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[[File:Surface equivalence principle in electromagnetics.png|thumb|365px|Illustration of the equivalence principle for an imaginary closed surface with impressed [[Electric current|electric]] <math>(J_s)</math> and [[magnetic current]] <math>(M_s)</math> sources: original (a) and equivalent (b) problems over the imaginary surface, <math>S</math>. <math>J_1</math> and <math>M_1</math> represent the original source distributions inside the surface. The equivalent formulation yields the same external [[electric field|electric]] and [[magnetic field]]s distribution as in the original problem. The internal fields and surface currents are chosen to enforce the [[Interface conditions for electromagnetic fields|boundary conditions]].]]
In [[electromagnetism]], '''surface equivalence principle''' or '''surface equivalence theorem''' relates an arbitrary [[electric current|current]] distribution within an imaginary [[closed surface]] with an equivalent source on the surface. It is also known as '''field equivalence principle''',<ref name="Rengarajan00"/> '''Huygens' equivalence principle'''<ref name="chew93"/> or simply as the '''equivalence principle'''.{{sfn|Harrington|2001|p=106-109}} Being a more rigorous reformulation of the [[Huygens–Fresnel principle]], it is often used to simplify the analysis of [[Electromagnetic radiation|radiating]] structures such as [[Antenna (radio)|antennas]]. 

Certain formulations of the principle are also known as '''Love equivalence principle''' and '''Schelkunoff equivalence principle''', after [[Augustus Edward Hough Love]] and [[Sergei Alexander Schelkunoff]], respectively.

==Physical meaning==
===General formulation===
The principle yields an equivalent problem for a [[Electromagnetic radiation|radiation]] problem by introducing an imaginary [[closed surface]] and fictitious [[current density|surface current densities]]. It is an extension of [[Huygens–Fresnel principle]], which describes each point on a [[wavefront]] as a [[spherical wave]] source. The equivalence of the imaginary surface currents are enforced by the [[Electromagnetism uniqueness theorem|uniqueness theorem in electromagnetism]], which dictates that a unique solution can be determined by fixing a [[boundary condition]] on a system. With the appropriate choice of the imaginary current densities, the fields inside the surface or outside the surface can be deduced from the imaginary currents.{{sfn|Balanis|2012|p=328-331}}  In a radiation problem with given current density sources, [[electric current]] density <math>J_1</math> and [[magnetic current]] density <math>M_1</math>, the [[Interface conditions for electromagnetic fields|tangential field boundary conditions]] necessitate that

:<math>J_s = \hat{n} \times (H_1 - H)</math>
:<math>M_s = -\hat{n} \times (E_1 - E)</math>

where <math>J_s</math> and <math>M_s</math> correspond to the imaginary current sources that are impressed on the closed surface. <math>E</math> and <math>H</math> represent the [[electric field|electric]] and [[magnetic field]]s inside the surface, respectively, while <math>E_1</math> and <math>H_1</math> are the fields outside of the surface. Both the original and imaginary currents should produce the same external field distributions.{{sfn|Balanis|2012|p=328-331}}

===Love and Schelkunoff equivalence principles===
[[File:Love and Schelkunoff equivalence.png|thumb|365px|Illustration of Love and Schelkunoff equivalence principles: a) The original problem, b) Love equivalent problem, c) Schelkunoff equivalent problem with perfect electric conductor, d) Schelkunoff equivalent problem with perfect magnetic conductor. All problems have the same external fields.]]
Per the boundary conditions, the fields inside the surface and the current densities can be arbitrarily chosen as long as they produce the same external fields.{{sfn|Harrington|2001|p=106-109}} Love's equivalence principle, introduced in 1901 by [[Augustus Edward Hough Love]],<ref name="Love01"/> takes the internal fields as zero:

:<math>J_s = \hat{n} \times H_1</math>
:<math>M_s = -\hat{n} \times E_1</math>

The fields inside the surface are referred as null fields. Thus, the surface currents are chosen as to sustain the external fields in the original problem. Alternatively, Love equivalent problem for field distributions inside the surface can be formulated: this requires the negative of surface currents for the external radiation case. Thus, the surface currents will radiate the fields in the original problem in the inside of the surface; nevertheless, they will produce null external fields.<ref name="Rengarajan00"/>

