[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki8\/kirchhoff-integral-theorem-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki8\/kirchhoff-integral-theorem-wikipedia\/","headline":"Kirchhoff integral theorem – Wikipedia","name":"Kirchhoff integral theorem – Wikipedia","description":"before-content-x4 Kirchhoff’s integral theorem (sometimes referred to as the Fresnel\u2013Kirchhoff integral theorem)[1] is a surface integral to obtain the value","datePublished":"2015-02-04","dateModified":"2015-02-04","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/wiki8\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/wiki8\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/30cb22bc92192e0048753d8432ade8ce9886f120","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/30cb22bc92192e0048753d8432ade8ce9886f120","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/en\/wiki8\/kirchhoff-integral-theorem-wikipedia\/","wordCount":11726,"articleBody":"     (adsbygoogle = window.adsbygoogle || []).push({});before-content-x4Kirchhoff’s integral theorem (sometimes referred to as the Fresnel\u2013Kirchhoff integral theorem)[1] is a surface integral to obtain the value of the solution of the homogeneous scalar wave equation at an arbitrary point P in terms of the values of the solution and the solution’s first-order derivative at all points on an arbitrary closed surface (on which the integration is performed) that encloses P.[2] It is derived by using the Green’s second identity and the homogeneous scalar wave equation that makes the volume integration in the Green’s second identity zero.[2][3]       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4Table of ContentsIntegral[edit]Monochromatic wave[edit]Non-monochromatic wave[edit]Integral derivation[edit]See also[edit]References[edit]Further reading[edit]Integral[edit]Monochromatic wave[edit]The integral has the following form for a monochromatic wave:[2][3][4]       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4U(r)=14\u03c0\u222bS[U\u2202\u2202n^(eikss)\u2212eikss\u2202U\u2202n^]dS,{displaystyle U(mathbf {r} )={frac {1}{4pi }}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,}where the integration is performed over an arbitrary closed surface S enclosing the observation point r{displaystyle mathbf {r} }, k{displaystyle k} in        (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4eiks{displaystyle e^{iks}} is the wavenumber, s{displaystyle s} in eikss{displaystyle {frac {e^{iks}}{s}}} is the distance from an (infinitesimally small) integral surface element to the point r{displaystyle mathbf {r} }, U{displaystyle U} is the spatial part of the solution of the homogeneous scalar wave equation (i.e., V(r,t)=U(r)e\u2212i\u03c9t{displaystyle V(mathbf {r} ,t)=U(mathbf {r} )e^{-iomega t}} as the homogeneous scalar wave equation solution), n^{displaystyle {hat {mathbf {n} }}} is the unit vector inward from and normal to the integral surface element, i.e., the inward surface normal unit vector, and \u2202\u2202n^{displaystyle {frac {partial }{partial {hat {mathbf {n} }}}}} denotes differentiation along the surface normal (i.e., a normal derivative) i.e., \u2202f\u2202n^=\u2207f\u22c5n^{displaystyle {frac {partial f}{partial {hat {mathbf {n} }}}}=nabla fcdot {hat {mathbf {n} }}} for a scalar field f{displaystyle f}. Note that the surface normal is inward, i.e., it is toward the inside of the enclosed volume, in this integral; if the more usual outer-pointing normal is used, the integral will have the opposite sign.This integral can be written in a more familiar formU(r)=14\u03c0\u222bS(U\u2207(eikss)\u2212eikss\u2207U)\u22c5dS\u2192,{displaystyle U(mathbf {r} )={frac {1}{4pi }}int _{S}left(Unabla left({frac {e^{iks}}{s}}right)-{frac {e^{iks}}{s}}nabla Uright)cdot d{vec {S}},}where dS\u2192=dSn^{displaystyle d{vec {S}}=dS{hat {mathbf {n} }}}.