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Research 3: C Aguergaray
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October
2011
Claude Aguergaray
with: V. I. Kruglov, J. D. Harvey
e-mail : [email protected]
Perturbations to self-similar propagation in optical
amplifiers
Physics Resear
ch3
October
2011
Context
Sech input pulse in DSF Raman amplifier.
Self-
sim
ilar p
ulse
s
Parabolic pulses are generated asymptotically in the fiber amplifier independent of the shape or the noise properties on the input pulse, and possess linear chirp.
A class of solution of the nonlinear Schrödinger equation (NLSE) with gain.
Self-similar evolution of the pulse intensity and chirp the intensity profile retains its parabolic shape and resists the deleterious effects of optical wave breaking.
October
2011
Under normal dispersion, nonlinearity and gain
↪ ↪ Any input pulse will evolve asymptotically into a similariton with a parabolic intensity profile and positive linear chirp.
Context
Finot et al. OE 11 (2003).
Self-
sim
ilar p
ulse
s
October
2011
The Self-Similar dynamics :- Peak power- Temporal width increase exponentially with propagation length.- Spectral width
Context
Billet et al. OE 13 (2005)Dudley et al. Nature 3 (2007).
Self-
sim
ilar p
ulse
s
October
2011 The common point between Self-similar and the well-known technique of chirped-pulse
amplification (CPA) is that they aim at avoiding the pulse break-up due to excessive nonlinear phase shifts accumulated through the fiber.
- When CPA avoids nonlinearity by stretching the pulse before amplification,
- Self-similar amplifier actively exploits nonlinearity, (possibility of obtaining output pulses shorter than the initial input pulse).
For energies >µJ similariton amplifiers can be limited by the available gain-bandwidth. But they are a good alternative to more complex CPA systems below this limit.
Self-similar amplifiers have been demonstrated:- Several types of gain medium: ytterbium, erbium and Raman- For a broad range seed pulses in the range 180 fs–10 ps- Fiber lengths in the range 1.2 m to 5.3 km - Gains varying from 14 to 32 dB.
Self-similar pulse in amplifiers:
Self-
sim
ilar p
ulse
s
Context
October
2011
Amplification to the μJ level in an environmentally stable and polarization-maintaining configuration has been a demonstrated.
Self-similar pulse in amplifiers:
Compressed duration: 240 fsRepetition rate: 27 MHzAverage power: 21 WPeak power: 5 MW
Schreiber et al. OL 31 (2006).
Self-
sim
ilar p
ulse
s
Context
October
2011
Context
Billet et al. OE 13 (2005).
After 7m of propagation
All fiber compression stage by use of photonic bandgap optical fibre to replace bulk gratings lead to the realization of an all-fiber source delivering pulses in the 100 fs range at 1550nm.
Self-similar pulse in amplifiers:
FWHM136 fs
FROG measurement of compressed pulses
Self-
sim
ilar p
ulse
s
October
2011
Context
Self-similar pulse in lasers:
The net GVD of the cavity can be normal or anomalous.
- With large net anomalous GVD, soliton like pulses. These lasers (1st developed) have stringent limitation in energy (nJ) and pulse duration (ps) due to excessive nonlinear phase shift accumulated by the pulse.
To overcome this limitation researcher have developed laser cavity with dispersion map.
Self-
sim
ilar p
ulse
s
October
2011
Context
Self-similar pulse in lasers:
Aguergaray et al. OE 18 (2010).
- GVD ≈ 0, stretched-pulse operation occurs. The pulse energy can be an order of magnitude higher than in a soliton laser.
- GVD >> 0, higher pulse energies can be achieved directly from an oscillator.
Among these are the self-similar laser and the so-called chirped pulse oscillator (CPO). Pulse shaping in such a laser is based on spectral filtering of the chirped pulse, which cuts off the temporal wings of the pulse.
