Nonlinear pulse propagation - the Keller Group

Ultrafast Laser Physics
Ursula Keller / Lukas Gallmann
ETH Zurich, Physics Department, Switzerland
www.ulp.ethz.ch
Chapter 4: Nonlinear pulse propagation
Ultrafast Laser
Physics
ETH Zurich

Kerr effect and self-phase modulation (SPM)
⎡ cm
( ) = n + n 2
I 2
n 2 ⎢
n I
⎣
W
⎤
⎥
⎦
= 4.19 × n 2
10−3
[ esu]
n
Material Refractive index n n 2 [ esu] n 2
cm 2 / W
Sapphire (Al 2 O 3 ) 1.76 @ 850 nm 1.25 × 10 −13 [89Ada] 3 ×10 −16
Fused quartz 1.45 @ 1.06 m 0.85 ×10 −13 [89Ada] 2.46 ×10 −16
Glass (Schott LG-
760)
1.5 @ 1.06 m 1.04 × 10 −13 [93Aza] 2.9 × 10 −16
YAG (Y 3 Al 5 O 12 ) 1.82 @ 1.064 m 3.47 × 10 −13 [93Aza] 6.2 × 10 −16
YLF (LiYF 4 )
n e
= 1.47 @ 1.047
m
⎡⎣
1.72 × 10 −16 [93Aza]
⎤ ⎦
Typical order of magnitude for the nonlinear index coefficient: n 2 ≈ 10 –16 cm 2 /W
Self-phase modulation (SPM):
( ) = −kn( I ) L K
= −k n + n 2
I ( t)
φ t
⎡⎣
⎤⎦ L K
SPM coefficient:
δ ≡ kn 2
L K
φ 2
( t) = −kn 2
I ( t) L K
= −kn 2
L K
A( t) 2 ≡ −δ A( t) 2
ULP, Chap. 4, p. 1

Kerr effect and self-phase modulation (SPM)
n 2
> 0
I(t)
( )
I t
φ 2
( t) = −kn 2
I ( t) L K
= −kn 2
L K
A( t) 2 ≡ −δ A( t) 2
leading edge
SPM: red
Pulsfront
ω ( 2 t)
ω 0
Gaussian Pulse
Zeitabhängige Intensität
trailing Pulsflanke edge
SPM: blue
( t)
t
Spectral broadening
Verbreiterung des Spektrums
t
δ ≡ kn 2
L K
ω 0
ω 2
t
( ) = dφ 2 ( t)
dt
= −δ dI ( t)
dt
Spectral broadening of a transform-limited pulse:
“red before blue”
ULP, Chap. 4, p. 2

Number of oscillations in SPM-broadened spectrum
⎛
φ 2,max
= kn 2
I p
L K
≈ M − 1 ⎞
⎝
⎜
2⎠
⎟ π
Theory: Parameter
φ 2,max
Experiment: Gaussian pulse in 99 m
fiber.
R. H. Stolen, C. Lin, Phys. Rev. A, 17, 1448, 1978
ULP, Chap. 4, p. 3

SPM
• Instantaneous change of refractive index:
Δ n( t) = n I( t)
2
• Consequences for a sech 2 pulse (without dispersion):
• Small phase changes: weak spectral broadening;
approximately parabolic phase in frequency domain
(can be compensated by constant GDD!)
• Large phase changes:
complicated spectral
broadening
(complete compression
is difficult)
4
2
0
phase (rad)
intensity (a. u.)
-2
-4
-400 -200 0 200 400
frequency offset (GHz)

Pure SPM in the Wigner picture
Initially 10 fs long Gaussian pulse at 800 nm, SPM (n 2 >0) only
n( I ) = n + n 2
I
• Temporal pulse shape remains unchanged
• Spectrum broadens
• Oscillatory spectral features due to interference in frequency domain

