Transcript 投影片 1 - National Taiwan University

Topic report 2012 March 15
Raman gain & stimulated emission gain
張俊霖
Solid-State Laser Crystal and Device Laboratory
Institute of Photonics and Optoelectronics
National Taiwan University
Outlines
• Model SRS & signal in an pulsed Yb:fiber amplifier
Know the different contributed behaviors between Raman & signal
• Definition of stimulated emission gain & Raman gain
Know the difference between them and then hard to compare with each other
• Define a SRS threshold to estimate SRS influence
Introduce the classic model & the modification of LMA core & propagation distance
• Strategy of suppressing SRS in fiber amplifier
Basically, optimize active & passive length, pumping scheme
by trade-off between Signal & Raman
In advance, pulse shaping & LP-FBG
by stripping the Raman signal during amplification.
2017/4/10
2
Motivation of this talk
2.5
5
SRS+signal (W)
Signal (W)
Raman average power (W)
3
2
Turning
point
1
120
SRS (W)
Signa (uJ)
2.0
100
Turning
point
1.5
80
1.0
60
0.5
40
energy (J)
Average power (W)
4
140
20
0.0
0
0
-0.5
0
2
4
6
8
10
12
0
1
2
3
4
5
6
7
8
9
10
11
Pump power (W)
Pump power (W)
(1)
Now in our third amplifier 15/130 DC PM,
we have far enough seed energy but the 1064-nm output energy is limited by signal
depletion of 1115-nm SRS generation.
(2)
We want to find the strategy to move the turning point right as can as possible.
For 15/130, target is 600 uJ
For 30/250, target is > 2mJ
2017/4/10
3
Signal & Raman in pulsed fiber amp simulation
Upper
population
CW
pump
CW
ASE
Pulsed
signal
Ith Stoke
Raman
SRS Stoke受激放大
Stoke 光纖損耗
反向瑞里散射
擷取前一order能量
反向瑞里散射
擷取前一order能量
給予後一order能量
Ith Anti-Stoke
Raman
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4
Outlines
• Model SRS & signal in an pulsed Yb:fiber amplifier
Know the different contributed behaviors between Raman & signal
• Definition of stimulated emission gain & Raman gain
Know the difference between them and then hard to compare with each other
• Define a SRS threshold to estimate SRS influence
Introduce the classic model & the modification of LMA core & propagation distance
• Strategy of suppressing SRS in fiber amplifier
Basically, optimize active & passive length, pumping scheme
by trade-off between Signal & Raman
In advance, pulse shaping & LP-FBG
by stripping the Raman signal during amplification.
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5
Signal stimulated gain for two doped conc.
g s  N 2 ( z, t ) es  N1 ( z, t ) as
30/250 Conc.=8.5E25 m-3
15/130 Conc.=4E25 m-3
300
300
small signal gain (1/m)
200
100
0
1020
1040
1060
Wavelength (nm)
2017/4/10
100
0
-100
-100
-200
1000
15/130 N2 (%)
25
20
15
10
5
2
200
small signal gain (1/m)
N2 percentage(%)
25
20
15
10
5
2
1080
1100
-200
1000
1020
1040
1060
1080
1100
Wavelength (nm)
By South Ampton Univ.
6
Raman gain spectrum
Measured Raman gain spectrum
Raman gain coefficient vs. wavelength
107
 9398.5 cm -1
(1064 nm)
Raman gain coefficient (m/W)
1.00E-013
8.00E-014
Raman at 1064 nm
6.00E-014
4.00E-014
2.00E-014
0.00E+000
1060
1080
1100
1120
1140
1160
1180
1200
1220
1240
Wavelength (nm)
107
-1


