Editing Casimir effect (section)

=== More recent theory ===
Casimir's analysis of idealized metal plates was generalized to arbitrary dielectric and realistic metal plates by [[Evgeny Lifshitz]] and his students.<ref name=":2">{{cite journal |last1=Dzyaloshinskii |first1=I E |last2=Lifshitz |first2=E M |last3=Pitaevskii |first3=Lev P |title=General Theory of van der Waals' Forces |journal=Soviet Physics Uspekhi |volume=4 |pages=153 |year=1961 |doi=10.1070/PU1961v004n02ABEH003330 |issue=2 |bibcode=1961SvPhU...4..153D }}</ref><ref>{{cite journal |last1=Dzyaloshinskii |first1=I E |last2=Kats |first2=E I |title=Casimir forces in modulated systems |arxiv=cond-mat/0408348 |journal=Journal of Physics: Condensed Matter |volume=16 |pages=5659 |year=2004 |doi=10.1088/0953-8984/16/32/003 |issue=32 |bibcode=2004JPCM...16.5659D |s2cid=250897415 }}</ref> Using this approach, complications of the bounding surfaces, such as the modifications to the Casimir force due to finite conductivity, can be calculated numerically using the tabulated complex dielectric functions of the bounding materials. Lifshitz's theory for two metal plates reduces to Casimir's idealized {{math|{{sfrac|1|''a''<sup>4</sup>}}}} force law for large separations {{mvar|a}} much greater than the [[skin depth]] of the metal, and conversely reduces to the {{math|{{sfrac|1|''a''<sup>3</sup>}}}} force law of the [[London dispersion force]] (with a coefficient called a [[Hamaker constant]]) for small {{mvar|a}}, with a more complicated dependence on {{mvar|a}} for intermediate separations determined by the [[Dispersion (optics)|dispersion]] of the materials.<ref>V. A. Parsegian, ''Van der Waals Forces: A Handbook for Biologists, Chemists, Engineers, and Physicists'' (Cambridge Univ. Press, 2006).</ref>

Lifshitz's result was subsequently generalized to arbitrary multilayer planar geometries as well as to anisotropic and magnetic materials, but for several decades the calculation of Casimir forces for non-planar geometries remained limited to a few idealized cases admitting analytical solutions.<ref name=Rodriguez11-review /> For example, the force in the experimental sphere–plate geometry was computed with an approximation (due to Derjaguin) that the sphere radius {{mvar|R}} is much larger than the separation {{mvar|a}}, in which case the nearby surfaces are nearly parallel and the parallel-plate result can be adapted to obtain an approximate {{math|{{sfrac|''R''|''a''<sup>3</sup>}}}} force (neglecting both skin-depth and [[Orders of approximation|higher-order]] curvature effects).<ref name=Rodriguez11-review /><ref>B. V. Derjaguin, I. I. Abrikosova, and E. M. Lifshitz, ''Quarterly Reviews, Chemical Society'', vol. 10, 295–329 (1956).</ref> However, in the 2010s a number of authors developed and demonstrated a variety of numerical techniques, in many cases adapted from classical [[computational electromagnetics]], that are capable of accurately calculating Casimir forces for arbitrary geometries and materials, from simple finite-size effects of finite plates to more complicated phenomena arising for patterned surfaces or objects of various shapes.<ref name=Rodriguez11-review>{{cite journal |first1=A. W. |last1=Rodriguez |first2=F. |last2=Capasso |title=The Casimir effect in microstructured geometries |journal=Nature Photonics |volume=5 |pages=211–221 |year=2011 |doi=10.1038/nphoton.2011.39 |last3=Johnson |first3=Steven G. |issue=4 |bibcode=2011NaPho...5..211R }} Review article.</ref><ref name=Reid2013>{{cite journal |first1=M. T. H. |last1=Reid |first2=J. |last2=White |first3=S. G. |last3=Johnson |title=Computation of Casimir interactions between arbitrary three-dimensional objects with arbitrary material properties |journal=Physical Review A |volume=84 |pages=010503(R) |year=2011 |doi=10.1103/PhysRevA.84.010503 |issue=1 |arxiv=1010.5539 |bibcode=2011PhRvA..84a0503R |s2cid=197461628 }}</ref>