Comparison with effect of TOD
Everything calculated for an initially 10-fs long Gaussian pulse
After 1000 fs 3 of TOD:
ϕ(ω ) = 1 6 ⋅1000 fs3 ⋅( ω − ω 0 ) 3
• “Beating of simultaneous frequencies”
causes post-(pre-)pulses
• Interference in time domain

Comparison of SPM and GDD
Everything calculated for an initially 10-fs long Gaussian pulse
φ 2
After SPM (n 2 >0):
( t) = −kn 2
I ( t) L K
= −kn 2
L K
A( t) 2 ≡ −δ A( t) 2
After 100 fs 2 of GDD:
ϕ(ω ) = 1 2 ⋅100 fs2 ⋅( ω − ω 0 ) 2
• “Red” before “blue”
• Chirp is (mostly) linear in center
⇒ Negative GDD can compensate linear chirp in center of SPM broadened pulse

Fiber grating pulse compressor
ULP, Chap. 4, p. 4

World-record pulse duration in 1987
6 fs FWHM
Fiber-grating-prism-pulse compressor
for the compression of 50 fs to 6 fs at 8 kHz
center wavelength 620 nm
SPM broadened spectrum: quartz fiber with core diameter
of ≈ 4 µm and a length of 0.9 cm, peak intensity 1-2 x 10 12 W/cm 2
Measured interferometric autocorrelation
ULP, Chap. 4, p. 5

World-record pulse duration in 1999
ULP, Chap. 4, p. 5

World-record pulse duration in 1999
6 fs FWHM
(1987)
ULP, Chap. 4, p. 5

Compressed pulses from a thin-disk laser
ASSP 2005
Incident on fiber
After fiber
P avg = 60 W
P avg = 42 W
τ p = 760 fs
launch efficiency:
τ p = 24 fs
I peak = 1.2 TW/cm 2 70%
P rej = 10 W (PBS)
After compression
P avg = 32 W
ULP, Chap. 4, p. 6

Compressed pulses from a thin-disk laser
large mode area fiber
Incident on fiber
P avg = 60 W
E p ≈ 1 µJ
τ p = 760 fs
I peak = 1.2 TW/cm 2
After compression
P avg = 32 W
E p = 0.56 µJ
τ p = 24 fs
P peak = 16 MW
A eff ≈ 200 µm 2 (mode area)
d ≈ 2.7 µm (hole Ø)
Λ ≈ 11 µm (spacing)
but fiber damage after 10-20 minutes
T. Südmeyer, et al., Opt. Lett. 28, 1951 (2003) and E. Innerhofer, TuA3, ASSP 2004!
ORC Southampton
!

Compressed pulses from a thin-disk laser
optical spectrum (not symmetric - other nonlinearities, self-steepening)
Compression output
P avg = 32 W f rep = 57 MHz
P peak = 16 MW τ p = 24 fs
E p = 0.56 µJ
• 73% of energy in central pulse
• Fourier limit: 20 fs
• fiber damage after ≈ 15 minutes
autocorrelation
retrieved pulse
ASSP 2005
ULP, Chap. 4, p. 6

Nonlinear pulse compression
• Approach: SPM in a fiber for spectral broadening, grating or
prisms for dispersion compensation"
• Established technique, but used for much lower power"
• High power in fiber requires large mode area with single-mode
operation"
Microstructured fiber with
large mode area!
Used fiber: # "
effektive mode area ≈ 205 µm 2!
#
K. Furusawa, J. C. Baggett, T. M. Monro,
and D. J. Richardson, ORC Southampton"
100 µm!

