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Soliton self-frequency blueshift in Kagome hollow-core photonic crystal fibers LENCOS 2012, Seville, Spain Fabio Biancalana , Mohammed Saleh , Philipp Hoelzer, KaFai Mak, Francesco Tani, John Travers, Wonkeun Chang and Philip St.J. Russell

slide-1
SLIDE 1

Soliton self-frequency blueshift in Kagome hollow-core photonic crystal fibers

Fabio Biancalana, Mohammed Saleh, Philipp Hoelzer, KaFai Mak, Francesco Tani, John Travers, Wonkeun Chang and Philip St.J. Russell

Max Planck Institute for the Science of Light Erlangen, Germany

mpl.mpg.de/mpf/php/abteilung3/jrg/

fabio.biancalana@mpl.mpg.de

LENCOS 2012, Seville, Spain

Tuesday, July 10, 12

slide-2
SLIDE 2

Outline

Hollow-core photonic crystal fibers (Kagome) New set of equations for light propagation in a photoionizable gas Numerical and analytical results: soliton self-frequency blueshift, spectral and temporal clustering, asymmetric SPM, universal modulational instability and ‘blue’ solitonic shower Comparison with experimental results performed at MPL

Tuesday, July 10, 12

slide-3
SLIDE 3

Photonic Crystal Fibers - PCFs

Solid-core PCF: All-silica fibers with an arrangement of air holes. Can be designed to be endlessly single mode Unprecedented control of the dispersion!

Tuesday, July 10, 12

slide-4
SLIDE 4

Raman self-frequency redshift

GNLSE&

Raman effect continuously pushes solitons to the red

Skryabin et al., Science 301, 5640 (2003). Mitschke and Mollenauer, Opt. Lett. (1986).

Tuesday, July 10, 12

slide-5
SLIDE 5

Hollow-core PCF

Photonic(bandgap(mechanism,(guides( light(in(the(central(hole!( Possibility(to(explore(light8ma9er( interac:ons(

  • Tweezers(
  • Sensing(
  • Gas8laser(interact.(

P.StJ.Russell,(1999(

Tuesday, July 10, 12

slide-6
SLIDE 6

Photonic-bandgap HC-PCFs

  • Low loss (ca. 1 dB/km)
  • Anomalous dispersion

Used for many experiments: low-threshold Raman scattering, particle guidance experiments, nanoliter chemistry

  • F. Benabid, J. C. Knight, G. Antonopoulos, and
  • P. St.J. Russell, Science 298, 399 (2002).
  • P. St.J. Russell, Science 299, 358 (2003).

Tuesday, July 10, 12

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SLIDE 7

Photonic-bandgap HC-PCFs

  • Low loss (ca. 1 dB/km)
  • Anomalous dispersion
  • Narrow guidance

(in the transverse bandgap)

  • Large variations of

the GVD

  • P. St.J. Russell, Science 299, 358 (2003).

Tuesday, July 10, 12

slide-8
SLIDE 8

Photonic-bandgap HC-PCFs

  • Low loss (ca. 1 dB/km)
  • Anomalous dispersion
  • Narrow guidance

(in the transverse bandgap)

  • Large variations of

the GVD

  • P. St.J. Russell, Science 299, 358 (2003).

Tuesday, July 10, 12

slide-9
SLIDE 9

Kagome HC-PCFs

  • High but tolerable loss

(ca. 1 dB/m)

  • Anomalous dispersion
  • P. St.J. Russell, J. Lightwave Technol. 24, 4729 (2006).

Tuesday, July 10, 12

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SLIDE 10

Kagome HC-PCFs

  • High but tolerable loss

(ca. 1 dB/m)

  • Anomalous dispersion
  • Very broad

guidance

  • Small GVD and

low dispersion slope

  • P. St.J. Russell, J. Lightwave Technol. 24, 4729 (2006).