Laser output pulse energy: 21 nJ
Self-
sim
ilar p
ulse
s
October
2011
Motivation
Linearly chirped parabolic pulse is an asymptotic class of solution of the GNLSE with constant gain, normal dispersion and in the presence of non-linearities.
↪ ↪ Any input pulse with right energy will evolve into a similariton regardless of its shape and duration.
Until recently most theoretical descriptions of the self-similar (SS.) propagation had been done assuming that only low-order nonlinear effects and low-order dispersion effects dominate the pulse evolution.
But experiment have shown that SS. propagation can be perturbed by effect like:
- Gain bandwidth limitation,- Third-order dispersion,- Gain saturation.Se
lf-si
mila
r pul
ses
October
2011
Similariton propagation and break-up with third-order
dispersion influence
October
2011
Motivation
↪ ↪ Asymmetric temporal pulse shape,
↪ ↪ Peak shifted towards the edges of the pulse,
(direction depends on TOD sign).
Lead to generation of shock wave instability and pulse break-up.
TOD can have a detrimental effect on parabolic pulse propagation.
Sim
ilarit
on b
reak
-up
with
3
October
2011
Anterior work
z = 1 km z = 779 m
3 = 0.025 ps3/km3 = 0 ps3/km
Wabnitz and Finot have observed pulse break-up in Dispersion Decreasing Fibers (increased TOD influence with distance)
Wabnitz and Finot JOSA B 25 (2008).
10 % accuracy for pulse break-up critical distance.
z = 755 m
3 = 0.025 ps3/km
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October
2011
Bale and Boscolo:
↪↪ Partial analytical description of the pulse before and at break-up,
↪↪ No theoretical prediction of critical distance.
z/zc = (a) 0.25, (b) 0.5,
(c) 0.75, (d) 1.
Bale and Boscolo, J. Opt. 12 (2010).
16% error between analytical and numerical simulations.
Sim
ilarit
on b
reak
-up
with
3
Anterior work
October
2011
Theoretical study.
Our Analytical model :1
2 2 60
2
21 1( , ) exp ( , )
2 3
g EU z gz Q z
Sim
ilarit
on b
reak
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3
October
2011
Theoretical study.
Numerical model :
(z) 3 controls pulse shape
(asymmetry of the pulse)
Critical parameter is given by
T2(z) = T3(z) (3 >0), orT1(z) = T2(z) (3 <0).
Condition gives critical length zc at which pulse breaks down.
Sim
ilarit
on b
reak
-up
with
3
October
2011
Theoretical study.
Yields to the critical distance parameter :
where
Sim
ilarit
on b
reak
-up
with
3
October
2011
Numerical results.
Analytical expression of critical distance :
2=0.13 ps2/m3=10-3 ps3/mg=2 m-1
=2.10-3 w-1m-1
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ilarit
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October
2011
Numerical definition of pulse break-up :
↪ ↪ Pulse experiences growth of side peak under 3 influence
Numerical results.Si
mila
riton
bre
ak-u
p w
ith
3
October
2011
Numerical results.
Numerical simulations varying energy E0 :
3 = 0.96x10-3 ps3/m0.1pJ<E0<10pJ
lnc
<1% error for <1% error for critical length critical length
predictionprediction
33 0 3 4
0 52
10gE
3
2 2
g
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ilarit
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reak
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3
October
2011
0 10E pJ3 3
3 10 /ps m
3 < 0 3 > 0
Numerical results.
Very good agreement between analytical prediction and numerical solution.
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ilarit
on b
reak
-up
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3
33 0 3 6
0 52
1.078 10gE
October
2011
Summary of β3 study
Novel analytical theory for propagating pulses in normal dispersion fiber amplifier with TOD.
Found the critical length zc at which the TOD generate pulse break-up for constant gain.
Shown numerically the limitations for input value providing a highly accurate analytical description of the quasi-similariton and the critical length: ≤ 10-4.
Critical distance zc does not depend on the sign of TOD.