Nonlinear Compression
Principle 1. Generation of additional spectral bandwidth (SPM)
2. Compression with prisms, gratings or chirped mirrors
Soliton pulse
compression in PBGF
+ Self compression
- Large third order
dispersion
for silica
n 2 ≈ 2.7·10 -20 m 2 /W
SPM in glass (e.g. large
mode area fiber)
+ High nonlinearity,
- Damage of the fiber, self
focusing
D.Ouzounov et al.,Opt..Exp.13,16 1951(2003)
T. Südmeyer, et al., Opt. Lett. 28, 1951 (2003)
for xenon
n 2 ≈ 8.1·10 -23 m 2 /W
SPM in gas filled hollow-core photonic crystal fiber (HC-PCF)
+ High damage threshold, flexible (type of gas, pressure)
+/- Low nonlinearity (long fiber, freedom of adjustment)
Guiding Compression of > 10 of µJ 1.9 1.2 and µJ 30 /1100 860 fs demonstrated fs (0.9 (1.9 MW) pulses (> 310 (13 (7.3 MW)
W) to
0.7 1.1 µJ / 250 48 fs pulses (7.7 (4.2 W)
O. H. Heckl, et al., Appl. Phys. B 97, 369-373 (2009).
O. H. Heckl, et al., Opt. Exp., sub-50fs, 97, 2010MW (2011) pulses at MHz repetition rate
F. Emaury et al., ALT 2012

15 fs, 16 nJ
Ti:Sa
Fiber compressor for 5.5 fs pulses
SPM broadening in a microstructure fiber (MF), length 5 mm
Ti:sapphire laser with prism pair and DCMs: f rep = 19 MHz (for higher pulse energy)
AS
SLM
MF
OC
AS DCMs
0.2nJ
SPIDER
Intensity (a. u.)
1.0
0.5
0.0
500
750
Wavelength (nm)
Microstructure fiber (MF):
2.6 µm core diameter
5 mm long
zero GDD at 940 nm
0
-200
-400
-600
1000
Dispersion (ps/nm/km)
SM
G
G
SM
B. Schenkel et al., JOSA B 22, 687, 2005

Broadband pulse shaper with SLM
A. M. Weiner, Rev. Sci. Instrum. 71, 1929 (2000)
spatial light modulator (640 pixel liquid crystal, each pixel ≈100 µm wide, 3 µm gap)
f = 300 mm
knife-edge
f = 300 mm
SLM
640 pixels
300 l/mm grating 300 l/mm grating
Possible bandwidth through Spatial Light Modulator:
400 - 1050 nm!

Intensity (a. u.)
1.0
0.5
5.5 fs
0.0
-40 -20 0 20 40
Time (fs)
Fiber compressor for 5.5 fs pulses
Power density (a. u.)
1.0
0.5
0.0
500
750
Wavelength (nm)
1000
4
0
-4
Spectral phase (rad)
Intensity (a. u.)
1.0
0.5
5.5 fs
0.0
-40 -20 0 20 40
Time (fs)
Interferogram
1.0
0.5
0.0
320
370
420
Wavelength (nm)
B. Schenkel et al., JOSA B 22, 687, 2005
5.5 fs, 0.2 nJ!
• Good fringe visibility:
reliable SPIDER measurement "
• Microstructure fiber 2.6-µm core
diameter, 5 mm long, zero GDD
at 940 nm"

Dual stage hollow fiber compressor for 3.8 fs
25 fs, 0.5 mJ
Ti:Sa Amp
100 µJ
Shaper
continuum generation
15 µJ
SPIDER
B. Schenkel et al.: Opt. Lett. 28, 1987 (2003)"

Dual stage hollow fiber compressor for 3.8 fs
Interferogram
1.0
0.5
0.0
340
400
Wavelength (nm)
460
Spectral Power Density
1.0
0.5
0.0
500
750
Wavelength (nm)
4
2
0
-2
-4
1000
Spectral Phase (rad)
2π phase shift!
1.0
→ pre- and post-pulses!
3.8 fs, 15 µJ!
B. Schenkel et al.: Opt. Lett. 28, 1987 (2003)"
Intensity
0.5
0.0
3.8 fs
-40 -20 0 20 40
Time (fs)

