate or one of its values. The two main aims of look-ahead are to choose a variable to evaluate next and the order of values to assign to it.

==Constraint satisfaction==
In a general [[constraint satisfaction problem]], every variable can take a value in a domain. A backtracking algorithm therefore iteratively chooses a variable and tests each of its possible values; for each value the algorithm is [[recursion|recursively]] run. Look ahead is used to check the effects of choosing a given variable to evaluate or to decide the order of values to give to it.

===Look ahead techniques===
[[Image:Forward-arc-0.svg|thumb|left|200px|In this example, ''x''1=2 and the tentative assignment ''x''2=1 is considered.]]
[[Image:Forward-arc-1.svg|thumb|200px|Forward checking only checks whether each of the unassigned variables ''x''3 and ''x''4 is [[consistent]] with the partial assignment, removing the value 2 from their domains.]]
The simpler technique for evaluating the effect of a specific assignment to a variable is called '''forward checking'''. Given the current partial solution and a candidate assignment to evaluate, it checks whether another variable can take a consistent value. In other words, it first extends the current partial solution with the tentative value for the considered variable; it then considers every other variable <math>x_k</math> that is still unassigned, and checks whether there exists an evaluation of <math>x_k</math> that is consistent with the extended partial solution. More generally, forward checking determines the values for <math>x_k</math> that are consistent with the extended assignment.

[[Image:Forward-arc-2.svg|thumb|200px|Arc consistency look ahead also checks whether the values of ''x''3 and ''x''4 are consistent with each other (red lines) removing also the value 1 from their domains.]]
A look-ahead technique that may be more time-consuming but may produce better results is based on [[arc consistency]]. Namely, given a partial solution extended with a value for a new variable, it enforces arc consistency for all unassigned variables. In other words, for any unassigned variables, the values that cannot consistently be extended to another variable are removed. The difference between forward checking and arc consistency is that the former only checks a single unassigned variable at time for consistency, while the second also checks pairs of unassigned variables for mutual consistency.

Two other methods involving arc consistency are full and partial look ahead. They enforce arc consistency, but not for every pair of variables. In particular, full look considers every pair of unassigned variables <math>x_i,x_j</math>, and enforces arc consistency between them. This is different than enforcing global arc consistency, which may possibly require a pair of variables to be reconsidered more than once. Instead, once full look ahead has enforced arc consistency between a pair of variables, the pair is not considered any more. Partial look ahead is similar, but a given order of variables is considered, and arc consistency is only enforced once for every pair <math>x_i,x_j</math> with <math>i < j</math>.

Look ahead based on arc consistency can also be extended to work with path consistency and general i-consistency or relational arc consistency.

===Use of look ahead===
The results of look ahead is used to decide the next variable to evaluate and the order of values to give to this variable. In particular, for any unassigned variable and value, look-ahead estimates the effects of setting that variable to that value.

The choice of the next variable and the choice of the next value to give it are complementary, in that the value is typically chosen in such a way a solution (if any) is found as quickly as possible, while the next variable is typically chosen in such a way unsatisfiability (if the current partial solution is unsatisfiable) is proven as quickly as possible.

The choice of the next variable to evaluate is particularly important, as it may produce exponential differences in running time. In order to prove unsatisfiability as quickly as possible, variables leaving few alternatives after assigned are the preferred ones. This idea can be implemented by checking only satisfiability or unsatisfiability of variable/value pairs. In particular, the next variable that is chosen is the one having a minimal number of values that are consistent with the current partial solution. In turn, consistency can be evaluated by simply checking partial consistency, or by using any of the considered look ahead techniques discussed above.

The following are three methods for ordering the values to tentatively assign to a variable:

# min-conflicts: the preferred values are those removing the least total values from the domain of unassigned variables as evaluated by look ahead;
# max-domain-size: the preference of a variable is inversely the number of values in the smallest domain they produce for the unassigned variables, as evaluated by look ahead;
# estimate solutions: the preferred values are those producing the maximal number of solutions, as evaluated by look ahead making the assumption that all values left in the domains of unassigned variables are consistent with each other; in other words, the preference for a value is obtain by multiplying the size of all domains resulting from look ahead.

Experiments proved that these techniques are useful for large problems, especially the min-conflicts one.

Randomization is also sometimes used for choosing a variable or value. For example, if two variables are equally preferred according to some measure, the choice can be done randomly.

==References==
*{{cite book
| first=Rina
| last=Dechter
| title=Constraint Processing
| publisher=Morgan Kaufmann
| year=2003
| url=http://www.ics.uci.edu/~dechter/books/index.html
}} ISBN 1-55860-890-7
*{{cite journal
| first=Ming
| last=Ouyang
| title=How Good Are Branching Rules in DPLL?
| journal=Discrete Applied Mathematics
| volume=89
| issue=1–3
| pages=281–286
| year=1998
| doi=10.1016/S0166-218X(98)00045-6}}

[[Category:Constraint programming]]
[[Category:Search algorithms]]

Integer broom topology

2013-04-15T12:44:52Z

72.33.245.197: /* Definition of the integer broom topology */

{{For|other articles bearing the name of the physicist|Fraunhofer (disambiguation)}}
{{mergefrom|N-slit_interferometric_equation|discuss=Talk:N-slit_interferometric_equation|date=June 2013}}
In [[optics]], the '''Fraunhofer diffraction''' equation is used to model the [[diffraction]] of waves when the diffraction pattern is viewed at a long distance from the diffracting object, and also when it is viewed at the [[focal plane]] of an imaging [[Lens (optics)|lens]].<ref>Born & Wolf, 1999, p 427.</ref><ref>Jenkins & White, 1957, p 288</ref>

The equation was named in honour of [[Joseph von Fraunhofer]] although he was not actually involved in the development of the theory.<ref>http://scienceworld.wolfram.com/biography/Fraunhofer.html</ref>

This article gives the equation in various mathematical forms, and provides detailed calculations of the Fraunhofer diffraction pattern for several different forms of diffracting apertures. A qualitative discussion of Fraunhofer diffraction can be found [[Fraunhofer diffraction|elsewhere]].

==Fraunhofer diffraction equation==

When a beam of light is partly blocked by an obstacle, some of the light is scattered around the object, and light and dark bands are often seen at the edge of the shadow – this effect is known as diffraction.<ref>Heavens & Ditchburn, 1996, p 62</ref> The [[Kirchhoff's diffraction formula|Kirchhoff diffraction equation]] provides an expression, derived from the [[wave equation]], which describes the wave diffracted by an aperture; analytical solutions to this equation are not available for most configurations.<ref>Born & Wolf, 2002, p 425</ref>

The Fraunhofer diffraction equation is an approximation which can be applied when the diffracted wave is observed in the [[far field]], and also when a lens is used to focus the diffracted light; in many instances, a simple analytical solution is available to the Fraunhofer equation – several of these are derived below.

==Cartesian co-ordinates==

[[File:Diffraction geometry 2.svg|thumb|250px|Diffraction geometry, showing aperture (or diffracting object) plane and image plane, with coordinate system.]] If the aperture is in {{math|''x'y'''}} plane, with the origin in the aperture and is illuminated by a [[monochromatic]] wave, of [[wavelength]] λ, [[wavenumber]] {{math|''k''}} with complex amplitude {{math|''A''(''x'' ',''y'' ')}}, and the diffracted wave is observed in the {{math|''x,y,z''}} plane where {{math|''l'',''m''}} are the [[direction cosines]] of the point {{math|''x,y''}} with respect to the origin, the complex amplitude {{math|''U''(''x'',''y'')}} of the diffracted wave is given by the Fraunhofer diffraction equation as:<ref>Lipson et al, 2011, eq(8.8) p 231</ref>

:<math>
\begin{align}
U(x,y,z)
&\propto \iint_\text{Aperture} \,A(x',y') e^{-i \frac{2\pi}{\lambda}(lx' + my')}dx'\,dy'\\
&\propto \iint_\text{Aperture} \,A(x',y') e^{-i k(lx' + my')}dx'\,dy'

\end{align}
</math>

It can be seen from this equation that the form of the diffraction pattern depends only on the direction of viewing, so the diffraction pattern changes in size but not in form with change of viewing distance.