k
(cm
)   (nm)
-1 2
(9398.5 cm )
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7
Equation of Raman gain coefficient (m/W)
Generalized equation of Raman Gain
(From Agrawal : Nonlinear fiber optics 2007)
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Three kinds of model to determine the transfer function :
Instantaneous & single-damped harmonic oscillator &
Multi-vibrational-mode mode
8
Two model for transfer function for gR
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9
Instantaneous model by Stolen et al.
Exp
Remove
structured
640 cm-1
750 cm-1
Lorentizian-fit
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10
Single-damped-oscillator model
Original
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11
Intermediate-broadening model (1)
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12
Intermediate-broadening model (2)
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13
Outlines
• Model SRS & signal in an pulsed Yb:fiber amplifier
Know the different contributed behaviors between Raman & signal
• Definition of stimulated emission gain & Raman gain
Know the difference between them and then hard to compare with each other
• Define a SRS threshold to estimate SRS influence
Introduce the classic model & the modification of LMA core & propagation distance
• Strategy of suppressing SRS in fiber amplifier
Basically, optimize active & passive length, pumping scheme
by trade-off between Signal & Raman
In advance, pulse shaping & LP-FBG
by stripping the Raman signal during amplification.
2017/4/10
14
Raman threshold at Forward
In fact, there is no SRS threshold physically.
In practice, we define a SRS threshold to identify the SRS influence.
The first work to define the SRS threshold is R.G. smith at 1972 AO.
Small core & long length fiber for application of
optical communication for forward wave interaction
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15
Raman threshold at Backward
Two orders
lower than SRS
typically
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16
Modification of propagation length effect
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17
SRS threshold modified by LMA fibers
(high power fiber laser, not opt. comm.)
In Smith’s work, SRS intensity threshold is
independent on core size, it is not true in real.
Start from Smith relation
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18
Rewrite analytical equation by new assumption
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19
1.65 times SRS threshold in 30/250 LMA fiber
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20
Outlines
• Model SRS & signal in an pulsed Yb:fiber amplifier
Know the different contributed behaviors between Raman & signal
• Definition of stimulated emission gain & Raman gain
Know the difference between them and then hard to compare with each other
• Define a SRS threshold to estimate SRS influence
Introduce the classic model & the modification of LMA core & propagation distance
• Strategy of suppressing SRS in fiber amplifier
Basically, optimize active & passive length, pumping scheme
by trade-off between Signal & Raman
In advance, pulse shaping & LP-FBG
by stripping the Raman signal during amplification.
2017/4/10
21
It can improve the output energy by suppressing peak power for signal saturation &
Raman (~20% improvements)
(~50% improvements
of output energy)
Third amplifier 15/130 – SRS-limited energy
2.5
5
Raman average power (W)
2.0
3
2
轉折點
1
120
SRS (W)
Signa (uJ)
100
1.5
80
1.0
60
0.5
40
energy (J)
Average power (W)
轉折點
SRS+signal (W)
Signal (W)
4
140
20
0.0
0
0
-0.5
0
2
4
6
Pump power (W)
8
10
12
0
1
2
3
4
5
6
7
8
9
10
11
Pump power (W)
目前到20A約10W, 1115-nm SRS + 1064-nm signal 約各占一半,
但我發現量測平均功率時, fluctuation並沒有很大的差別,波形也是,
我非常確定現在能作的事只有減短active fiber, 試著將轉折點儘量往右移
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Third amplifier 15/130 – Raman spectra
-30
20A
18A
16A
14A
12A
10A
8A
-35
-40
Signal (dBm)
-45
-50
目前轉折點約
在12~14A
-55
-60
-65
-70
-75
-80
960
1000
1040
1080
1120
1160
1200
Wavelength (nm)
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Third amplifier 15/130 – Raman spectra
3.5
轉折點 energy >100 uJ, peak power 約> 20 kW
9
8
7
6
5
4
2
1
0
3.0
Signal (V)
2.5
2.0
1.5
1.0
0.5
0.0
-20
-10
0
10
20
Time (ns)
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Backward-pumped 15/130 & 30/250
Old
Splices
Passive length after
15/19
Splices
Passive length after
15/19
New
Active fibers
length
before
After
15
0
1
6/125-1
1.6
0.75
16
1
2
6/125-2
1.6
0.75
0.5
17
2
0.5
15/130
5
18
0.5
0
18
0.5
0
Total
8.2
19
0
1.5
19
0
1.5
20
3
0.5
20
1.5
0.5
before
After
15
0
1
16
1
17
Total
2017/4/10
12 /5
Total
10.5 /3.5
27
MOPA configuration 2012/02/16
30/250 – 35A x4 (95.8W)
30./250-gain fiber
30/250 – 143W
15/130 –
gain fiber
30/250 – 10A x2 (47.1W)
15/130 – 10A x2 (39.8 W)
15/130 – 39.4W
15/130end cap
15/130PCX lens
2017/4/10
15/130HWP
15/130PCX lens
30/2500-deg.
endfacet
30/250passive
30/2508deg.
endfacet
Measure
coupling
efficiency
(1) 準直的部份有達到2.4mm
(2) 無光纖旋轉夾具,
先用HWP解決
(3) 15/130功率不敢開太高, 現在
沒有BPF, 沒法濾掉Raman,
故不敢開高功率
(4) 好消息Semrock BPF 下周到
可解決此問題
(5) 有看到30/250出口有光, 預計
明天才會well optimize好與
製作30/250 出口的end cap
(6) 第4級30/250已架好, 預計下
星期可試低功率的30/250初
步放大成果, 該需要freespace isolator了
28
Nonlinear optics in Fiber (2)
For intense electromagnetic fields, the response of any dielectric to light becomes
nonlinear, including optical fiber. On a fundamental level, the origin of the nonlinear
response is related to anharmonic motion of bound electrons under the influence of an
applied field. As a result, the dipole moment per unit volume P (polarization) induced
by electric dipole (charge separation) is not linear in the electric field E, but satisfies the
more general relation in one-dimensional isotropic medium :