Optical pulse cleaner
Optical pulse cleaner based on nonlinear birefringence
Optics Letters, vol. 17, pp. 136-138, 1992
ULP, Chap. 4, p. 9-10

Self-focusing
Kerr medium
length L K
( ) = I p
exp −2 x2 + y 2
I x,y
⎛
⎝
⎜
w 2
⎞
⎠
⎟
⎛
≈ I p
1 − 2 x2 + y 2 ⎞
⎝
⎜
w 2
⎠
⎟
↓ ( x 2 + y 2
)

B-integral
B ≡ 2π λ
L
∫
0
n 2
I ( z)dz
To prevent material damage: B should be smaller than 3 to 5
ULP, Chap. 4, p. 10

Critical power for beam collapse
P cr
≡ 3.72λ 0 2 / 8π n 0
n 2
L c
L c
=
P in
/ P cr
0.376L DF
⎡( ) 1 2 − 0.852
⎣
⎤
⎦ 2 − 0.0219
L DF
= π n w 2
0 0
λ 0
Rayleigh length
Argon at 800 nm (atmospheric pressure):
n 0 = 1.0, n 2 = 3 10 –19 cm 2 /W, P cr = 3.2 GW
Fused quartz at 1.06 µm:
n 0 = 1.45, n 2 = 2.46 10 –16 cm 2 /W, P cr = 3.8 MW
ULP, Chap. 4, p. 14

Filamentation
ULP, Chap. 4, p. 14

Filamentation
Filamentation of mJ-level, 30-fs pulses at 800 nm in Ar
During propagation SPM continues to broaden
spectrum of pulse ⇒ white light

Fundamental Soliton Pulses
• Basic idea: nonlinear phase change from Kerr effect is
compensated by dispersive phase change,
apart from a constant phase shift.
• Conditions (for constant GDD):
• Negative (anomalous) GDD, if n 2 > 0
• Unchirped sech 2 pulse shape, fulfilling the condition
k n
τ p
= 1.7627 × 4 D = 1.7627 × 2 ′′
δ e p
kn 2
e p
• Remarkable stability of soliton pulses:
particle character in collision
pulse automatically “finds“ the exact
required shape
(may shed some energy into a background pulse)

Nonlinear pulse propagation
Linear pulse propagation:
GDD and no SPM
k n
′′ ≠ 0
n 2
= 0
Nonlinear pulse propagation:
no GDD and SPM
k n
′′ = 0
n 2
≠ 0
ULP, Chap. 4, p. 18

Nonlinear pulse propagation
Nonlinear pulse propagation:
GDD > 0 and SPM > 0
k n
′′ > 0
n 2
> 0
Nonlinear pulse propagation:
Soliton pulses
GDD < 0 and SPM > 0
k n
′′ < 0
n 2
> 0
ULP, Chap. 4, p. 19

Nonlinear Schrödinger Equation (NSE)
Slowly varying envelope approximation:
( ) = e −i k n ω 0 +Δω
A L d
,Δω
( )−k n ω 0
( )
⎡⎣ ⎤ ⎦ L d A(0,Δω )
k n
( ω ) ≅ k n ( ω 0 ) + k n
′ Δω + 1 2 k′′
nΔω 2
k n
′ = ∂k n
k n
′′ = ∂ 2 k n
∂ω ω0
∂ω 2 ω 0
( ) = exp −i ′
A Ld ,Δω
⎧
⎨
⎩
⎛
k n
Δω + 1 2 k′′
nΔω 2
⎝
⎜
⎞
⎠
⎟ L d
⎫
⎬
⎭
A(0,Δω )
Dispersion first order:
Linearized operator in the time domain
k n
′ Δω L d