Tuesday, July 10, 12

slide-11
SLIDE 11

Kagome HC-PCFs

  • High but tolerable loss

(ca. 1 dB/m)

  • Anomalous dispersion
  • Very broad

guidance

  • Small GVD and

low dispersion slope

  • P. St.J. Russell, J. Lightwave Technol. 24, 4729 (2006).

Tuesday, July 10, 12

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SLIDE 12

The hollow-core can be filled with gases and liquids Ideal for exploring extremely nonlinear light-matter interactions Filling with noble gases kills completely the Raman effect

Filling the hollow-core

Typical intensities: hundreds of TW/cm^2

Tuesday, July 10, 12

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SLIDE 13

Kagome dispersion

10 µm radius 0.1 bar 20 bar Argon

Tuesday, July 10, 12

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SLIDE 14

Kagome dispersion

10 µm radius 0.1 bar 20 bar Argon

nmn(λ, p, T) = r n2

gas − u2 mn

k2a2

Almost the same dispersion of a capillary, but with much smaller losses!

Marcatili & Schmeltzer, Bell Syst. Tech. J. 43, 1783 (1964).

Tuesday, July 10, 12

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SLIDE 15

Photoionization

E x

  • E

x c

  • s

ω t

multiphoton tunneling

a) b)

  • Ex cosωt

Keldysh parameter of typical experiments: about 1

γK << 1 γK >> 1

Tuesday, July 10, 12

slide-16
SLIDE 16

Photoionization

E x

  • E

x c

  • s

ω t

multiphoton tunneling

a) b)

  • Ex cosωt

Keldysh parameter of typical experiments is > 1

γK << 1 γK >> 1

D

  • m

i n a n t e f f e c t

Tuesday, July 10, 12

slide-17
SLIDE 17

Photoionization dynamics

The pulse leaves a tail of free electrons behind

time [fs] ionization fraction [%]

15 10 5

  • 5
  • 10
  • 15
  • 30
  • 20
  • 10

10 20 30

Ionization threshold

Ne ∼ 1017cm−2

10 fs pulse 1 bar argon 10^15 W/cm^2

Tuesday, July 10, 12

slide-18
SLIDE 18

Photoionization dynamics

The pulse leaves a tail of free electrons behind

time [fs] ionization fraction [%]

15 10 5

  • 5
  • 10
  • 15
  • 30
  • 20
  • 10

10 20 30

Ionization threshold

I n t r a p u l s e d y n a m i c s

10 fs pulse 1 bar argon 10^15 W/cm^2

Tuesday, July 10, 12

slide-19
SLIDE 19

intensity [TW/cm2]

200 150 100 50 2 4 6 8 10

time [fs]

5

  • 5

refractive index change [10-5]

Refractive index change

Plasma decreases the refractive index inside the pulse n ¼ n2I !2

p=ð2n0!2

  • Þ;

0 ¼

!p ¼ ½e2ne=ð0meÞ1=2

Tuesday, July 10, 12

slide-20
SLIDE 20

Photoionization rate

W(I) = d (IH/I)1/4 exp[°b (IH/I)1/2],

∂Ne ∂t = W(I)(NT − Ne) − rN 2

e

time [fs] ionization fraction [%]

15 10 5

  • 5
  • 10
  • 15
  • 30
  • 20
  • 10

10 20 30

  • P. Sprangle, J. R. Penano, and B. Hafizi, Phys. Rev. E 66, 046418 (2002).
  • M. Ammosov, N. Delone, and V. Krainov, Sov. Phys. JETP 64, 1191 (1986).
  • G. L. Yudin and M. Y. Ivanov, Phys. Rev. A, 64, 013409 (2001).