2
3
P(t )   0  (1)  E (t )   (2) : E (t )   (3) E (t ) 
0
8.85 10
 (1)  n2  1
 (2)
 (3)
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12
F
( )
m

Vacuum permittivity
1st order optical susceptibility (linear)
2nd order optical susceptibility
3rd order optical susceptibility
29
Fiber nonlinear optics (3)
 (1)
1st order susceptibility － linear propagation giving rise to propagation speed
through medium (real part : refractive index) and absorption in medium
(imaginary part : attenuation coefficient)
 (2)
2nd order and even order susceptibility is ~ 0 for silica glass because SiO2 is a
symmetric molecule.
 (3)
3rd order susceptibility is the lowest-order nonlinear effect in an optical fiber as
an isotropic glass medium, .
Local ordering
2017/4/10
Amorphous silicon (a-Si or α-Si) is the
non-crystalline allotropic form of silicon.
The amorphous structure of glassy Silica
(SiO2) in two dimensions. No long range
order is present, however there is local
ordering with respect to the tetrahedral
arrangement of Oxygen (O) atoms around
the Silicon (Si) atoms
Nonlinearity limits the performance of rare-earth-doped fiber lasers !
30
Fiber nonlinear effect induced by χ(3) (4)
There are three categories of nonlinear process for χ(3)
Phase matching condition is poor in fiber rather than that in crystal .
Define what we care in nanosecond fiber MOPA here
(1) Frequency mixing process – (phase matching)
Third harmonic generation (THG) : poor phase matching
(2) Optical Kerr effect – (Intensity dependent refractive index)
Self-phase modulation (SRS) : short duration & pulse shape & chirp
Self-focusing (SF) : bulk damage threshold
Four-wave mixing (FWM) : poor phase matching
Degenerate four-wave (or Three-wave) mixing (DFWM or TWM) : poor phase matching
Cross-polarized wave generation (XPW) : Not femtosecond time scale
Cross-phase modulation (XPM) : Not femtosecond time scale
Optical solitons : No GVD issue for long-distance propagation
(3) Light scattering – (Active process without phase matching)
Stimulated Raman scattering (SRS) : peak power threshold
Stimulated Briilouin scattering (SBS) : peak power & linewidth threshold
Stimulated Rayleigh scattering (SRS) : ?
(4) Two Photon aborption – (simultaneous absorption of two photons for single electron)
Two photon absorption (TPA) : explain well for the green light from active fiber
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31
Nonlinear effect & Photo-darkening effect to limite C & L
When the signal parameters is beyond the nonlinear threshold, especially SRS in our
case, SRS effect is dominant by peak power x fiber interaction length. For this condition,
we must choose short highly-doped length with tolerant incomplete pump absorption.
2017/4/10
For power amplifier, the length need artificially
short, doapnt conc. as high as possible
without nonlinear & photo-darkening and with
enough heat dissipation.
32
SRS Stokes threshold for Yb-doped LMA fiber
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33
Outlines
• Saturation
no real amplifier operating without saturation limit. We need to find it.
• Extractable energy
Use saturation as utilized unit to find the maximum extractable energy you can achieve.
• FDTD rate equation for repetition rate issue
If quantitative results to compare with experimental results is necessary, it can not be avoided.
• Fiber nonlinear optics
In fiber, it cannot be avoided. It is advantage/disadvantage to enhance/suppress it.
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Signal saturation for gain reduction
CW regime
s
g0
E
g
; Psats  sat
Ps
g
1 s
Psat
Pulse regime
g  g0e

