A beam passing through a saturable absorber reaches a self-consistent transmitted…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Optics & Photonics · Nonlinear optics

Error, Chance and the Machine, Ch Iteration: nonlinear systems and eigenvalues

Problem

A beam passing through a saturable absorber reaches a self-consistent transmitted intensity II (in W mm−2\mathrm{W\,mm^{-2}}) obeying I=Iine−αI,I = I_{\rm in} e^{-\alpha I}, with Iin=e=1.6487 W mm−2I_{\rm in} = \sqrt{e} = 1.6487\ \mathrm{W\,mm^{-2}} and α=0.5 mm2 W−1\alpha = 0.5\ \mathrm{mm^2\,W^{-1}}. The iteration In+1=g(In)=Iine−αInI_{n+1} = g(I_n) = I_{\rm in} e^{-\alpha I_n} is run repeatedly starting from a measured value I0=1.2 W mm−2I_0 = 1.2\ \mathrm{W\,mm^{-2}}. ((a)) Show that I∗=1 W mm−2I^\ast = 1\ \mathrm{W\,mm^{-2}} is a fixed point of gg. ((b)) Compute g′(I∗)g'(I^\ast) and state whether the contraction condition for convergence is satisfied. ((c)) Using the contraction factor found in (b) as the per-step error ratio, estimate how many iterations from I0I_0 are needed before the error ∣In−I∗∣|I_n - I^\ast| falls below 0.01 W mm−20.01\ \mathrm{W\,mm^{-2}}.

Related theory

Browse the problem library