Palaiseau - FRANCE A nonlinear envelope …reseau-femto.cnrs.fr/IMG/pdf/ARNOLD.pdf UMR 7639 A nonlinear envelope equation for ultrashort pulse propagation in guided structures. Comparison

http://loa.ensta.fr/ UMR 7639

A nonlinear envelope equation for ultrashort pulse

propagation in guided structures.

Comparison with experimental results


C.L. Arnold1, S. Akturk1,2, B. Zhou1, M. Franco1,A.Couairon3 and A. Mysyrowicz1

1 Laboratoire d’Optique Appliquée, École Nationale Supérieure des Techniques Avancées - École Polytechnique, Palaiseau, France

2 Department of Physics, Istanbul Technical University, Maslak 34469 Istanbul, Turkey

3 Centre de Physique Théorique, CNRS, École Polytechnique, Palaiseau, France

[email protected]

Palaiseau - FRANCE


Generation of near single-cycle laser pulses

Typical CPA lasers provide high energy pulses of >20fs duration. However, shorter

pulses are preferable for many applications.

Negativedispersion

Hollow capillary

measurement

spectrum

temporal profile

pulse compression with hollow fibers

Gas cell

Negativedispersion

measurement

pulse compression via filamentation

spectrum

temporal profileGas cell


Limitations of these schemes

• Hollow Core Fibers

• Filamentation

5fs

mJ1

<≈

τE

5fs

mJ1

<≈

τE

Limited by ionization and damage of the fiber, in particular at high repetition rate. The scheme can be extended to an output of some mJ by using pressure gradients.

Limited by multiple filamentationand intensity clamping.


Pulse compression in planar hollow waveguides

The nonguiding direction is used to adjust the intensity to below

the threshold for nonlinear photoionization.

Modeling supports compression of pulses at the 100mJ level

down to the few-cycle regime.

side view

top view

glass slabs (separation 127µm)cylindrical lens (or mirror)

M. Nurhuda et al., Phys. Rev. Lett. 97, 153902 (2006)


Experimental Setup

fused silica slabs(19x6x210 mm)

Ein < 15 mJ, Τin< 50 fs

cylindrical mirror (f=1.3m)

cylindrical mirror (f=1.3m)

argon, krypton or xenon

Chirped mirrors

Diagnostics:

FROGSpectrometerEnergy-meterPhase sensor


Nonlinear Helmholtz Equation

• Second order equation

• Small step size

• Computationally expensive

Bidirectional propagation


Nonlinear unidirectional envelope equation

Nonlinear Contributions:

- Self-focusing

- Plasma defocusing

- Nonlinear losses

- Shock terms

Nonlinear Contributions:

- Self-focusing

- Plasma defocusing

- Nonlinear losses

- Shock terms

Linear Contributions:

- Diffraction

- Dispersion

- Linear losses

- Space-time focusing


- Diffraction

- Dispersion

- Linear losses


Wavenumber in propagation direction

Transverse wavenumber

Unidirectional propagation


Glass plates

Guided dimension

Free dimension

Waveguide coordinate system

• The guided modes satisfy the Helmholtz equation.

• We assume vanishing fields on the glass surfaces.


-1.0 -0.5 0.0 0.5 1.0-1.0

-0.5

0.0

0.5

1.0

Nor

mal

ized

Fie

ld

Position x/a

p=1 p=2 p=3 p=4

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

Nor

mal

ized

Pow

er

Waveguide Length (m)

p=1 p=2 p=3 p=4

Shape of guided modes Attenuation TE-modes

Basic waveguide features


Nonlinear

Contributions:

- Self-focusing

- Plasma

defocusing

- Nonlinear losses

- Shock terms

Nonlinear

Contributions:

- Self-focusing

- Plasma

defocusing

- Nonlinear losses

- Shock terms


- Mode selective dispersion

-Mode selective Absorption

- Diffraction in free dimension



- Mode selective dispersion

-Mode selective Absorption

- Diffraction in free dimension


Nonlinear envelope equation for waveguides

Mode projection

Field reconstruction


Mode-Dynamics


Experiment Modeling

Parameters:8 mJ output energy127 µm waveguide separation7.2 mm transverse beam size1.5 atm argon pressure3.35x1013 Wcm-2 Intensity

Comparison

CL Arnold, S Akturk et al., accepted to Opt. Express


Output spatial mode quality

linear propagation regime

argon krypton xenon

S Akturk, CL Arnold et al., Opt. Lett. 34(9):1462 (2009)


The maximum spectral broadening and thus the shortest

possible pulse duration after compression is mostly

limited by the transverse dynamics.

• Spectral broadening

• Shortest duration

• Energy scalability

• Transverse stability

• Transverse homogeniety

• Transverse phase

• Focusability

Things to compare between experiment and

simulations

z

y

x


Propagation stability vs. energy scalability

The propagation is determined by the intensity,the pressure, and the length of the waveguide.