Schelkunoff equivalence principle, introduced by [[Sergei Alexander Schelkunoff]],<ref name="Schelkunoff38"/><ref name="Schelkunoff39"/><ref name="Schelkunoff51"/> substitutes the closed surface with a [[Perfect conductor|perfectly conducting]] material body. In the case of a perfect electrical conductor, the electric currents that are impressed on the surface won't radiate due to [[Reciprocity (electromagnetism)|Lorentz reciprocity]]. Thus, the original currents can be substituted with surface magnetic currents only. A similar formulation for a perfect magnetic conductor would use impressed electric currents.<ref name="Rengarajan00"/>

The equivalence principles can also be applied to conductive [[Half-space (geometry)|half-spaces]] with the aid of [[method of image charges]].<ref name="Rengarajan00"/>{{sfn|Balanis|2012|p=328-331}}

==Applications==
The surface equivalence principle is heavily used in the analysis of [[antenna (radio)|antenna]] problems to simplify the problem: in many of the applications, the close surface is chosen as so to encompass the conductive elements to alleviate the limits of [[Numerical integration|integration]].{{sfn|Balanis|2012|p=328-331}} Selected uses in antenna theory include the analysis of [[Antenna aperture|aperture antennas]]{{sfn|Balanis|2005|p=653-660}} and the cavity model approach for [[microstrip antenna|microstrip]] [[patch antenna]]s.{{sfn|Kinayman|Aksun|2005|p=300}} It has also been used as a [[Domain decomposition methods|domain decomposition method]] for [[Method of moments (electromagnetics)|method of moments]] analysis of complex antenna structures.<ref name="Li07"/> Schelkunoff's formulation is employed particularly for [[scattering]] problems.<ref name="chew93"/>{{sfn|Balanis|2012|p=333}}<ref name="barber-osa"/><ref name="fem-mom"/>

The principle has also been used in the analysis design of [[metamaterial]]s such as Huygens’ metasurfaces<ref name="Huygens-metasurface"/><ref name="active-cloak"/> and [[plasmonics|plasmonic]] scatterers.<ref name="array-plasmonic"/>

==See also==
* [[Babinet's principle]]
* [[Electromagnetism uniqueness theorem]]
* [[Huygens–Fresnel principle]]
* [[Reciprocity (electromagnetism)]]

==References==
{{Reflist|30em|refs=

<ref name="active-cloak">{{cite journal |last1=Selvanayagam |first1=Michael |last2=Eleftheriades |first2=George. V. |author2-link=George V. Eleftheriades |title=An Active Electromagnetic Cloak Using the Equivalence Principle |journal=[[IEEE Antennas and Wireless Propagation Letters]] |date=October 2012 |volume=22 |pages=1226–1229 |doi=10.1109/LAWP.2012.2224840|bibcode=2012IAWPL..11.1226S |s2cid=45470309 }}</ref>

<ref name="array-plasmonic">{{cite journal |last1=Memarzadeh |first1=Babak |last2=Mosallaei |first2=Hossein |title=Array of planar plasmonic scatterers functioning as light concentrator |journal=[[Optics Letters]] |date=2011 |volume=36 |issue=13 |pages=2569–2571 |doi=10.1364/OL.36.002569|pmid=21725482 |bibcode=2011OptL...36.2569M }}</ref>

<ref name="barber-osa">{{cite journal |last1=Barber |first1=P. |last2=Yeh |first2=C. |title=Scattering of electromagnetic waves by arbitrarily shaped dielectric bodies |journal=[[Applied Optics]] |date=1975 |volume=14 |issue=12 |pages=2864–2872 |doi=10.1364/AO.14.002864|pmid=20155124 |bibcode=1975ApOpt..14.2864B }}</ref>

<ref name="chew93">{{cite journal |last1=Chew |first1=W.C. |last2=Lu |first2=C.C. |author1-link=Weng Cho Chew |title=The use of Huygens' equivalence principle for solving the volume integral equation of scattering |journal=[[IEEE Transactions on Antennas and Propagation]] |date=July 1993 |volume=41 |issue=7 |pages=897–904 |doi=10.1109/8.237620|bibcode=1993ITAP...41..897C }}</ref>

