File:Anderson Localization 1D dynamics.gif
Summary
| Description |
English: In 1D, a sufficiently long random structure has exponentially localized eigenmodes (Anderson localization). If you shine a pulse on it, several of these eigenmodes (at different frequencies) will be excited, and the energy will beat between them. |
| Date | |
| Source | https://twitter.com/j_bertolotti/status/1126488445101199360 |
| Author | Jacopo Bertolotti |
| Permission (Reusing this file) |
https://twitter.com/j_bertolotti/status/1030470604418428929 |
Mathematica 11.0 code
c = 3 10^8; (*speed of light*)
M[n_, k_,
d_] := {{Cos[n k d], I c/n Sin[n k d]}, {I n/c Sin[n k d],
Cos[n k d]}}; (*transfer matrix*)
Mi[n_, k_,
d_] := {{Cos[d k n], -((I c Sin[d k n])/n)}, {-((I n Sin[d k n])/
c), Cos[d k n]}}; (*Inverse of a transfer matrix*)
t[m_, n0_, n2_] := (2 n0/c)/(
n2/c m[[1, 1]] - (n0 n2)/c^2 m[[1, 2]] - m[[2, 1]] +
n0/c m[[2, 2]]); (*transmission coefficient*)
dim = 200; (*number of layers in the Bragg mirror*)
s = Join[Table[1., 100], RandomReal[{1., 3.}, dim],
Table[1.,
5]];(*Reflective indices of each layer (including some space to \
show the pulse arrive*)
dim = Dimensions[s][[1]];
d = RandomReal[{0.95 10^-6, 1.05 10^-6}, dim]; (*layer thickness in m*)
nstep = 4000;
\[Omega]min = 2.2 10^15;
\[Omega]max = 2.3 10^15;
trasm = Reap[
For[\[Omega] = \[Omega]min, \[Omega] <= \[Omega]max, \[Omega] = \[Omega] + (\[Omega]max - \[Omega]min)/nstep,
tm = Apply[Dot, Table[M[s[[j]], \[Omega]/c, d[[j]]], {j, 1, dim}]];
Sow[N[t[tm, 1, 1]] ];
];][[2, 1]];
source = E^(-(1/2) (w - w0)^2 \[Sigma]^2) /. {w0 -> 2.255 10^15, \[Sigma] -> (40 10^-6)/c, a -> 10^12};
sourcel = Table[source, {w, \[Omega]min, \[Omega]max, (\[Omega]max - \[Omega]min)/nstep}];
field = trasm*sourcel;
freq = Table[j, {j, \[Omega]min, \[Omega]max, (\[Omega]max - \[Omega]min)/nstep}];
fn = Transpose[{field, field/c}];
tmp0 = fn;
ssm = Reap[For[i = dim, i > 0, i--,
tmp =
Table[((Mi[s[[i]], freq/c, d[[i]] ])[[All, All, j]].tmp0[[j]]), {j, 1, nstep}];
Sow[tmp[[All, 1]]];
tmp0 = tmp;
];][[2, 1]];
fssm = Map[Fourier, ssm]; (*Fourier transform with respect of frequncy to get the time evolution*)
p1 = Table[
Show[
ListPlot[2 (s - 1), PlotStyle -> {Red, Thick}, Joined -> True, PlotRange -> {-10, 10}, InterpolationOrder -> 0], ListPlot[{Re@Reverse@fssm[[All, -j]], Abs@Reverse@fssm[[All, -j]], -Abs@Reverse@fssm[[All, -j]]}, Joined -> True, PlotRange -> {-10, 10}, PlotStyle -> {Directive[Orange], Directive[Thick, Black], Directive[Thick, Black]}], Axes -> False, PlotRange -> {-10, 10},
Epilog -> {Text[Style["Random structure", Medium, Bold], {190, 5}], Text[Style["n", Red, Medium, Bold], {307, 3.5}]} ], {j, -15, 400, 1}];
ListAnimate[Drop[p1, {16}], 10]
Licensing
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