The Fraunhofer diffraction equation can be expressed in a variety of mathematically equivalent forms. For example:<ref>Hecht, 2002, eq (11.63), p 529</ref>
:<math>
\begin{align}
U(x,y,z)
&\propto \iint_\text{Aperture} \,A(x',y') e^{-i \frac{2\pi}{\lambda z}(x' x + y' y)}\,dx'\,dy'\\
&\propto \iint_\text{Aperture} \,A(x',y') e^{-i \frac {k(x' x + y' y)}{z}}\,dx'\,dy'
\end{align}
</math>

It can be seen that the integral in the above equations is the [[Fourier transform]] of the aperture function evaluated at frequencies<ref>Hecht, 2002, eq(11.67), p 540</ref>

:<math> f_x=x/(\lambda z)=l/\lambda</math>
:<math> f_y= y/(\lambda z)= m/\lambda</math>

Thus, we can also write the equation in terms of a [[Fourier transform]] as:

:<math>U(x,y,z) \propto \hat f[A(x',y')]_{f_x f_y} </math>

where {{math|''Â''}} is the Fourier transform of {{math|''A''}}. The Fourier transform formulation can be very useful in solving diffraction problems.

Another form is:

:<math>U(\mathbf r)\propto { \int_\text{Aperture} A(\mathbf {r'}) e^{-i\mathbf{k} \cdot (\mathbf{r'} -\mathbf r)} dr'}= { \int_\text{Aperture} a_0 (\mathbf{r'}) e^{i\mathbf{(k_0-k)} \cdot (\mathbf{r'}-\mathbf r)} dr' }</math>

where {{math|'''r''' and '''r' '''}} represent the observation point and a point in the aperture respectively, {{math|'''k0'''}} and {{math|'''k'''}} represent the [[wave vector]]s of the disturbance at the aperture and of the diffracted waves respectively, and {{math|a0('''r' ''')}} represents the [[Euclidean vector|magnitude]] of the disturbance at the aperture.

==Polar co-ordinates==

When the diffracting aperture has circular symmetry, it is useful to use [[polar coordinate system|polar]] rather than [[Cartesian coordinate system|Cartesian]] co-ordinates.<ref>Born & Wolf, 2002, Section 8.5.2, eqs (6–8), p 439</ref>

A point in the aperture has co-oordinates {{math|ρ,ω}} giving:

:<math>~x'=\rho' \cos \omega'; y'=\rho' \sin \omega'</math>

and

:<math>~x=\rho \cos \omega; y=\rho \sin \omega</math>

The complex amplitude at {{math|ρ'}} is given by {{math|A(ρ)}}, and the area {{math|d''x'' d''y''}} converts to [[Polar coordinate system#Generalization|ρ' dρ' dω']], giving

:<math>
\begin{align}
U(\rho,\omega,z)
&\propto \int_0^\infty \int_0^{2 \pi} A(\rho') e^{-i \frac{2\pi}{\lambda z}(\rho \rho' \cos \omega \cos \omega' + \rho \rho' \sin \omega \sin \omega' )} \rho' d \rho' d \omega'\\
&\propto \int_0^{2 \pi} \int_0^{\infty} A(\rho') e^{-i \frac{2\pi}{\lambda z}\rho \rho' \cos (\omega - \omega') } d \omega' \rho' d \rho'
\end{align}
</math>

Using the integral representation of the [[Bessel function]]:<ref>Abramowitz & Stegun, 1964, Section 9.1.21, p 360</ref>
:<math>J_0(p)=\frac {1}{2 \pi} \int_0^{2 \pi} e^{ip \cos \alpha} d \alpha </math>

we have

:<math>
\begin{align}
U(\rho,z)
&\propto 2 \pi \int_0^{\infty} A(\rho')J_0\left(\frac{2 \pi \rho' \rho}{\lambda z}\right) \rho' d \rho'
\end{align}
</math>

where the integration over {{math|ω}} gives {{math|2π}} since the equation is circularly symmetric, i.e. there is no dependence on {{math|ω}}.

In this case, we have {{math|U(ρ,z)}} equal to the [[Hankel transform|Fourier–Bessel or Hankel transform]] of the aperture function, {{math|''A''(''ρ'')}}

==Examples of Fraunhofer diffraction with a normally incident monochromatic plane wave==

In each case, the diffracting object is located in the z=0 plane, and the complex amplitude of the incident [[plane wave]] is given by

:<math>~A(x',y')= a e^{i 2 \pi c t/\lambda} = a e^{ik c t}</math>

where

:{{math|''a''}} is the [[Euclidean vector|magnitude]] of the wave disturbance,
:{{math|λ}} is the wavelength,
:{{math|''c''}} is the velocity of light,
:{{math|''t''}} is the time
:{{math|''k''}} = {{math|2 π/λ}} is the [[wave number]]

and the [[Phase (waves)|phase]] is zero at time {{math|''t}} = 0.

The time dependent factor is omitted throughout the calculations, as it remains constant, and is averaged out when the [[Amplitude|intensity]] is calculated. The intensity at {{math|'''r'''}} is proportional to the amplitude times its [[complex conjugate]]
:<math>I(\mathbf{r}) \propto U(\mathbf{r}) \overline{U} (\mathbf{r})</math>

These derivations can be found in most standard optics books, in slightly different forms using varying notations. A reference is given for each of the systems modelled here. The Fourier transforms used can be found [[Fourier transforms#Tables of important Fourier transforms|here]].

===Slit of infinite depth===

[[File:diffraction1.png|right|thumb|Graph and image of single-slit diffraction]]The aperture is a slit of width {{math|''W''}} which is located along the {{math|''y''}}-axis,

====Solution by integration====

Assuming the centre of the slit is located at {{math|''x ''[[=]] 0}}, the first equation above, for all values of {{math|''y''}}, is:<ref>Born & Wolf, 1999, Section 8.5.1 p 436</ref>

:<math>
\begin{align}
U(x,z)
&= a \int_ {-W/2}^{W/2} e^{ {-2 \pi ixx'}/(\lambda z)} dx'\\
&= -\frac{a \lambda z}{2 \pi i x} |e^{ {-2 \pi ixx'}/(\lambda z)} |_{-W/2}^{W/2}
\end{align}
</math>

Using [[Euler's formula]], this can be simplified to:

:<math>
\begin{align}
U(x,z)
&= aW \frac {\sin \left [\frac {\pi Wx} {\lambda z} \right ]} {\frac {\pi Wx} {\lambda z}}\\
&= aW ~\mathrm{sinc} \frac {\pi Wx}{\lambda z}
\end{align}
</math>

where {{math|[[sinc]](''p''){{=}} sin(''p'')/''p''}}. It should be noted that the [[sinc]] function is sometimes defined as {{math|sin(π''p'')/π''p''}} and this may cause confusion when looking at derivations in different texts.

This can also be written as:

:<math>U(\theta) = aW ~\mathrm {sinc} \left [\frac {\pi W \sin \theta} {\lambda} \right]</math>

where {{math|''θ''}} is the angle between ''z''-axis and the line joining x to the origin and {{math|sin ''θ'' ≈ ''x''/''z''}} when {{math|''θ'' << 1}}.