<ref name="fem-mom">{{cite journal |last1=Yuan |first1=X. |last2=Lynch |first2=D.R. |last3=Strohbehn |first3=J.W. |title=Coupling of finite element and moment methods for electromagnetic scattering from inhomogeneous objects |journal=[[IEEE Transactions on Antennas and Propagation]] |date=March 1990 |volume=38 |issue=3 |pages=386–393 |doi=10.1109/8.52246|bibcode=1990ITAP...38..386Y }}</ref>

<ref name="Huygens-metasurface">{{cite journal |last1=Epstein |first1=Ariel |last2=Eleftheriades |first2=George V. |author2-link=George V. Eleftheriades |title=Huygens' metasurfaces via the equivalence principle: design and applications |journal=[[Journal of the Optical Society of America|Journal of the Optical Society of America B]] |date=2016 |volume=33 |issue=2 |pages=A31–A50 |doi=10.1364/JOSAB.33.000A31|bibcode=2016JOSAB..33A..31E }}</ref>

<ref name="Li07">{{cite journal |last1=Li |first1=Mao-Kun |last2=Chew |first2=Weng Cho |author2-link=Weng Cho Chew |title=Wave-Field Interaction With Complex Structures Using Equivalence Principle Algorithm |journal=[[IEEE Transactions on Antennas and Propagation]] |date=2007 |volume=55 |issue=1 |pages=130–138 |doi=10.1109/TAP.2006.888453|bibcode=2007ITAP...55..130L |s2cid=46424215 }}</ref>

<ref name="Love01">{{cite journal |last1=Love |first1=A. E. H. |author1-link=Augustus Edward Hough Love |title=The Integration of the Equations of Propagation of Electric Waves |journal=[[Philosophical Transactions of the Royal Society A]] |date=1901 |volume=197 |issue=287–299 |pages=1–45 |doi=10.1098/rsta.1901.0013|bibcode=1901RSPTA.197....1L |doi-access=free }}</ref>

<ref name="Rengarajan00">{{cite journal |last1=Rengarajan |first1=S.R. |last2=Rahmat-Samii |first2=Y. |author-link2=Yahya Rahmat-Samii
|title=The field equivalence principle: illustration of the establishment of the non-intuitive null fields |journal=IEEE Antennas and Propagation Magazine |date=2000 |volume=42 |issue=4 |pages=122–128 |doi=10.1109/74.868058 |bibcode=2000IAPM...42..122R |url=http://www.csun.edu/~hceen002/paper1.pdf}}</ref>

<ref name="Schelkunoff38">{{cite journal |last1=Schelkunoff |first1=S. A. |author1-link=Sergei Alexander Schelkunoff |title=Some Equivalence Theorems of Electromagnetics and Their Application to Radiation Problems |journal=[[Bell Labs Technical Journal]] |date=1936 |volume=15 |issue=1 |pages=92–112 |doi=10.1002/j.1538-7305.1936.tb00720.x}}</ref>

<ref name="Schelkunoff39">{{cite journal |last1=Schelkunoff |first1=S. A. |title=On Diffraction and Radiation of Electromagnetic Waves |journal=[[Physical Review]] |date=1939 |volume=56 |issue=4 |pages=308–316 |doi=10.1103/PhysRev.56.308 |bibcode=1939PhRv...56..308S |author1-link=Sergei Alexander Schelkunoff}}</ref>

<ref name="Schelkunoff51">{{cite journal |last1=Schelkunoff |first1=S. A. |title=Kirchhoof's formula, its vector analogue, and other field equivalence theorems |journal=[[Communications on Pure and Applied Mathematics]] |date=1951 |volume=4 |issue=1 |pages=43–59 |doi=10.1002/cpa.3160040107 |author1-link=Sergei Alexander Schelkunoff}}</ref>
}}

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[[Category:Electromagnetism]]
[[Category:Antennas]]
[[Category:Diffraction]]
[[Category:Electromagnetic radiation]]