Nonlinear Schrödinger Equation (NSE)
Slowly varying envelope approximation:
( ) = e −i k n ω 0 +Δω
A L d
,Δω
( )−k n ω 0
( )
⎡⎣ ⎤ ⎦ L d A(0,Δω )
k n
( ω ) ≅ k n ( ω 0 ) + k n
′ Δω + 1 2 k′′
nΔω 2
k n
′ = ∂k n
k n
′′ = ∂ 2 k n
∂ω ω0
∂ω 2 ω 0
( ) = exp −i ′
A Ld ,Δω
⎧
⎨
⎩
⎛
k n
Δω + 1 2 k′′
nΔω 2
⎝
⎜
⎞
⎠
⎟ L d
⎫
⎬
⎭
A(0,Δω )
Dispersion second order:
Linearized operator in the time domain
F −1
{ Δω 2 A ( z,Δω ) } = − ∂ 2
∂t A z,t 2
Dispersion parameter D
k n
Δω 2 L d

Nonlinear Schrödinger Equation (NSE)
A L d
,t
⎛
⎝
⎜
( ) ≅ 1 − ′
k n
L d
∂
∂t
⎞
⎠
⎟ A 0,t
( ) , for ′
k n
Δω L d

E ( L K
,t) = A 0,t
δ ≡ kn 2
L K
SPM operator
( )exp ⎡⎣ iω 0
t + iφ ( t)
⎤⎦ = A( 0,t)exp
⎡ iω t − ik 0 n
ω
⎣
0
A( L K
,t) = e −iδ A 2 A( 0,t)e −ik n ω 0
( )L K δ A 2

Nonlinear Schrödinger Equation (NSE)
( ) ≅ 1 − ′
A L d
,t
⎛
⎝
⎜
k n
L d
∂
∂t
⎞
⎠
⎟ A 0,t
( ) , for ′
k n
Δω L d

Nonlinear Schrödinger Equation (NSE)
∂
∂z A z, t′
k n
( ) = i ′′
2
∂ 2
∂ t′
A z, t′
2
Solution: a fundamental soliton
A s ( z, t′
) = A 0
sech
τ p
= 1.7627 ⋅τ
Δν p
τ p
= 0.3148
( ) − ikn A( z, t′
) 2 2
A( z, t′
) Nonlinear Schrödinger Equation (NSE)
⎛
⎝
⎜
t′ ⎞
τ ⎠
⎟ e −iφ 0
φ 0
= φ 2 max
2
φ 2 max
= kn 2
I p
z , I p
= A 0
2
The pulse as a whole experiences a homogeneous
phase shift (not like SPM alone!)
ULP, Chap. 4, p. 20

Nonlinear Schrödinger Equation (NSE)
∂
∂z A z, t′
k n
( ) = i ′′
2
∂ 2
∂ t′
A z, t′
2
( ) − ikn A( z, t′
) 2 2
A( z, t′
) Nonlinear Schrödinger Equation (NSE)
Solution: a fundamental soliton
A s ( z, t′
) = A 0
sech
τ p
= 1.7627 ⋅τ
⎛
⎝
⎜
t′ ⎞
τ ⎠
⎟ e −iφ 0
k n
τ p
= 1.7627 × 4 D = 1.7627 × 2 ′′
δ e p
kn 2
e p
∝ 1 e p
Δν p
τ p
= 0.3148
φ 0
= φ 2 max
2
φ 2 max
= kn 2
I p
z , I p
= A 0
2
The pulse as a whole experiences a homogeneous
phase shift (not like SPM alone!)
ULP, Chap. 4, p. 21