====Fourier transform solution====

The slit can be represented by the [[rect]] function as:<ref>Hecht, 2002, p 540</ref>

:<math>~\mathrm{rect}(x/W)</math>

The [[Fourier transform#Square-integrable functions|Fourier transform]] of this function is given by

:<math>\hat f(\mathrm{rect}(ax)) = \displaystyle \frac{1}{|a|}\cdot \operatorname{sinc}\left(\frac{\xi}{a}\right)</math>

where {{math|ξ}} is the Fourier transform frequency, and the {{math|sinc}} function is here defined as sin(''πx'')/(''πx'')

The Fourier transform frequency here is {{math|''x/λz''}}, giving

:<math>
\begin{align}
U(x,z)
&\propto \frac {\sin{ \frac {\pi Wx} {\lambda z}}}{ \frac {\pi Wx} {\lambda z}}\\
&\propto W \mathrm{sinc} { \frac {\pi Wx} {\lambda z}}\\
& \propto W \mathrm{sinc} { \frac { \pi W \sin \theta} {\lambda }} \\
& \propto W \mathrm{sinc} (kW \sin \theta /2)
\end{align}
</math>

Note that the {{math|sinc}} function is here defined as sin(''x'')/(''x'') to maintain consistency.

===Intensity===

The [[Intensity (physics)|intensity]] is proportional to the square of the amplitude, and is then<ref>Hecht, 2002, eqs (10.17) (10.18), p 453</ref>

:<math>
\begin{align}
I(\theta)
&\propto \operatorname{sinc}^2 \left [\frac { \pi W \sin \theta} {\lambda} \right]\\
&\propto \operatorname{sinc}^2 \left [\frac {kW \sin \theta} {2} \right]
\end{align}
</math>

==Rectangular aperture==

[[File:rectangular diffraction.jpg|right|200px|thumbnail|Fraunhofer diffraction by a rectangular aperture]]When a slit of width ''W'' and height ''H'' is illuminated normally by a [[monochromatic]] [[plane wave]] of wavelength λ, the complex amplitude can be found using similar analyses to those in the previous section, applied over two independent dimensions as:<ref>Longhurst, 1967, p 217</ref><ref>Goodman, eq(4.28), p 76</ref>
:<math>
\begin{align}
U(\theta, \phi)
&\propto \operatorname{sinc}\left(\frac{ \pi W \sin\theta}{\lambda}\right)\operatorname{sinc}\left(\frac{ \pi H \sin\phi}{\lambda}\right)\\
&\propto \operatorname{sinc}\left(\frac{ k W \sin\theta}{2}\right)\operatorname{sinc}\left(\frac{ kH \sin\phi}{2}\right)
\end{align}
</math>

The intensity is given by

:<math>
\begin{align}
I(\theta, \phi)
& \propto \operatorname{sinc}^2\left(\frac{\pi W \sin\theta}{\lambda}\right)\operatorname{sinc}^2\left(\frac{ \pi H \sin\phi}{\lambda}\right)\\
& \propto \operatorname{sinc}^2\left(\frac{k W \sin\theta}{2}\right)\operatorname{sinc}^2\left(\frac{k H \sin\phi}{2}\right)
\end{align}
</math>

where {{math|θ}} and {{math|φ}} are the angles between the {{math|''x''}} and {{math|''z''}} axes and the {{math|''y''}} and {{math|''z''}} axes, respectively.

In practice, all slits are of finite length and will therefore produce diffraction on both directions. If the length of the slit is much greater than its width, then the spacing of the horizontal diffraction fringes will be much less than the spacing of the vertical fringes. If the illuminating beam does not illuminate the whole length of the slit, the spacing of the horizontal fringes is determined by the dimensions of the laser beam. Close examination of the two-slit pattern below shows that there are very fine horizontal diffraction fringes above and below the main spot, as well as the more obvious vertical fringes.

==Circular aperture==
[[File:Airy-pattern.svg|thumb|400px|right|Airy diffraction pattern]]The aperture has diameter {{math|''W''}}. The complex amplitude in the observation plane is given by

:<math>
\begin{align}
U(\rho,z)&=2 \pi a \int_0^{W/2} J_0(2 \pi \rho' \rho/\lambda z) \rho' d \rho'

\end{align}
</math>

===Solution by integration===

Using the recurrence relationship<ref>Whittaker and Watson, example 2, p 360</ref>
:<math>\frac {d} {dx} \left[x^{n+1}J_{n+1}(x) \right] =x^{n+1}J_n(x)</math>

to give

:<math>\int_0^x x'J_0(x')dx'=xJ_1(x)</math>

If we substitute
:<math>x'= \frac{2 \pi \rho}{\lambda z} \rho'</math>

and the limits of the integration become 0 and {{math|πρW/λz}}, we get

:<math>U(\rho,z) \propto \frac{J_1(\pi W \rho/ \lambda z)}{\pi W \rho / \lambda z}</math>

Putting {{math|ρ/''z''}} = sin {{math|θ}}, we get

:<math>U(\theta) \propto \frac{J_1(\pi W \sin \theta/ \lambda)}{\pi W \sin \theta / \lambda}</math>

===Solution using Fourier–Bessel transform===

We can write the aperture function as a [[step function]]

:<math>~\Pi (W/2)</math>

The Fourier–Bessel transform for this function is given by the relationship

:<math>~F[\Pi(r/a)] = \frac {2 \pi J_1 (qa)} {q} </math>

where {{math|q/2π}} is the transform frequency which is equal to {{math|ρ/λ''z''}} and {{math|''a''}} = {{math|''W''/2}}.

Thus, we get
:<math>
\begin{align}
U(\rho)
&= \frac {2 \pi J_1(\pi W \rho / \lambda z)}{2 \pi W \rho/\lambda z}\\
&= \frac { 2 \pi J_1(\pi W \sin \theta /\lambda)}{W \sin \theta/\lambda}\\
&=\frac { 2 \pi J_1(k W \sin \theta /2)}{ kW \sin \theta/2}

\end{align}
</math>

===Intensity===

The intensity is given by:<ref>Hecht, 2002, eq (10.56), p 469</ref>

:<math>
\begin{align}
I(\theta)
&\propto \left [\frac{J_1(\pi W \sin \theta/ \lambda)}{\pi W \sin \theta/\lambda)} \right]^2\\
&\propto \left [\frac{J_1(k W \sin \theta/2)}{(k W \sin \theta/2)} \right]^2
\end{align}
</math>

===Form of the diffraction pattern===

This known as the [[Airy disk|Airy diffraction pattern]]

The diffracted pattern is symmetric about the normal axis.

==Aperture with a Gaussian profile==

[[File:Gaussian profile.svg|thumb|Intensity of a plane wave diffracted through an aperture with a Gaussian profile]]An aperture with a Gaussian profile, for example, a photographic slide whose transmission has a Gaussian variation, so that the amplitude at a point in the aperture located at a distance ''r''' from the origin is given by

:<math>A(\rho') = \exp { \left [ -\frac {\rho'} {\sigma} \right]^2}</math>

giving

:<math>
U(\rho,z)=2 \pi a \int_0^\infty \exp { \left [-\frac {\rho'} {\sigma} \right]^2} J_0(2 \pi \rho' \rho/\lambda z) \rho' \, d \rho'
</math>

===Solution using Fourier-Bessel transform===

The [[Hankel transform|Fourier-Bessel or Hankel]] transform is defined as

:<math>
F_\nu(k) = \int_0^\infty f(r)J_\nu(kr)\,r\,dr
</math>
where ''J''ν is the [[Bessel function]] of the first kind of order ν with ν ≥ −1/2.

