Difference between revisions of "Kirchhoff integral theorem"

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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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'''[[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 that encloses '''P'''.<ref name = "Born and Wolf">Max Born and Emil Wolf, Principles of Optics, 1999, Cambridge University Press, Cambridge, pp. 417-420</ref>
  
 
==Equation==
 
==Equation==
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===Monochromatic waves===
 
===Monochromatic waves===
  
The integral has the following form for a [[monochromatic]] wave:<ref name = "Born and Wolf"/>
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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>
<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.
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:<math>U(\mathbf{r}) = \frac{1}{4\pi} \int_S \left[ U \frac{\partial}{\partial\hat{\mathbf{n}}} \left( \frac{e^{iks}}{s} \right) - \frac{e^{iks}}{s} \frac{\partial U}{\partial\hat{\mathbf{n}}} \right] dS,</math>
  
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.
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where the integration is performed over an arbitrary [[closed surface]] ''S'' (enclosing '''r'''), ''s'' is the distance from the surface element to the point '''r''', and ∂/∂'''n''' denotes differentiation along the surface normal. Note that in this equation the normal points inside the enclosed volume; if the more usual [[Normal (geometry)#Uniqueness of the normal|outer-pointing normal]] is used, the integral has the opposite sign.
  
 
===Non-monochromatic waves===
 
===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:
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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>
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:<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:
 
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>
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:<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"/>
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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>
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:<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''.
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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.
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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 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==
 
==See also==

Latest revision as of 01:14, 13 October 2014

Kirchhoff's integral theorem (sometimes referred to as the Fresnel-Kirchhoff integral theorem)[1] 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 that encloses P.[2]

Equation

Monochromatic waves

The integral has the following form for a monochromatic wave:[2][3]

where the integration is performed over an arbitrary closed surface S (enclosing r), s is the distance from the surface element to the point r, and ∂/∂n denotes differentiation along the surface normal. Note that in this equation the normal points inside the enclosed volume; if the more usual outer-pointing normal is used, the integral has the opposite sign.

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:

where, by Fourier inversion, we have:

The integral theorem (above) is applied to each Fourier component Uω, and the following expression is obtained:[2]

where the square brackets on V terms denote retarded values, i.e. the values at time ts/c.

Kirchhoff showed the above equation can be approximated in many cases to a simpler form, known as the Kirchhoff, or Fresnel–Kirchhoff diffraction formula, which is equivalent to the 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

References

  1. G. Kirchhoff, Ann. d. Physik. 1883, 2, 18, p663
  2. 2.0 2.1 2.2 Max Born and Emil Wolf, Principles of Optics, 1999, Cambridge University Press, Cambridge, pp. 417-420
  3. Introduction to Fourier Optics J. Goodman sec. 3.3.3

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