Published in: Optics Letters / Vol. 35, No. 18, p. 3084 (2010)
Physical Review A / Vol. 84, No. 2, 023823 (2011)
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3
October
2011
Motivation
Until recently most theoretical descriptions of the self-similar (SS.) propagation had been done assuming that only low-order nonlinear effects and low-order dispersion effects dominate the pulse evolution.
But experiment have shown that SS. propagation can be perturbed by effect like:
- Gain bandwidth limitation, - Third-order dispersion,- Gain saturation.
Self-
sim
ilar p
ulse
s
October
2011
Parabolic and Hyper-Gaussian similaritons
propagating in fiber with saturation effect.
October
2011
Gain saturation effect is important for the pulse evolution in normal fiber ring lasers.
It is negligible over the duration of a single pulse but cannot be neglected for a long pulse train since the amplifier gain will saturate over long time scale (> population relaxation time).
We use here the standard model equation for saturation effect obtained by averaging the gain dynamics in the presence of the pulse train.
Sim
ilarit
on w
ith g
ain
satu
ratio
n
0
1
ss
S
gg z
E z
E
Gain saturation model
Dependence of laser gain on the optical power at the steady state
October
2011
Sim
ilarit
on w
ith g
ain
satu
ratio
n
1
2
0
11 ,SS
S
g z g z dE
2 02
2 2z
g zi i
, , exp ,z t A z t z t
Analytical solution
The analytical solution is an exact asymptotical solution of the NLSE .
(From differential equation for the propagating pulses in optical amplifiers with an arbitrary gain function).
Slowly varying envelope:
Gain function:
October
2011
Sim
ilarit
on w
ith g
ain
satu
ratio
n
1
3 13
3ln
2s
p
13223
4 3sp
0g z
0
2
g
0 2
SS
E
g
0
SE
E
Analytical solution
3ln ln
Peak power:
Pulse duration:
Analytical solution with saturation effect takes the form:Dimensionless variables
Dimensionless parameters
1 22
1 2
2, 1p
u p
October
2011
Sim
ilarit
on w
ith g
ain
satu
ratio
n
Parabolic similaritons
= 4000= 400
Simulation parameters: β2 = 0.02 ps2m-1 / = 2 10-5 W-1m-1 / g0 = 2 m-1.Input energy: E0 = 200 pJ 0 = 0.02.
Saturation energy: ES = 20 nJ S = 2.
Temporal profile and the chirp of the pulses for two different propagation distances:
Input parameters:
October
2011
Sim
ilarit
on w
ith g
ain
satu
ratio
n
= 600= 100
Input energy: E0 = 10 pJ 0 = 0.001.
Saturation energy: ES = 10 µJ S = 100.
Results for increased saturation energy:
Input parameters:
Parabolic similaritons
October
2011
Sim
ilarit
on w
ith g
ain
satu
ratio
n
Hyper-Gaussian similaritons
For low amplification regimes, S < 0.3 (in our case ES < 3 nJ), the gain seen by the pulse goes very quickly to zero along the fiber.
The input pulses evolve into a different similariton regime with a linear chirp but a non parabolic shape. The pulse develops an Hyper-Gaussian shape.
It propagates through the fiber self-similarly with a linear chirp !
0
1
ss
S
gg z
z
October
2011
Sim
ilarit
on w
ith g
ain
satu
ratio
n
Hyper Gaussian similaritons
2 4
, expE
pW W W
This function is a product of a Gaussian and a super-Gaussian therefore we named it Hyper-Gaussian pulse (HG pulse).
Two asymptotic non-linear attractors:
which route depends on S
October
2011
Sim
ilarit
on w
ith g
ain
satu
ratio
n
Hyper-Gaussian similaritons
The spectral density of differs significantly parabolic shape.
HG pulses undergoe small spectral broadening due to weak non-linear effects what leads to a very smooth spectral shape (interest for fiber based amplification systems).
Linear chirp proving the self-similar aspect of the HG pulse propagation.
October
2011
Sim
ilarit
on w
ith g
ain
satu
ratio
n
Hyper-Gaussian similaritons
Test for different input pulse shape:
All the pulses converge towards a HG shape pulse
with linear chirp !