Es
s
Esat
(1) CW & pulse are totally different stories by definition quantitatively.
(2) CW pump & ASE becomes important during energy storage period
and NOT important during signal depletion period.
(3) Let’s examine all kinds of saturation effect.
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http://www.rp-photonics.com/gain_saturation.html
35
Signal saturation fluence (1)
J sat
hvs
 s
 e   as
hvs  6.626 10
34
J sat
μJ
 0.738 2
μm
 es  2.5 1025  m2
 as  2.95 1027  m 2
 299792458(ms 1 ) 
19
(J  s)  
  1.867 10  J
1064 (nm)


Why do we use fluence & energy instead of intensity & power.
1. When does gain saturation set in during pulse build-up (signal depletion period)?
2. Many people believe that saturation becomes strong when the optical intensity
in the gain medium reaches the saturation intensity.
3. For most pulsed lasers, however, this guess is far off, since the used rule holds
only for the steady state, and the time during pulse build-up is far too short for
the steady state to be reached.
4. In reality, saturation in pulse regime sets when the time-integrated intensity
reaches the saturation fluence. Therefore the upper-state lifetime is not relevant:
it does not influence the saturation fluence, being the relevant quantity for
saturation in that situation.
5. Finally, it is material property function of wavelength.
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http://www.rp-photonics.com/spotlight_2008_01_06.html
36
Signal saturation fluence (2)
1.2
Fluence
2
Saturation fluence (Jm )
1.0
0.8
0.6
0.4
0.2
0.0
960
990
1020
1050
1080
Wavelength (nm)
Signal saturation fluence in Yb-doped silica vs. wavelength
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37
Signal saturation energy
Esat  J sat 
Adoped
s
 hvs 
hvs AMFD
 s
 AMFD  s
s 
s






a 
e
a
 e
s 
Adoped
AMFD
(1) Saturation energy of a laser gain medium is defined as the pulse energy of a short
signal pulse which leads to a reduction in the gain to 1/e (~37%) of it initial value.
(2) For fiber laser or amplifier, the light is guided stably in the waveguide (fiber).
Therefore we can directly use “fluence x MFD” to obtain “signal saturation energy”
(3) Therefore, we need to obtain the MFD in different fibers function of wavelength.
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38
Mode field diameter in fiber
V parameter
Marcuse equation
35
40
30
Wavelength -NA
1064nm-0,07
1064nm-0,08
1031nm-0,07
1031nm-0,08
Wavelength (nm) - NA
1064-0.07
1064-0.08
1064-0.11
30
25
V parameter
mode field diameters (m)
35
25
20
15
20
10
15
5
10
0
10
20
30
40
50
0
20
40
Core diameter (m)
DMFD
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
1.619 2.879 
 Ddoped   0.65  1.5 
6 
V
V
s
s


60
80
100
Core diameter (m)
Vs 
Marcuse, J. Opt. Soc. Am 68, 103 (1978).
2
s
 NAdoped 
Ddoped
2
39
Signal saturation energy
500
Signal N.A.
Esat (μJ)g
6/125 SM
0.11
32.6
6/125 DC LMA
0.15
20.87
10/125 DC LMA
0.08
71.74
12/125 DC LMA
0.08
85.82
15/130 DC LMA
0.08
104.29
20/125 DC LMA
0.07
164.84
25/250 DC LMA
0.07
224.95
30/250 DC LMA
0.07
297.28
50/400 DC LMA
0.07
707.49
80/400 DC LMA
0.07
1685.55
Fiber \ Items
Wavelength -NA
1031 nm - 0.07
1064 nm - 0.07
Dots are for NA=0.08
Signal saturation energy (J)
400
300
200
100
0
10
20
30
Core diameter (m)
40
50
(1) When your input pulse energy starts to approach this value,
it can be considered as saturated input to exhaust “Gain” during signal
depletion period to decrease to “Gain” threshold after this period.
(2) For larger MFD, high saturation energy can be obtained.
Beyond the signal saturation energy, pulse shape will be deformed.
(3) Signal saturation energy is limited by material & geometry.
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Pump saturation power (1)
Definition : For fixed mode field along the gain fiber, this is the pump power that
pumps one-half of population in a three- or four-level system, and about one-half in
a quasi-two-level system, to upper laser level. At this level and above, absorption of
pump power is significantly saturated due to depletion of the ground-state
population by pumping when pump rate is equal to fluorescence transition rate .
Pump saturation occurs when pumping rate = spontaneous emission rate (1/s)
Wp 
1
 sp
In this way, we can describe “pump saturation intensity” by lifetime
I
p
sat