Nonlinear Schrödinger Equation (NSE)
∂
∂z A z, t′
k n
( ) = i ′′
2
∂ 2
∂ t′
A z, t′
2
( ) − ikn A( z, t′
) 2 2
A( z, t′
) Nonlinear Schrödinger Equation (NSE)
Solution: a fundamental soliton
A s ( z, t′
) = A 0
sech
τ p
= 1.7627 ⋅τ
Δν p
τ p
= 0.3148
⎛
⎝
⎜
t′ ⎞
τ ⎠
⎟ e −iφ 0
τ p
= 1.7627 × 4 D = 1.7627 × 2 ′′ ∝ 1 δ e p
kn 2
e p
e p
Balance between negative GDD and
positive SPM:
φ 0
= D τ 2
= 1 2 δ I p
= δ e p
4τ
k n
= kn 2
e p
4τ
z
D ≡ 1 2 k′′
L n d
δ ≡ kn 2
L K
φ 0
= φ 2 max
2
φ 2 max
= kn 2
I p
z , I p
= A 0
2
( )
( ) 2 d ′
e p
= E p
2
= I z, t′
A eff
∫ d t ′ = ∫ A s
z, t′
t = 2 A 0
τ
The pulse as a whole experiences a homogeneous
phase shift (not like SPM alone!)
ULP, Chap. 4, p. 21

Nonlinear Schrödinger Equation (NSE)
∂
∂z A z, t′
k n
( ) = i ′′
2
∂ 2
∂ t′
A z, t′
2
( ) − ikn A( z, t′
) 2 2
A( z, t′
) Nonlinear Schrödinger Equation (NSE)
Solution: a fundamental soliton
k n
τ p
= 1.7627 × 4 D = 1.7627 × 2 ′′
δ e p
kn 2
e p
∝ 1 e p
τ p
∝ 1 e p
τ p
∝ k n
′′
⎛
Soliton area = ∫ A 0
sech t ⎞
⎝
⎜
τ ⎠
⎟ dt = π A 0
τ
“Solitons have constant area”
ULP, Chap. 4, p. 22

Nonlinear Schrödinger Equation (NSE)
∂
∂z A z, t′
k n
( ) = i ′′
2
∂ 2
∂ t′
A z, t′
2
Solution: a fundamental soliton
( ) − ikn A( z, t′
) 2
2
A( z, t′
) Nonlinear Schrödinger Equation (NSE)
ULP, Chap. 4, p. 22

Nonlinear Schrödinger Equation (NSE)
∂
∂z A z, t′
k n
( ) = i ′′
2
∂ 2
∂ t′
A z, t′
2
Solution: a fundamental soliton
( ) − ikn A( z, t′
) 2
2
A( z, t′
) Nonlinear Schrödinger Equation (NSE)
ULP, Chap. 4, p. 24

Nonlinear Schrödinger Equation (NSE)
∂
∂z A z, t′
k n
( ) = i ′′
2
∂ 2
∂ t′
A z, t′
2
( ) − ikn A( z, t′
) 2 2
A( z, t′
) Nonlinear Schrödinger Equation (NSE)
Solution: higher order soliton (example: second order)
Soliton Period φ 0 ( z = z 0 ) = π 4
⇒ z 0
= π 2
τ 2
k n
′′
ULP, Chap. 4, p. 23

Higher-Order Soliton Pulses
• Inject a pulse with N 2 –times the fundamental soliton energy:
periodically evolving higher-order soliton pulse (N is an integer).
t′ ⎞
• Initial condition:
N = 2 for second-order soliton
A( 0, t′
) = N A 0
sech
⎛
⎝
⎜
τ ⎠
⎟
• Soliton period:
φ 0 ( z = z 0 ) = π 4
⇒ z 0
= π 2
τ 2
′′ k n
becomes short for short pulses and strong dispersion
• At certain locations, significantly shorter (but not sech 2 -shaped) pulses
occur
important for pulse compression
• Note: soliton period is an important parameter
also for fundamental solitons:
length scale on which the interaction is significant
(periodic perturbation)

Optical communication with repeaters
ULP, Chap. 4, p. 25

Optical communication with solitons
ULP, Chap. 4, p. 26

Optical communication with solitons
−Δω 0
Δω
ULP, Chap. 4, p. 27

∂
∂z A z, t′
( ) = i ′′
Periodic perturbation of solitons
k n
2
∂ 2
∂ t′
A z, t′
2
( ) − ikn 2
A( z, t′
) 2 A( z, t′
) + iξ δ ( z − nz a )
∞
∑ A z, t′
n=−∞
( )
NSE + periodic perturbation
period z a
Important for modelocked lasers:
periodic perturbation per round-trip through output coupler, gain crystal …
ULP, Chap. 4, p. 27-31

