nonlinear pulse propagation

ETH ZurichUltrafast Laser Physics

Ursula Keller / Lukas Gallmann

ETH Zurich, Physics Department, Switzerlandwww.ulp.ethz.ch

Chapter 4: Nonlinear pulse propagation

Ultrafast Laser Physics

Kerr effect and self-phase modulation (SPM)

n I( ) = n + n2 I n2cm2

W⎡

⎣⎢

⎤

⎦⎥ = 4.19 ×10

−3 n2 esu[ ]n

Material Refractive index n n2 esu[ ] n2 cm2 / W⎡⎣ ⎤⎦

Sapphire (Al2O3) 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 (Y3Al5O12) 1.82 @ 1.064 m 3.47 ×10−13 [93Aza] 6.2 ×10−16

YLF (LiYF4) ne = 1.47@ 1.047 m

1.72 ×10−16 [93Aza]

ULP, Chap. 4, p. 1

Typical order of magnitude for the nonlinear index coefficient: n2 ≈ 10–16 cm2/W

Self-phase modulation (SPM):

SPM coefficient:

φ t( ) = −kn I( )LK = −k n + n2 I t( )⎡⎣ ⎤⎦ LK

δ ≡ kn2LK

φ2 t( ) = −kn2 I t( )LK = −kn2LK A t( ) 2 ≡ −δ A t( ) 2

Kerr effect and self-phase modulation (SPM)

ULP, Chap. 4, p. 2

t

I(t)

Zeitabhängige Intensität

tω0

ω ( t)

Verbreiterung des Spektrums

Pulsfront

Pulsflanke

2

Gaussian Pulse

Spectral broadening

leading edgeSPM: red

trailing edgeSPM: blue

I t( )

t

t

ω2 t( )

ω0

ω2 t( ) = dφ2 t( )dt

= −δdI t( )dt


δ ≡ kn2LK

Spectral broadening of a transform-limited pulse:

“red before blue”

n2 > 0

Number of oscillations in SPM-broadened spectrum

ULP, Chap. 4, p. 3

φ2,max = kn2I pLK ≈ M −12

⎛⎝⎜

⎞⎠⎟π

Theory: Parameter Experiment: Gaussian pulse in 99 mfiber.

R. H. Stolen, C. Lin, Phys. Rev. A, 17, 1448, 1978

φ2,max

SPM• Instantaneous change of refractive index:

• Consequences for a sech2 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 spectralbroadening(complete compressionis difficult)

4

2

0

-2

-4-400 -200 0 200 400

frequency offset (GHz)

intensity (a. u.)phase (rad)

2( ) ( )n t n I tΔ =

Pure SPM in the Wigner picture

• Temporal pulse shape remains unchanged• Spectrum broadens• Oscillatory spectral features due to interference in frequency domain

Initially 10 fs long Gaussian pulse at 800 nm, SPM (n2>0) only

n I( ) = n + n2 I

Comparison with effect of TODEverything calculated for an initially 10-fs long Gaussian pulse

After 1000 fs3 of TOD:

ϕ(ω ) = 16⋅1000 fs3 ⋅ ω −ω 0( )3

• “Beating of simultaneous frequencies”causes post-(pre-)pulses

• Interference in time domain

Comparison of SPM and GDDEverything calculated for an initially 10-fs long Gaussian pulse

After 100 fs2 of GDD:

ϕ(ω ) = 12⋅100 fs2 ⋅ ω −ω 0( )2

• “Red” before “blue”

• Chirp is (mostly) linear in center

After SPM (n2>0):


⇒ 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

Fiber-grating-prism-pulse compressor for the compression of 50 fs to 6 fs at 8 kHzcenter 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 1012 W/cm2

Measured interferometric autocorrelation

6 fs FWHM

ULP, Chap. 4, p. 5


ULP, Chap. 4, p. 5


6 fs FWHM

(1987)

ULP, Chap. 4, p. 5

Compressed pulses from a thin-disk laser

After compressionPavg = 32 Wτ p = 24 fs

After fiberPavg = 42 Wlaunch efficiency: 70%Prej = 10 W (PBS)

Incident on fiberPavg = 60 Wτ p = 760 fsIpeak = 1.2 TW/cm2

ASSP 2005

ULP, Chap. 4, p. 6

large mode area fiber

Aeff≈ 200 µm2 (mode area)d ≈ 2.7µm (hole Ø)Λ ≈ 11 µm (spacing)