Full field ionization

Tuesday, July 10, 12

slide-21
SLIDE 21

Photoionization rate

W(I) = d (IH/I)1/4 exp[°b (IH/I)1/2],

∂Ne ∂t = W(I)(NT − Ne) − rN 2

e

time [fs] ionization fraction [%]

15 10 5

  • 5
  • 10
  • 15
  • 30
  • 20
  • 10

10 20 30

  • P. Sprangle, J. R. Penano, and B. Hafizi, Phys. Rev. E 66, 046418 (2002).
  • M. Ammosov, N. Delone, and V. Krainov, Sov. Phys. JETP 64, 1191 (1986).
  • G. L. Yudin and M. Y. Ivanov, Phys. Rev. A, 64, 013409 (2001).

Full field ionization

Tuesday, July 10, 12

slide-22
SLIDE 22

Photoionization rate

W(I) = d (IH/I)1/4 exp[°b (IH/I)1/2],

∂Ne ∂t = W(I)(NT − Ne) − rN 2

e

time [fs] ionization fraction [%]

15 10 5

  • 5
  • 10
  • 15
  • 30
  • 20
  • 10

10 20 30

W(I) ≈ ˜ σ (I − Ith) Θ (I − Ith)

  • P. Sprangle, J. R. Penano, and B. Hafizi, Phys. Rev. E 66, 046418 (2002).
  • M. Ammosov, N. Delone, and V. Krainov, Sov. Phys. JETP 64, 1191 (1986).

Full field ionization

  • G. L. Yudin and M. Y. Ivanov, Phys. Rev. A, 64, 013409 (2001).

Tuesday, July 10, 12

slide-23
SLIDE 23

Photoionization rate

W(I) = d (IH/I)1/4 exp[°b (IH/I)1/2],

∂Ne ∂t = W(I)(NT − Ne) − rN 2

e

time [fs] ionization fraction [%]

15 10 5

  • 5
  • 10
  • 15
  • 30
  • 20
  • 10

10 20 30

W(I) ≈ ˜ σ (I − Ith) Θ (I − Ith)

  • P. Sprangle, J. R. Penano, and B. Hafizi, Phys. Rev. E 66, 046418 (2002).
  • M. Ammosov, N. Delone, and V. Krainov, Sov. Phys. JETP 64, 1191 (1986).

Full field ionization

  • G. L. Yudin and M. Y. Ivanov, Phys. Rev. A, 64, 013409 (2001).

Tuesday, July 10, 12

slide-24
SLIDE 24

" i@z + ˆ D(i@t) + ∞KR(t) ⊗ |Ψ(t)|2 − !2

p

2k0c2 + iÆ # Ψ = 0 @tne = [˜ æ/Aeff] [nT − ne] ∆|Ψ|2 Θ ° ∆|Ψ|2¢

  • M. F. Saleh, W. Chang, P. Hölzer, A. Nazarkin, J. C. Travers, N. Y. Joly, P. St.J. Russell, and F. Biancalana, Phys. Rev. Lett.

Our new envelope model

Tuesday, July 10, 12

slide-25
SLIDE 25

" i@z + ˆ D(i@t) + ∞KR(t) ⊗ |Ψ(t)|2 − !2

p

2k0c2 + iÆ # Ψ = 0 @tne = [˜ æ/Aeff] [nT − ne] ∆|Ψ|2 Θ ° ∆|Ψ|2¢

Solvable by using the split-step Fourier method! No full field is required,

  • nly the envelope
  • M. F. Saleh, W. Chang, P. Hölzer, A. Nazarkin, J. C. Travers, N. Y. Joly, P. St.J. Russell, and F. Biancalana, Phys. Rev. Lett.

Our new envelope model

Tuesday, July 10, 12

slide-26
SLIDE 26

" i@z + ˆ D(i@t) + ∞KR(t) ⊗ |Ψ(t)|2 − !2

p

2k0c2 + iÆ # Ψ = 0 @tne = [˜ æ/Aeff] [nT − ne] ∆|Ψ|2 Θ ° ∆|Ψ|2¢

z the longitudinal coordinate alon y, ˆ D(i@t) ≡ P

m∏2 Øm(i@t)m/m! i

at an arbitrary reference frequenc

Solvable by using the split-step Fourier method! No full field is required,

  • nly the envelope
  • M. F. Saleh, W. Chang, P. Hölzer, A. Nazarkin, J. C. Travers, N. Y. Joly, P. St.J. Russell, and F. Biancalana, Phys. Rev. Lett.

