An array of semiconductor laser diodes is coupled through a shared feedback cavity;…

PHYSICS · Subjective · Fermi University Physics · Thermal, Statistical & Complex Systems · Complex and collective systems · Complex and collective systems

Many Parts, One Behaviour, Ch Synchronisation, cascades and the resilience of networks

Problem

An array of

N=300

semiconductor laser diodes is coupled through a shared feedback cavity; assume the cavity mixes every diode's optical phase equally with every other diode's, so the array is well described as an all-to-all Kuramoto model

\dot\theta_i=\omega_i+\dfrac{K}{N}\sum_{j=1}^N\sin(\theta_j-\theta_i)

with coupling

K

(units

\mathrm{rad\,ns^{-1}}

). The diodes' frequency detunings

\omega_i

from the array's mean lasing frequency are independently drawn from a Gaussian density of standard deviation

\sigma=0.50\ \mathrm{rad\,ns^{-1}}

. (a) Rewrite the coupling sum for diode

i

in terms of the complex order parameter

re^{i\psi}=N^{-1}\sum_j e^{i\theta_j}

, showing that each diode's equation of motion reduces to

\dot\theta_i=\omega_i+Kr\sin(\psi-\theta_i)

. (b) Compute the synchronisation threshold

K_{\rm c}

for this Gaussian detuning distribution. (c) So close to threshold, a simulation of only

300

diodes scatters by about

\pm0.14

r

from one draw of frequencies to the next, too much to read a growth law from. Solving the self-consistency equation numerically for this Gaussian (the

N\to\infty

limit) gives instead

r=0.287

K=0.82\ \mathrm{rad\,ns^{-1}}

r=0.425

K=0.85\ \mathrm{rad\,ns^{-1}}

and

r=0.516

K=0.88\ \mathrm{rad\,ns^{-1}}

. Using the two most widely separated couplings, estimate the exponent of the growth of

r

just past threshold and compare it with the value expected for a smooth frequency density with a rounded peak.

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