T. Südmeyer, et al., Opt. Lett. 28, 1951 (2003) and E. Innerhofer, TuA3, ASSP 2004ORC Southampton

After compressionPavg = 32 WEp = 0.56 μJτp = 24 fsPpeak = 16 MW

Incident on fiberPavg = 60 WEp ≈ 1 μJτp = 760 fsIpeak = 1.2 TW/cm2

but fiber damage after 10-20 minutes


autocorrelation

optical spectrum (not symmetric - other nonlinearities, self-steepening)

retrieved pulse

Compression outputPavg = 32 W frep = 57 MHzPpeak = 16 MW τp = 24 fsEp= 0.56 µJ• 73% of energy in central pulse• Fourier limit: 20 fs• fiber damage after ≈ 15 minutes

ASSP 2005ULP, Chap. 4, p. 6


Nonlinear pulse compression

100 µm

Microstructured fiber withlarge mode area

Used fiber:#

effektive mode area ≈ 205 µm2

# K. Furusawa, J. C. Baggett, T. M. Monro, and D. J. Richardson, ORC Southampton

• 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

−2000 −1200 −400 400 1200 20000

102030405060708090

100110

peak

pow

er /

MW

Delay / fs

−2000 −1200 −400 400 1200 20000481216202428323640

spec

tral p

hase

/ ra

d

retrieved pulse

88 fs

Nonlinear compression (in gas-filled hollow fiber) Fiber (7-cell 3-ring hypocycloid core designed for 1030 nm)mode diameter: 30 µm length: 66 cm filling: 13 bar Argon

Compressed output

Pav = 112 Wτp = 88 fsPpeak = 105 MWfrep = 7 MHzEp = 16 µJMain Peak = 59%

T = 92% T > 95% T > 99.7%

Total equivalent efficiency > 88%Laser input

Pav = up to 127 Wτp = 740 fsPpeak = 21 MWfrep = 7 MHzEp = 18 µJ

20 bounces

≈ - 11 kfs2 of dispersion

F. Emaury et al., Opt. Lett. 39, 6843 (2014)

Fiber compressor for 5.5 fs pulses

SPIDER

Ti:Sa

SLM

G GSM SM

OCDCMsAS ASMF

15 fs, 16 nJ

0.2nJ

SPM broadening in a microstructure fiber (MF), length 5 mm

Ti:sapphire laser with prism pair and DCMs: frep = 19 MHz (for higher pulse energy)

Microstructure fiber (MF):2.6 µm core diameter5 mm longzero GDD at 940 nm

B. Schenkel et al., JOSA B 22, 687, 2005

1.0

0.5

0.0

Inte

nsity

(a. u

.)

1000750500Wavelength (nm)

-600

-400

-200

0

Dis

pers

ion

(ps/

nm/k

m)

Broadband pulse shaper with SLM

Possible bandwidth through Spatial Light Modulator: 400 - 1050 nm

SLM640 pixels

300 l/mm grating 300 l/mm grating

f = 300 mm f = 300 mm

knife-edge

A. M. Weiner, Rev. Sci. Instrum. 71, 1929 (2000)spatial light modulator (640 pixel liquid crystal, each pixel ≈100 µm wide, 3 µm gap)

1.0

0.5

0.0

Inte

rfer

ogra

m


1.0

0.5

0.0

Inte

nsity

(a. u

.)

-40 -20 0 20 40Time (fs)

5.5 fs

• Good fringe visibility: reliable SPIDER measurement

• Microstructure fiber 2.6-µm corediameter, 5 mm long, zero GDDat 940 nm

1.0

0.5

0.0Pow

er d

ensi

ty (a

. u.)


-4

0

4

Spec

tral

pha

se (r

ad)

5.5 fs, 0.2 nJ

Fiber compressor for 5.5 fs pulses

B. Schenkel et al., JOSA B 22, 687, 2005

Hollow-core fiber compression of ~mJ level pulses

(image source: fdglass.com)

(M. Nisoli et al., Appl. Phys. B 65, 189-196 (1997))

• Typical input pulse parameters:~0.5-1 mJ, 20-30 fs (Ti:sapphire amplifiers!)

