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Maxwell's Equations in Fourier Space

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Maxwell's Equations in Fourier Space 5/14/2020 Advanced Computation: Computational Electromagnetics Maxwell’s Equations in Fourier Space Outline • What is Fourier space? • Complex Fourier series in terms of the reciprocal lattice vectors • Maxwell’s equations in Fourier space • Visualizing the plane wave expansion Slide 2 1 5/14/2020 What is Fourier Space? Slide 3 Fourier‐Space Vs. Frequency‐Domain H E t Fourier transform x, y, E and z to k , k , and k . H Fourier transform t to . x y z t H jk E t EjH E H jE jk H t Fourier Space Frequency Domain Real‐Space Fourier‐Space Time‐Domain FDTD, Discontinuous Galerkin Pseudo‐spectral FDTD Frequency‐Domain FDFD, FEM, MoM, MoL RCWA, SAM, PWEM, spectral domain Slide 4 2 5/14/2020 What is Fourier Space? Real Space So far, fields and devices were represented on an x‐y‐z grid where field values and material properties are definedReal at Space discrete points. Fourier Space In Fourier‐space, fields are represented as a sum of plane r Fourier Space waves at different angles and different wavelengths called spatial harmonics. Devices are also represented as the sum of sinusoidal gratings at different angles and periods. q p Slide 5 Visualizing the Spatial Harmonics kpqr,, pTqTrT123 p, q, and r are the indices of the spatial harmonics. p integer , , 2, 1,0,1,2, , q integer , , 2, 1,0,1,2, , r integer , , 2, 1,0,1,2, , , , and are the reciprocal lattice vectors. Each of these plane waves will be assigned its own complex amplitude to convey its magnitude and phase. Slide 6 3 5/14/2020 Complex Fourier Series in Terms of the Reciprocal Lattice Vectors Slide 7 Conventional Complex Fourier Series Periodic functions can be expanded into a Fourier series. For 1D periodic functions, this is 22px 2 px jj1 f xapeapfxedx p 2 For 2D periodic functions, this is 22px qy 22 px qy jj 1 f xy, apqe ,xy apq, fxye , xy dA pq A A For 3D periodic functions, this is 222px qy rz 222 px qy rz j j 1 f xyz, , a pqre , ,xyz a pqr, , f xyze , , xyz dV pqr V V Slide 8 4 5/14/2020 Generalized Complex Fourier Series Fourier series can be written in terms of the reciprocal lattice vectors. For 1D periodic functions, this is 12 2 fx ape jpTx ap fxe jpTx dx T p 2 For 2D periodic functions, this is jpTqT r1 jpTqT r f xy, a pqe ,12 a pq, f xye , 12 dA pq A A For 3D periodic functions, this is jpTqTrTr 1 jpTqTrTr f rapqreapqrfredV , , 123 , , 123 pqr V V For rectangular, tetrahedral, or orthorhombic 222 TxTyTz12ˆˆ 3ˆ geometries, the reciprocal lattice vectors are: xyz Slide 9 Maxwell’s Equations in Fourier Space Slide 10 5 5/14/2020 Starting Point Start with Maxwell’s equations in the following form… E E H z y H z y kH0r x kE yz yz0r x E E x z H x H z kH0r y kE zx zx0r y Ey E x H y H kH0r z x kE