Our new envelope model

Tuesday, July 10, 12

slide-27
SLIDE 27

" i@z + ˆ D(i@t) + ∞KR(t) ⊗ |Ψ(t)|2 − !2

p

2k0c2 + iÆ # Ψ = 0 @tne = [˜ æ/Aeff] [nT − ne] ∆|Ψ|2 Θ ° ∆|Ψ|2¢

z the longitudinal coordinate alon y, ˆ D(i@t) ≡ P

m∏2 Øm(i@t)m/m! i

at an arbitrary reference frequenc

⊗ y, !p = [e2ne/(≤0me)]1/2

  • n charge and mass, and

Solvable by using the split-step Fourier method! No full field is required,

  • nly the envelope
  • M. F. Saleh, W. Chang, P. Hölzer, A. Nazarkin, J. C. Travers, N. Y. Joly, P. St.J. Russell, and F. Biancalana, Phys. Rev. Lett.

Our new envelope model

Tuesday, July 10, 12

slide-28
SLIDE 28

" i@z + ˆ D(i@t) + ∞KR(t) ⊗ |Ψ(t)|2 − !2

p

2k0c2 + iÆ # Ψ = 0 @tne = [˜ æ/Aeff] [nT − ne] ∆|Ψ|2 Θ ° ∆|Ψ|2¢

z the longitudinal coordinate alon y, ˆ D(i@t) ≡ P

m∏2 Øm(i@t)m/m! i

at an arbitrary reference frequenc

⊗ y, !p = [e2ne/(≤0me)]1/2

  • n charge and mass, and

s, Æ2 = AeffUI

2|Ψ|2 @tne

= , Ψ =

Solvable by using the split-step Fourier method! No full field is required,

  • nly the envelope
  • M. F. Saleh, W. Chang, P. Hölzer, A. Nazarkin, J. C. Travers, N. Y. Joly, P. St.J. Russell, and F. Biancalana, Phys. Rev. Lett.

Our new envelope model

Tuesday, July 10, 12

slide-29
SLIDE 29

" i@z + ˆ D(i@t) + ∞KR(t) ⊗ |Ψ(t)|2 − !2

p

2k0c2 + iÆ # Ψ = 0 @tne = [˜ æ/Aeff] [nT − ne] ∆|Ψ|2 Θ ° ∆|Ψ|2¢

z the longitudinal coordinate alon y, ˆ D(i@t) ≡ P

m∏2 Øm(i@t)m/m! i

at an arbitrary reference frequenc

⊗ y, !p = [e2ne/(≤0me)]1/2

  • n charge and mass, and

s, Æ2 = AeffUI

2|Ψ|2 @tne

= , Ψ =

a, ∆|Ψ|2 = |Ψ|2 − |Ψ|2

th,

he fiber, associated with

Solvable by using the split-step Fourier method! No full field is required,

  • nly the envelope
  • M. F. Saleh, W. Chang, P. Hölzer, A. Nazarkin, J. C. Travers, N. Y. Joly, P. St.J. Russell, and F. Biancalana, Phys. Rev. Lett.