Stable Propagation, if:-Self-focusing length > waveguide-Length for mod. Instab. > waveguide

pzInkB 20≈

CL Arnold, S Akturk et al., accepted to Opt. Express


Numerical predictions


Transverse homogeneity

-10 -8 -6 -4 -2 0 2 4 6 8 106

8

10

12

14

16

920

880

840

800

760

720

Wav

elen

gth

(nm

)

0

0.01800

0.03600

0.05400

0.07200

0.09000

0.1080

0.1260

0.1440

0.1620

0.1800

SpectralIntensity (a.u.)

-12

-8

-4

0

4

8

12

Gui

ded

Dire

ctio

n (m

m)

Pul

se D

urat

ion

(FW

HM

) (f

s)

Free Dimension (mm)

-40 -20 0 20 40

0.0

0.2

0.4

0.6

0.8

1.0

Nor

mal

ized

Inte

nsity

Time (fs)

-6-4-20246

Pha

se (

rad)

Experimental Numerical


Free Dim. (a.u.) Guided Dim. (a.u.)

Pha

se S

hift

(λ)

20%

Pha

se S

hift

(λ)


40%

Pha

se S

hift

(λ)


60%P

hase

Shi

ft (λ

)


80%P

hase

Shi

ft (λ

)


100%

Transverse Phase


Guided Dimension Free Dimension

-4 -3 -2 -1 0 1 20

100

200

300

400 2σ (Vacuum) 2σ (800 mbar) 2σ (1200 mbar) 2σ (1600 mbar)

Siz

e (µ

m)

Position optical axis (cm)

-4 -3 -2 -1 0 1 20

100

200

300

400

Siz

e (µ

m)

Position optical axis (cm)

2σ (Vacuum) 2σ (800 mbar) 2σ (1200 mbar) 2σ (1600 mbar)

Astigmatism and Focusability


-200-100

0100200

Fre

e ax

is (

µm) z=95cm z=96cm z=97cm

-200-100

0100200

z=98cm

Fre

e ax

is (

µm) z=99cm

-50 -25 0 25 50 Time (fs)

z=100cm

-50 -25 0 25 50

-200-100

0100200

z=101cm

Time (fs)

Fre

e ax

is (

µm)

-50 -25 0 25 50

z=102cm

Time (fs)90 95 100 105 110

-200-100

0100200

Position opt. axis (cm)

Fre

e ax

is (

µm)

-200-100

0100200

Fre

e ax

is (


-200-100

0100200

z=98cm

Fre

e ax

is (

µm) z=99cm

-50 -25 0 25 50 Time (fs)

z=100cm

-50 -25 0 25 50

-200-100

0100200

z=101cm

Time (fs)F

ree

axis

(µm

)

-50 -25 0 25 50

z=102cm

Time (fs)90 95 100 105 110

-200-100

0100200


Fre

e ax

is (

µm)

Numerical analysis of focusability.

Impact of adaptive optics.

Uncorrected Corrected (adaptive optics)


-50 -25 0 25 500.0

0.2

0.4

0.6

0.8

1.0N

orm

aliz

ed In

tens

ity

Time (fs)

-4

-2

0

2

4

6

Pha

se (

rad)

10.0 fs

Pathway to relativistic intensity

dy

dx

Parameters:10.6 mJ Output energy10.1+-0.33 fs Compressed duration127 µm Waveguide separation7.2 mm Transverse beam size1.5 atm Argon pressure>80% Throughput

Rough estimation

The focal spot size is nonuniform due to the different focusabilities in x- and y- direction.


- Planar hollow waveguides are well suited for the compression of high

power ultrashort laser pulses.

- The NEE model presented, nicely reproduces experimental pulse

compression and the essential waveguide dynamics

- Transverse dynamics (small-scale self-focusing) determines the limit

of the method.

- Stable propagation can be achieved by a proper combination of

parameters, such as intensity, gas, pressure, separation, and length

of the waveguide.

- The energy can be up-scaled by using the stable propagation criterion.

- Relativistic intensity is possible with relatively simple table-top lasers.

Conclusions


[email protected]


-10 -8 -6 -4 -2 0 2 4 6 8 106

8

10

12

14

16

920

880

840

800

760

720

Wav

elen

gth

(nm

)0

0.01800

0.03600

0.05400

0.07200

0.09000

0.1080

0.1260

0.1440

0.1620

0.1800

SpectralIntensity (a.u.)

-12

-8

-4

0

4

8

12

Gui

ded

Dire

ctio

n (m

m)

Pul

se D

urat

ion

(FW

HM

) (f

s)

Free Dimension (mm)

-40 -20 0 20 40

0.0

0.2

0.4

0.6

0.8

1.0

Nor

mal

ized

Inte

nsity

Time (fs)

-6-4-20246

Pha

se (

rad)

(a)

(b)

(c)(d)


-8

-4

0

4

8

Fre

e D

im. (

mm

)

Fluence / Jcm-2

0.0 0.1 0.2 0.3 0.4 0.5-8

-4

0

4

8

Propagation Direction z (m)

Fre

e D

im. (

mm

)

0

0.4000

0.8000

1.200

1.600

2.000

0.0 0.1 0.2 0.3 0.4 0.5-8

-4

0

4

8

Fre

e D

im. y

(m

m)