• Typical parameters of glass capillary:~1 m long, 250 – 300 µm inner diameter

• Nonlinear medium:Rare gases, today mostly Ar and Ne, pressure on the order of 1 bar

• Output pulses in few-fs range, pulse energies of several 100 µJ

Dual stage hollow fiber compressor for 3.8 fs

SPIDER

Ti:Sa Amp

continuum generation

25 fs, 0.5 mJ

100 µJ

15 µJShaper

B. Schenkel et al.: Opt. Lett. 28, 1987 (2003)

Dual stage hollow fiber compressor for 3.8 fs

3.8 fs, 15 µJ

2π phase shift→ pre- and post-pulses

1.0

0.5

0.0Sp

ectr

al P

ow

er D

ensi

ty


-4

-2

0

2

4

Sp

ectr

al P

has

e (r

ad)1.0

0.5

0.0

Inte

rfer

ogra

m


1.0

0.5

0.0

Inte

nsity

-40 -20 0 20 40Time (fs)

3.8 fs

B. Schenkel et al.: Opt. Lett. 28, 1987 (2003)

Choice of compression medium• Available input pulse parameters determine best medium for pulse

compression• Limits peak power: bulk damage due to self-focusing (see next slides)• However, damage in fibers usually occurs at facets first ⇒ Want highest available nonlinearity without damage

Pulse energy pJ-nJ multi-nJ µJ mJ

Peak power <MW MW 10s-100s of MW GW

Nonlinearity few 10-16 cm2/W few 10-16 cm2/W few 10-19 cm2/W few 10-19 cm2/W

Mode size ~µm ~10 µm 10s of µm 100s of µm

Medium Microstructure fiber

Standard fiber Hollow micro-structure fiber

Hollow-core fiber (capillary)

• Not much choice with regards to compression medium• Compression of femtosecond pulses with more than a few mJ is very

challenging

Optical pulse cleaner

Optical pulse cleaner based on nonlinear birefringenceOptics Letters, vol. 17, pp. 136-138, 1992

ULP, Chap. 4, p. 9-10

Self-focusing

Kerr mediumlength LK

n x,y( ) = n + n2 I x,y( )

≅ np − 2Δnpx2 + y2

w2

f = w2

4ΔnpLK

I x,y( ) = I pexp −2 x2 + y2

w2

⎛⎝⎜

⎞⎠⎟

↓ x2 + y2( ) << w2

≈ I p 1− 2 x2 + y2

w2

⎛⎝⎜

⎞⎠⎟

ULP, Chap. 4, p. 11

B-integral

ULP, Chap. 4, p. 10

B ≡ 2πλ

n2 I z( )dz0

L

∫

To prevent material damage: B should be smaller than 3 to 5

Critical power for beam collapse

ULP, Chap. 4, p. 14

Pcr ≡ 3.72λ02 / 8π n0n2

Lc =0.376LDF

Pin / Pcr( )1 2 − 0.852⎡⎣

⎤⎦2 − 0.0219

LDF = π n0w02

λ0Rayleigh length

Lc

Argon at 800 nm (atmospheric pressure):

n0 = 1.0, n2 = 3 10–19 cm2/W, Pcr = 3.2 GW

Fused quartz at 1.06 µm:

n0 = 1.45, n2 = 2.46 10–16 cm2/W, Pcr = 3.8 MW

Filamentation

ULP, Chap. 4, p. 14

Filamentation

During propagation SPM continues to broaden spectrum of pulse ⇒ white light

Filamentation of mJ-level, 30-fs pulses at 800 nm in Ar

Filamentation pulse compression

660 μJ

900 mbar340 μJ

820 mbar

120 μJ

30 fs

10 fs

5 fs

A. Guandalini et al., J. Phys. B 39, S257 (2006)C.P. Hauri et al., Opt. Exp. 13, 7541 (2005)

Comparison with hollow-core fiber

Filamentation• SPM and plasma generation• Self-compression possible• Beam is spatially less

homogeneous (spatio-temporal coupling)

• More robust to beam pointing fluctuations at input

• More instability of beam pointing at output

Hollow-core fiber• SPM is dominant nonlinear

process• Self-compression typically not

possible• Capillary acts as spatial filter,

homogenizes output• Sensitive to in-coupling of beam

0.001

2

46

0.01

2

46

0.1

2

46

1

Spec

trum


Hollow core-fiber Filament

1.0

0.8

0.6

0.4

0.2

0.0

Spec

trum

900800700600500W avelength (nm)

Hollow core-fiber Filament

(L. Gallmann et al., Appl. Phys. B 86, 561 (2007))

