Towards Multi-Scale Models of Intense Laser-Plasma...

Towards Multi-Scale Models of IntenseLaser-Plasma Interactions

B. A. Shadwick

Department of Physics and Astronomy

University of Nebraska – Lincoln

— November 22, 2008 —

2008 US-Japan Workshop on Progress of Multi-Scale Simulation Models

B. A. Shadwick Towards Multi-Scale Models of Laser-Plasma Interactions

Acknowledgements

Collaborators

C. B. Schroeder, LBNL

E. Esarey, LBNL

Support

US DoE

UNL


Technological Applications of Laser-PlasmaInteractions

Accelerators

Electron sources

Light sources



Accelerators

Electron sources

Light sources

Particle physics has insatiable appetitefor higher energy colliders.

Conventional metal cavities are limitedto gradients of 10 − 100 MeV=m.

Higher energy means (larger) longermachines and higher cost.

Plasma wave can easily supportgradients greater than 10 GeV=m.



Accelerators

Electron sources

Light sources



Accelerators

Electron sources

Light sources

High-quality electron bunches arenecessary for high energy acceleratorsand light-sources.

Electron bunches can be produced byplasma instabilities.

Electron bunches can be produceddirect manipulation of phase space(LILAC: Umstadter et al. PRL 96; CPI:Esarey et al. PRL 97.).



Accelerators

Electron sources

Light sources



Accelerators

Electron sources

Light sources Plasma based alternatives to traditionalFELs.

THz sources based on plasma-vacuuminterface (OTR).

Xrays from betatron (synchrotron)radiation or Thomson scattering.

“Exotic” schemes: laser undulators,Raman amplifiers, . . . .


Recent Experimental Results

Pre 2004

2004

2006

Post 2006



Pre 2004

2004

2006

Post 2006

Self-injection due to growth of an instability.

100% energy spread.

Bunch energy > 100 MeV.

Bunch charge ∼ 10 nC.

Laser power . 10 TW.

Gas jet target.

Plasma densities 1019 − 1020 cm−3.

Many experiment in this regime: Modena etal. (95); Nakajima et al. (95); Umstadter etal. (96); Ting et al. (97); Gahn et al. (99);Leemans et al. (01); Malka et al. (01).



Pre 2004

2004

2006

Post 2006



Pre 2004

2004

2006

Post 2006

Three groups report high-quality electronbunches, Nature, September (2004):

– Geddes et al. (LBNL);– Mangles et al. (RAL); and– Faure et al. (LOA).

Narrow energy spread and small divergence.

Bunch energy ∼ 100 MeV.

Bunch charge 100s pC.

Laser power 10s TW.

Gas jet targets.

Plasma densities 1018–1019 cm−3.



Pre 2004

2004

2006

Post 2006


– Geddes et al. (LBNL);

– Mangles et al. (RAL); and– Faure et al. (LOA).




Laser power 10s TW.

Gas jet targets.




Pre 2004

2004

2006

Post 2006


– Geddes et al. (LBNL);– Mangles et al. (RAL); and

– Faure et al. (LOA).




Laser power 10s TW.

Gas jet targets.




Pre 2004

2004

2006

Post 2006


– Geddes et al. (LBNL);– Mangles et al. (RAL); and– Faure et al. (LOA).




Laser power 10s TW.

Gas jet targets.




Pre 2004

2004

2006

Post 2006



Pre 2004

2004

2006

Post 2006

High quality electron beam production at1 GeV. Leemans et al., Nature Physics,September (2006).


Bunch energy ∼ 1 GeV.

Bunch change ∼ 100 pC.

Laser power ∼ 40 TW.

Plasma channel ∼ 3 cm long, formed bycapillary discharge.

Plasma density 3� 1018 cm−3.



Pre 2004

2004

2006

Post 2006

Colliding pulse injection. Faure et al., Nature,October (2006).

Narrow energy spread, �E=E = 11% andsmall divergence.

Bunch energy 117 MeV.

Bunch change ∼ 20 pC.

Laser power ∼ 50 TW.

Plasma density 7:5� 1018 cm−3.



Pre 2004

2004

2006

Post 2006



Pre 2004

2004

2006

Post 2006 High-quality beams in the “bubble regime.”


Reproducible.

Many groups in US, Europe and Asia.


Introduction - Typical Example

Quasi-neutral plasma.Intense, short laser pulse.Transverse parabolic profile.Plasma acts like optical fiber guiding the laser pulse.Laser generates a large amplitude (nonlinear) plasma wave.



Quasi-neutral plasma.