∂
∂z A z, t′
( ) = i ′′
Periodic perturbation of solitons
k n
2
∂ 2
∂ t′
A z, t′
2
( ) − ikn 2
A( z, t′
) 2 A( z, t′
) + iξ δ ( z − nz a )
∞
∑ A z, t′
n=−∞
( )
( ) = A 0
sech
A s
z, t′
NSE + periodic perturbation
period z a
Assuming small perturbation: Ansatz
Solution without perturbation
Soliton pulse:
( ) = A s ( z, t′
) + u ( z, t′
)
A z, t′
u z, t′
⎛ t′ ⎞
⎝
⎜
τ ⎠
⎟ e −iφ 0
( )

∂
∂z A z, t′
( ) = i ′′
Periodic perturbation of solitons
k n
2
∂ 2
∂ t′
A z, t′
2
( ) − ikn 2
A( z, t′
) 2 A( z, t′
) + iξ δ ( z − nz a )
∞
∑ A z, t′
n=−∞
( )
NSE + periodic perturbation
period z a
Periodic perturbation has no resonance effects:
u ( z,ω )

∂
∂z A z, t′
( ) = i ′′
Periodic perturbation of solitons
k n
2
∂ 2
∂ t′
A z, t′
2
( ) − ikn 2
A( z, t′
) 2 A( z, t′
) + iξ δ ( z − nz a )
∞
∑ A z, t′
n=−∞
( )
NSE + periodic perturbation
period z a
Periodic perturbation has no resonance effects:
u ( z,ω )

Delayed Nonlinear Response
• Intensity-dependent phase change is not always instantaneous:
• Electronic contribution (usually dominating): response time

Delayed Raman Response
Optical phonons have high frequencies (e.g. around
13 THz for silica), only weakly dependent on wave
vector:
w optical phonons!
range of interest"
acoustical phonons!
Consequence: phase matching possible for
forward and backward direction:
k p "
k p "
k s "
k"
k s "
k phonon "
k phonon "

Raman Gain Spectrum of Silica
g R
(Δω)
Raman gain (a. u.)
1.4
1.2
1.0
0.8
0.6
0.4
0.2
gain spectrum
1000-nm pump"
0.0
1000
1040
1080
1120
wavelength (nm)
• Maximum gain at ≈40-50 nm wavelength offset
(depends on pump wavelength)
• Gain rises ≈linearly for small offsets
• Influence of composition of the fiber core (Ge, P etc.)

Intra-Pulse Raman Scattering
Principle:
J. P. Gordon, Opt. Lett. 11 (10), 662 (1986)"
• Energy transfer within the pulse spectrum:
⇒ center wavelength drifts
towards longer values
• Soliton interaction preserves the pulse shape
l
Note: Raman gain is small for small frequency offsets
⇒ effect is significant only for femtosecond (soliton) pulses
shift per meter: roughly prop. to (1/τ ) 4 "
Examples:
• Wavelength shift from 1.56 µm to 1.78 µm:
N. Nishizawa et al., IEEE Photon. Technol. Lett. 11 (3), 325 (1999)
• Wavelength shift from 1.06 µm to 1.33 µm (in holey fiber):
J. H. V. Price et al., JOSA B 19 (6), 1286 (2002)

Raman Response of Silica
R. H. Stolen et al., JOSA B 6 (6), 1159 (1989)"
4
3
2
Response function"
1
0
-1
0
200
400
600
800
1000
delay time τ (fs)
Damped oscillation with ≈13 THz:
strong contribution, if two optical waves
with ≈13 THz frequency difference beat