Our new envelope model

Tuesday, July 10, 12

slide-30
SLIDE 30

Our new envelope model

" i@z + ˆ D(i@t) + ∞KR(t) ⊗ |Ψ(t)|2 − !2

p

2k0c2 + iÆ # Ψ = 0 @tne = [˜ æ/Aeff] [nT − ne] ∆|Ψ|2 Θ ° ∆|Ψ|2¢

z the longitudinal coordinate alon y, ˆ D(i@t) ≡ P

m∏2 Øm(i@t)m/m! i

at an arbitrary reference frequenc

⊗ y, !p = [e2ne/(≤0me)]1/2

  • n charge and mass, and

s, Æ2 = AeffUI

2|Ψ|2 @tne

= , Ψ =

a, ∆|Ψ|2 = |Ψ|2 − |Ψ|2

th,

he fiber, associated with

, R(t) = (1 − Ω)±(t) + Ω h(t) a function, is the relative str

Solvable by using the split-step Fourier method! No full field is required,

  • nly the envelope
  • M. F. Saleh, W. Chang, P. Hölzer, A. Nazarkin, J. C. Travers, N. Y. Joly, P. St.J. Russell, and F. Biancalana, Phys. Rev. Lett.

Tuesday, July 10, 12

slide-31
SLIDE 31

Our new envelope model

" i@z + ˆ D(i@t) + ∞KR(t) ⊗ |Ψ(t)|2 − !2

p

2k0c2 + iÆ # Ψ = 0 @tne = [˜ æ/Aeff] [nT − ne] ∆|Ψ|2 Θ ° ∆|Ψ|2¢

z the longitudinal coordinate alon y, ˆ D(i@t) ≡ P

m∏2 Øm(i@t)m/m! i

at an arbitrary reference frequenc

⊗ y, !p = [e2ne/(≤0me)]1/2

  • n charge and mass, and

s, Æ2 = AeffUI

2|Ψ|2 @tne

= , Ψ =

a, ∆|Ψ|2 = |Ψ|2 − |Ψ|2

th,

he fiber, associated with

, R(t) = (1 − Ω)±(t) + Ω h(t) a function, is the relative str

  • M. F. Saleh, W. Chang, P. Hölzer, A. Nazarkin, J. C. Travers, N. Y. Joly, P. St.J. Russell, and F. Biancalana, Phys. Rev. Lett.

Θ(x) = Heaviside function

Tuesday, July 10, 12

slide-32
SLIDE 32

Long pulse approx.

i∂ξψ + ˆ D(i∂τ)ψ + |ψ|2ψ ° τRψ∂τ|ψ|2 ° ηψ Z τ

°1

|ψ|2dτ 0 = 0

Tuesday, July 10, 12

slide-33
SLIDE 33

Long pulse approx.

i∂ξψ + ˆ D(i∂τ)ψ + |ψ|2ψ ° τRψ∂τ|ψ|2 ° ηψ Z τ

°1

|ψ|2dτ 0 = 0 (⇠, ⌧) =

2 τ

| | A(⇠) sech [A(⇠)(⌧ ⌧p(⇠))] eiδ(ξ)τ, w

@A @⇠ = 2 T

  • A tanh # | |2

th T

  • @

@⇠ = ⌘ A2 2 3 tanh3 # + | |2

th

A2

  • # sech2# tanh #
  • (14)

# = AT.

Tuesday, July 10, 12

slide-34
SLIDE 34

i∂ξψ + ˆ D(i∂τ)ψ + |ψ|2ψ ° τRψ∂τ|ψ|2 ° ηψ Z τ

°1

|ψ|2dτ 0 = 0

Long pulse approx.

Tuesday, July 10, 12

slide-35
SLIDE 35

Solitons propagate just above the threshold

  • F. M. Mitschke and L. F. Mollenauer,
  • Opt. Lett. 11, 659 (1986).

i∂ξψ + ˆ D(i∂τ)ψ + |ψ|2ψ ° τRψ∂τ|ψ|2 ° ηψ Z τ

°1

|ψ|2dτ 0 = 0

R d g = gred + gblue, w

i

  • e gred = +(8/15)⌧RA4

d gblue = (2/3)⌘A2

Long pulse approx.

  • M. F. Saleh et al,, Phys. Rev. Lett.

Tuesday, July 10, 12

slide-36
SLIDE 36

Solitons propagate just above the threshold

i∂ξψ + ˆ D(i∂τ)ψ + |ψ|2ψ ° τRψ∂τ|ψ|2 ° ηψ Z τ

°1

|ψ|2dτ 0 = 0

Long pulse approx.