hvp
p
e
   p
p
a
Wp 

hvp
p
e
  ap  p sp
Moreover, for CW pumped fiber laser and amplifier, the light is guided stably in the
waveguide (fiber). Therefore we can directly use “intensity x pump area” to obtain
“signal saturation energy”. For single-clad fiber, pump area is MFD by Marcuse
equation. For double-clad fiber (clad NA is >0.46), pump area can be considered
as clad area directly. (wave optics -> geometrical optics)
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Pump saturation power (2)
Pump saturation power
Launched pump power in
clad for pump saturation
 Adoped
P  p
 Ap  p

p
p
 e   a p sp
 e   a p sp  p
p
sat
hvp  6.626 10
34
hvp
hvp
 299792458(ms 1 ) 
19
(J  s)  
  2.035 10  J
976 (nm)


p 



Adoped
 ep  2.44 1024  m 2
Ap
 ap  2.5 1024  m 2
 sp  0.85 ms
Ytterbium-doped fiber amplifier pumped at 975 nm can operated far above its pump
saturation power and the particularly strong pumping phenomena occurs in cases
where both the absorption cross sections and the applied pump power are rather high.
It can lead to peculiar saturation behavior: the pump power decays about linearly
rather than exponentially in the fiber, and the slope of this decays can be substantially
increased if the signal power is also high. Furthermore, the saturation characteristics
for signal wave can also be strongly modified because ions undergoing stimulated
emission can be quickly returned to the upper laser level.
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Pump saturation power (3)
(1) For extracted energy from amplifier as possible as we can “efficiently”, basically
we need to make “the utilized pump power” to approach the “pump saturation
power to reach “strongly pumping”. It is very easy for Yb-doped fiber.
(2) If the fiber SPEC can not be chosen and you still want to extract more energy
from this fiber, you may use pump power much higher than saturated one but
you will loss the efficiency. In this way, core-pumped pre-amp can use it.
However, clad-pumped power amp is not preferred unless pulse pumping.
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Transparent Pump power
I
p
trans
hvp
 as
p

I

sat
  ep  s
  es   ap
s
p
 e   p  a



e   sp  p
s


a
 a 


p
p
Ptrans
 I trans
 Ap 
hvp  Ap
  es   ap
p



e   sp  p
s

a


(1) Transparency pump power is the required minimum pump power when the
GAIN COEFFICIENT is ZERO.
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Optimized fiber length
g  N 2 ( z , t ) es  N1 ( z , t ) as
CW-pump definition :
Therefore in end-pump scheme, the residual pump power at another fiber end should be
higher than transparent pump power for no negative GAIN COEFFICIENT along the entire
fiber as completely absorption.
p
trans_in
P
p
hvp  Ap
Ptrans


p
  es   ap
p



e   sp  p   p
s
 a

Pulsed operation definition : At the fiber end, the population is fixed at threshold.
For pulsed amplifier, the optimal fiber length is a range depending on operation
parameters.
=
(1) 1064-nm Yb-doped fiber or amplifier is quasi-four level system, length is not critical.
(2) Choose a set of concentration & length to completely absorb pump power along fiber.
(3) However, too high Conc. makes Photodarking effect easy to happen. Too low Conc.
needs longer fiber length and it makes nonlinear threshold (SRS & SBS) decrease.
2017/4/10
45
Outlines
• Saturation
no real amplifier operating without saturation limit. We need to find it.
• Extractable energy
Use saturation as utilized unit to find the maximum extractable energy you can achieve
• FDTD rate equation for repetition rate issue
If quantitative results to compare with experimental results is necessary, it can not be avoided.
• Fiber nonlinear optics
In fiber, it cannot be avoided. It is advantage/disadvantage to enhance/suppress it.
2017/4/10
46
Model of pulsed amplification
2017/4/10
We start from the signal depletion period only.
47
Frantz-Nodvick model (1)


N 2 ( z, t )
s
s Ps ( z , t ) s s
s
s
s Ps ( z , t ) s s
 ( N 2 ( z, t ) e  N1 ( z, t ) a )
   N 2 ( z, t )( e   a )  N a 
t
hcAs
hcAs
Ps ( z, t ) 1 Ps ( z, t )