The [[Hankel transform#Some Hankel transform pairs|Hankel transform]] is

:<math>F_\nu [e^{(ar)^2/2}] = \frac {e^{-k^2/2a^2}}{a^2}</math>

giving

:<math>
\begin{align}
U(\rho,z)
&\propto e^{-[\frac{\pi \rho \sigma}{\lambda z}]^2}
\end{align}
</math>
and
:<math> U(\theta) \propto e^{-[\frac{\pi \sigma \sin \theta}{\lambda}]^2}</math>

===Intensity===

The intensity is given by:<ref>Hecht, 2002, eq (11.2), p 521</ref>

: <math> I(\theta) \propto e^{-[\frac{2\pi \sigma \sin \theta}{\lambda}]}</math>

This function is plotted on the right, and it can be seen that, unlike the diffraction patterns produced by rectangular or circular apertures, it has no secondary rings. This can be used in a process called [[apodization]] - the aperture is covered by a filter whose transmission varies as a Gaussian function, giving a diffraction pattern with no secondary rings.:<ref>Heavens & Ditchburn, 1991, p 68</ref><ref>Hecht, 2002, Figure (11.33), p 543</ref>

==Two slits==
The pattern which occurs when light diffracted from two slits overlaps is of considerable interest in physics, firstly for its importance in establishing the wave theory of light through [[Young's interference experiment]], and secondly because of its role as a thought experiment in [[double-slit experiment]] in quantum mechanics.

===Narrow slits===
[[File:Two slit diagram3.svg|thumb|Geometry of two slit diffraction]][[File:Laserdiffraction.jpg|right|thumb|Two slit interference using a red laser]]Assume we have two long slits illuminated by a plane wave of wavelength {{math|λ}}. The slits are in the {{math|''z'' {{=}} 0}} plane, parallel to the {{math|''y''}} axis, separated by a distance {{math|''S''}} and are symmetrical about the origin. The width of the slits is small compared with the wavelength.

====Solution by integration====

The incident light is diffracted by the slits into uniform spherical waves. The waves travelling in a given direction {{math|θ}} from the two slits have differing phases. The phase of the waves from the upper and lower slits relative to the origin is given by {{math|(2π/λ)(S/2)sin θ}} and {{math|-(2π/λ)(S/2)sin θ}}

The complex amplitude of the summed waves is given by:<ref>Jenkins & White, 1957, eq (16c), p 312</ref>

:<math>
\begin{align}
U(\theta)
&= a e^{\frac { i\pi S \sin \theta }{\lambda}} + a e^{- \frac { i \pi S \sin \theta} {\lambda}}\\
&=a (\cos {\frac { \pi S \sin \theta }{\lambda}} +i \sin {\frac { \pi S \sin \theta }{\lambda}} )+a (\cos {\frac { \pi S \sin \theta }{\lambda}} -i \sin {\frac { \pi S \sin \theta }{\lambda}} )\\
&=2a \cos {\frac { \pi S \sin \theta }{\lambda}}
\end{align}
</math>

====Solution using Fourier transform====

The aperture can be represented by the function:<ref>Hecht, 2002, eq (11.4328), p 5</ref>

:<math>~a[\delta {(x-S/2)}+ \delta {(x+S/2)}]</math>

where {{math|δ}} is the [[Dirac delta function|delta function]].

We [[Fourier transform#Tables of important Fourier transforms|have]]

:<math>\hat {f}[\delta (x)] =1</math>

and

:<math>\hat {f} [g(x-a)] = e^{-2 \pi i a f_x} \hat {f} [g(x)]</math>

giving

:<math>
\begin{align}
U(x,z)
&=\hat {f} [\delta {(x-W/2)}+ \delta {(x+W/2)}]\\
&= e^{- i \pi Sx/\lambda z}+e^{ i \pi Sx/\lambda z}\\
&= 2 \cos \frac {\pi S x}{\lambda z}
\end{align}
</math>

:<math>U(\theta)= 2 \cos \frac {\pi S \sin \theta} {\lambda}</math>

This is the same expression as that derived above by integration.

====Intensity====

This gives the intensity of the combined waves as:<ref>Lipson et al, 2011, eq (9.3), p 280</ref>

:<math>
\begin{align}
I(\theta)
& \propto \cos^2 \left[\frac { \pi S \sin \theta}{\lambda} \right]\\
& \propto \cos^2 {[ \frac {kS \sin \theta}{2}]}

\end{align}
</math>

===Slits of finite width===

[[File:Single slit and double slit3.jpg|thumb|Single and double slit diffraction – slit separation is 0.7mm and the slit width is 0.02mm]] The width of the slits, {{math|''W''}} is finite.

====Solution by integration====

The diffracted pattern is given by:<ref>Hecht, 2002, Section 10.2.2, p 451</ref>

:<math>
\begin{align}
U(\theta)
&= a \left [e^{\frac { i\pi S \sin \theta }{\lambda}} + e^{- \frac { i \pi S \sin \theta} {\lambda}} \right]\int_ {-W/2}^{W/2} e^{ {-2 \pi ix' \sin \theta}/(\lambda)} dx'\\
&= 2a \cos {\frac { \pi S \sin \theta }{\lambda}} W ~\mathrm{sinc} \frac { \pi W \sin \theta}{\lambda}
\end{align}
</math>

====Solution using Fourier transform====

The aperture function is given by:<ref>Hecht, 2002, p 541</ref>

:<math>a \left[\mathrm {rect} \left (\frac{x-S/2}{W} \right) + \mathrm {rect} \left (\frac{x+S/2}{W} \right) \right]</math>

The [[Fourier transform#Square-integrable functions|Fourier transform]] of this function is given by

:<math>\hat f(\mathrm{rect}(ax)) = \displaystyle \frac{1}{|a|}\cdot \operatorname{sinc}\left(\frac{\xi}{a}\right)</math>

where {{math|ξ}} is the Fourier transform frequency, and the {{math|sinc}} function is here defined as sin(''πx'')/(''πx'')

and

:<math>\hat {f} [g(x-a)] = e^{-2 \pi i a f_x} \hat {f} [g(x)]</math>

We have

:<math>
\begin{align}
U(x,z)
&= \hat {f} \left [a \left[\mathrm {rect} \left (\frac{x-S/2}{W} \right) + \mathrm {rect} \left (\frac{x+S/2}{W} \right) \right ] \right ]\\
&= 2W \left[ e^{- i \pi Sx/\lambda z}+e^{ i \pi Sx/\lambda z} \right] \frac {\sin { \frac {\pi Wx} {\lambda z}}}{ \frac {\pi Wx} {\lambda z}}\\
&= 2a \cos {\frac { \pi S x }{\lambda z}} W ~\mathrm{sinc} \frac { \pi Wx}{\lambda z}
\end{align}
</math>

or

<math>U(\theta)= 2a \cos {\frac { \pi S \sin \theta }{\lambda}} W ~\mathrm{sinc} \frac { \pi W \sin \theta}{\lambda}</math>

This is the same expression as was derived by integration.

====Intensity====

The intensity is given by:<ref>Jenkins and White, 1967, eq (16c), p 313</ref>

:<math>
\begin{align}
I(\theta)
&\propto \cos^2 \left [{\frac {\pi S \sin \theta}{\lambda}}\right]~\mathrm{sinc}^2 \left [ \frac {\pi W \sin \theta}{\lambda} \right]\\
&\propto \cos^2 \left [\frac{k S \sin \theta}{2}\right] \mathrm{sinc}^2 \left [ \frac {kW \sin \theta}{2} \right]
\end{align}
</math>

It can be seen that the form of the intensity pattern is the product of the individual slit diffraction pattern, and the interference pattern which would be obtained with slits of negligible width. This is illustrated in the image at the right which shows single slit diffraction by a laser beam, and also the diffraction/interference pattern given by two identical slits.

==Grating==

A grating is defined in Born and Wolf as "any arrangement which imposes on an incident wave a periodic variation of amplitude or phase, or both".<ref>Born & Wolf, 1999, Section 8.6.1, p 446</ref>

===Narrow slit grating===

A simple grating consists of a screen with N slits whose width is significantly less than the wavelength of the incident light with slit separation of {{math|''S''}}.

