Calkin–Wilf tree: Difference between revisions

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Undid revision 597432046 by Xelnx (talk) you could do it that way, but it's unnecessarily convoluted, and you have to change the 2x at the start of the formula
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'''[[Gustav Kirchhoff|Kirchhoff]]'s integral theorem''' (sometimes referred to as the Fresnel-Kirchhoff integral theorem)<ref>G. Kirchhoff, Ann. d. Physik. 1883, 2, 18, p663</ref> uses [[Green's identities]] to derive the solution to the homogeneous [[wave equation]] at an arbitrary point '''P''' in terms of the values of the solution of the wave equation and its first order derivative at all points on an arbitrary surface which encloses '''P'''.<ref name = "Born and Wolf">Max Born and Emil Wolf, Principles of Optics, 1999, Cambridge University Press, Cambridge, pp. 417-420</ref>
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==Equation==
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===Monochromatic waves===
 
The integral has the following form for a [[monochromatic]] wave:<ref name = "Born and Wolf"/>
<ref>Introduction to Fourier Optics J. Goodman sec. 3.3.3</ref>
:<math>U(r)  = \frac {1}{4 \pi} \int_{S} \left[ U \frac {\partial}{\partial n} \left( \frac {e^{iks}}{s} \right) - \frac {e^{iks}}{s} \frac {\partial U}{\partial n} \right]dS </math>
 
where the integration is performed over the whole of the arbitrary surface ''S'', ''s'' is the distance between the point ''r'' and the surface ''S'', and ∂/∂''n'' denotes differentiation along the normal on the surface with direction into the surface.
 
Please note: It may be confusing because most used to normal direction pointing outwards of the surface. In that case the eq. shall be multiplied by minus.
 
===Non-monochromatic waves===
 
A more  general form can be derived for non-monochromatic waves. The complex amplitude of the wave can be represented by a Fourier integral of the form:
 
:<math> V(r,t)= \frac {1}{\sqrt{2 \pi}} \int U_ \omega(r) e^{-i \omega t}d \omega</math>
 
where, by [[Fourier inversion theorem|Fourier inversion]], we have:
 
:<math> U_\omega (r)= \frac {1}{\sqrt{2 \pi}} \int V(r,t) e^{i \omega t}dt</math>
 
The integral theorem (above) is applied to each Fourier component {{math|U<sub>ω</sub>}}, and the following expression is obtained<ref name = "Born and Wolf"/>
 
:<math>V(r,t)  = \frac {1}{4 \pi} \int_{S}  \left\{[V] \frac {\partial}{\partial n} \left(\frac {1}{s}\right) - \frac {1}{cs} \frac {\partial s}{\partial n} \left[\frac{\partial V}{\partial t}\right] - \frac{1}{s} \left[\frac{\partial V}{\partial n} \right] \right\} dS </math>
 
where the square brackets on ''V'' terms denote retarded values, i.e. the values at time ''t − s/c''.
 
Kirchhoff  showed the above equation can be approximated in many cases to a simpler form, known as the [[Kirchhoff's diffraction formula|Kirchhoff, or Fresnel-Kirchhoff diffraction formula]], which is equivalent to the [[Huygens-Fresnel principle|Huygens-Fresnel equation]], but provides a formula for the inclination factor which is not defined in the latter. The diffraction integral can be applied to a wide range of problems in optics.
 
==See also==
 
*[[Kirchhoff's diffraction formula]]
*[[Vector calculus]]
*[[Integral]]
*[[Huygens–Fresnel principle]]
*[[Wavefront]]
*[[Surface]]
 
== References ==
<references />
 
==Further reading==
* ''The Cambridge Handbook of Physics Formulas'', G. Woan, Cambridge University Press, 2010, ISBN 978-0-521-57507-2.
* ''Introduction to Electrodynamics (3rd Edition)'', D.J. Griffiths, Pearson Education, Dorling Kindersley, 2007, ISBN 81-7758-293-3
* ''Light and Matter: Electromagnetism, Optics, Spectroscopy and Lasers'', Y.B. Band, John Wiley & Sons, 2010, ISBN 978-0-471-89931-0
* ''The Light Fantastic – Introduction to Classic and Quantum Optics'', I.R. Kenyon, Oxford University Press, 2008, ISBN 978-0-19-856646-5
* ''Encyclopaedia of Physics (2nd Edition)'', R.G. Lerner, G.L. Trigg, VHC publishers, 1991, ISBN (Verlagsgesellschaft) 3-527-26954-1, ISBN (VHC Inc.) 0-89573-752-3
* ''McGraw Hill Encyclopaedia of Physics (2nd Edition)'', C.B. Parker, 1994, ISBN 0-07-051400-3
 
[[Category:Diffraction| ]]
[[Category:Optics]]

Revision as of 03:41, 28 February 2014

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