Laser propagation in plasma

Stuart Mangles

Learning Objectives

• By the end of this lecture you should – be familiar with basic concepts of laser propagation in vacuum – be able to describe the non-linear refractive index of plasmas – be able to use the non-linear refractive index to study key phenomena

including: »self-focussing; guiding, pulse-compression, photon “deceleration” »self-modulation instability; hosing; wakefield evolution

• Gaussian beam approximation – Electric field of propagating laser pulse in vacuum:

Laser propagation in vacuum

zR =⇡w2

0

�w(z) = w0

s

1 +

✓z

zR

◆2

R(z) = z

1 +

⇣zRz

⌘2�

– Rayleigh range – transverse beam size – wavefront curvature

transverse longitudinal phaseE(r, z, t) = E0x̂

w0

w(z)exp

✓� r2

w(z)2

◆exp

� (z � vgt)2

(c⌧)2

�exp

�i

✓kz � !t+ k

r2

2R(z)� (z)

◆�

• The f-number f# – Ratio of focal length to (collimated) beam diameter

•controls how tightly focussed the laser is

•controls distance over which laser stays intense

• f/2 (0.8 µm laser): »w0 = 1.4 µm; ZR = 8.1 µm

•f/20 (0.8 µm laser) »w0 = 14.4 µm; ZR = 815 µm

Laser propagation in vacuum

w0 =2p2

⇡�f# ⇡ 0.9�f#

f/2

f/4

f/10

f/20

zR =⇡w2

0

�=

!0w20

2c⇡ 2.5�f2

#

•NB real high power lasers are not gaussian!! – near field beam is closer to flat top – wavefront curvature and intensity profile are not perfect – spatio-temporal couplings: errors in chirped pulse amplification system

»all affect the propagation of lasers, especially due to non-linear effects in plasma

Laser propagation in vacuum

R Shalloo PhD Thesis 2019

Laser propagation in plasma

• dispersion relation for low intensity EM waves in plasma

!2 = !2p + k2c2

!p =

snee2

✏0me

vg =@!

@k

= c

1�

!2p

!2

! 12

⇡ c

1� 1

2

!2p

!2

!vp =

!

k

= c

1�

!2p

!2

!� 12

⇡ c

1 +

1

2

!2p

!2

!

• phase velocity • group velocity

Non-linear refractive index in plasma

• High intensity laser modifies the refractive index

• Depends on » local plasma density » local laser frequency » local laser intensity

• Ignoring 2nd order terms this gives

⌘ =c

vp=

1�

!2p

�!20

! 12

n = n0 +�n

n0n0

!L = !0 +�!L

!0!0

h�i = 1 +a204

⌘ = 1� 1

2

!2p

!20

✓1 +

�n

n0� 2�!L

!0� a20

4

◆

WB Mori IEEE J Quantum Elec. 33, 1942 (1997)

Wave frame

• Non-linear refractive index due to high intensity laser pulse moves with the laser

– change in density due to ponderomotive force

– change in electron “mass” due to quiver motion in laser field

• Introduce “wave frame” variables

laser a0

plasma density

refractive index

⇠ = z � ct

⌧ = t

Self-focusing

• Transverse gradient of refractive index leads to tilting of wavefront

vp1

vp2

vp2

z

z1’= z + vp1∆t

z2’= z + vp2∆t θ

w

WB Mori IEEE J Quantum Elec. 33, 1942 (1997)

✓ ' �vp�t/w

�vp ⇡ w@vp@r

= �wc

⌘2@⌘

@r✓ ' � c

⌘2@⌘

@r�t

Self-focusing

•Energy flows perpendicular to the wavefront – rate of energy flow inwards (or outwards) is equal to rate of change of spot

size

– in plasma vg is related to refractive index through: – so we get:

– ‘acceleration’ of spot size due to radial variation in refractive index is therefore

@w

@⌧=

c2

⌘

@⌘

@r�t

@2w

@⌧2=

c2

⌘

@⌘

@r

vg = c⌘

WB Mori IEEE J Quantum Elec. 33, 1942 (1997)

@w

@⌧= �vg,r = �vg sin ✓ ⇡ �vg✓

• High laser intensity on axis creates focusing effect through the relativistic (a0) term in the non-linear refractive index

• To get self-focusing need rate of focussing to be faster than the rate of defocussing from diffraction

Relativistic self-focussing

@2w

@⌧2

����plasma

+@2w

@⌧2

����di↵raction

0

Relativistic self-focusing

•The diffraction term can be found from gaussian waist equation

– near focus (z < zR), can differentiate this to get

w(z) = w0

s

1 +

✓z

zR

◆2

@2w

@⌧2

����di↵raction

=4c4

!20w

30

@w

@⌧=

@z

@⌧

@w

@z⇡ c

@w

@z

= c@

@z

✓1 +

z2

2z2R

◆

=cz

z2R=

4c3z

w30!