[3]Non-monochromatic wave[edit]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 formV(r,t)=12\u03c0\u222bU\u03c9(r)e\u2212i\u03c9td\u03c9,{displaystyle V(r,t)={frac {1}{sqrt {2pi }}}int U_{omega }(r)e^{-iomega t},domega ,}where, by Fourier inversion, we haveU\u03c9(r)=12\u03c0\u222bV(r,t)ei\u03c9tdt.{displaystyle U_{omega }(r)={frac {1}{sqrt {2pi }}}int V(r,t)e^{iomega t},dt.}The integral theorem (above) is applied to each Fourier component U\u03c9{displaystyle U_{omega }}, and the following expression is obtained:[2]V(r,t)=14\u03c0\u222bS{[V]\u2202\u2202n(1s)\u22121cs\u2202s\u2202n[\u2202V\u2202t]\u22121s[\u2202V\u2202n]}dS,{displaystyle V(r,t)={frac {1}{4pi }}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,}where the square brackets on V terms denote retarded values, i.e. the values at time t \u2212 s\/c.Kirchhoff showed that the above equation can be approximated to a simpler form in many cases, known as the Kirchhoff, or Fresnel\u2013Kirchhoff diffraction formula, which is equivalent to the Huygens\u2013Fresnel equation, except that it provides the inclination factor, which is not defined in the Huygens\u2013Fresnel equation. The diffraction integral can be applied to a wide range of problems in optics.Integral derivation[edit]Here, the derivation of the Kirchhoff’s integral theorem is introduced. First, the Green’s second identity as the following is used.\u222bV(U1\u22072U2\u2212U2\u22072U1)dV=\u222e\u2202V(U2\u2202U1\u2202n^\u2212U1\u2202U2\u2202n^)dS,{displaystyle int _{V}left(U_{1}nabla ^{2}U_{2}-U_{2}nabla ^{2}U_{1}right)dV=oint _{partial V}left(U_{2}{partial U_{1} over partial {hat {mathbf {n} }}}-U_{1}{partial U_{2} over partial {hat {mathbf {n} }}}right)dS,}where the integral surface normal unit vector n^{displaystyle {hat {mathbf {n} }}} here is toward the volume V{displaystyle V} closed by an integral surface \u2202V{displaystyle partial V}. Scalar field functions U1{displaystyle U_{1}} and U2{displaystyle U_{2}} are set as solutions of the Helmholtz equation, \u22072U+k2U=0{displaystyle nabla ^{2}U+k^{2}U=0} where k=2\u03c0\u03bb{displaystyle k={frac {2pi }{lambda }}} is the wavenumber (\u03bb{displaystyle lambda } is the wavelength), that gives the spatial part of a complex-valued monochromatic (single frequency in time) wave expression. (The product between the spatial part and the temporal part of the wave expression is a solution of the scalar wave equation.) Then, the volume part of the Green’s second identity is zero, so only the surface integral is remained as\u222e\u2202V(U2\u2202U1\u2202n^\u2212U1\u2202U2\u2202n^)dS=0.{displaystyle oint _{partial V}left(U_{2}{partial U_{1} over partial {hat {mathbf {n} }}}-U_{1}{partial U_{2} over partial {hat {mathbf {n} }}}right)dS=0.}Now U2{displaystyle U_{2}} is set as the solution of the Helmholtz equation to find and U1{displaystyle U_{1}} is set as the spatial part of a complex-valued monochromatic spherical wave U1=eikss{displaystyle U_{1}={frac {e^{iks}}{s}}} where s{displaystyle s} is the distance from an observation point P{displaystyle P} in the closed volume V{displaystyle V}. Since there is a singularity for U1=eikss{displaystyle U_{1}={frac {e^{iks}}{s}}} at P{displaystyle P} where s=0{displaystyle s=0} (the value of eikss{displaystyle {frac {e^{iks}}{s}}} not defined at s=0{displaystyle s=0}), the integral surface must not include P{displaystyle P}. (Otherwise, the zero volume integral above is not justified.) A suggested integral surface is an inner sphere S1{displaystyle S_{1}} centered at P{displaystyle P} with the radius of s1{displaystyle s_{1}} and an outer arbitrary closed surface S2{displaystyle S_{2}}.Then the surface integral becomes\u222eS1(U2\u2202\u2202n^eikss\u2212eikss\u2202\u2202n^U2)dS+\u222eS2(U2\u2202\u2202n^eikss\u2212eikss\u2202\u2202n^U2)dS=0.