Intense, short laser pulse.Transverse parabolic profile.Plasma acts like optical fiber guiding the laser pulse.Laser generates a large amplitude (nonlinear) plasma wave.



Quasi-neutral plasma.Intense, short laser pulse.

Transverse parabolic profile.Plasma acts like optical fiber guiding the laser pulse.Laser generates a large amplitude (nonlinear) plasma wave.



Quasi-neutral plasma.Intense, short laser pulse.Transverse parabolic profile.

Plasma acts like optical fiber guiding the laser pulse.Laser generates a large amplitude (nonlinear) plasma wave.



Quasi-neutral plasma.Intense, short laser pulse.Transverse parabolic profile.Plasma acts like optical fiber guiding the laser pulse.

Laser generates a large amplitude (nonlinear) plasma wave.



Quasi-neutral plasma.Intense, short laser pulse.Transverse parabolic profile.Plasma acts like optical fiber guiding the laser pulse.Laser generates a large amplitude (nonlinear) plasma wave.



Typical parameters:

Domain size:

kp x ∼ 20 – 40kp � ∼ 20 – 60

Time scale:

!0=!p ∼ 20 – 100

Propagation time:

!pt ∼ 1000 – 50;000distance: mm to 10’s of cm

Laser field:

a0 =q

mc2 A ∼ 1 – 3

Initial temperature:

T0 � 10 eV

Momentum space:

max |px |, max |pz |� mc�th



Typical parameters:Domain size:

kp x ∼ 20 – 40kp � ∼ 20 – 60

Time scale:

!0=!p ∼ 20 – 100

Propagation time:


Laser field:

a0 =q

mc2 A ∼ 1 – 3


T0 � 10 eV

Momentum space:





kp x ∼ 20 – 40

kp � ∼ 20 – 60

Time scale:

!0=!p ∼ 20 – 100

Propagation time:


Laser field:

a0 =q

mc2 A ∼ 1 – 3


T0 � 10 eV

Momentum space:





kp x ∼ 20 – 40kp � ∼ 20 – 60

Time scale:

!0=!p ∼ 20 – 100

Propagation time:


Laser field:

a0 =q

mc2 A ∼ 1 – 3


T0 � 10 eV

Momentum space:





kp x ∼ 20 – 40kp � ∼ 20 – 60

Time scale:!0=!p ∼ 20 – 100

Propagation time:


Laser field:

a0 =q

mc2 A ∼ 1 – 3


T0 � 10 eV

Momentum space:





kp x ∼ 20 – 40kp � ∼ 20 – 60

Time scale:!0=!p ∼ 20 – 100

Propagation time:!pt ∼ 1000 – 50;000

distance: mm to 10’s of cm

Laser field:

a0 =q

mc2 A ∼ 1 – 3


T0 � 10 eV

Momentum space:





kp x ∼ 20 – 40kp � ∼ 20 – 60

Time scale:!0=!p ∼ 20 – 100

Propagation time:!pt ∼ 1000 – 50;000distance: mm to 10’s of cm

Laser field:

a0 =q

mc2 A ∼ 1 – 3


T0 � 10 eV

Momentum space:





kp x ∼ 20 – 40kp � ∼ 20 – 60

Time scale:!0=!p ∼ 20 – 100


Laser field:a0 =

qmc2 A ∼ 1 – 3


T0 � 10 eV

Momentum space:





kp x ∼ 20 – 40kp � ∼ 20 – 60

Time scale:!0=!p ∼ 20 – 100


Laser field:a0 =

qmc2 A ∼ 1 – 3

Initial temperature:T0 � 10 eV

Momentum space:





kp x ∼ 20 – 40kp � ∼ 20 – 60

Time scale:!0=!p ∼ 20 – 100


Laser field:a0 =

qmc2 A ∼ 1 – 3

Initial temperature:T0 � 10 eV

Momentum space:max |px |, max |pz |� mc�th


Multi-Scale Method for the Vlasov Equation

Asymptotics studies: if the width of f starts small it stayssmall.

By an exact transformation we can remove the bulk motion inmomentum space.

Putp = eP + ep

where Pb is arbitrary.

If we choose eP to be cold fluid momentum, then f is non-zeroover a small range of ep .

Specialize to three phase-space dimensions: (z;px ;pz).

Laser electric field is polarized in the x-direction.


Multi-Scale Method for the Vlasov Equation

Asymptotics studies: if the width of f starts small it stayssmall.

By an exact transformation we can remove the bulk motion inmomentum space.Put

p = eP + epwhere Pb is arbitrary.

If we choose eP to be cold fluid momentum, then f is non-zeroover a small range of ep .

Specialize to three phase-space dimensions: (z;px ;pz).

