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Advanced Computation: Computational Electromagnetics Maxwell’s Equations in Fourier Space

Outline

• What is Fourier space? • Complex in terms of the reciprocal vectors • Maxwell’s equations in Fourier space • Visualizing the plane wave expansion

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What is Fourier Space?

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Fourier‐Space Vs. Frequency‐Domain   H E  t  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

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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

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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, ,

𝑇, 𝑇, and 𝑇 are the reciprocal lattice vectors.

Each of these plane waves will be assigned its own complex amplitude to convey its magnitude and phase.

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Complex Fourier Series in Terms of the Reciprocal Lattice Vectors

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Conventional Complex Fourier Series Periodic functions can be expanded into a Fourier series. For 1D periodic functions, this is

22px 2 px  jj1  f xapeapfxedx       p   2

For 2D periodic functions, this is

22px qy  22  px qy  jj  1   f xy, apqe  ,xy apq , fxye , xy dA pq  A A

For 3D periodic functions, this is

222px qy rz  222  px qy rz  j j 1   f xyz, , a  pqre , ,xyz a pqr , , f xyze , , xyz dV pqr   V V

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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   TxTyTz12ˆˆ 3ˆ geometries, the reciprocal lattice vectors are: xyz

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Maxwell’s Equations in Fourier Space

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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.

   jpTqTrT123  r  r rapqre  ,,  pqr    1  jpTqTrT r apqr,,   re123 dV  r  V V

   jpTqTrT123  r r rbpqre  ,,  pqr    1  jpTqTrT123 r bpqr,,   re dV  r  V V

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Fourier Expansion of the Fields

The expansions are slightly different for fields because a wave could be travelling in any direction 𝛽⃗. 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 𝑆 𝑝, 𝑞, 𝑟 .

    Spqre,, jkxyz 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 rzzzz pT1,  qT 2,  rT 3, Slide 13

Substitute Expansions into Maxwell’s Equations

    jpTqTrT123  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  jk 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 yzpqr   pqr  

    jpTqTrT123  r j kxyz pqr,, x k pqr ,, y k pqr ,, z  k0  a pqr , , e Sx  pqr,, e pq r pqr    Slide 14

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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 yypqr   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  ,, zzpqr   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  

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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   nnn000 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   

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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 yzpqr  pqr   

    jpTqTrTr123  j kxyz pqr,, x k pqr ,, y k pqr ,, z  kapqre0  , , Sx  pqr,, e pqr pqr 

    jkxyz 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 ,, ej 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   

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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 exyz 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.

In Fourier‐space, amplitudes are known of discrete plane waves.

A less clear, but more accurate picture is when all the plane waves overlap. r

q p

Slide 23

Visualizing Expansions for Cubic Symmetry     k p,, q r pT123  qT  rT   0    

   0    

Slide 24

12 5/14/2020

Visualizing Expansions with Different Symmetries    k p,, q r pT123  qT  rT

Simple‐Cubic Face‐Centered‐Cubic Hexagonal

Square Triangular

Slide 25

13