Coordinate Transformation & Invariance in Electromagnetism
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Coordinate Transformation & Invariance in Electromagnetism Steven G. Johnson, notes for the course 18.369 at MIT Created April 2007; updated March 10, 2010 It is a remarkable fact [1] that Maxwell’s equa- We will show that Maxwell’s equations take on tions under any coordinate transformation can the same form (1–4) in the primed coordinate be written in an identical “Cartesian” form, if system, with r replaced by r0, if we make the simple transformations are applied to the ma- transformations: terials (" and µ), the fields (E and H), and the sources (ρ and J). This result has numerous use- E0 = (J T )−1E; (6) ful and/or beautiful consequences, from designs of “invisibility cloaks” [2], to a simple derivation of PML absorbing boundaries [3], to enabling H0 = (J T )−1H; (7) analyses of bent and twisted waveguides in terms analogous to a quantum Stark effect [4] , to pro- J "J T viding a simple way of applying numerical meth- "0 = ; (8) ods designed for Cartesian coordinates to other det J coordinate systems [1]. Here, we review the proof in a compact form J µJ T (from [5]), generalized to arbitrary anisotropic µ0 = ; (9) det J media. (Most previous derivations seem to have been for isotropic media in at least one coordi- nate frame [1], or for coordinate transformations J J J0 = ; (10) with purely diagonal Jacobians J where Jii de- det J pends only on xi [3], or for constant affine coor- dinate transforms [6].) ρ ρ0 = ; (11) det J Summary of the Result where J T is the transpose. Maxwell’s equations in Cartesian coordinates x Note that, even if we start out with isotropic are written (in natural units "0 = µ0 = 1): materials (scalar " and µ), after a coordinate @E transformation we in general obtain anisotropic r × H = " + J (1) 0 0 @t materials (tensors " and µ ). @H For example, if x0 = sx for some scale factor r × E = −µ (2) 0 0 @t s 6= 0, then " = "=s and µ = µ=s, which is pre- r · ("E) = ρ (3) cisely the material scaling required to keep e.g. r · (µH) = 0; (4) the eigenfrequencies fixed under a rescaling of a structure. Note also that if s = −1, i.e. a coordi- where J and ρ are the usual free current and nate inversion, then we set E0 = −E, H0 = −H; charge densities, respectively, and "(x) and µ(x) "0 = −" and µ0 = −µ, and the system switches are the 3 × 3 relative permittivity and perme- “handed-ness” (flipping the sign of the refractive ability tensors, respectively. Now, suppose that index). [A more common alternative choice in we make some (differentiable) coordinate trans- that case would be to set H0 = H, transform- formation x 7! x0 (usually chosen to be non- ing H as a pseudovector [7], while keeping " and singular, with some exceptions [2]). Let J de- µ unchanged. This corresponds to sprinkling a note the 3 × 3 Jacobian matrix: few factors of sign(det J ) in the above equations, 0 @xi which we are free to do as long as the sign is con- Jij = : (5) @xj stant.] 1 Proof coordinates is also straightforward. Gauss’ Law, eq. (3), becomes We will proceed in index notation, employing 0 0 the Einstein convention whereby repeated in- ρ = @a"abEb = Jia@i"abJjbEj dices are summed over. Eq. (1) is now expressed: 0 −1 0 0 = Jia@i(det J )Jak "kjEj @E 0 0 0 −1 0 0 d = (det J )@i"ijEj + (@aJak det J )"kjEj @aHbabc = "cd + Jc (12) @t 0 0 0 = (det J )@i"ijEj; (21) where abc is the usual Levi-Civita permutation which gives r0 · ("0E0) = ρ0 for ρ0 = ρ/ det J , tensor and @a = @=@xa. Under a coordinate 0 0 @xa corresponding to eq. (11). Similarly for eq. (4). change x 7! x , if we let Jab = be the @xb Here, we have used the fact that (non-singular) Jacobian matrix associated with the coordinate transform (which may be a func- −1 @aJak det J = @aanmkijJinJjm=2 = 0; (22) tion of x), we have from the cofactor formula for the matrix inverse, @ = J @0 : (13) a ba b and recalling that @aJjbabc = 0 from above. In particular, note that ρ = 0 () ρ0 = 0 and Furthermore, as in eqs. (6–7), let J = 0 () J0 = 0, so a non-singular coordinate 0 transformation preserves the absence (or pres- Ea = JbaEb; (14) 0 ence) of sources.d Ha = JbaHb: (15) Hence, eq. (12) becomes References 0 0 0 @El Jia@iJjbHjabc = "cdJld + Jc: (16) [1] A. J. Ward and J. B. Pendry, “Refraction and @t geometry in Maxwell’s equations,” J. Modern 0 Here, the Jia@i = @a derivative falls on both Optics, vol. 43, no. 4, pp. 773–793, 1996. the J and H0 terms, but we can eliminate the jb j [2] J. B. Pendry, D. Schurig, and D. R. Smith, former thanks to the : @ J = 0 because abc a jb abc “Controlling electromagnetic fields,” Science, @ J = @ J . Then, again multiplying both a jb b ja vol. 312, pp. 1780–1782, 2006. sides by the Jacobian Jkc, we obtain [3] F. L. Teixeira and W. C. Chew, “Gen- @E0 J J J @0H0 = J " J l + J J eral closed-form PML constitutive tensors to kc jb ia i j abc kc cd ld @t kc c (17) match arbitrary bianisotropic and dispersive linear media,” IEEE Microwave and Guided Noting that JiaJjbJkcabc = ijk det J by defi- nition of the determinant, we finally have Wave Lett., vol. 8, no. 6, pp. 223–225, 1998. 0 [4] S. G. Johnson, M. Ibanescu, M. Skorobo- 0 0 1 @El JkcJc @ H ijk = Jkc"cdJld + (18) gatiy, O. Weisberg, T. D. Engeness, M. Sol- i j det J @t det J jačić, S. A. Jacobs, J. D. Joannopoulos, or, back in vector notation, and Y. Fink, “Low-loss asymptotically single- mode propagation in large-core OmniGu- J "J T @E0 r0 × H0 = + J0; (19) ide fibers,” Optics Express, vol. 9, no. 13, det J @t pp. 748–779, 2001. 0 where J = J J= det J according to (10). Thus, [5] C. Kottke, A. Farjadpour, and S. G. Johnson, we see that we can interpret Ampere’s Law in “Perturbation theory for anisotropic dielec- arbitrary coordinates as the usual equation in tric interfaces, and application to sub-pixel Euclidean coordinates, as long as we replace the smoothing of discretized numerical meth- materials etc. by eqs. (6–8). By an identical ods,” Phys. Rev. E, vol. 77, p. 036611, 2008. argument, we obtain [6] I. V. Lindell, Methods for Electromagnetic J µJ T @H0 Fields Analysis. Oxford, U.K.: Oxford Univ. r0 × E0 = − ; (20) det J @t Press, 1992. which yields the transformation (9) for µ. [7] J. D. Jackson, Classical Electrodynamics. The transformation of the remaining diver- New York: Wiley, third ed., 1998. gence equations into equivalent forms in the new 2.