Laser electric field is polarized in the x-direction.


Moving Phase-Space Grid for the Vlasov Equation. . .

Cold fluid orbits:

@ ePz@t+

ePz

m 0

@ ePz@z= Ez +

ePx

mc 0By

ePx = −qc

Ax

where 0 =q

1 + eP2=(m2c2).

Transformed distribution function

F (z; ep ; t) = f (r ;p ; t) = f (r ; ep + eP; t)

Derivatives transform as

@@z→ @@z

+@ ePx@z

@@px+

@ ePz@z@@pz@@t

→ @@t+

@ ePx@t@@px

+@ ePz@t

@@pz



Cold fluid orbits:@ ePz@t+

ePz

m 0

@ ePz@z= Ez +

ePx

mc 0By

ePx = −qc

Ax

where 0 =q

1 + eP2=(m2c2).




@@z→ @@z

+@ ePx@z

@@px+

@ ePz@z@@pz@@t

→ @@t+

@ ePx@t@@px

+@ ePz@t

@@pz




ePz

m 0

@ ePz@z= Ez +

ePx

mc 0By

ePx = −qc

Ax

where 0 =q

1 + eP2=(m2c2).


F (z; ep ; t) = f (r ;p ; t) = f (r ; ep + eP; t)Derivatives transform as

@@z→ @@z

+@ ePx@z

@@px+

@ ePz@z@@pz@@t

→ @@t+

@ ePx@t@@px

+@ ePz@t

@@pz




ePz

m 0

@ ePz@z= Ez +

ePx

mc 0By

ePx = −qc

Ax

where 0 =q

1 + eP2=(m2c2).




@@z→ @@z

+@ ePx@z

@@px+

@ ePz@z@@pz@@t

→ @@t+

@ ePx@t@@px

+@ ePz@t

@@pz




ePz

m 0

@ ePz@z= Ez +

ePx

mc 0By

ePx = −qc

Ax

where 0 =q

1 + eP2=(m2c2).


F (z; ep ; t) = f (r ;p ; t) = f (r ; ep + eP; t)Derivatives transform as

@@z→ @@z

+@ ePx@z

@@px+

@ ePz@z@@pz@@t

→ @@t+

@ ePx@t@@px

+@ ePz@t

@@pz




ePz

m 0

@ ePz@z= Ez +

ePx

mc 0By

ePx = −qc

Ax

where 0 =q

1 + eP2=(m2c2).


F (z; ep ; t) = f (r ;p ; t) = f (r ; ep + eP; t)Derivatives transform as@@z

→ @@z+

@ ePx@z@@px

+@ ePz@z

@@pz

@@t→ @@t

+@ ePx@t

@@px+

@ ePz@t@@pz




ePz

m 0

@ ePz@z= Ez +

ePx

mc 0By

ePx = −qc

Ax

where 0 =q

1 + eP2=(m2c2).


F (z; ep ; t) = f (r ;p ; t) = f (r ; ep + eP; t)Derivatives transform as@@z

→ @@z+

@ ePx@z@@px

+@ ePz@z

@@pz@@t→ @@t

+@ ePx@t

@@px+

@ ePz@t@@pz



The transformed Vlasov equation

@F@t+ vz

@F@z+

@F@epzmc2 @@z

( 0 − ) = 0 :Speed up comes from greatly reduced grid size

|ep | ∼ mc� th compared to

|Px | ∼ a0

|Pz | ∼ a20 (linear)

|Pz | ∼ a0 (nonlinear)

For T0 = 5 eV, a0 = 1:5 savings ∼ 104

Details: Shadwick et al. Phys. Plasmas 12, 056710 (2005).




@F@t+ vz

@F@z+

@F@epzmc2 @@z

( 0 − ) = 0 :

Speed up comes from greatly reduced grid size


|Px | ∼ a0



For T0 = 5 eV, a0 = 1:5 savings ∼ 104





@F@t+ vz

@F@z+

@F@epzmc2 @@z

( 0 − ) = 0 :Speed up comes from greatly reduced grid size


|Px | ∼ a0



For T0 = 5 eV, a0 = 1:5 savings ∼ 104



Example: Maxwellian Initial Distribution

Ignore laser evolution. Concentrate on plasma response

Co-moving coordinate

(z; t) −→ (� = t − z=c; t)

Method of lines

4th order in momentum.Evolution using RK4.Not optimal; purpose to demonstrate algorithm.





(z; t) −→ (� = t − z=c; t)Method of lines






(z; t) −→ (� = t − z=c; t)

Method of lines







4th order in momentum.

Evolution using RK4.Not optimal; purpose to demonstrate algorithm.






