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{{Wikify|date=January 2012}}
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.


{{Refimprove|Date|date=August 2011}}
If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]]
{{Orphan|Date|date=August 2011}}
* Only registered users will be able to execute this rendering mode.
* Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.


The '''T-matrix method''' is a computatational technique of [[light scattering]] by nonspherical particles originally formulated by [http://www.giss.nasa.gov/staff/mmishchenko/t_matrix.html P. C. Waterman] (1928-2012) in 1965.<ref>M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, T-matrix computations of light scattering by nonspherical particles: A review, J. Quant. Spectrosc. Radiat. Transfer, 55, 535-575 (1996).</ref> The technique is also known as null field method and extended boundary technique method (EBCM). In the method, matrix elements are obtained by matching boundary conditions for solutions of [[Maxwell equations]].
Registered users will be able to choose between the following three rendering modes:  


== Definition of the T-Matrix ==
'''MathML'''
:<math forcemathmode="mathml">E=mc^2</math>


The incident and scattered electric field are expanded into spherical vector wave functions (SVWF), which are also encountered in [[Mie scattering]]. They are the fundamental solutions of the vector [[Helmholtz equation]] and
<!--'''PNG'''  (currently default in production)
can be generated from the scalar fundamental solutions in spherical coordinates, the spherical [[Bessel functions]] of the first kind and the spherical Hankel Functions. Accordingly, there are two linearly independent sets of solutions
:<math forcemathmode="png">E=mc^2</math>
denoted as <math>\mathbf{M}^1,\mathbf{N}^1</math> and <math>\mathbf{M}^3,\mathbf{N}^3</math>, respectively. They are also called regular and propagating SVWFs, respectively. With this, we can write the incident field as


<math>\mathbf{E}_{inc}= \sum_{n=1}^\infty \sum_{m=-n}^n a_{mn} \mathbf{M}^1_{mn}+ b_{mn} \mathbf{N}^1_{mn}.</math>
'''source'''
:<math forcemathmode="source">E=mc^2</math> -->


The scattered field is expanded into radiating SVWFs:
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].


<math>\mathbf{E}_{scat}= \sum_{n=1}^\infty \sum_{m=-n}^n f_{mn} \mathbf{M}^3_{mn}+ g_{mn} \mathbf{N}^3_{mn}.</math>
==Demos==


The T-Matrix relates the expansion coefficients of the incident field to those of the scattered field.  
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


<math>\begin{pmatrix} a_{mn}\\ b_{mn}\end{pmatrix} = T \begin{pmatrix} f_{mn} \\ g_{mn} \end{pmatrix}</math>


The T-Matrix is determined by the scatterer shape and material and for given indcient field allows to calculate the scattered
* accessibility:
field.  
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)]
** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]].
** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.


== Calculation of the T-Matrix ==
==Test pages ==


The standard way to actually calculate the T-Matrix method is the Null-Field Method, that relies on the Stratton-Chu equations <ref>J.A. Stratton & L.J. Chu: Diffraction Theory of Electromagnetic Waves Phys. Rev., American Physical Society, 56 (1939)</ref>. They basically state that the electromagnetic fields outside a given volume can be expressed as integrals over the surface enclosing the volume involving only the tangential components of the fields on the surface. If the observation point is located inside this volume, the integrals vanish.
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
*[[Displaystyle]]
*[[MathAxisAlignment]]
*[[Styling]]
*[[Linebreaking]]
*[[Unique Ids]]
*[[Help:Formula]]


By making use of the boundary conditions for the tangential field components on the scatterer surface
*[[Inputtypes|Inputtypes (private Wikis only)]]
<math>\mathbf{n} \times (\mathbf{E}_{scat} + \mathbf{E}_{inc}) = \mathbf{E}_{int}</math> and <math>\mathbf{n} \times (\mathbf{H}_{scat} + \mathbf{H}_{inc}) = \mathbf{H}_{int}</math>, where <math>\mathbf{n}</math> is the normal vector to the scatterer surface, one can derive an integral representation of the scattered field in terms of the tangential components of the internal fields on the scatterer surface. A similar representation can be derived for the incident field.
*[[Url2Image|Url2Image (private Wikis only)]]
 
==Bug reporting==
By expanding the internal field in terms of SVWFs and exploiting their orthogonality on spherical surfaces, one arrives at an expression for the T-Matrix. Numerical codes for the evaluation of the T-Matrix can be found online [http://www.scattport.org/index.php/light-scattering-software/t-matrix-codes/list].
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .
 
== References ==
{{reflist|2}}
 
{{DEFAULTSORT:T-matrix method}}
[[Category:Computational physics]]
[[Category:Electromagnetism]]
[[Category:Electrodynamics]]
[[Category:Scattering, absorption and radiative transfer (optics)]]

Latest revision as of 22:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

If you would like use the MathML rendering mode, you need a wikipedia user account that can be registered here [[1]]

  • Only registered users will be able to execute this rendering mode.
  • Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.

Registered users will be able to choose between the following three rendering modes:

MathML

E=mc2


Follow this link to change your Math rendering settings. You can also add a Custom CSS to force the MathML/SVG rendering or select different font families. See these examples.

Demos

Here are some demos:


Test pages

To test the MathML, PNG, and source rendering modes, please go to one of the following test pages:

Bug reporting

If you find any bugs, please report them at Bugzilla, or write an email to math_bugs (at) ckurs (dot) de .

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