  N 2 ( z, t ) es  N1 ( z, t ) as  Ps ( z, t ) s   N 2 ( z, t )( es   as )  N as  Ps ( z, t ) s
z
cs
t
We can use FDTD to solve these two rate equations.
N  N1  N2
Give the two initial conditions N 2 ( z , 0) & Ps (0, 0)
(1) You can obtain temporal shape & pulse energy by Ps+ ( L, t )
(2) You can obtain gain coefficient along the fiber & Amp GAIN by N2 ( z, T )
+
(3) You can obtain signal amplification along fiber by Ps ( z , T )
(4) No spectrum in this model
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48
Frantz-Nodvick model (2)
(1) To know the relation between input & output energy with small-signal
gain, F-N assume a square shape and solve it analytically as follows:
Ps (t )  {
+
Ps ( z , t ) 
P0s , 0  t   s
(2) Assume square pulse shape
0, otherwise
P0+
+

G
G

1  1  e

 ( es  as ) N 2 ( z ,t )  N1 ( z ,t )  z
 e  ( e  a ) P0  c0t  z 

s
s
+
(3) Analytical solution of
signal peak power
P0+
1  1  e

1
P0s s



 ( es  as ) 2 N 2 ( z ,t )  N  z
 e  ( e  a ) P0  c0t  z 

s
s
(4) Define the amplifier gain
Ps+ ( L, t ) dt

1
+
ln 1  e( e  a ) P0  s c0  1 e( N2 ( z ,t ) e  N1 ( z ,t ) a ) L
s
s
s


( e   a ) P0  s c0
1

s
s
s
s
s
ln 1  e( e  a ) P0  s c0  1 e( N2 ( z ,t )( e  a )  N a ) L
s
s
s


( e   a ) P0  s c0
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s
s
s
s
s
s


(5) Integrate it and obtain
the amplifier gain
49
Frantz-Nodvick model (3)
(2) Arrange the GAIN solution by energy fluence, saturated fluence and
small signal gain, you may obtain the well-know F-N equation.
J
s
out


 J ins
 J  ln 1  G0 exp  s
 J sat


s
sat
  
  1 
  
(3) For fiber laser, signal propagates in a light waveguide. Therefore, we
can transform the fluence into energy directly as follows:
s
out
E


 Eins
 E  ln 1  G0 exp  s
 Esat


s
sat
  
  1 
  
(4) Finally, The derivation shows there is no pump & ASE & repetition
rate and considers signal depletion only. The G0 can not be known in
this model (If you want G0, calculation of energy storage period is
necessary).
(5) According to the model, F-N results are usually considered as the
upper limit.
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50
Maximum extractable energy (A)
(1) According to F-N curve, the max extractable energy can be obtained
s
by letting Eins
Esat
s
s
 Esat
Eout
G  s  1  s
Ein
 Ein

 ln G0

s
s
s
Eext
 Eout
 Eins  Esat
ln G0
(2) In practice, small signal gain will be limited by the onset of parasitic
lasing or amplified spontaneous emission to < ~30dB. It means the
maximum extractable energy is
s
Eext_max
 6.9078  Esat
(3) Moreover, there is a journal paper on JOSA B to use a definition to
solve analytically and obtain max. extractable energy during energy
storage period only under the assumption of clad-pumped scheme
and saturated input. In this way, the repetition rate issue can be
considered in it also.
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51
Maximum extractable energy (B-1)
Assume the signal is beyond the
saturation level and then the threshold
population can be assumed at the
beginning of the energy storage period.
(Blue point)
We focus on the population at the end
of energy storage period (red point) to
obtain the max energy where the ASE
is just saturated (yellow point) in terms
of fiber spec, pump & ASE.
Assume NO ASE depletion duration
signal depletion period.
An analytical model for high energy YDFA to calculate the maximum extractable energy & optimal length before ASE is
saturated and depletes the inversion.
2017/4/10
Y. Sintov et al., J. Opt. Soc. Am. B 23, 218 (2005).
52
Maximum extractable energy (B-2)
Assume the time of pump passage through the fiber << energy storage period
Pump power distribution along the fiber with existed inversion is :
dPp± ( z , t )
  p ( ep   ap )n2 ( z , t )   p ap n   p   Pp ( z , t )
dz
ASE power evolution in vASE band along the fiber during energy storage period is
±
dPASE
( z , t , vASE )