4th order in momentum.Evolution using RK4.

Not optimal; purpose to demonstrate algorithm.


Example: Maxwellian Initial Distribution. . .

Parametersa0 = 1, !0 = 10!p,!p � = 2, T0 = 50 eV

f (�; epx ; epz)

Momentum Space Orbit

Pz

Px

-1.0 -0.5 0.0 0.5 1.0-0.4

-0.2

0.0

0.2

0.4BAS-2-0114 Copyright © 2004, B. A. Shadwick. All rights reserved.

�(�)

-10 0 10 20

-0.45

-0.30

-0.15

0.00

0.15

0.30BAS-2-0114 Copyright © 2004, B. A. Shadwick. All rights reserved.

kpξ


Time Scale Separation: !0 � !p

Quasi-Static approximation.

Family of approximations with a common theme.Can be applied to fluids, PIC, Vlasov.Essential ingredients:

Moving window.Slow evolution of the laser pulse compared to the transit timein the window.Potentials.

Non-essential ingredients:

Averaging.Envelopes.

History

NRL fluid theory and code. (Whittam for beams.)WAKEUNL/LBNL Fluid codes & Vlasov codesQUICKPIC (UCLA/USC). . .



Quasi-Static approximation.Family of approximations with a common theme.

Can be applied to fluids, PIC, Vlasov.Essential ingredients:




History




Quasi-Static approximation.Family of approximations with a common theme.Can be applied to fluids, PIC, Vlasov.

Essential ingredients:




History




Quasi-Static approximation.Family of approximations with a common theme.Can be applied to fluids, PIC, Vlasov.Essential ingredients:




History





Moving window.

Slow evolution of the laser pulse compared to the transit timein the window.Potentials.



History





Moving window.Slow evolution of the laser pulse compared to the transit timein the window.

Potentials.Non-essential ingredients:


History








History






Non-essential ingredients:Averaging.

Envelopes.History






Non-essential ingredients:Averaging.Envelopes.

History







HistoryNRL fluid theory and code. (Whittam for beams.)

WAKEUNL/LBNL Fluid codes & Vlasov codesQUICKPIC (UCLA/USC). . .






HistoryNRL fluid theory and code. (Whittam for beams.)WAKE

UNL/LBNL Fluid codes & Vlasov codesQUICKPIC (UCLA/USC). . .






HistoryNRL fluid theory and code. (Whittam for beams.)WAKEUNL/LBNL Fluid codes & Vlasov codes

QUICKPIC (UCLA/USC). . .






HistoryNRL fluid theory and code. (Whittam for beams.)WAKEUNL/LBNL Fluid codes & Vlasov codesQUICKPIC (UCLA/USC). . .


Quasi-Static Plasma – Details

Introduce a moving window: (z; t) −→ (� = t − z=c; t)Conservation of transverse canonical momentum: px + ax = 0

Drop time derivative in plasma equations.

First integral from the momentum equation: pz = � + − 1

Equation for �:

@2�@ �2 =12

"1 −

1 + a2x

(1 − �)2

#

Wave equation for ax :

@2@ t2 + 2@2@t @�

!ax =

ax� − 1

The only approximation is in the (slow) plasma response.

We have retained the complete wave operator.



Introduce a moving window: (z; t) −→ (� = t − z=c; t)

Conservation of transverse canonical momentum: px + ax = 0



Equation for �:

@2�@ �2 =12

"1 −

1 + a2x

(1 − �)2

#


@2@ t2 + 2@2@t @�

!ax =

ax� − 1








Equation for �:

@2�@ �2 =12

"1 −

1 + a2x

(1 − �)2

#


@2@ t2 + 2@2@t @�

!ax =

ax� − 1








Equation for �:

@2�@ �2 =12

"1 −

1 + a2x

(1 − �)2

#Wave equation for ax :

@2@ t2 + 2@2@t @�

!ax =

ax� − 1








Equation for �:

@2�@ �2 =12

"1 −

1 + a2x

(1 − �)2

#


@2@ t2 + 2@2@t @�

!ax =

ax� − 1








Equation for �:

@2�@ �2 =12

"1 −

1 + a2x

(1 − �)2

#Wave equation for ax :

@2@ t2 + 2@2@t @�

!ax =

ax� − 1








Equation for �:

@2�@ �2 =12

"1 −

1 + a2x

(1 − �)2

#Wave equation for ax : @2@ t2 + 2

@2@t @�!

ax =ax� − 1




Why use the Quasi-Static Approximation?

Since we are dropping physics there must be some benefit.

Analytical?Computational?