====Solution by integration====

The complex amplitude of the diffracted wave at an angle {{math|θ}} is given by:<ref>Jenkins & White, 1957, eq (17a), p 330</ref>

<math>
\begin{align}
U(\theta)
&= a\sum_{n=1}^N e^{ \frac {-i 2 \pi nS \sin \theta} {\lambda}}\\
&= \frac {1-e^{ -i 2 \pi NS \sin \theta/\lambda}} {1-e^{-i 2 \pi D \sin \theta / \lambda}}
\end{align}
</math>

since this is the sum of a [[Geometric series#Formula|geometric series]].

====Solution using Fourier transform====

The aperture is given by

:<math> \sum _{n=0}^{N} \delta(x-nS) </math>

The Fourier transform of this function is:<ref>Lipson et al, 2011, eq(4.41), p 106</ref>

:<math>
\begin{align}
\hat {f} \left[\sum _{n=0}^{N} \delta(x-nS) \right]
&=\sum _{n=0}^{N} e^{-i f_x nS}\\
&= \frac {1-e^{ -i 2 \pi NS \sin \theta/\lambda}} {1-e^{-i 2 \pi S \sin \theta / \lambda}}
\end{align}
</math>

====Intensity====

[[File:Grating50.svg|thumb|200px|right|Diffraction pattern for 50 narrow-slit grating]][[File:Grating20and 50.svg|thumb|200px|right|Detail of main maximum in 20 and 50 narrow slit grating diffraction patterns]]The intensity is given by:<ref>Born & Wolf, 1999, eq(5a), p 448</ref>

<math>
\begin{align}
I(\theta)
&\propto \frac {1 - \cos (2 \pi N S\sin \theta/\lambda)}{1-\cos (2 \pi S \sin \theta / \lambda)}\\
&\propto \frac{ \sin^2 (\pi N S \sin \theta/\lambda)}{ \sin^2 (\pi S \sin \theta/\lambda)}
\end{align}
</math>

This function has a series of maxima and minima. There are regularly spaced "principal maxima", and a number of much smaller maxima in between the principal maxima. The principal maxima occur when

:<math>\pi S \sin_n \theta/\lambda =n \pi, n = 0, \pm 1, \pm 2,..... </math>

and the main diffracted beams therefore occur at angles:

:<math>\sin \theta_n = \frac {n \lambda} {S}, n=0, \pm 1 \pm 2, ....</math>

This is the [[grating equation]] for normally incident light.

The number of small intermediate maxima is equal to the number of slits, {{math|''N''}} − 1 and their size and shape is also determined by {{math|''N''}}.

The form of the pattern for {{math|''N''}}=50 is shown in the first figure .

The detailed structure for 20 and 50 slits gratings are illustrated in the second diagram.

===Finite width slit grating===

[[File:gratingwide.svg|thumb|200px|right|Diffraction pattern from grating with finite width slits]]The grating now has N slits of width {{math|''W''}} and spacing {{math|''S}}

====Solution using integration====

The amplitude is given by:<ref>Born & Wolf, Section 8.6.1, eq (5), p 448</ref>
:<math>
\begin{align}
U(\theta, \phi)
&\propto a\sum_{n=1}^N e^{ \frac {-i 2 \pi nS \sin \theta} {\lambda}}\int_ {-W/2}^{W/2} e^{ {-2 \pi ixx'}/(\lambda z)} dx' \\
&\propto a\mathrm{sinc}\left(\frac{ W \sin\theta}{\lambda}\right)\frac {1-e^{ -i 2 \pi NS \sin \theta/\lambda}} {1-e^{-i 2 \pi D \sin \theta / \lambda}}
\end{align}
</math>

====Solution using Fourier transform====

The aperture function can be written as:<ref>Hecht, The array theorem, p 543</ref>

:<math>\sum_{n=1}^{N} \mathrm {rect} \left[ \frac {x'-nS} {W} \right] </math>

Using the [[Fourier transform#Convolution theorem|convolution theorem]] which says that if we have two functions {{math|''f(x)''}} and {{math|''g(x)'')}}, and we have

:<math>h(x) = (f*g)(x) = \int_{-\infty}^\infty f(y)g(x - y)\,dy,</math>

where ∗ denotes the convolution operation, then we also have

:<math>\hat{h}(\xi) = \hat{f}(\xi)\cdot \hat{g}(\xi).</math>

we can write the aperture function as

:<math>\mathrm {rect} (x'/W)* \sum_{n=0}^N \delta (x'-nS)</math>

The amplitude is then given by the Fourier transform of this expression as:

:<math>
\begin{align}
U(x,z)
&=\hat {f} [\mathrm {rect} (x'/W)] \hat {f} [ \sum_{n=0}^N \delta (x'-nS)]\\
&=aW ~\mathrm{sinc} \frac {\pi Wx}{\lambda z}{1-e^{-i 2 \pi S \sin \theta / \lambda}}
\end{align}
</math>

====Intensity====

The intensity is given by:<ref>Born & Wolf, 2002, Section 8.6, eq (10), p 451</ref>
:<math>
\begin{align}
I(\theta)
& \propto\mathrm{sinc}^2\left(\frac{ W \sin\theta}{\lambda}\right)\frac{ \sin^2 (\pi N S \sin \theta/\lambda)}{ \sin^2 (\pi S \sin \theta/\lambda)}
\end{align}
</math>

The diagram shows the diffraction pattern for a grating with 20 slits, where the width of the slits is 1/5th of the slit separation. The size of the main diffracted peaks is modulated with the diffraction pattern of the individual slits.

===Other gratings===

The Fourier transform method above can be used to find the form of the diffraction for any periodic structure where the Fourier transform of the structure is known. Goodman<ref>Goodman, 2005, Sections 4.4.3 and 4.4.4, p 78</ref> uses this method to derive expressions for the diffraction pattern obtained with sinsoidal amplitude and phase modulation gratings. These are of particular interest in [[holography]].

==Non-normal illumination==

If the aperture is illuminated by a mono-chromatic plane wave incident in a direction {{math|(''l''0,''m''0, ''n''0)}}, the first version of the Fraunhofer equation above becomes:<ref>Lipson et al, 2011, Section 8.2.2, p 232</ref>

:<math>
\begin{align}
U(x,y,z)
&\propto \iint_\text{Aperture} \,A(x',y') e^{-i \frac{2\pi}{\lambda}[(l-l_0)x' + (m-m_0)y']}dx'\,dy'\\
&\propto \iint_\text{Aperture} \,A(x',y') e^{-i k[(l-l_0)x' + (m-m_0)y']}dx'\,dy'

\end{align}
</math>

The equations used to model each of the systems above are altered only by changes in the constants multiplying {{math|''x''}} and {{math|''y''}}, so the difffracted light patterns will have the form, except that they will now be centred around the direction of the incident plane wave.

The grating equation becomes<ref>Born & Wolf, 1999, eq (8), p 449</ref>

:<math>\sin \theta_n = \frac {n \lambda} {S} + \sin \theta_0, n=0, \pm1, \pm2.... </math>.

==Non-monochromatic illumination==

In all of the above examples of Fraunhofer diffraction, the effect of increasing the wavelength of the illuminating light is to reduce the size of the diffraction structure, and conversely, when the wavelength is reduced, the size of the pattern increases. If the light is not mono-chromatic, i.e. it consists of a range of different wavelengths, each wavelength is diffracted into a pattern of a slightly different size to its neighbours. If the spread of wavelengths is significantly smaller than the mean wavelength, the individual patterns will vary very little in size, and so the basic diffraction will still appear with slightly reduced contrast. As the spread of wavelengths is increased, the number of "fringes" which can be observed is reduced.