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 n2 > 0

• Unchirped sech2 pulse shape, fulfilling the condition

• Remarkable stability of soliton pulses:

particle character in collision

pulse automatically “finds“ the exact required shape (may shed some energy into a background pulse)

τ p = 1.7627 ×4 Dδ ep

= 1.7627 ×2 ′′kn kn2ep

Nonlinear pulse propagation Linear pulse propagation:GDD and no SPM

Nonlinear pulse propagation:no GDD and SPM

′′kn ≠ 0

n2 = 0

′′kn = 0

n2 ≠ 0

ULP, Chap. 4, p. 18

Nonlinear pulse propagation Nonlinear pulse propagation: GDD > 0 and SPM > 0

Nonlinear pulse propagation:Soliton pulsesGDD < 0 and SPM > 0

ULP, Chap. 4, p. 19

′′kn > 0

n2 > 0

n2 > 0

′′kn < 0

Nonlinear Schrödinger Equation (NSE)

ULP, Chap. 4, p. 15

Slowly varying envelope approximation:

Dispersion first order:Linearized operator in the time domain

A Ld ,Δω( ) = e− i kn ω0+Δω( )−kn ω0( )⎡⎣ ⎤⎦Ld A(0,Δω )

kn ω( ) ≅ kn ω0( ) + ′knΔω + 12

′′knΔω2

′kn =∂kn∂ω ω0

′′kn =∂ 2kn∂ω 2

ω0

A Ld ,Δω( ) = exp −i ′knΔω + 12

′′knΔω2⎛

⎝⎜⎞⎠⎟ Ld

⎧⎨⎩

⎫⎬⎭A(0,Δω )

′knΔωLd << 1

A Ld ,Δω( ) = e− i ′knΔωLd A 0,Δω( ) ≅ 1− i ′knΔωLd( ) A 0,Δω( )

F−1 Δω A z,Δω( ){ } = −i ∂

∂tA z,t( )

A Ld ,t( ) ≅ 1− ′kn Ld∂∂t

⎛⎝⎜

⎞⎠⎟ A 0,t( ) , for ′knΔωLd << 1


ULP, Chap. 4, p. 16

Slowly varying envelope approximation:

Dispersion second order:Linearized operator in the time domain

Dispersion parameter D

A Ld ,Δω( ) = e− i kn ω0+Δω( )−kn ω0( )⎡⎣ ⎤⎦Ld A(0,Δω )

kn ω( ) ≅ kn ω0( ) + ′knΔω + 12

′′knΔω2

′kn =∂kn∂ω ω0

′′kn =∂ 2kn∂ω 2

ω0

A Ld ,Δω( ) = exp −i ′knΔω + 12

′′knΔω2⎛

⎝⎜⎞⎠⎟ Ld

⎧⎨⎩

⎫⎬⎭A(0,Δω )

F−1 Δω 2 A z,Δω( ){ } = − ∂ 2

∂t 2A z,t( )

′′knΔω2Ld << 1

A Ld ,Δω( ) = e− i

12

′′knΔω2Ld A 0,Δω( ) ≅ 1− i 1

2′′knΔω

2Ld⎛⎝⎜

⎞⎠⎟A 0,Δω( )

A Ld ,t( ) ≅ 1+ i 12

′′kn Ld∂ 2

∂t 2⎛⎝⎜

⎞⎠⎟

A 0,t( ) ≡ 1+ iD ∂ 2

∂t 2⎛⎝⎜

⎞⎠⎟

A 0,t( ) , for ′′knΔω2Ld << 1D ≡ 1

2′′knLd


ULP, Chap. 4, p. 17

A Ld ,t( ) ≅ 1+ i 12

′′kn Ld∂ 2

∂t 2⎛⎝⎜

⎞⎠⎟

A 0,t( ) ≡ 1+ iD ∂ 2

∂t 2⎛⎝⎜

⎞⎠⎟

A 0,t( ) , for ′′knΔω2Ld << 1

A Ld ,t( ) ≅ 1− ′kn Ld∂∂t

⎛⎝⎜

⎞⎠⎟ A 0,t( ) , for ′knΔωLd << 1 1. Order Dispersion

2. Order Dispersion

∂∂z

A z,t( ) = limLd→0

A Ld ,t( ) − A 0,t( )Ld

≈ − ′kn∂∂tA z,t( ) + i 1

2′′kn∂ 2

∂t 2A z,t( )

∂∂z

A z,t( ) + 1υg

∂∂tA z,t( ) = i ′′kn

2∂ 2

∂t 2A z,t( )