If we make no other approximations, there is nocomputational advantage.

Generically, the wave equation will impose the same stabilityconstraint (�t = K��) as for the full plasma model.

We only see a computational gain by dropping the@2@ t2 term

in the wave equation.

In the moving window, this term relates to the slow laserevolution.



Since we are dropping physics there must be some benefit.Analytical?

Computational?








Since we are dropping physics there must be some benefit.Analytical?Computational?








Since we are dropping physics there must be some benefit.Analytical?Computational?




in the wave equation.In the moving window, this term relates to the slow laserevolution.


Reduced Wave Operator

Introduce an envelope representation for the laser:

ax = Re h(ar + i ai) ei k0 �i

We can average the plasma response (optional).

Wave equation becomes:

@2ar@t @� − k0@ ai@ t

=12

ar� − 1@2ai@t @� + k0@ ar@ t

=12

ai� − 1

The potential satisfies

@2�@ �2 =12

"1 −

1 + 12

�a2

r + a2i

�(1 − �)

2

#





We can average the plasma response (optional).Wave equation becomes:

@2ar@t @� − k0@ ai@ t

=12

ar� − 1@2ai@t @� + k0@ ar@ t

=12

ai� − 1The potential satisfies

@2�@ �2 =12

"1 −

1 + 12

�a2

r + a2i

�(1 − �)

2

#





We can average the plasma response (optional).Wave equation becomes:@2ar@t @� − k0

@ ai@ t=

12

ar� − 1@2ai@t @� + k0@ ar@ t

=12

ai� − 1

The potential satisfies

@2�@ �2 =12

"1 −

1 + 12

�a2

r + a2i

�(1 − �)

2

#





We can average the plasma response (optional).Wave equation becomes:@2ar@t @� − k0

@ ai@ t=

12

ar� − 1@2ai@t @� + k0@ ar@ t

=12

ai� − 1The potential satisfies

@2�@ �2 =12

"1 −

1 + 12

�a2

r + a2i

�(1 − �)

2

#


Test Problem

Physical parameters:

a0 = 1!0 = 20!p!p � = 2

Compare the averaged, reduced wave-equation quasi-staticmodel to the full-fluid model.

Numerical parameters:

k0�� = 0:0621 (�0=�� 100)�t = 14 ��


Cold Fluid with Full Time-Dependence

ωp t = 0

Laser Plasma

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

-5 0 5 10 15 20-0.50

0.00

0.50

1.00

1.50

-5 0 5 10 15 20 25 30kp ξ kp ξ

Ax

Ne

Ez



ωp t = 250

Laser Plasma

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

-5 0 5 10 15 20-0.50

0.00

0.50

1.00

1.50

-5 0 5 10 15 20 25 30kp ξ kp ξ

Ax

Ne

Ez



ωp t = 500

Laser Plasma

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

-5 0 5 10 15 20-0.50

0.00

0.50

1.00

1.50

-5 0 5 10 15 20 25 30kp ξ kp ξ

Ax

Ne

Ez



ωp t = 750

Laser Plasma

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

-5 0 5 10 15 20-0.50

0.00

0.50

1.00

1.50

-5 0 5 10 15 20 25 30kp ξ kp ξ

Ax

Ne

Ez



ωp t = 1000

Laser Plasma

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

-5 0 5 10 15 20-0.50

0.00

0.50

1.00

1.50

-5 0 5 10 15 20 25 30kp ξ kp ξ

Ax

Ne

Ez



ωp t = 1250

Laser Plasma

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

-5 0 5 10 15 20-0.50

0.00

0.50

1.00

1.50

-5 0 5 10 15 20 25 30kp ξ kp ξ

Ax

Ne

Ez



ωp t = 1500

Laser Plasma

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

-5 0 5 10 15 20-0.50

0.00

0.50

1.00

1.50

-5 0 5 10 15 20 25 30kp ξ kp ξ

Ax

Ne

Ez



ωp t = 1750

Laser Plasma

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

-5 0 5 10 15 20-0.50

0.00

0.50

1.00

1.50

-5 0 5 10 15 20 25 30kp ξ kp ξ

Ax

Ne

Ez



ωp t = 2000

Laser Plasma

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

-5 0 5 10 15 20-0.50

0.00

0.50

1.00

1.50

-5 0 5 10 15 20 25 30kp ξ kp ξ

Ax

Ne

Ez



ωp t = 2250

Laser Plasma

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

-5 0 5 10 15 20-0.50

0.00

0.50

1.00

1.50

-5 0 5 10 15 20 25 30kp ξ kp ξ

Ax

Ne

Ez



ωp t = 2500

Laser Plasma

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

-5 0 5 10 15 20-0.50

0.00

0.50

1.00

1.50

-5 0 5 10 15 20 25 30kp ξ kp ξ

Ax

Ne

Ez



ωp t = 2750

Laser Plasma

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

-5 0 5 10 15 20-0.50

0.00

0.50

1.00

1.50

-5 0 5 10 15 20 25 30kp ξ kp ξ

Ax

Ne

Ez



ωp t = 3000

Laser Plasma

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

-5 0 5 10 15 20-0.50

0.00

0.50

1.00

1.50

-5 0 5 10 15 20 25 30kp ξ kp ξ

Ax

Ne

Ez



ωp t = 3250

Laser Plasma

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

-5 0 5 10 15 20-0.50

0.00

0.50

1.00

1.50

-5 0 5 10 15 20 25 30kp ξ kp ξ

Ax

Ne

Ez



ωp t = 3500

Laser Plasma

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

-5 0 5 10 15 20-0.50

0.00

0.50

1.00

1.50

-5 0 5 10 15 20 25 30kp ξ kp ξ

Ax

Ne

Ez



ωp t = 4000

Laser Plasma

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

-5 0 5 10 15 20-0.50

0.00

0.50

1.00

1.50

-5 0 5 10 15 20 25 30kp ξ kp ξ

Ax

Ne

Ez


Laser Evolution

Normalized Laser Energy

1

2

3

45

0.5

0.6

0.7

0.8

0.9

1.0

0 1000 2000 3000 4000ωp t

E(t)

E(0)

2: Minimum pulse length.

3: Maximum wake amplitude.

4: Rapid pulse lengthening.

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-5 0 5 10 15 20kp ξ

1

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-5 0 5 10 15 20kp ξ

2

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-5 0 5 10 15 20kp ξ

3

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-5 0 5 10 15 20kp ξ

4

!p t = 0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-5 0 5 10 15 20kp ξ


Laser Evolution


1

2

3

45

0.5

0.6

0.7

0.8

0.9

1.0

0 1000 2000 3000 4000ωp t

E(t)

E(0)




-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-5 0 5 10 15 20kp ξ

1

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-5 0 5 10 15 20kp ξ

2

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-5 0 5 10 15 20kp ξ

3

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-5 0 5 10 15 20kp ξ

4

!p t = 1000

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-5 0 5 10 15 20kp ξ


Laser Evolution


1

2

3

45

0.5

0.6

0.7

0.8

0.9

1.0

0 1000 2000 3000 4000ωp t

E(t)

E(0)




-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-5 0 5 10 15 20kp ξ

1

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-5 0 5 10 15 20kp ξ

2

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-5 0 5 10 15 20kp ξ

3

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-5 0 5 10 15 20kp ξ

4

!p t = 2000

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-5 0 5 10 15 20kp ξ


Laser Evolution


1

2

3

4

5

0.5

0.6

0.7

0.8

0.9

1.0

0 1000 2000 3000 4000ωp t

E(t)

E(0)




-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-5 0 5 10 15 20kp ξ

1

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-5 0 5 10 15 20kp ξ

2

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-5 0 5 10 15 20kp ξ

3

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-5 0 5 10 15 20kp ξ

4

!p t = 3000

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-5 0 5 10 15 20kp ξ


Laser Evolution


1

2

3

45

0.5

0.6

0.7

0.8

0.9

1.0

0 1000 2000 3000 4000ωp t

E(t)

E(0)