== See also ==
*[[Kirchhoff's diffraction formula]]
*[[Fresnel diffraction]]
*[[Huygens principle]]
*[[Airy disc]]
*[[Fourier optics]]

== References ==
{{reflist}}

=Reference sources=
*Abramowitz Milton & Segun Irene A,1964, Dover Publications Inc, New York.
*[[Max Born|Born M]] & Wolf E, Principles of Optics, 1999, 7th Edition, Cambridge University Press, ISBN 978-0-521-64222-4
*Goodman Joseph, 2005, Introduction to Fourier Optics, Roberts & Co. ISBN 0-9747077-2-4 or online [http://books.google.com/books?id=ow5xs_Rtt9AC&dq=Introduction+to+Fourier+Optics&pg=PP1&ots=GUo2xN9FRK&sig=g9oSrJ8avm3Ea-jRmN60uKkftPo&prev=http://www.google.com/search%3Fsourceid%3Dnavclient-ff%26ie%3DUTF-8%26rls%3DGGGL,GGGL:2006-43,GGGL:en%26q%3DIntroduction%2Bto%2BFourier%2BOptics&sa=X&oi=print&ct=title#PPA33,M1 here]
*Heavens OS and Ditchburn W, 1991, Insight into Optics, Longman and Sons, Chichester ISBN 978-0-471-92769-3
*Hecht Eugene, Optics, 2002, Addison Wesley, ISBN 0-321-18878-0
*Jenkins FA & White HE, 1957, Fundamentals of Optics, 3rd Edition, McGraw Hill, New York
*Lipson A, Lipson SG, [[Henry Lipson|Lipson H]], 2011, ''Optical Physics'', 4th ed., Cambridge University Press, ISBN=978-0-521-49345-1
*Longhurst RS, 1967, Geometrical and Physical Optics, 2nd Edition, Longmans, London
*Whittaker and Watson, 1962, Modern Analysis, Cambridge University Press.

[[Category:Diffraction]]
[[Category:Fourier analysis]]

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Protein adsorption

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72.33.235.126: /* Ionic or Electrostatic Interactions */

[[Sonoluminescence]] is a phenomenon that occurs when a small gas bubble is acoustically suspended and periodically driven in a liquid solution at ultrasonic frequencies, resulting in bubble collapse, [[cavitation]], and light emission. The thermal energy that is released from the bubble collapse is so great that it can cause weak light emission.<ref name="Single-bubble sonoluminescence">{{cite journal | author=Michael P. Brenner| title=Single-bubble sonoluminescence| doi=10.1103/RevModPhys.74.425|journal=Reviews of Modern Physics| year=2002| volume=74| pages=425–484| url=http://rmp.aps.org/abstract/RMP/v74/i2/p425_1| bibcode=2002RvMP...74..425B}}</ref> The mechanism of the light emission remains uncertain, but some of the current theories, which are categorized under either thermal or electrical processes, are [[Bremsstrahlung]] radiation, [[Rectifier#Argon gas electron tube|argon rectification hypothesis]],<ref name="Evidence for Gas Exchange in Single-Bubble Sonoluminescence">{{cite journal |title=Evidence for Gas Exchange in Single-Bubble Sonoluminescence |pages=865–868 |author=Thomas J. Matula, Lawrence A. Crum |journal=Physical Review Letters |year=1998 |volume=80 |issue=4 |url=http://prl.aps.org/abstract/PRL/v80/i4/p865_1 | doi = 10.1103/PhysRevLett.80.865 |bibcode=1998PhRvL..80..865M}}</ref> and hot spot. People are beginning to lean more towards thermal processes as temperatures have consistently been proven with different methods of spectral analysis.<ref name="Temperature of Cavitation">{{cite journal | author=K.S. Suslick, W.B. McNamara III, Y. Didenko| title=Hot Spot Conditions During Multi-Bubble Cavitation| journal=Sonochemistry and Sonoluminescence| year=1999| pages=191–205| url=http://www.scs.illinois.edu/suslick/documents/natoasi.conditions.rv1.pdf}}</ref> In order to understand the light emission mechanism, it is important to know what is happening in the bubble's interior and at the bubble's surface. [[File:Sonoluminescence Setup.png|thumb|400px|A setup similar to the following is required to create a bubble that can sonoluminesce.]]

== Current competing theories ==
Prior to the early 1990s, the studies on different chemical and physical variables of sonoluminescence were all conducted using multi-bubble sonoluminescence (MBSL).<ref name="Interfacial Effects">{{cite book |title=Interfacial Effects on Aqueous Sonochemistry and Sonoluminescence |pages=1–252 |author=Joe Zeljko Sostaric |year=1999}}</ref> This was a problem since all of the theories and bubble dynamics were based on single bubble sonoluminescence (SBSL) and researchers believed that the bubble oscillations of neighboring bubbles could affect each other.<ref name="Interfacial Effects" /> Single bubble sonoluminescence wasn't achieved until the early 1990s and allowed the study of the effects of various parameters on a single cavitating bubble.<ref name="Interfacial Effects" /> After many of the early theories were disproved, the remaining plausible theories can be classified into two different processes: electrical and thermal.<ref name="Single-bubble sonoluminescence" /><ref name="Interfacial Effects" />

===Single-bubble sonoluminescence (SBSL)===
SBSL emits more light than MBSL due to fewer interactions between neighboring bubbles.<ref name="Interfacial Effects" /> Another advantage for SBSL is that a single bubble collapses without being affected by other surrounding bubbles, allowing more accurate studies on acoustic cavitation and sonoluminescence theories.<ref name="Interfacial Effects" /> Some exotic theories have been made, for example from Schwinger in 1992 who hinted the dynamical [[Casimir effect]] as a potential photon-emission process. Several theories say that the location of light emission is in the liquid instead of inside the bubble. Other SBSL theories explain, that the emission of photons due to the high temperatures in the bubble are analogical to the hot spot theories of MBSL. Regarding the thermal emission a large variety of different processes are prevalent. Because temperatures are increasing from several hundred to many thousand kelvin during collapse, the processes can be molecular recombination, collision-induced emission, molecular emission, excimers, atomic recombination, radiative attachments of ions, neutral and ion [[Bremsstrahlung]], or emission from confined electrons in voids. Which of these theories applies depends on accurate measurements and calculations of the temperature inside the bubble.<ref name="Single-bubble sonoluminescence" />

===Multi-bubble sonoluminescence (MBSL)===
Unlike single-bubble sonoluminescence, multi-bubble sonoluminescence is the creation of many oscillating and collapsing bubbles. Typically in MBSL, the light emission from each individual bubble is weaker than in SBSL because the neighboring bubbles can interact and affect each other.<ref name="Interfacial Effects" /> Because each neighboring bubble can interact with each other, it can make it more difficult to produce accurate studies and to characterize the properties of the collapsing bubble.[[File:MBSLred.JPG|thumb|300px|alt= Mutli-bubble sonoluminescence creates many oscillating and collapsing bubbles that will emit light. Typically the light emission is weaker than with single-bubble sonoluminescence. The bright blue dots, which are viewable when the image is at 100% are the sonoluminescencing bubbles.|Mutli-bubble sonoluminescence creates many oscillating and collapsing bubbles that will emit light. Typically the light emission is weaker than with single-bubble sonoluminescence. The bright blue dots, which are viewable when the image is at 100% are the sonoluminescencing bubbles.]]

== Bubble interior ==
One of the greatest obstacles in sonoluminescence research has been trying to obtain measurements of the interior of the bubble. Most measurements, like temperature and pressure, are indirectly measured using models and bubble dynamics.<ref name="Single-bubble sonoluminescence" />

===Temperature===
Some of the developed theories about the mechanism of SBSL result in prognoses for the peak temperature from 6000 K to 20,000 K. What they all have in common is, a) the interior of the bubble heats up and becomes at least as hot as that measured for MBSL, b) water vapor is the main temperature-limiting factor and c) the averaged temperature over the bubble does not rise higher than 10,000 K.<ref name="Single-bubble sonoluminescence" />

==Bubble dynamics==
These equations were made using five major assumptions,<ref name="Gaitan" /> with four of them being common to all the equations:
# The bubble remains spherical
# The bubble contents obey the [[ideal gas law]]
# The internal pressure remains uniform throughout the bubble
# No [[evaporation]] or [[condensation]] occurs inside the bubble
The fifth assumption, which changes between each formulation, pertains to the thermodynamic behavior of the liquid surrounding the bubble. These assumptions severely limit the models when the pulsations are large and the wall velocities reach the [[speed of sound]].