  ASE ( eASE   aASE )n2 ( z , t )   ASE aASE n   ASE   PASE
( z , t , vASE )
dz
g (vASE )
NA2
 mhvASE  ASE
n2 ( z , t )
A
2
4nc 2


In the absence of spurious reflections at both fiber ends, PASE (0, t , vASE )  PASE ( L, t , vASE )  0
copropagating ASE signal vASE band at the fiber end is

(the same form for counter-propagating ASE signal) PASE (0, t )
ASE
ASE
ASE
SE
(
t
,
v
)
exp[

(



)
N
(
L
,
t
)

(


n   ASE ) L]  1

ASE
ASE
e
a
2
ASE
a
+
PASE ( L, t ) 
N ( L, t )
 ASE ( eASE   aASE ) 2
  ASE aASE n   ASE
L
L
SE (t ) ASE ( eASE   aASE )
N 2 ( L, t ) NA2 ASE
2
ASE
N
(
L
,
t
)

 m ASE
(



)
g
(
v
)
2
0 n2 ( z, t )dz
e
a
ASE
hvASE A
L 2 4nc 2
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53
Maximum extractable energy (B-3)
The rate equation describes the evolution of the upper-state population density
Pp ( z , t )  Pp ( z , t )
dn2 ( z , t )
p
p
p
  p a n   p ( e   a )n2 ( z , t )  
dt
hvp A

n2 ( z , t )
2. (ASE spontaneous emission)
2
 n2 ( z , t )  
B.W.
Gaussian
n  
B.W.
Gaussian
1. (pump absorption & depletion)
Beyond a certain inversion, pump will
be lost as ASE.
 ASE (
 ASE
ASE
a
ASE
e

ASE
a


PASE
( z , t , vASE )  PASE
( z , t , vASE )
)
 dvASE 3. (ASE stimulated emission)
hvASE A


PASE
( z , t , vASE )  PASE
( z , t , vASE )
 dvASE 4. (ASE absorption)
hvASE A
ASE becomes dominant to deplete inversion when ASE stimulated emission is comparable to
ASE spontaneous emission.

B.W.
Gaussian
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 ASE (
ASE
e

ASE
a


PASE
( z, t , vASE )  PASE
( z, t , vASE )
1
)
 dvASE 
hvASE A
2
54
Maximum extractable energy (B-4)
The max. allowed upper-state population integral at the end of energy storage period is
B.W.
4nc 2
exp[ ASE ( eASE   aASE ) N 2 ( L, Tes )  ( ASE aASE n   ASE ) L]  1

g (vASE )dvASE
nthASE L
NA2 ASE m Gaussian
1
N 2 ( L, Tes )
ASE
th
n
 ASE aASE n   ASE

The upper-state population at which the net gain at ASE wavelength is 0
 ASE ( eASE   aASE )
We can use numerical solution to obtain N2 ( L, Tes ) in terms of fiber SPEC, pump & rep. rate
Finally, the maximum extractable energy can be transformed by the following relation,
s
Eext  S ( es   as ) N2 ( L, Tes )  (ASE aASE n   ASE ) L  Esat
Where the saturation energy is
2017/4/10
s
Esat