  • F. M. Mitschke and L. F. Mollenauer,
  • Opt. Lett. 11, 659 (1986).

R d g = gred + gblue, w

i

  • e gred = +(8/15)⌧RA4

d gblue = (2/3)⌘A2

E x a c t l y

  • p

p

  • s

i t e t

  • t

h e R a m a n e f f e c t !

  • M. F. Saleh et al,, Phys. Rev. Lett.

Tuesday, July 10, 12

slide-37
SLIDE 37

Soliton self-frequency blueshift

  • M. F. Saleh, W. Chang, P. Hoelzer, A. Nazarkin, J. C. Travers, N. Y. Joly, P. St.J. Russell, and F. Biancalana, Phys. Rev. Lett.

Tuesday, July 10, 12

slide-38
SLIDE 38

‘Floating’ or ‘iceberg’ solitons

Tuesday, July 10, 12

slide-39
SLIDE 39

‘Floating’ or ‘iceberg’ solitons

Tuesday, July 10, 12

slide-40
SLIDE 40

Experimental demonstration

Input pulse duration: 65 fsec

20 µm

  • P. Hoelzer, W. Chang, J. C. Travers, A. Nazarkin, J. Nold, N. Y. Joly, M. F. Saleh, F. Biancalana, and P. St.J. Russell, Phys. Rev. Lett.

Tuesday, July 10, 12

slide-41
SLIDE 41

Experimental demonstration

  • P. Hoelzer, W. Chang, J. C. Travers, A. Nazarkin, J. Nold, N. Y. Joly, M. F. Saleh, F. Biancalana, and P. St.J. Russell, Phys. Rev. Lett.

Input pulse duration: 65 fsec

Tuesday, July 10, 12

slide-42
SLIDE 42

Experimental demonstration

  • P. Hoelzer, W. Chang, J. C. Travers, A. Nazarkin, J. Nold, N. Y. Joly, M. F. Saleh, F. Biancalana, and P. St.J. Russell, Phys. Rev. Lett.

Input pulse duration: 65 fsec

Tuesday, July 10, 12

slide-43
SLIDE 43

Temporal and spectral clustering

  • M. F. Saleh, W. Chang, P. Hoelzer, A. Nazarkin, J. C. Travers, N. Y. Joly, P. St.J. Russell, and F. Biancalana, Phys. Rev. Lett.

time [fs] ionization fraction [%]

15 10 5

  • 5
  • 10
  • 15
  • 30
  • 20
  • 10

10 20 30

Peculiar nonlocal intersolitonic forces are present due to the plasma tail

Tuesday, July 10, 12

slide-44
SLIDE 44

Temporal and spectral clustering

Tuesday, July 10, 12

slide-45
SLIDE 45

Universal modulational instability

Input pulses > 500 fs

threshold

Tuesday, July 10, 12

slide-46
SLIDE 46

Asymmetric SPM and MI

No energy goes into the red part of the spectrum!

Tuesday, July 10, 12

slide-47
SLIDE 47

XFROG for long input pulses

Tuesday, July 10, 12

slide-48
SLIDE 48

XFROG for long input pulses

N

  • e

n e r g y g

  • e

s i n t

  • t

h e r e d !

Tuesday, July 10, 12

slide-49
SLIDE 49

References

Bright Spatially Coherent Wavelength-Tunable Deep-UV Laser Source Using an Ar-Filled Photonic Crystal Fiber

  • N. Y. Joly,2,1 J. Nold,2 W. Chang,1 P. Ho

¨lzer,1 A. Nazarkin,1 G. K. L. Wong,1 F. Biancalana,1 and P. St. J. Russell1,2

1Max Planck Institute for the Science of Light, Gu

¨nther-Scharowsky-Strasse 1/Bau 24, 91058 Erlangen, Germany

2Department of Physics, University of Erlangen-Nuremberg, Germany

(Received 12 October 2010; published 16 May 2011)