===Keller-Miksis formulation===
The Keller-Miksis formulation is an equation derived for the large, radial oscillations of a bubble trapped in a sound field. When the frequency of the sound field approaches the natural frequency of the bubble, it will result in large amplitude oscillations. The Keller-Miksis equation takes into account the viscosity, surface tension, incident sound wave, and acoustic radiation coming from the bubble, which was previously unaccounted for in Lauterborn's calculations. Lauterborn solved the equation that Plesset, ''et al.'' modified from Rayleigh's original analysis of large oscillating bubbles.<ref name="Keller and Miksis">{{cite journal|last=Keller|first=Joseph B.|coauthors=Michael Miksis|title=Bubble oscillations of large amplitude|journal=Journal of the Acoustical Society of America|date=August 1980|volume=68|issue=2|pages=628–633|url=http://asadl.org/jasa/resource/1/jasman/v68/i2/p628_s1|accessdate=30 May 2011|bibcode = 1980ASAJ...68..628K |doi = 10.1121/1.384720 }}</ref> Keller and Miksis obtained the following formula:<ref name="Gaitan" />

<math>\left ( 1 - \frac{\dot{R}}{c} \right ) R \ddot{R} + \frac{3}{2} \dot{R^2} \left ( 1 - \frac{\dot{R}}{3c} \right ) = \left ( 1 + \frac{\dot{R}}{c} \right ) \frac{1}{\rho_l} \left [ p_B(R,t) - p_A\left(t + \frac{R}{c}\right) - P_\infty \right ] + \frac{R dp_B(R,t)}{\rho_l c dt}</math>

where <math>\scriptstyle R</math> is the radius of the bubble, the dots indicate first and second time derivatives, <math>\scriptstyle \rho_l</math> is the density of the liquid, <math>\scriptstyle c</math> is the speed of sound through the liquid, <math>\scriptstyle dp_B(R,t)</math> is the pressure on the liquid side of the bubble's interface, <math>\scriptstyle t</math> is time, and <math>\scriptstyle p_A(t+\frac{R}{c})</math> is the time-delayed driving pressure.

===Prosperetti formulation===
Prosperetti found a way to accurately determine the internal pressure of the bubble using the following equation.<ref name=Prosperetti>{{cite journal|last=Prosperetti|first=Andrea|coauthors=Lawrence A. Crum, Kerry W. Commander|title=Nonlinear bubble dynamics|journal=Journal of the Acoustical Society of America|date=February 1988|volume=83|issue=2|pages=502–514|url=http://asadl.org/jasa/resource/1/jasman/v83/i2/p502_s1|accessdate=30 May 2011|bibcode = 1988ASAJ...83..502P |doi = 10.1121/1.396145 }}</ref>

<math> \dot{p} = \frac{3}{r} \left ( (\gamma - 1) K \frac {\partial T}{\partial r}\Bigg|_R - \gamma p \dot{R} \right )</math>

where <math>T</math> is the temperature, <math>K</math> is the thermal conductivity of the gas, and <math>r</math> is the radial distance.

===Flynn's formulation===
This formulation allows the study of the motions and the effects of heat conduction, shear viscosity, compressibility, and surface tension on small cavitation bubbles in liquids that are set into motion by an acoustic pressure field. The effect of vapor pressure on the cavitation bubble can also be determined using the interfacial temperature. The formulation is specifically designed to describe the motion of a bubble that expands to a maximum radius and then violently collapses or contracts.<ref name=Flynn>{{cite journal|last=Flynn|first=H.G.|title=Cavitation dynamics. I. A mathematical formulation|journal=Journal of the Acoustical Society of America|date=June 1975|volume=57|issue=6|pages=1379–1396|url=http://asadl.org/jasa/resource/1/jasman/v57/i6/p1379_s1|accessdate=30 May 2011|bibcode = 1975ASAJ...57.1379F |doi = 10.1121/1.380624 }}</ref> This set of equations was solved using an improved [[Euler method]].

<math>\left ( 1 - \frac{\dot{R}}{c} \right ) R \ddot{R} + \frac{3}{2} \dot{R^2} \left ( 1 - \frac{\dot{R}}{3c} \right ) = \left ( 1 + \frac{\dot{R}}{c} \right ) \frac{1}{\rho_l} \left [ p_B(R,t) - p_A(t) - P_\infty \right ] + \frac{R}{\rho_l c} \left ( 1 - \frac{\dot{R}}{c} \right ) \frac{dp_B(R,t)}{dt}</math>

where <math>\scriptstyle R</math> is the radius of the bubble, the dots indicate first and second time derivatives, <math>\scriptstyle \rho_l</math> is the density of the liquid, <math>\scriptstyle c</math> is the speed of sound through the liquid, <math>\scriptstyle dp_B(R,t)</math> is the pressure on the liquid side of the bubble's interface, <math>\scriptstyle t</math> is time, and <math>\scriptstyle p_A(t)</math> is the driving pressure.

=== Rayleigh-Plesset equation ===
The theory of bubble dynamics was started in 1917 by [[Lord Rayleigh]] during his work with the Royal Navy to investigate cavitation damage on ship propellers. Over several decades his work was refined and developed by [[Milton Plesset]], [[Andrea Prosperetti]], and others.<ref name="Single-bubble sonoluminescence" /> The [[Rayleigh-Plesset equation]]<ref name="Single-bubble sonoluminescence" /> is:

<math>R \ddot{R} + \frac{3}{2} \dot{R^2} = \frac{1}{\rho_l} \left ( p_g - P_0 - P \left ( t \right ) - 4\mu \frac{\dot{R}}{R} - \frac{2\gamma}{R} \right )</math>

where <math>\scriptstyle R</math> is the bubble radius, <math>\scriptstyle \ddot{R}</math> is the second order derivative of the bubble radius with respect to time, <math>\scriptstyle \dot{R}</math> is the first order derivative of the bubble radius with respect to time, <math>\scriptstyle \rho_l</math> is the density of the liquid, <math>\scriptstyle p_g</math> is the pressure in the gas (which is assumed to be uniform), <math>\scriptstyle P_0</math> is the background static pressure, <math>\scriptstyle P(t)</math> is the sinusoidal driving pressure, <math>\scriptstyle \mu</math> is the [[viscosity]] of the liquid, and <math>\scriptstyle \gamma</math> is the [[surface tension]] of the gas-liquid interface.

== Bubble surface ==
The surface of a collapsing bubble like those seen in both SBSL and MBSL serves as a boundary layer between the liquid and vapor phases of the solution.

===Generation===
MBSL has been observed in many different solutions under a variety of conditions. Unfortunately it is more difficult to study as the bubble cloud is uneven and can contain a wide range of pressures and temperatures. SBSL is easier to study due to the predictable nature of the bubble. This bubble is sustained in a [[standing acoustic wave]] of moderate pressure, approximately 1.5 atm.<ref>Flannigan DJ, Suslick KS. 2008. Inside a Collapsing Bubble: Sonoluminescence and the Conditions During Cavitation. Annu. Rev. Phys. Chem. 59:659–83</ref> Since cavitation does not normally occur at these pressures the bubble may be seeded through several techniques:
# Transient boiling through short current pulse in nichrome wire.
# A small jet of water perturbs the surface to introduce air bubbles.
# A rapidly formed vapor cavity via focused laser pulse.
The standing acoustic wave, which contains pressure antinodes at the center of the containment vessel, causes the bubbles to quickly coalesce into a single radially oscillating bubble.

