Nonlinear stability analysis of a time-delay
opto-electronic oscillator
Dan Gauthier, Kristine Callan, Lucas Illing,*
Zheng Gao, Eckehard Schöll†
Duke University, Department of Physics,
Fitzpatrick Institute for Photonics,
Center for Nonlinear and Complex Systems
* Reed College
† Technische Universität Berlin
University of Maryland, Applied Nonlinear Dynamics Seminar, Dec. 10, 2009
1
Opt-electronic oscillator basics
Uses commercial telecommunication components
τ
Feedback loop
generates
t time
ti
delay
d l
Ikeda, et al., PRL 49, 1467 (1982)
Goedgebuer, et al., IEEE J. Quantum Electron. 38, 1178 (2002)
Meucci et al., PRE 66, 026216 (2002)
Gastaud, et al., Electron. Lett. 40, 898 (2004)
Pouut/Pin
MZ:
cosine squared
transfer function
VMZ [V]
FAST DYNAMICS
Bandwidth: 10-20 GHz
2
Experimental setup
MZM
AMP
PD
LD
3
Quick summary of observations
• ultra-fast, large-amplitude pulsing
• featureless
f t l
power spectrum
t
above
b
a critical
iti l lloop gain
i
• “immediate” transition from V=0 state to broadband chaos (through complex
transient) as loop gain increases
4
More details of what we did and what we observe
5
“Standard” MZM bias
τ
Pout/Pin
Bias to center
edge of fringe
m=π/4
VMZ [V]
∝m
“Most” people bias under these conditions (Meucci et al. is a counter example)
IIncrease loop
l
gain
i γ from
f
zero until
til an iinstability
t bilit iis observed
b
d ((proportional
ti
l tto
laser power, amplifier gain, photodiode response, etc.)
6
Pout//Pin
Bifurcation sequence: “Standard” bias
VMZ [V]
FP
P
QP?
Q
C?
PD S
Signal (mV
V)
Increase γ
15
0
-15
0
15
Time (ns)
7
“Non-standard” MZM bias
τ
Pout/Pin
Bias to top
of fringe
VMZ [V]
System should be linearly stable!
IIncrease loop
l
gain
i γ from
f
zero until
til an iinstability
t bilit iis observed
b
d ((proportional
ti
l tto
laser power, amplifier gain, photodiode response, etc.)
8
Poutt/Pin
Bifurcation Sequence: “Non-standard” bias
VMZ [V]
FP
FP
FP
BBC
Increase γ
9
Mathematical Model of the Opto-Electronic Oscillator
10
Time delays are important!
τ
Signals take a finite time to propagate through a high-speed
photonic/electrical circuits
In a typical optical waveguide or coaxial cable, v~0.6c
v 0.6c
Signal take about 1 ns to travel 1 ft.
Dynamics described by time-delay differential equations
Bandpass filtering in loop (typical of high-speed devices)
( 10 kH
(~10
kHz – 20 GH
GHz ffor our system))
Amplifier saturates for high input voltage
11
Dimensionless model
high frequency cut-off
MZM bias
loop time delay
x& ( s ) = − x( s ) − y ( s ) + γ cos 2 {m + d tanh[
h[ x( s − τ )]} − γ cos 2 m
y& ( s ) = ε x( s )
normalized filter (bandwidth)-1
amplifier saturation
ε = 2 ×10−6 d = 2.1 τ = 1,820
12
Linear Stability Analysis
x& ( s ) = − x( s ) − y ( s ) + γ cos 2 {m + d tanh[ x( s − τ )]} − γ cos 2 m
y& ( s ) = ε x( s )
fixed point
( x*, y*) = (0, 0)
becomes unstable for increasing γ via a Hopf bifurcation
γH = −
b±
d sin(2m)
b± ≈ ±1
γH → ∞ m = 0
System linearly stable at “non-standard” bias (top of the fringe)!
Not what we observe!
13
Observed Instability Boundary
γH = −
b±
d sin(2m)
b± ≈ ±1
typical system noise
add broadband noise to system
• system unstable when m=0
• depends
d
d on system
t
noise
i
• asymmetric function of m
→
fi it i perturbations,
finite-size
t b ti
nonlinear
li
analysis
l i needed
d d
14
Observed Transient Behavior with m=0
Catch the system during slow increase in g and near instability boundary
ps ((near measurement limit of oscilloscope)
p )
Pulse width ~200 p
τ
Why does the
system want
to pulse with
bandwidth
limited
pulses?