∂∂z

A z, ′t( ) = i ′′kn2

∂ 2

∂ ′t 2A z, ′t( )

′t = t − zυg

retarded time

SPM operator

ULP, Chap. 4, p. 2

E LK ,t( ) = A 0,t( )exp iω0t + iφ t( )⎡⎣ ⎤⎦ = A 0,t( )exp iω0t − ikn ω0( )LK − iδ A t( ) 2⎡⎣

⎤⎦

A LK ,t( ) = e− iδ A 2

A 0,t( )e− ikn ω0( )LK δ A 2<<1⎯ →⎯⎯⎯ ≈ 1− iδ A t( ) 2( )A 0,t( )e− ikn ω0( )LK

SPM Operator (linearized)

δ ≡ kn2LK

δ A t( ) 2 = δ I t( ) << 1


ULP, Chap. 4, p. 17

A Ld ,t( ) ≅ 1+ i 12

′′kn Ld∂ 2

∂t 2⎛⎝⎜

⎞⎠⎟

A 0,t( ) ≡ 1+ iD ∂ 2

∂t 2⎛⎝⎜

⎞⎠⎟

A 0,t( ) , for ′′knΔω2Ld << 1

A Ld ,t( ) ≅ 1− ′kn Ld∂∂t

⎛⎝⎜

⎞⎠⎟ A 0,t( ) , for ′knΔωLd << 1 1. Order Dispersion

2. Order Dispersion

∂∂z


A Ld ,t( ) − A 0,t( )Ld

≈ − ′kn∂∂tA z,t( ) + i 1

2′′kn∂ 2

∂t 2A z,t( )

′t = t − zυg

retarded time

A LK ,t( ) ≈ 1− iδ A t( ) 2( )A 0,t( ) δ ≡ kn2LK

∂∂z


A L,t( ) − A 0,t( )L

≈ − ′kn∂∂tA z,t( ) + i 1

2′′kn∂ 2

∂t 2A z,t( ) − ikn2 A t( ) 2 A z,t( )

+ SPM

∂∂z

A z, ′t( ) = i ′′kn2

∂ 2

∂ ′t 2A z, ′t( ) − ikn2 A z, ′t( ) 2 A z, ′t( ) Nonlinear Schrödinger Equation (NSE)


∂∂z

A z, ′t( ) = i ′′kn2

∂ 2


As z, ′t( ) = A0sech′tτ

⎛⎝⎜

⎞⎠⎟ e− iφ0

τ p = 1.7627 ⋅τ

Δν pτ p = 0.3148

Solution: a fundamental soliton

φ0 =φ2max2

φ2max = kn2 I pz , I p = A02

The pulse as a whole experiences a homogeneousphase shift (not like SPM alone!)

ULP, Chap. 4, p. 20


∂∂z

A z, ′t( ) = i ′′kn2

∂ 2



⎛⎝⎜

⎞⎠⎟ e− iφ0

τ p = 1.7627 ⋅τ

Δν pτ p = 0.3148


φ0 =φ2max2



ULP, Chap. 4, p. 21

τ p = 1.7627 ×4 Dδ ep

= 1.7627 ×2 ′′kn kn2ep

∝ 1ep


∂∂z

A z, ′t( ) = i ′′kn2

∂ 2



⎛⎝⎜

⎞⎠⎟ e− iφ0

τ p = 1.7627 ⋅τ

Δν pτ p = 0.3148


φ0 =φ2max2



ULP, Chap. 4, p. 21

Balance between negative GDD and positive SPM:

φ0 =Dτ 2

= 12δ I p =

δ ep

4τ=

kn2 ep

4τz

δ ≡ kn2LK D ≡ 12

′′knLd

ep =Ep

Aeff= I z, ′t( )∫ d ′t = As z, ′t( ) 2 d ′t∫ = 2 A0

2 τ

τ p = 1.7627 ×4 Dδ ep

= 1.7627 ×2 ′′kn kn2ep

∝ 1ep


∂∂z

A z, ′t( ) = i ′′kn2

∂ 2



ULP, Chap. 4, p. 22

τ p = 1.7627 ×4 Dδ ep

= 1.7627 ×2 ′′kn kn2ep

∝ 1ep

τ p ∝1ep

τ p ∝ ′′kn

Soliton area = A0∫ sech tτ

⎛⎝⎜

⎞⎠⎟ dt = πA0τ “Solitons have constant area”