HG pulse is a local asymptotic attractor.
October
2011
Sim
ilarit
on w
ith g
ain
satu
ratio
n
Our analytical solution for similariton pulses in a fiber amplifier with gain saturation allows an accurate predictions of the pulse temporal shape and chirp for a wide range of the saturation energy parameter.
A limit of s > 0.3 setting the lower boundary has been found
No upper limit on s… (computation time is restrictive).
A new local non-linear attractor leading to self-similar HG pulses has been identified.
Summary of ESAT study
October
2011
Conclusion
Two analytical solutions able to predict accurately the perturbations to the self-similar propagation caused by the TOD and the gain saturation.
Since low amplification is required for a pulse to evolve into a HG pulse,
Could be implemented in low energy laser systems delivering linearly chirped pulses.
Potential for pre-amplification stage of ultra-short pulse CPA systems to obtain linearly chirped
pulses with no spectral structure.
October
2011
October
2011
October
2011
Bric a brac
October
2011
0 expss
S
E zg z g
E
October
2011
Numerical simulations
The HG similaritons may form when:
- The energy E(z) of the pulse is a slowly growing function of distance,
- The peak power of the pulse is a constant or decreasing function of z.
October
2011
Overview
Motivations
Theoretical study
Numerical results
Conclusion
October
2011
Motivation
How to predict accurately the
critical distance and the pulse shape?
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October
2011
Analytical pulse energy coincides with exact energy 0gzE z E e
Numerical results.
October
2011
OWN1 / Finot OSA OFC 2009
Context
October
2011
Theoretical study.
Renormalisation procedure : k
E zE z
z n
kz z
12 2 6
0
2
21 1( , ) exp ( , ) ( , )
2 3
g EU z gz Q z I z
October
2011
Motivation
However self-similar propagation is severally affected by Third Order Dispersion (TOD).
Novel features observed due to TOD in fiber amplifiers.
ΦNL accumulated in amplifier (SPM) compensated by TOD of
fiber stretcher + grating compressor
ΦNL = 0.4 π
ΦNL = 1.9 π
Zhou et al. (Wise) OE 13, 4869 (2005)
Grating stretcher and compressor best result.
ΦNL = 1.9 π
October
2011
Theoretical study.
No renormalisation With renormalisationprocedure applied
0 10E pJ3 3
3 10 /ps m
3 60 1.078 10
October
2011
Motivation
Novel features observed due to TOD in mode-locked lasers.
Logvin et al. OE 15, 985 (2007)
October
2011
Motivation
Net cavity GVD= 0.005 ps2
SMF and Yb fibers TOD (Negligible TOD)Similariton regime: symmetric pulse, top spectrum tilted.
PBF TOD = 500 fs3/mmCubicon-like features: asymmetric pulse, triangular shape spectrum.
PBF TOD = 1200 fs3/mm.Stretched Pulse regime: narrower pulse, broader spectrum with asymmetric sidebands.
Logvin et al. OE 15, 985 (2007)
October
2011 E0 3.5 3.5 8
0.029 0.1 0.1
Numerical results.
Numerical model :
with
Normalised variable
October
2011
Theoretical study.
Respecting < 10-4 condition :
52.44 10
0 1E 0.029
3
0E
October
2011
Theoretical study.
-8 -6 -4 -2 0 2 4 6 80.0
0.5
1.0
1.5
2.0
Ga
us
sia
n in
itia
l pu
lse
Normalised time
Pulse propagation in fibers with TOD is described by the following NLSE :
Analytical solution found using a first order perturbation theory :
October
2011
MotivationIn
tens
ity
Time [ps]
Dis
tanc
e [k
m]
Inte
nsity
Wavelength [nm]
Dis
tanc
e [k
m]
Latkin et al. OE 32, 331 (2007)
October
2011
Th
e A
ust
rali
an O
pti
cal S
ocie
ty