-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-5 0 5 10 15 20kp ξ

1

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-5 0 5 10 15 20kp ξ

2

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-5 0 5 10 15 20kp ξ

3

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-5 0 5 10 15 20kp ξ

4

!p t = 4000

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-5 0 5 10 15 20kp ξ


Quasi-Static Reduced Wave Operator, Averaged

ωp t = 0

Laser Plasma

-0.50

0.00

0.50

-0.10

0.00

0.10

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

-5 0 5 10 15 20-0.50

0.00

0.50

1.00

1.50

-5 0 5 10 15 20 25 30

kp ξ kp ξ

∆Ax ∆Ne

∆Ez

Ax

Ne

Ez



ωp t = 250

Laser Plasma

-0.50

0.00

0.50

-0.10

0.00

0.10

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

-5 0 5 10 15 20-0.50

0.00

0.50

1.00

1.50

-5 0 5 10 15 20 25 30

kp ξ kp ξ

∆Ax ∆Ne

∆Ez

Ax

Ne

Ez



ωp t = 500

Laser Plasma

-0.50

0.00

0.50

-0.10

0.00

0.10

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

-5 0 5 10 15 20-0.50

0.00

0.50

1.00

1.50

-5 0 5 10 15 20 25 30

kp ξ kp ξ

∆Ax ∆Ne

∆Ez

Ax

Ne

Ez



ωp t = 750

Laser Plasma

-0.50

0.00

0.50

-0.10

0.00

0.10

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

-5 0 5 10 15 20-0.50

0.00

0.50

1.00

1.50

-5 0 5 10 15 20 25 30

kp ξ kp ξ

∆Ax ∆Ne

∆Ez

Ax

Ne

Ez



ωp t = 1000

Laser Plasma

-0.50

0.00

0.50

-0.10

0.00

0.10

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

-5 0 5 10 15 20-0.50

0.00

0.50

1.00

1.50

-5 0 5 10 15 20 25 30

kp ξ kp ξ

∆Ax ∆Ne

∆Ez

Ax

Ne

Ez



ωp t = 1250

Laser Plasma

-0.50

0.00

0.50

-0.10

0.00

0.10

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

-5 0 5 10 15 20-0.50

0.00

0.50

1.00

1.50

-5 0 5 10 15 20 25 30

kp ξ kp ξ

∆Ax ∆Ne

∆Ez

Ax

Ne

Ez



ωp t = 1500

Laser Plasma

-0.50

0.00

0.50

-0.10

0.00

0.10

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

-5 0 5 10 15 20-0.50

0.00

0.50

1.00

1.50

-5 0 5 10 15 20 25 30

kp ξ kp ξ

∆Ax ∆Ne

∆Ez

Ax

Ne

Ez



ωp t = 1750

Laser Plasma

-0.50

0.00

0.50

-0.10

0.00

0.10

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

-5 0 5 10 15 20-0.50

0.00

0.50

1.00

1.50

-5 0 5 10 15 20 25 30

kp ξ kp ξ

∆Ax ∆Ne

∆Ez

Ax

Ne

Ez



ωp t = 2000

Laser Plasma

-0.50

0.00

0.50

-0.10

0.00

0.10

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

-5 0 5 10 15 20-0.50

0.00

0.50

1.00

1.50

-5 0 5 10 15 20 25 30

kp ξ kp ξ

∆Ax ∆Ne

∆Ez

Ax

Ne

Ez



ωp t = 2250

Laser Plasma

-0.50

0.00

0.50

-0.10

0.00

0.10

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

-5 0 5 10 15 20-0.50

0.00

0.50

1.00

1.50

-5 0 5 10 15 20 25 30

kp ξ kp ξ

∆Ax ∆Ne

∆Ez

Ax

Ne

Ez



ωp t = 2500

Laser Plasma

-0.50

0.00

0.50

-0.10

0.00

0.10

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

-5 0 5 10 15 20-0.50

0.00

0.50

1.00

1.50

-5 0 5 10 15 20 25 30

kp ξ kp ξ

∆Ax ∆Ne

∆Ez

Ax

Ne

Ez



ωp t = 2750

Laser Plasma

-0.50

0.00

0.50

-0.10

0.00

0.10

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

-5 0 5 10 15 20-0.50

0.00

0.50

1.00

1.50

-5 0 5 10 15 20 25 30

kp ξ kp ξ

∆Ax ∆Ne

∆Ez

Ax

Ne

Ez



ωp t = 3000

Laser Plasma

-0.50

0.00

0.50

-0.10

0.00

0.10

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

-5 0 5 10 15 20-0.50

0.00

0.50

1.00

1.50

-5 0 5 10 15 20 25 30

kp ξ kp ξ

∆Ax ∆Ne

∆Ez

Ax

Ne

Ez



ωp t = 3250

Laser Plasma

-0.50

0.00

0.50

-0.10

0.00

0.10

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

-5 0 5 10 15 20-0.50

0.00

0.50

1.00

1.50

-5 0 5 10 15 20 25 30

kp ξ kp ξ

∆Ax ∆Ne

∆Ez

Ax

Ne

Ez



ωp t = 3500

Laser Plasma

-0.50

0.00

0.50

-0.10

0.00

0.10

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

-5 0 5 10 15 20-0.50

0.00

0.50

1.00

1.50

-5 0 5 10 15 20 25 30

kp ξ kp ξ

∆Ax ∆Ne

∆Ez

Ax

Ne

Ez



ωp t = 3750

Laser Plasma

-0.50

0.00

0.50

-0.10

0.00

0.10

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

-5 0 5 10 15 20-0.50

0.00

0.50

1.00

1.50

-5 0 5 10 15 20 25 30

kp ξ kp ξ

∆Ax ∆Ne

∆Ez

Ax

Ne

Ez



ωp t = 4000

Laser Plasma

-0.50

0.00

0.50

-0.10

0.00

0.10

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

-5 0 5 10 15 20-0.50

0.00

0.50

1.00

1.50

-5 0 5 10 15 20 25 30

kp ξ kp ξ

∆Ax ∆Ne

∆Ez

Ax

Ne

Ez


Quasi-Static Phenomenology

Averaged model requires similar resolution to the full model.