xy xy0r z Recall that the magnetic field was normalized according to H jH0 0 Slide 11 Fourier Expansion of the Materials Assuming the device is infinitely periodic in all directions, the permittivity and permeability functions can be expanded into a generalized Fourier Series. jpTqTrT123 r r rapqre ,, pqr 1 jpTqTrT r apqr,, re123 dV r V V jpTqTrT123 r r rbpqre ,, pqr 1 jpTqTrT123 r bpqr,, re dV r V V Slide 12 6 5/14/2020 Fourier Expansion of the Fields The expansions are slightly different for fields because a wave could be travelling in any direction ⃗. The field must obey Bloch’s theorem. j pT qT rT r jr 123 Think of as k , but be careful with that analogy. E r e S pqr,, e inc pqr j r j pT123 qT rT r e was brought inside summation Spqre,, and combined with second exponential. pqr Let this be kpqr ,, Spqre,, jk pqr,, r This is clearly a set of plane waves ⃗ pqr with amplitudes , , . Spqre,, jkxyz pqr,, x k pqr ,, y k pqr ,, z kxxxxx pqr,, pT1, qT 2, rT 3, pqr k p,, q r pT123 qT rT kyyyyy p,, q r pT1, qT 2, rT 3, kz p,, q rzzzz pT1, qT 2, rT 3, Slide 13 Substitute Expansions into Maxwell’s Equations jpTqTrT123 r j kxyz pqr,, x k pqr ,, y k pqr ,, z r a pqr,, e H r U pqr,, e r yy pqr pqr j kxyz pqr,, x k pqr ,, y k pqr ,, z jk pqr,, x k pqr ,, y k pqr ,, z H r U pqr,, e xyz zz Erxx S pqre,, pqr pqr H H z y kE yz0r x j kxyz pqr,, x k pqr ,, y k pqr ,, z j kxyz pqr ,, x k pqr ,, y k pqr ,, z Uz pqr,, e Uy pqr ,, e yzpqr pqr jpTqTrT123 r j kxyz pqr,, x k pqr ,, y k pqr ,, z k0 a pqr, , e Sx pqr,, e pq r pqr Slide 14 7 5/14/2020 Algebra for the Left Side Terms First ugly term… j kxyz pqr,, x k pqr ,, y k pqr ,, z j kxyz pqr ,, x k pqr ,, y k pqr ,, z Uz pqr,, e Uz pqr ,, e yypqr pqr j kxyz pqr,, x k pqr ,, y k pqr ,, z Upqrjkpqrezy,, ,, pqr j kxyz pqr,, x k pqr ,, y k pqr ,, z jkyz p,, q r U p ,, q r e pqr Second ugly term… j kxyz pqr,, x k pqr ,, y k pqr ,, z j kxyz pqr ,, x k pqr ,, y k pqr ,, z Upqrey ,, Upqrey ,, zzpqr pqr j kxyz pqr,, x k pqr ,, y k pqr ,, z Upqrjkpqreyz,, ,, pqr j kxyz pqr,, x k pqr ,, y k pqr ,, z jkzy p,, q r U p ,, q r e pqr Slide 15 Algebra for the Right‐Side Term Third ugly term… This term as the product of two triple summations. j pT123 qT rT r j kxyz pqr,, x k pqr ,, y k pqr ,, z a pqr,, e Sx pqr ,, e pqr pqr This is called a Cauchy product and is handled as follows. n abccabnnnnmnm nnn000 m 0 Applying this rule to the triple summations, gives j kxyz pqr,, x k pqr ,, y k pqr ,, z e ap pq,, qr r Sx pqr ,, pqr pqr Slide 16 8 5/14/2020 Combine the Terms Inside Summation j kxyz pqr,, x k pqr ,, y k pqr ,, z j kxyz pqr ,, x k pqr ,, y k pqr ,, z Uz pqr,, e Uy pqr ,, e yzpqr pqr jpTqTrTr123 j kxyz pqr,, x k pqr ,, y k pqr ,, z kapqre0 , , Sx pqr,, e pqr pqr jkxyz pqr,, x k pqr ,, y k pqr ,, z j kxyz pqr,, x k pqr ,, y k pqr ,, z jkyz pqrU,, pqr ,, e jkzy pqrU ,, pqr ,, e pqr pqr j kxyz pqr,, x k pqr ,, y k pqr ,, z ke0 ap pq,, qr r Sx pqr ,, pqr p q r The equation can now be brought inside a single triple summation. jk pqrU,, pqr ,, ej kxyz pqr,, x k pqr ,, y k pqr ,, z jk pqrU ,, pqr ,, e j kxyz pqr ,, x k pqr ,, y k pqr ,, z yz zy j kxyz pqr,, x k pqr ,, y k pqr ,, z pqr ke0 ap pq, qr , r Sx pqr , , pqr Slide 17 Final Equation for (p,q,r)th Harmonic j k pqr,, x k pqr ,, y k pqr ,, z j k pqr ,, x k pqr ,, y k pqr ,, z jk p,, q r U p ,, q r exyz jk p ,, q r U p ,, q r e xyz yz zy j kxyz pqr,, x k pqr ,, y k pqr ,, z pqr ke0 ap pq, qr , r Sx pqr , , pqr The equation inside the braces much be satisfied for each combination of (p,q,r). j kxyz pqr,, x k pqr ,, y k pqr ,, z j kxyz pqr ,, x k pqr ,, y k pqr ,, z jkyz pqrU,, pqr ,, e jk zy pqrU ,, pqr ,, e j kxyz pqr,, x k pqr ,, y k pqr ,, z ke0 ap pq, qr , r Sx pqr , , pqr Last, divide both sides by the common exponential term and move the j to the right‐hand side. kyz pqrU,, pqr ,, k zy pqrU ,, pqr ,, jk0 a p p , q q , r r Sx p , q , r pqr Slide 18 9 5/14/2020 Alternate Derivation Start with H H z y kE0r x yz Point‐by‐point multiplication in real‐space… Fourier‐transform this equation in x, y, and z resulting in kyz pqrU,, pqr ,, k zy pqrU ,, pqr ,, jka0 S x …becomes a convolution in Fourier‐space. a FT r SExx FT It can now be seen that the strange triple summation remaining in the Fourier‐space equation is actually a 3D convolution in Fourier space! a Sxx ap pq,, qr r S pqr ,, pqr Slide 19 Summary of Maxwell’s Equations in Fourier Space Real‐Space Fourier‐Space H H z y kE yz0r x k pqrU,, pqr ,, k pqrU ,, pqr ,, jka pqr ,, S pqr ,, yz zy0 x H x H z kE kzx pqrU,, pqr ,, k xz pqrU ,, pqr ,, jka0 pqr ,, S y pqr ,, zx0r y k pqrU,, pqr ,, k pqrU ,, pqr ,, jka pqr ,, S pqr ,, xy yx0 z H y H x kE0r z xy k pqr,, kxxyyzz pqr ,, aˆˆˆ k pqr ,, a k pqr ,, a pT123 qT rT p , , 2, 1,0,1,2, , q , , 2, 1,0,1,2, , r , , 2, 1,0,1,2, , E E z y kH yz0r x kyz pqr,, S pqr ,, k zy pqr ,, S pqr ,, jkb0 pqr ,, U x pqr ,, Ex Ez kH kzx pqr,, S pqr ,, k xz pqr ,, S pqr ,, jkb0 pqr ,, U y pqr ,, zx0r y k pqr,, S pqr ,, k pqr ,, S pqr ,, jkb pqr ,, U pqr ,, E E xy yx0 z y x kH xy0r z Slide 20 10 5/14/2020 Vector Form of Maxwell’s Equations in Fourier Space kyz pqrU,, pqr ,, k zy pqrU ,, pqr ,, jka0 pqr ,, S x pqr ,, kzx pqrU,, pqr ,, k xz pqrU ,, pqr ,, jka0 pqr ,, S y pqr ,, kxy pqrU,, pqr ,, k yx pqrU ,, pqr ,, jka0 pqr ,, S z pqr ,, k pqr,, U pqr ,, jk0r pqr ,, S pqr ,, kyz pqr,, S pqr ,, k zy pqr ,, S pqr ,, jkb0 pqr ,, U x pqr ,, kzx pqr,, S pqr ,, k xz pqr ,, S pqr ,, jkb0 pqr ,, U y pqr ,, kxy pqr,, S pqr ,, k yx pqr ,, S pqr ,, jkb0 pqr ,, U z pqr ,, k pqr,, S pqr ,, jk0r pqr ,, U pqr ,, Slide 21 Visualizing the Plane Wave Expansion Slide 22 11 5/14/2020 Visualizing Maxwell’s Equations in Fourier Space In real‐space, the field values are known at discrete points.
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