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Asymptotic Analysis
c[ x( s − τ )]
x& ( s ) = − x( s ) − y ( s ) + γ cos 2 {m + d tanh[[ x( s − τ )]} − γ cos 2 m
y& ( s ) = ε x( s )
ε = 2 ×10−6
T k
Take
( x*, y*) = (0, 0)
y=0
When y = 0 , a perturbation
p
in the form of a pulse
p
cycles
y
around the loop
p to
give another pulse at time τ, which cycles around the loop to given another
pulse at time 2τ, etc.
Discrete map
xn +1 = c[ xn ]
(pulse amplitude)
16
Fixed points of discrete map with m=0
xn +1 = c[ xn ]
For m=0 and our value of d, there exist one or three fixed points of the map,
depending on γ
xs*1 = 0
always stable (identical to linear analysis)
xs*2
exits for γ
≥ γ c and is stable
*
u
exits for γ
≥ γ c and is unstable
x
≈ −1/(γ d 2 )
17
Visualize fixed points of discrete map
xs*2
x
*
u
x
*
s1
x
γc
γ
18
Finite-size perturbation destabilizes fixed
point
xs*2
x
*
u
x
perturbation
amplitude
lit d
*
s1
x
γc
γ th
pulsing state
≈ −1/(γ d 2 )
γ
19
Inject pulses into the oscillator
τ
inject electrical pulses
20
Map out perturbation stability boundary
full DDE's
experiment
map predictions
21
Instability boundaries for other m
γ th determined for one particular size of the perturbation amplitude
expect instability boundary to be given by
min(γ H , γ th )
22
Compare with observed instability boundary
qualitatively
q
y similar, explains
p
asymmetry
y
y about m=0
23
Fixed points for m = 0.3
m=0.3
-x
~0.305
γH
γc
γ
Expect to see Hopf bifurcation
24
Fixed points for m = -0.2
x
m = -0.2
γc
γH
γ
expect to see sub
sub-critical
critical saddle-node
saddle node bifurcation
…. except near m ~ -π/4
25
But why ultrafast pulsing for m=0 and
featureless power spectrum????
26
Continuous mapping
Extend asymptotic analysis
Fi d a mapping
Find
i off th
the iinterval
t
l s-τ,s onto
t the
th interval
i t
l s,s+τ
xs , s +τ = c[ xs −τ , s ]
Start with an initial Gaussian pulse
look at one iteration of continuous mapping
27
Strong pulse compression for m=0
-x
X 1.0
0.8
initial pulse
0.6
after
ft one round
d trip
ti
0.4
0.2
0.2
0.4
0.6
0.8
1.0
t s
Conclusion: After one round trip, strong pulse compression. Pulse will continue
to compress
p
until the p
pulse bandwidth is comparable
p
to filter bandwidth
(asymptotic analysis then breaks down). Such ultrafast pulses at irregular time
intervals will have a flat power spectrum!
28
Consistent with observations
… ultrafast pulses initiated by noise
… featureless power spectrum
29
Continuous mapping for m = π/4
X-x
1.0
initial pulse
0.8
after one round trip
0.6
0.4
02
0.2
0.2
0.4
0.6
0.8
1.0
ts
For "standard" MZM bias, pulse perturbations expand in time. Therefore, pulses
smoothed out.
out Likely to see periodic waveforms consistent with a Hopf
bifurcation.
30
Summary
• Opto-electronic oscillator shows surprising complex behavior that has been
largely missed by other researchers
• Finite-size perturbations (even from noise) forces the system to access
states far from the trivial fixed point
• A full
f ll nonlinear
li
stability
t bilit analysis
l i iis needed
d d tto understand
d t db
behavior
h i
• A continuous mapping demonstrates that the system wants to produce
pulsing behavior at "non
non-standard
standard" MZM bias
Question: Can feedback control stabilize ultrafast pulses???
• True Hopf bifurcations only expected near "standard" bias. This is a
surprise to at least one theorist who has studied this system for many years!
31