∂A( z, t′
)
∂z
− i 2 k′′
n
⎡
= −iγ ⎢ A z, t′
⎣⎢
( )
Generalized NSE
second and third-order dispersion
( )
∂ 2 A z, t′
− 1 ∂ t′
2 6 k′′′
∂ 3 A z, t′
n
∂ t′
3
( ) 2 A( z, t′
) − i
ω 0
∂
∂ t′
( A( z, t ′) 2 A( z, t ′)) − T R
A z, t′
( ) ∂ A( z, t′
)
∂ t′
SPM self-steepening Raman
T R sets slope of
Raman gain
intensity dependence
of phase velocity
intensity dependence
of group velocity
γ = n 2ω 0
cA eff
2
⎤
⎥
⎦⎥
shock formation
self-frequency shift
Raman and self-steepening lead to asymmetry in SPM broadened spectra

Self-steepening and SPM (without GDD and Raman)
Example: 50 fs, center wavelength 800 nm, fiber core diameter 1.7 µm and
n 2
= 2.5 ⋅10 −20 m 2 / W
s = 1
ω 0
τ = T
2πτ = 0.01
Self-steeping means that group velocity is intensity dependent:
peaks moves at a lower speed than the wings
40
16
Power [kW]
30
20
10
Energy/Wavelength [pJ/nm]
14
12
10
8
6
4
2
0
-80 -60 -40 -20 0 20 40 60 80
Time [fs]
0
0.4
0.5
0.6
0.7 0.8
Wavelength [um]
0.9
1.0
1.1
Input: Gaussian pulse at z = 0
z = 3 mm (dashed) z = 6 mm
z = 6 mm (solid)
asymmetry in SPM broadened spectrum

Self-steepening, SPM and GDD>0 (no Raman)
Power [kW]
40
30
20
10
0
Example: 50 fs, center wavelength 800 nm, fiber core diameter 1.7 µm and
GDD 3.5 fs 2 /100 µm
s = 1
n 2
= 2.5 ⋅10 −20 m 2 / W
ω 0
τ = T
2πτ = 0.01
z = 6 mm
-80 -60 -40 -20 0 20 40 60 80
Time [fs]
Energy/Wavelength [pJ/nm]
16
14
12
10
8
6
4
2
0
0.4
0.5
0.6 0.7 0.8 0.9
Wavelength [um]
1.0
1.1
Power [kW]
14
12
10
8
6
4
2
0
-200 -100 0 100 200
Time [fs]
Energy/Wavelength [pJ/nm]
16
14
12
10
8
6
4
2
0
0.6
0.7 0.8 0.9
Wavelength [um]
1.0
z = 12 mm
Power [kW]
8
6
4
2
0
Energy/Wavelength [pJ/nm]
14
12
10
8
6
4
2
0
-200
0
Time [fs]
200
0.6
0.7 0.8 0.9
Wavelength [um]
1.0

Assuming: two-level system
Saturable gain and absorber
Dichte der Atome
Fläche A
Dicke Δz
Einfallende
Lichtintensität
Wirkungsquerschnitt σ
( ) ≡ N V σ α 0 = N 0
g z
g =
g 0
1+ I I sat ,L
V σ
σ A
= σ L
= σ
α
α = 0
1+ I I sat ,A
I sat ,L
= hν
στ L
I sat ,A
= hν
στ A

Semiconductor saturable absorber
Appl. Phys. B 73, 653, 2001"

Nonlinear transmission of cw beam
Assume homogeneous two-level absorption saturation:
dI ( z)
dz
= −2α ( I ) I ( z) = −
2α 0
1 + I z
( )
I sat
I ( z)
1
I
⎛
⎝
⎜
1 + I
I sat
⎞
⎠
⎟ dI = − 2α 0
dz
T ≡ I out
I in
I in
I sat
lnT + I in
( T − 1) = −2α 0
d
I sat
I in