{displaystyle oint _{S_{1}}left(U_{2}{partial  over partial {hat {mathbf {n} }}}{frac {e^{iks}}{s}}-{frac {e^{iks}}{s}}{partial  over partial {hat {mathbf {n} }}}U_{2}right)dS+oint _{S_{2}}left(U_{2}{partial  over partial {hat {mathbf {n} }}}{frac {e^{iks}}{s}}-{frac {e^{iks}}{s}}{partial  over partial {hat {mathbf {n} }}}U_{2}right)dS=0.}For the integral on the inner sphere S1{displaystyle S_{1}},\u2202\u2202n^eikss=\u2207eikss\u22c5n^=(iks\u22121s2)eiks,{displaystyle {frac {partial }{partial {hat {mathbf {n} }}}}{frac {e^{iks}}{s}}=nabla {frac {e^{iks}}{s}}cdot {hat {mathbf {n} }}=left({frac {ik}{s}}-{frac {1}{s^{2}}}right)e^{iks},}and by introducing the solid angle d\u03a9{displaystyle dOmega } in dS=s2d\u03a9{displaystyle dS=s^{2}dOmega },\u222eS1(U2\u2202\u2202n^eikss\u2212eikss\u2202\u2202n^U2)dS=\u222eS1(U2(iks\u22121s2)eiks\u2212eikss\u2202\u2202n^U2)s2d\u03a9=\u222eS1(iksU2\u2212U2\u2212s\u2202\u2202n^U2)eiksd\u03a9{displaystyle oint _{S_{1}}left(U_{2}{partial  over partial {hat {mathbf {n} }}}{frac {e^{iks}}{s}}-{frac {e^{iks}}{s}}{partial  over partial {hat {mathbf {n} }}}U_{2}right)dS=oint _{S_{1}}left(U_{2}left({frac {ik}{s}}-{frac {1}{s^{2}}}right)e^{iks}-{frac {e^{iks}}{s}}{partial  over partial {hat {mathbf {n} }}}U_{2}right)s^{2}dOmega =oint _{S_{1}}left(iksU_{2}-U_{2}-s{frac {partial }{partial {hat {mathbf {n} }}}}U_{2}right)e^{iks}dOmega }due to \u2202\u2202n^U2=\u2207U2\u22c5n^=\u2202\u2202sU2{displaystyle {frac {partial }{partial {hat {mathbf {n} }}}}U_{2}=nabla U_{2}cdot {hat {mathbf {n} }}={frac {partial }{partial s}}U_{2}}. (The spherical coordinate system which origin is at P{displaystyle P} can be used to derive this equality.)By shrinking the sphere S1{displaystyle S_{1}} toward the zero radius (but never touching P{displaystyle P} to avoid the singularity), eiks\u21921{displaystyle e^{iks}to 1} and the first and last terms in the S1{displaystyle S_{1}} surface integral becomes zero, so the integral becomes \u22124\u03c0U2{displaystyle -4pi U_{2}}. As a result, denoting U2{displaystyle U_{2}}, the location of P{displaystyle P}, and S2{displaystyle S_{2}} by U{displaystyle U}, the position vector r{displaystyle mathbf {r} }, and S{displaystyle S} respectively,U(r)=14\u03c0\u222eS(U\u2202\u2202n^eikss\u2212eikss\u2202\u2202n^U)dS.{displaystyle U(mathbf {r} )={frac {1}{4pi }}oint _{S}left(U{partial  over partial {hat {mathbf {n} }}}{frac {e^{iks}}{s}}-{frac {e^{iks}}{s}}{partial  over partial {hat {mathbf {n} }}}Uright)dS.}See also[edit]References[edit]^ G. Kirchhoff, Ann. d. Physik. 1883, 2, 18, p. 663.^ a b c d Max Born and Emil Wolf, Principles of Optics, 7th edition, 1999, Cambridge University Press, Cambridge, pp. 418\u2013421.^ a b c Hecht, Eugene (2017). “Appendix 2: The Kirchhoff Diffraction Theory”. Optics (5th and Global\u00a0ed.). Pearson Education. p.\u00a0680. ISBN\u00a0978-1292096933.^ Introduction to Fourier Optics J. Goodman sec. 3.3.3Further reading[edit]The Cambridge Handbook of Physics Formulas, G. Woan, Cambridge University Press, 2010, ISBN\u00a0978-0-521-57507-2.Introduction to Electrodynamics (3rd Edition), D.J. Griffiths, Pearson Education, Dorling Kindersley, 2007, ISBN\u00a081-7758-293-3Light and Matter: Electromagnetism, Optics, Spectroscopy and Lasers, Y.B. Band, John Wiley & Sons, 2010, ISBN\u00a0978-0-471-89931-0The Light Fantastic \u2013 Introduction to Classic and Quantum Optics, I.R. Kenyon, Oxford University Press, 2008, ISBN\u00a0978-0-19-856646-5Encyclopaedia of Physics (2nd Edition), R.G. Lerner, G.L. Trigg, VHC publishers, 1991, ISBN (Verlagsgesellschaft) 3-527-26954-1, ISBN (VHC Inc.) 0-89573-752-3McGraw Hill Encyclopaedia of Physics (2nd Edition), C.B. Parker, 1994, ISBN\u00a00-07-051400-3     (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki8\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki8\/kirchhoff-integral-theorem-wikipedia\/#breadcrumbitem","name":"Kirchhoff integral theorem – Wikipedia"}}]}]