∂∂z

A z, ′t( ) = i ′′kn2

∂ 2



ULP, Chap. 4, p. 22


∂∂z

A z, ′t( ) = i ′′kn2

∂ 2



ULP, Chap. 4, p. 24


∂∂z

A z, ′t( ) = i ′′kn2

∂ 2


Solution: higher order soliton (example: second order)

ULP, Chap. 4, p. 23

Soliton Period φ0 z = z0( ) = π4

⇒ z0 =π2

τ 2

′′kn

Higher-Order Soliton Pulses• Inject a pulse with N2–times the fundamental soliton energy:

periodically evolving higher-order soliton pulse (N is an integer).

• Initial condition: N = 2 for second-order soliton

• Soliton period:

becomes short for short pulses and strong dispersion

• At certain locations, significantly shorter (but not sech2-shaped) pulses occurimportant for pulse compression

• Note: soliton period is an important parameteralso for fundamental solitons:length scale on which the interaction is significant (periodic perturbation)

φ0 z = z0( ) = π4

⇒ z0 =π2

τ 2

′′kn

A 0, ′t( ) = N A0sech′tτ

⎛⎝⎜

⎞⎠⎟

Optical communication with repeaters

ULP, Chap. 4, p. 25

Optical communication with solitons

ULP, Chap. 4, p. 26

Optical communication with solitons

Δω−Δω 0

ULP, Chap. 4, p. 27

Periodic perturbation of solitons

ULP, Chap. 4, p. 27-31

∂∂z

A z, ′t( ) = i ′′kn2

∂ 2

∂ ′t 2A z, ′t( ) − ikn2 A z, ′t( ) 2 A z, ′t( ) + iξ δ z − nza( )

n=−∞

∞

∑ A z, ′t( )

NSE + periodic perturbation period za

Important for modelocked lasers:

periodic perturbation per round-trip through output coupler, gain crystal …

Periodic perturbation of solitons∂∂z

A z, ′t( ) = i ′′kn2

∂ 2


n=−∞

∞

∑ A z, ′t( )


A z, ′t( ) = As z, ′t( ) + u z, ′t( )


⎛⎝⎜

⎞⎠⎟ e− iφ0

u z, ′t( ) << As z, ′t( )

Assuming small perturbation: AnsatzSolution without perturbationSoliton pulse:

It can be shown that:

with the solution:

∂∂zu z, ′t( ) ≈ i ′′kn

2∂ 2

∂ ′t 2u z, ′t( ) + iξ δ z − nza( )

n=1

∞

∑ As z, ′t( )

u z,ω( ) =1za

ξA0πτsechπ2τω⎛

⎝⎜⎞⎠⎟

nka −′′kn 2

1τ 2

+ω 2⎛⎝⎜

⎞⎠⎟

ei nka−

12τ 2

′′kn ⎛⎝⎜

⎞⎠⎟

z

n=1

∞

∑Resonance effects!

ULP, Chap. 4, p. 27-31


ULP, Chap. 4, p. 27-31

∂∂z

A z, ′t( ) = i ′′kn2

∂ 2


n=−∞

∞

∑ A z, ′t( )


Periodic perturbation has no resonance effects:

In this regime periodic perturbation can be considered as a continuous perturbation.

Soliton period

… the perturbation period has to be made smaller for shorter pulses.

u z,ω( ) << As z,ω( ) ⇔ za << 8z0

z0 =π2

τ 2

′′kn ∝τ 2


ULP, Chap. 4, p. 27-31

∂∂z

A z, ′t( ) = i ′′kn2

∂ 2


n=−∞

∞

∑ A z, ′t( )


Periodic perturbation has no resonance effects:

u z,ω( ) << As z,ω( ) ⇔ za << 8z0

Dλ = 17 pskm ⋅nm

Delayed Nonlinear Response

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

<1 fs (nearly instantaneous) in typical solids• Lattice contribution: excitation of phonons:

• Optical phonons: Raman effect(positive and negative charges oscillating in anti-phase)

• Acoustical phonons: Brillouin effect(positive and negative charges oscillating in phase)

• Beating between different optical frequency componentsexcites phonons.

• Phonons create moving index gratings which can couple light waves with different propagation directions and frequencies. Pump photon can split into Stokes photon and a phonon.