PRL 106, 203901 (2011) P H Y S I C A L R E V I E W L E T T E R S

week ending 20 MAY 2011

Theory of Photoionization-Induced Blueshift of Ultrashort Solitons in Gas-Filled Hollow-Core Photonic Crystal Fibers

Mohammed F. Saleh,1 Wonkeun Chang,1 Philipp Ho ¨lzer,1 Alexander Nazarkin,1 John C. Travers,1 Nicolas Y. Joly,1,2 Philip St. J. Russell,1,2 and Fabio Biancalana1

1Max Planck Institute for the Science of Light, Gu

¨nther-Scharowsky Strasse 1, 91058 Erlangen, Germany

2Department of Physics, University of Erlangen-Nuremberg, 91054 Erlangen, Germany

(Received 27 June 2011; published 7 November 2011)

PRL 107, 203902 (2011) P H Y S I C A L R E V I E W L E T T E R S

week ending 11 NOVEMBER 2011

Femtosecond Nonlinear Fiber Optics in the Ionization Regime

  • P. Ho

¨lzer,1 W. Chang,1 J. C. Travers,1 A. Nazarkin,1 J. Nold,1 N. Y. Joly,2,1 M. F. Saleh,1

  • F. Biancalana,1 and P. St. J. Russell1,2

1Max Planck Institute for the Science of Light, Gu

¨nther-Scharowsky-Strasse 1, 91058 Erlangen, Germany

2Department of Physics, University of Erlangen-Nuremberg, 91054 Erlangen, Germany

(Received 27 June 2011; published 7 November 2011)

PRL 107, 203901 (2011) P H Y S I C A L R E V I E W L E T T E R S

week ending 11 NOVEMBER 2011

  • M. Saleh et al, arxiv 1204:1892 (2012) ... to be published on PRL

Tuesday, July 10, 12

slide-50
SLIDE 50

Conclusions

A brand new (and rich!) area of nonlinear fiber

  • ptics is now accessible to experiments and theory

Envelope equations for light propagation in ionizable media: the ‘floating’ soliton is a key concept New effects explained: soliton self-frequency blueshift, spectral and temporal clustering, asymmetric SPM, universal modulational instability and solitonic shower New ways to push SCG into the blue

Tuesday, July 10, 12

slide-51
SLIDE 51

Thank you for your attention!

http://mpl.mpg.de/mpf/php/abteilung3/jrg/home/

Funding:

fabio.biancalana@mpl.mpg.de

Tuesday, July 10, 12

slide-52
SLIDE 52

Extra Slides

Tuesday, July 10, 12

slide-53
SLIDE 53

Gaeta’ s paper

VOLUME 84, NUMBER 16 P H Y S I C A L R E V I E W L E T T E R S 17 APRIL 2000

Catastrophic Collapse of Ultrashort Pulses

Alexander L. Gaeta

School of Applied and Engineering Physics, Cornell University, Ithaca, New York 14853

(Received 28 October 1999)

≠u ≠z i 4 µ 1 1 i vtp ≠ ≠t ∂21 =2

u 2 i Ldf

Lds ≠2u ≠t2 1 i µ 1 1 i vtp ≠ ≠t ∂ pnl ,

pnl Ldf Lnl juj2u 2 Ldf Lpl 1 2 ivtcru 1 i Ldf Lmp juj2m21u , where is the nonlinear length,

≠r ≠t arjuj2 1 juj2m, is the avalanche

Tuesday, July 10, 12

slide-54
SLIDE 54

MI

i∂ξa + s 2 ∂2

τa + ψ2 0 (a + a∗) − uψ0 = 0,

∂τu = −Fu + g (a + a∗) ,

κ = ±|Ω| 2

  • Ω2 − 4s
  • ψ2

0 − g ψ0 (F + i Ω)

F 2 + Ω2 1/2

Tuesday, July 10, 12