hvs A
S ( es   as )
55
Example for 3rd amp, 30/250 DC (1)
4.0
3.5
Amplified energy (J)
3.0
Pump (W)
150
130
110
90
70
50
30
10
FN
2.5
2.0
1.5
1.0
0.5
0.0
0
50
100
150
200
250
300
350
400
450
500
Seed energy (J)
Optimal fiber length is 2.95 meter.
Pump is150 W at 20 kHz
When seed energy is> 250 uJ, pulse energy of > 3 mJ can be obtained.
In real launched should be~130 W, input energy of approaching 400 uJ is safe
after energy raiser stage.
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56
Example for 3rd amp, 30/250 DC (2)
4.0
50
3.5
For 30/250 DC
Seed energy at 150 W, 20kHz
400 J
380 J
350 J
297 J
200 J
100 J
50 J
2.5
2.0
40
30
efficiency (%)
Amplified energy (mJ)
3.0
1.5
1.0
For 30/250 DC at 20kHz
seed energy
400 J
380 J
350 J
300 J
200 J
100 J
50 J
20
10
0.5
0
0.0
0
20
40
60
80
100
Pump power (W)
120
140
160
0
20
40
60
80
100
120
140
160
Pump power (W)
To approach saturated input signal , 150-W pump power can obtain 40% efficiency.
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57
Example for 3rd amp, 30/250 DC (3)
20
4.0
18
3.5
16
40
14
For 30/250 DC
at 0.38 mJ, 150 W
energy
efficiency
10
8
30
20
6
4
10
3.0
30
2.5
20
2.0
amplified energy (mJ)
efficiency (%)
1.5
efficiency (%)
40
12
Amplified energy (mJ)
50
efficiency (%)
Amplified energy (mJ)
50
60
10
2
0
0
0
20
40
60
Repetition rate (kHz)
80
100
1.0
0
50
100
150
200
250
300
350
400
0
450
seed energy (J)
(1) Due to CW pumping, lower repetition rate will decrease amplifier efficiency
but raise the output energy. 20 kHz is Mid-Low repetition rate, the amplifier
efficiency is ~40%
(2) Due to stimulated amplification, to increase seed energy will make amp
efficiency better under unsaturated input.
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58
Outlines
• Saturation
no real amplifier operating without saturation limit. We need to find it.
• Extractable energy
Use saturation as utilized unit to find the maximum extractable energy you can achieve
• FDTD rate equation for repetition rate issue
If quantitative results to compare with experimental results is necessary, it can not be avoided.
• Fiber nonlinear optics
In fiber, it cannot be avoided. It is advantage/disadvantage to enhance/suppress it.
2017/4/10
59
Dispersion in fiber
Index vs. wavelength
Time lag vs. fiber length
1.4506
200
refeactive index of fused silica
150
1.4502
100
1.4500
50
Time lag (ps)
Refractive index
1.4504
1.4498
1.4496
0
-50
-100
1.4494
-150
1.4492
-200
1000
1020
1040
1060
Wavelength (nm)
1080
1100
Wavelngth
1069 nm
1065 nm
1064 nm
1063 nm
1059 nm
0
200
400
600
800
1000
Fiber length (m)
For pulse broadening in 1~20 ns duration & 1~3 meter long fiber,
(1) Group-Velocity Dispersion (GVD) above is small (<0.01% duration).
(2) Polarization-Mode Dispersion (PMD) is relative small compared with GVD effects
(3) If pulse duration is < 50 ps, the dispersion length decreases and becomes
comparable to both the fiber length and nonlinear length.
2017/4/10
G. P. Agrawal, “Nonlinear fiber optics” 4 ed. Academia Press. (2007)
60
Final-version rate equations (1)
N2隨時變化
Pump受激吸收增加 N2
Raman受激放射消耗N2
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Signal & ASE 受激放射消耗N2
ASE 自發輻射消耗N2
61
Final-version rate equations (2)
Pump 隨時空
propagation
ASE 隨時空
propagation
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Pump受激吸收
ASE受激放大
考慮ASE
光纖傳播
損耗
Pump光纖
傳播損耗
考慮線寬所 反向
造成的受激 Rayleigh 散
與自發輻射 射的貢獻
貢獻
62
Final-version rate equations (3)
Signal受激放大
Signal隨時空
propagation
Signal光纖
傳輸損耗
2017/4/10
考慮線寬所
造成的受激
與自發輻射
貢獻
反向
Rayleigh 散
射的貢獻
拉曼損耗signal作
非線性轉換
63
Add considerations step by step
1. Rate equation considering N2 vs. Ps+ only (Step 1)
只考慮signal depletion period
2. Rate equation including pump (Step 2)
以下都需同時考慮energy storage與signal depletion period (重覆頻率)
3. Rate equation including ASE (Step 3)
4. Rate equation including linewidth (Step 4)
5. Rate equation including fiber propagation loss & Rayleigh scattering
contribution (Step 5)
6. Rate equation including SRS Stoke (Step 6)
7. Rate equation comparing SRS Anti-Stoke & SBS (Step 7)
PS. Step 1 將Seed改成Multi-wavelength 可驗證parasitic stimulated
emission
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64
Outlines
• Saturation
no real amplifier operating without saturation limit. We need to find it.
• Extractable energy
Use saturation as utilized unit to find the maximum extractable energy you can achieve
• FDTD rate equation for repetition rate issue
If quantitative results to compare with experimental results is necessary, it can not be avoided.
• Fiber nonlinear optics
In fiber, it cannot be avoided. It is advantage/disadvantage to enhance/suppress it.
2017/4/10
65