Delayed Raman ResponseOptical phonons have high frequencies (e.g. around

13 THz for silica), only weakly dependent on wave vector:

Consequence: phase matching possible forforward and backward direction:

kpks

kphonon

kp

kphononks

k

Ω optical phonons

acoustical phonons

range of interest

Raman Gain Spectrum of Silica

gain spectrum1000-nm pump

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

Ram

an g

ain

(a. u

.)

1120108010401000

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.)

gR (Δω )

Intra-Pulse Raman Scattering

Principle:• Energy transfer within the pulse spectrum:⇒ center wavelength drifts

towards longer values

• Soliton interaction preserves the pulse shape

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

λ

J. P. Gordon, Opt. Lett. 11 (10), 662 (1986)

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

4

3

2

1

0

-110008006004002000

delay time τ (fs)

Damped oscillation with ≈13 THz:strong contribution, if two optical waveswith ≈13 THz frequency difference beat

R. H. Stolen et al., JOSA B 6 (6), 1159 (1989)

Response function

Generalized NSE

∂A z, ′t( )∂z

− i2

′′kn∂2A z, ′t( )

∂ ′t 2− 16

′′′kn∂3A z, ′t( )

∂ ′t 3

= −iγ A z, ′t( ) 2 A z, ′t( ) − iω0

∂∂ ′t

A z, ′t( ) 2 A z, ′t( )( ) − TRA z, ′t( ) ∂ A z, ′t( )∂ ′t

2⎡

⎣⎢⎢

⎤

⎦⎥⎥

second and third-order dispersion

SPM self-steepening RamanTR sets slope of Raman gain

γ = n2ω0

cAeff

self-frequency shift

intensity dependence of group velocity

shock formation

intensity dependence of phase velocity

Raman and self-steepening lead to asymmetry in SPM broadened spectra

Self-steepening and SPM (without GDD and Raman)

40

30

20

10

0

Pow

er [k

W]

-80 -60 -40 -20 0 20 40 60 80Time [fs]

Example: 50 fs, center wavelength 800 nm, fiber core diameter 1.7 µm and

Self-steeping means that group velocity is intensity dependent: peak moves at a lower speed than the wings

n2 = 2.5 ⋅10−20 m2 / W

Input: Gaussian pulse at z = 0

z = 3 mm (dashed) z = 6 mm

z = 6 mm (solid)

s = 1ω0τ

= T2πτ

= 0.01

16

14

12

10

8

6

4

2

0

Ener

gy/W

avel

engt

h [p

J/nm

]1.11.00.90.80.70.60.50.4

Wavelength [um]

asymmetry in SPM broadened spectrum

Self-steepening, SPM and GDD>0 (no Raman)

14

12

10

8

6

4

2

0

Pow

er [k

W]

-200 -100 0 100 200Time [fs]

16141210

86420

Ener

gy/W

avel

engt

h [p

J/nm

]

1.00.90.80.70.6Wavelength [um]

8

6

4

2

0

Pow

er [k

W]

2000-200Time [fs]

Example: 50 fs, center wavelength 800 nm, fiber core diameter 1.7 µm andGDD 3.5 fs2/100 µm

n2 = 2.5 ⋅10−20 m2 / W

s = 1ω0τ

= T2πτ

= 0.01

14

12

10

8

6

4

2

0Ener

gy/W

avel

engt

h [p

J/nm

]

1.00.90.80.70.6Wavelength [um]

z = 6 mm

z = 12 mm

40

30

20

10

0

Pow

er [k

W]

-80 -60 -40 -20 0 20 40 60 80Time [fs]

16

14

12

10

8

6

4

2

0

Ener

gy/W

avel

engt

h [p

J/nm

]

1.11.00.90.80.70.60.50.4Wavelength [um]

Saturable gain and absorber

Fläche A

Wirkungsquerschnitt σ

Dicke Δz

Dichte der Atome

EinfallendeLichtintensität

g z( ) ≡ NVσ α0 =

N0

Vσ

α = α0

1+ I Isat ,A

σ A = σ L = σg = g0

1+ I Isat ,L

Isat ,L =hνστ L

Assuming: two-level system

Isat ,A = hνστ 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( ) = − 2α0

1+I z( )Isat

I z( )

1I

1+ IIsat

⎛⎝⎜

⎞⎠⎟dI = − 2α0 dz

lnT + IinIsat

T −1( ) = −2α0d

Iin << Isat

T ≈ e−2α0d

T ≡ IoutIin Iin Isat

nonlinear pulse propagation

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