Confirmed by a detailed examination of the numericaldispersion relation.Intuitively reasonable since k0 enters the envelope equationsin the same manner as the spatial derivative.The carrier can be removed by an exact (linear)transformation. It is not surprising that this does not alter theresolution requirements.

Where are the computational gains?

When the@2@ t2 term is dropped from the wave equation, the

numerical stability condition is greatly relaxed.For the reduced wave operator �t ∼ 600 − 1000� larger thanfor the full wave operator at �0=�� = 100.



Averaged model requires similar resolution to the full model.Confirmed by a detailed examination of the numericaldispersion relation.

Intuitively reasonable since k0 enters the envelope equationsin the same manner as the spatial derivative.The carrier can be removed by an exact (linear)transformation. It is not surprising that this does not alter theresolution requirements.






Averaged model requires similar resolution to the full model.Confirmed by a detailed examination of the numericaldispersion relation.Intuitively reasonable since k0 enters the envelope equationsin the same manner as the spatial derivative.

The carrier can be removed by an exact (linear)transformation. It is not surprising that this does not alter theresolution requirements.






Averaged model requires similar resolution to the full model.Confirmed by a detailed examination of the numericaldispersion relation.Intuitively reasonable since k0 enters the envelope equationsin the same manner as the spatial derivative.The carrier can be removed by an exact (linear)transformation. It is not surprising that this does not alter theresolution requirements.









numerical stability condition is greatly relaxed.

For the reduced wave operator �t ∼ 600 − 1000� larger thanfor the full wave operator at �0=�� = 100.


Multi-Scale?

What scales can we dispense with and which must we keep?

Multi-scale in momentum space for Vlasov.

Exact transformation.“Graceful” failure mode.Still very expensive; can not afford large domain

Multi-scale in time for wave equation.

Depending on propagation distance (and accuracy), we mayneed to keep “small” terms.Practical to treat E&B in laser differently?Sub-step wave equation?Ignore time derivative in collective plasma response.

Multi-physics

Optimize physics content (optimizing computational cost).Spatial localization of physical processes.Localized physics models.


Multi-Scale?


Multi-scale in momentum space for Vlasov.Exact transformation.

“Graceful” failure mode.Still very expensive; can not afford large domain



Multi-physics



Multi-Scale?


Multi-scale in momentum space for Vlasov.Exact transformation.“Graceful” failure mode.

Still very expensive; can not afford large domain



Multi-physics



Multi-Scale?


Multi-scale in momentum space for Vlasov.Exact transformation.“Graceful” failure mode.Still very expensive; can not afford large domain



Multi-physics



Multi-Scale?



Multi-scale in time for wave equation.Depending on propagation distance (and accuracy), we mayneed to keep “small” terms.

Practical to treat E&B in laser differently?Sub-step wave equation?Ignore time derivative in collective plasma response.

Multi-physics



Multi-Scale?



Multi-scale in time for wave equation.Depending on propagation distance (and accuracy), we mayneed to keep “small” terms.Practical to treat E&B in laser differently?

Sub-step wave equation?Ignore time derivative in collective plasma response.

Multi-physics



Multi-Scale?



Multi-scale in time for wave equation.Depending on propagation distance (and accuracy), we mayneed to keep “small” terms.Practical to treat E&B in laser differently?Sub-step wave equation?

Ignore time derivative in collective plasma response.

Multi-physics



Multi-Scale?



Multi-scale in time for wave equation.Depending on propagation distance (and accuracy), we mayneed to keep “small” terms.Practical to treat E&B in laser differently?Sub-step wave equation?Ignore time derivative in collective plasma response.

Multi-physics



Multi-Scale?




Multi-physicsOptimize physics content (optimizing computational cost).

Spatial localization of physical processes.Localized physics models.


Multi-Scale?




Multi-physicsOptimize physics content (optimizing computational cost).Spatial localization of physical processes.

Localized physics models.


Multi-Scale?




Multi-physicsOptimize physics content (optimizing computational cost).Spatial localization of physical processes.Localized physics models.


Towards Multi-Scale Models of Intense Laser-Plasma...

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Transcript of Towards Multi-Scale Models of Intense Laser-Plasma...