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Material Interfaces

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At a material interface, Maxwell’s equations require that the tangential components of and and the normal components of and must be continuous.

The tangential continuity of components and is automatically satisfied, because FlexPDE stores only a single value of variables at the interface.  Continuity of , which is always tangential, requires, using (3.12), .  Continuity of requires .

Now consider the integrals (3.15) to be taken over each material independently.  Each specifies in a general sense the “flux” of some quantity outward from the region.  The sum of the two integrands, taking into account the reversed sign of the outward normal, specifies the conservation of the “flux”.  In the usual case, the sum is zero, representing “flux” conservation.  In our case, we must specify a jump in the flux in order to satisfy the requirements of Maxwell’s equations.

For the component equation we have, using the outward normals from region 1,

   

But the continuity requirements above dictate that the numerators be continuous, so the internal natural boundary condition becomes

   

By a similar argument, the internal natural boundary condition for the component equation is

   

Clearly, at an internal interface where is continuous, the internal natural boundary condition reduces to zero, which is the default condition.

In the example which follows, we consider a simple 2x1 metal box with dielectric material in the left half.  Note that FlexPDE will compute the eigenvalues with lowest magnitude, regardless of sign, while negative eigenvalues correspond to modes with propagation constants below cutoff, and are therefore not physically realizable.

See also "Samples | Usage | Eigenvalues | Filledguide.pde"

Descriptor 3.2 Filledguide.pde

 

title "Filled Waveguide"

 

select

 modes = 8     { This is the number of Eigenvalues desired. }

 

variables

 Hx,Hy

 

definitions

 cm = 0.01           ! conversion from cm to meters

 b = 1*cm             ! box height

 L = 2*b               ! box width

 epsr

 epsr1=1        epsr2=1.5

 ejump = 1/epsr2-1/epsr1     ! the boundary jump parameter

 eps0 = 8.85e-12    

 mu0 = 4e-7*pi

 c =  1/sqrt(mu0*eps0) ! light speed

 k0b = 4

 k0 = k0b/b

 k02 = k0^2           ! k0^2=omega^2*mu0*eps0

 

 curlh = dx(Hy)-dy(Hx) ! terms used in equations and BC’s

 divh = dx(Hx)+dy(Hy)

 

equations

 Hx: dx(divh)/epsr - dy(curlh/epsr) + k02*Hx - lambda*Hx/epsr = 0

 Hy: dx(curlh/epsr) + dy(divh)/epsr + k02*Hy - lambda*Hy/epsr = 0

 

boundaries

region 1  epsr=epsr1

  start(0,0)

  natural(Hx) = 0 value(Hy)=0

  line to (L,0)

  value(Hx) = 0 value(Hy)=0 natural(Hy)=0

  line to (L,b)

  natural(Hx) = 0 value(Hy)=0

  line to (0,b)

  value(Hx) = 0 natural(Hy)=0

  line to close

 

region 2  epsr=epsr2

  start(b,b)

  line to (0,b) to (0,0) to (b,0)

  natural(Hx) = normal(-ejump*divh,ejump*curlh)

  natural(Hy) = normal(-ejump*curlh,-ejump*divh)

  line to close

 

monitors

    contour(Hx) range=(-3,3)

    contour(Hy) range=(-3,3)

 

plots

    contour(Hx) range=(-3,3) report(k0b)

report(sqrt(abs(lambda))/k0)

    surface(Hx) range=(-3,3) report(k0b)

report(sqrt(abs(lambda))/k0)

    contour(Hy) range=(-3,3) report(k0b)

report(sqrt(abs(lambda))/k0)

    surface(Hy) range=(-3,3) report(k0b)

report(sqrt(abs(lambda))/k0)

 

summary export

    report(k0b)

    report lambda

    report(sqrt(abs(lambda))/k0)

 

end

 

 


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