===Collapse===
Once a single bubble is stabilized in the pressure antinode of the standing wave, it can be made to emit pulses of light by driving the bubble into highly nonlinear oscillations. This is done by the increasing pressure of the acoustic wave to disrupt the steady, linear growth of the bubble which cause the bubble to collapse in a runaway reaction that only reverts due to the high pressures inside the bubble at its minimum radius.

===Afterbounces===
The collapsed bubble expands due to high internal pressure and experiences a diminishing effect until the high pressure antinode returns to the center of the vessel. The bubble continues to occupy more or less the same space due to the acoustic radiation force, the [[Bjerknes force]], and the [[buoyancy]] force of the bubble.

[[File:Acoustic pressure vs bubble radius.png|thumb|400px|upright=3.0|Bubble oscillations correspond to pressure anti-nodes]]

===Surface chemistry===
The effect that different chemicals present in solution have to the velocity of the collapsing bubble has recently been studied. Nonvolatile liquids such as [[sulfuric]] and [[phosphoric acid]] have been shown to produce flashes of light several nanoseconds in duration with a much slower bubble wall velocity,<ref>Flannigan DJ, Suslick KS. 2005. Plasma formation and temperature measurement
during single-bubble cavitation. Nature 434:52–55</ref> and producing several thousand-fold greater light emission. This effect is probably masked in SBSL in aqueous solutions by the absorption of light by water molecules and contaminants.

====Surface tension====
It can be inferred from these results that the difference in surface tension between these different compounds is the source of different spectra emitted and the time scales in which emission occur.

== Light emission ==
The inertia of a collapsing bubble generates high pressures and temperatures capable of ionizing a small fraction of the noble gas within the volume of the bubble. This small fraction of ionized gas is transparent and allows for volume emission to be detected. Free electrons from the ionized noble gas begin to interact with other neutral atoms causing thermal [[bremsstrahlung]] radiation. Surface emission emits a more intense flash of light with a longer duration and is dependent on wavelength. Experimental data suggest that only volume emission occurs in the case of sonoluminescence.<ref name="Single-bubble sonoluminescence" /> As the sound wave reaches a low energy trough the bubble expands and electrons are able to recombine with free ions and halt light emission. Light pulse time is dependent on the [[ionization energy]] of the noble gas with argon having a light pulse of 160 picoseconds.

{| class="wikitable" style="text-align:center"
|-
! Radiance (W/nm) !! Relative brightness<ref name="Single-bubble sonoluminescence" />
|-
| 0.50{{E|-12}} || Bright
|-
| 9.00{{E|-13}} || Semi-bright
|-
| 1.75{{E|-13}} || Dim
|-
| 7.00{{E|-14}} || Very dim
|-
| 2.00{{E|-14}} || Extremely dim
|}

{| class="wikitable" style="text-align:center"
|-
! Solution type !! Average max. radiance (W/nm)<ref name="Single-bubble sonoluminescence" /><ref name=Unknown>{{cite journal|last=Barber|first=Bradley P.|coauthors=Robert A. Hiller, Ritva Losfstedt, Seth K. Putterman, Keith R. Weninger|title=Defining the Unknowns of Sonoluminescence|journal=Physics Reports|year=1997|volume=281|pages=65–143|url=http://www.sciencedirect.com/science/article/pii/S0370157396000506 | doi=10.1016/S0370-1573(96)00050-6 | bibcode=1997PhR...281...65B}}</ref>
|-
| Xenon in water || 1.04{{E|-9}}
|-
| Krypton in water || 8.00{{E|-10}}
|-
| Argon in water || 7.75{{E|-10}}
|-
| Neon in water || 5.40{{E|-10}}
|-
| Helium in water || 4.45{{E|-11}}
|-
| 3He in water || 3.60{{E|-11}}
|}

===Electrical processes===
In 1937, the explanations for the light emission have been with the favoritism through electrical discharges. The first ideas have been about the charge separation in cavitation bubbles, which have been seen as spherical capacitors with charges at the center and the wall.
At the collapse, the capacitance decreases and voltage increases until electric breakdown occurs. A further suggestion was a charge separation by enhancing charge fluctuations on the bubble wall, however, a breakdown should take place during the expansion phase of the bubble dynamics.
These discharge theories have to assume that the emitting bubble undergoes an asymmetric collapse, because a symmetric charge distribution cannot radiate light.<ref name="Single-bubble sonoluminescence" />

===Thermal processes===
Because the bubble collapse occurs within microseconds,<ref name=Gaitan>{{cite journal|last=Gaitan|first=D. Felipe|coauthors=Lawrence A. Crum, Charles C. Church, Ronald A. Roy|title=Sonoluminescence and bubble dynamics for a single,stable, cavitation bubble|journal=Journal of the Acoustical Society of America|date=June 1992|volume=91|issue=6|pages=3166–3183|url=http://asadl.org/jasa/resource/1/jasman/v91/i6/p3166_s1|accessdate=29 May 2011|bibcode = 1992ASAJ...91.3166G |doi = 10.1121/1.402855 }}</ref> the hot spot theory states that the thermal energy results from an [[adiabatic]] bubble collapse. In 1950 it was assumed that the bubble internal temperatures were as high as 10,000 K at the collapse of a spherical symmetric bubble.<ref name="Single-bubble sonoluminescence" /> In the 1990s, sonoluminescence spectra were used by [[Kenneth S. Suslick|Suslick]] to measure effective emission temperatures in bubble clouds (multibubble sonoluminescence) of 5000 K,<ref>Flint, E. B.; Suslick, K. S. "The Temperature of Cavitation" Science 1991, 253, 1397-1399.</ref><ref>McNamara III, W. B.; Didenko, Y.; Suslick, K. S. "Sonoluminescence Temperatures During Multibubble Cavitation" Nature, 1999, 401, 772-775.</ref> and more recently temperatures as high as 20,000 K in single bubble cavitation.<ref>Didenko, Y.; McNamara III, W. B.; Suslick, K. S. "Molecular Emission from Single Bubble Sonoluminescence" Nature, 2000, 406, 877-879.</ref><ref>Didenko, Y.; Suslick, K. S. "The Energy Efficiency of Formation of Photons, Radicals, and Ions During Single Bubble Cavitation" Nature 2002, 418, 394-397.</ref><ref>Flannigan, D. J.; Suslick, K. S. "Plasma Formation and Temperature Measurement during Single-Bubble Cavitation" Nature, 2005, 434, 52-55.</ref>

[[File:Hot Spot Diagram.png|thumb|400px|alt= Upon the collapse of a bubble experiencing cavitation, a hot spot is produced for a small amount of time. That hot spot contains a high temperature core that is surrounded by a cooler outer shell.|Upon the collapse of a bubble experiencing cavitation, a hot spot is produced for a small amount of time. That hot spot contains a high temperature core that is surrounded by a cooler outer shell.]]

== Bubble shape stability ==
The limit for the ambient size of the bubble is set by the appearance of instabilities in the shape of the oscillating bubble.
The shape stability thresholds depend on changes in the radial dynamics, caused by different liquid viscosities or driving frequencies. If the frequency is decreased, the parametric instability is suppressed as the stabilizing influence of viscosity can appear longer to suppress perturbations. However, the collapses of low-frequency-driven bubbles favor and earlier onset of the Rayleigh-Taylor instability. Larger bubbles can be stabilized to show sonoluminescence when not too high forcing pressures are applied. At low-frequency the water vapor becomes more important. The bubbles can be stabilized by cooling the fluid, whereas more light is emitted.<ref name="Single-bubble sonoluminescence" />

==See also==
* [[Bubble fusion]]

==References==
{{Reflist}}

[[Category:Luminescence]]