Fluence / Jcm-2

0.0 0.1 0.2 0.3 0.4 0.5-8

-4

0

4

8


Fre

e D

im. y

(m

m)

0

0.2000

0.4000

0.6000

0.8000

1.000

1.200

1.400

1.600

1.800

2.000

A

B

-8

-4

0

4

8

Fre

e D

im. (

mm

)

Fluence / Jcm-2

0.0 0.1 0.2 0.3 0.4 0.5-8

-4

0

4

8


Fre

e D

im. (

mm

)

0

0.4000

0.8000

1.200

1.600

2.000

(a)

(b)


-200-100

0100200

Fre

e ax

is (


-200-100

0100200

z=98cm

Fre

e ax

is (

µm) z=99cm

-50 -25 0 25 50 Time (fs)

z=100cm

-50 -25 0 25 50

-200-100

0100200

z=101cm

Time (fs)

Fre

e ax

is (

µm)

-50 -25 0 25 50

z=102cm

Time (fs)90 95 100 105 110

-200-100

0100200


Fre

e ax

is (

µm)

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)


-20 -10 0 10 200.0

0.2

0.4

0.6

0.8

1.0

Nor

mal

ized

Inte

nsity

Transverse Position (mm)

p=1 p=2 p=3 p=4

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.00.0

0.2

0.4

0.6

0.8

1.0

Cou

plin

g E

ffici

ency

Transverse Beam Radius w/a (1/e2)

p=1 p=3 p=5

w~0.735a

-1.0 -0.5 0.0 0.5 1.0-1.0

-0.5

0.0

0.5

1.0

Nor

mal

ized

Fie

ld

Position x/a

p=1 p=2 p=3 p=4

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

Nor

mal

ized

Pow

er

Waveguide Length (m)

p=1 p=2 p=3 p=4

Basic waveguide features


Focusability

linear case nonlinear case

Measured focal spots behind a 1m focal length lens:

-50 -25 0 25 50

-200

-100

0

100

200

Time (fs)

y-ax

is fo

cus

(µm

)

z=94cm

-50 -25 0 25 50

-200

-100

0

100

200

z=96cm

Time (fs)

y-ax

is fo

cus

(µm

)

-50 -25 0 25 50

-200

-100

0

100

200

z=98cm

Time (fs)

y-ax

is fo

cus

(µm

)

-50 -25 0 25 50

-200

-100

0

100

200

z=100cm

Time (fs)

y-ax

is fo

cus

(µm

)

-50 -25 0 25 50

-200

-100

0

100

200

z=102cm

Time (fs)

y-ax

is fo

cus

(µm

)

-50 -25 0 25 50

-200

-100

0

100

200

z=104cm

Time (fs)

y-ax

is fo

cus

(µm

)

Numerical focusing with 1m focal length.


Higher pulse energy in the few-cycle regime can be

generated by OPCPA

Today’s optical chirped pulse amplification OPCPA provides

pulse energy of ~130 mJ at ~7 fs.

However, OPCPA requires additional pump lasers and very

complicated setups.

F. Tavella et al., Opt. Express 14, 12822 (2006)

N. Ishii et al., Opt. Lett. 30, 567 (2005).


( ) ( ) ( ) ( )( ){ }

( ) ( ) ( ) ( )( ) ( ) ( )

( )22

0

0020

2

2

1

,,,1

,,

..exp,,,,,

,,,,,~112,,~

−

−=

=

+−=

+

−−−−=∂

∫

∑

∑

−

ka

u

a

ni

ka

uk

dxxUtzyxa

tzy

cctizixUtzytzyx

tzyxNFzkvx

kiLzk

pnpp

p

a

a

p

ppp

pyp

gp

TpDfyz

ωβ

εκ

ωβκε

εωκωωββ

βωε

Numerical model based on the nonlinear envelope

equation (NEE)

Linear Part: mode selective dispersion and absorption,Space-time focusing

Nonlinear part: shock formation, plasma-defocusing, nonlinear ionization

Transformations betweenNormal space and mode space

Wave number, dispersion and absorption of TE waveguide modes


Numerical modeling

Quasi-guidedpropagation

Free propagation

y

x

z

Cosine modes

-1.0 -0.5 0.0 0.5 1.0

-1.0

-0.5

0.0

0.5

1.0

U3(x)

U5(x)

U(x)

x/a

U1(x)

0.0 0.1 0.2 0.3 0.4 0.50.0

0.2

0.4

0.6

0.8

1.0

Nor

mal

ized

pow

er

Propagation distance (m)

Attenuation2

1 )(zκ

2

3 )(zκ

2

5 )(zκ


700 750 800 850 900-10

-5

0

5

10

Wave length (nm)

-40 -20 0 20 40-10

-5

0

5

10

Fre

e w

aveg

uide

axi

s (m

m)

Time (fs)

10 20 30 40 50-10

-5

0

5

10

Duration (fs)

(a) (b) (c)

Documents

Palaiseau - FRANCE A nonlinear envelope …reseau-femto.cnrs.fr/IMG/pdf/ARNOLD.pdf UMR 7639 A nonlinear envelope equation for ultrashort pulse propagation in guided structures. Comparison