| Author | Message | ||
     Saulo Lekanda (lekanda) New member Username: lekanda Post Number: 1 Registered: 03-2007 |
I would like to find the transmission eigenmodes of a disordered 2D waveguide. My waveguide has a clean left section from x=-infinity, then a disordered middle section (in which the wave equation includes a certain disordered potential V(x,y)), and another clean section to x=+infinity. The solution can be written as an incoming wave plus a scattered wave. A transmission eigenmode is a solution in which the wave at x=+infinity (incoming plus scattered) is proportional to the incoming wave at x=-infinity. The (unknown) transmission is the proportionality factor. Can these modes/transmission for a given disordered V(x,y) be calculated with FlexPDE? What are the proper boundary conditions? If not, could it be done if 'infinity' above were actually a finite number? Thanks in advance! | ||
| Robert G. Nelson (rgnelson) Moderator Username: rgnelson Post Number: 787 Registered: 06-2003 |
I don't know what the equations would look like. Can you send me any documentation of the theory? | ||
| Saulo Lekanda (lekanda) New member Username: lekanda Post Number: 2 Registered: 03-2007 |
Thanks for the quick answer! I didn't want to go into many details to make the post more readable. Let me elaborate: I have a wave equation in the frequency domain. It is the Dirac equation, but it basically boils down to four coupled linear PDEs involving first derivatives in space of the four unknowns, Ar, Ai, Br, Bi. -dy(Br)+dx(Bi)=(omega+V(x,y))*Ar -dy(Bi)-dx(Br)=(omega+V(x,y))*Ai dy(Ar)+dx(Ai)=(omega+V(x,y))*Br dy(Ai)-dx(Ar)=(omega+V(x,y))*Bi (incidentally I tried FlexPDE with these equations and they would not converge for some reason. I tried a second order PDE set that is equivalent to these and then it would work, no clue why) Then there is the boundary conditions. I have a waveguide along the x direction of infinite length and width 1. In the y-direction I have Bi=Br=0 at y=0, and Ai=Ar=0 at y=1. Now, the disorder V(x,y) is zero for x < -L/2 and x > L/2. The incoming modes in these clean regions can be solved analytically, but are not so simple as those for a sound waveguide. In general, for a fixed frequency omega there are many modes. Any superposition of these modes naturally satisfies the linear equations as well. The goal: I would like to compute the precise superposition of modes that would have to be fed into the waveguide at x=-infinity so as to obtain an outgoing wave at x=infinity that is proportional to the same superposition of modes. In this way the proportionality factor (that is what I am actually after) will be the (complex) transmission amplitude of the scattering eigenmode. If I can compute the modulus squared of these transmissions (T) for the first few eigenmodes I have a good estimate for the conductance of the waveguide. The question: can I use FlexPDE to compute such transmissions? I would think that if I could set a boundary condition that related right-going current (defined as j=2(Ar*Ai+Br*Bi)) at x=-L/2 with that at x=L/2 j(x=L/2)=T j(x=-L/2) for some unkwown and real T this would do it. Alas, it doesn't seem to be possible to impose such non-local constraints in FlexPDE, am I right? Thanks a lot. |