20

Relativistic self-focusing

• The relativistic term from plasma refractive index becomes

– approximating the transverse gradient of a2 as:

⌘ = 1� 1

2

!2p

!20

✓1� a2(r, z)

4

◆

@2w

@⌧2= �c2

8

!2p

!20

@

@ra2(r, z)

@2w

@⌧2=

c2

⌘

@⌘

@r

@

@ra2 ⇡ a20

w0

@2w

@⌧2

����plasma

= �1

8

!2p

!20

a20w0

c2

Relativistic self-focusing

– Noting that w02a02 is related to the laser power (area x intensity) it can be shown….

w20a

20 = 32

c2

!2p

@2w

@⌧2

����plasma

= �1

8

!2p

!20

a20w0

c2@2w

@⌧2

����di↵raction

=4c4

!20w

30

@2w

@⌧2

����plasma

+@2w

@⌧2

����di↵raction

= 0

– Balancing the rate of relativistic focusing with diffraction:

P [GW] � 17.3!20

!2p

Plasma channel guiding

•A similar treatment can be applied to estimate the effect of a parabolic plasma channel:

– balancing focusing due to the channel with diffraction produces a “matched spot size”

ne(r) = ne0 +�ner2

r2ch

wm =

✓r2ch

⇡re�ne

◆ 14

Self-guiding

• In LWFA we have a lot going on

– relativistic: variation in a0

– ponderomotive: variation in ne due to laser pulse – pre-formed: variation in ne due to pre-formed channel

@2w

@⌧2

����relativistic

+@2w

@⌧2

����ponderomotive

+@2w

@⌧2

����pre-formed

+@2w

@⌧2

����di↵raction

0

Self-guiding

– Experimental measurements of self-guiding

– Laser profile at exit of 2 mm gas jet shows guided spot with 14 TW pulse for P > Pc

– guiding over approx 5–7 zR

M Streeter

AGR Thomas PRL 2007

f/12

f/3

f/12

– Experimental measurements of self-guiding

– Laser profile at exit of 15 mm gas jet shows guided spot with 180 TW pulse for P > Pc

– guiding over approx 15 zR

Self-guiding

M Streeter

MJV Streeter PhD Thesis 2013f/12

f/3

f/12

images of laser spot at exit 15 mm laser wakefield accelerator

plasma density x1018 cm-3

M Streeter

Pre-formed plasma channel guiding

• Pre-formed plasma channels very successful at guiding high power lasers over long distances:

– e.g LBNL successfully guided 0.85 PW laser pulse over 20 cm (15 ZR)

AJ Gonsalves PRL 2019

Pulse Compression

•Longitudinal variation in group velocity changes shape of pulse envelope as it propagates

– compression / stretching

z1, vg1z1, vg1

L L +∆L

z’1 = z1 + vg1 ∆t z’2 = z2 + vg2 ∆t

WB Mori IEEE J Quantum Elec. 33, 1942 (1997)

Pulse Compression

– consider point at front, z1 and back, z2, of laser pulse initially separated by distance L

– change in separation in time ∆t: – relate change in group velocity to gradient in group velocity – in wave frame rate of compression is

�vg ⇡ @vg@z

L

�L = (vg2 � vg1)�t

z1, vg1z1, vg1

L L +∆L

z’1 = z1 + vg1 ∆t z’2 = z2 + vg2 ∆t

1

L

@L

@t= �c

@⌘

@⇠

WB Mori IEEE J Quantum Elec. 33, 1942 (1997)

Pulse Compression in the Bubble Regime

• Simple model for compression in the bubble regime – front of pulse in plasma ne = n0; back of pulse ne = 0 – gives rate of compression

– measured rates of compression in non-linear plasma wave very close to this prediction

⌧ = ⌧0 �n0l

2cnc

JS Schreiber PRL 105, 235003 (2010)

Photon “acceleration”

• longitudinal variation in refractive index also means phase velocity varies, leading to a change in the wavelength

• For refractive index gradient caused by the laser pulse, the rate of change of frequency in the wave frame is:

�v� =@v�@z

�0

z1, vp1z2, vp2 z’1 = z1 + vp1∆tz’2 = z2 + vp2∆t

λ λ’

@�

@t=

@v�@z

�0

1

!

@!

@⌧=

c

⌘2@⌘

@⇠

WB Mori IEEE J Quantum Elec. 33, 1942 (1997)

Photon “deceleration” in the Bubble Regime

• Refractive index at front of bubble causes red-shifting of photons – these then slip back inside the pulse (lower group velocity) – leads to pulse front etching at – non-linear group velocity is approximately

K Poder PhD Thesis 2016

laser pulse

plasma density

red-shifting

vetch =!2p

!20

c

vg,nl = vg � vetch

⇡ c

1� 3

2

!2p

!20

!

Photon “deceleration” in the Bubble Regime

• Frequency Resolved Optical Gating (FROG) measurements of pulse shape show red-shifting at the front of the pulse

JS Schreiber PRL 105, 235003 (2010)

Photon “deceleration” in the Bubble Regime

•Photon deceleration / etching determines the pump depletion length in non-linear wakes

J Ralph PRL 102, 175003 (2009)

Letch ' c

vetchc⌧FWHM '

!2

p

!2

0

c⌧FWHM

Propagation instabilities: SM-LWFA

• Self-modulation instability – laser pulse longer than plasma wavelength – drives a low amplitude plasma wave

»compression / photon “acceleration” in longitudinal direction »and self focusing in transverse direction

»increases plasma wave amplitude in positive feedback look

»produces very large amplitude waves

c⌧ > �p

laser intensity plasma density perturbation

Propagation instabilities: SM-LWFA

• Signature of SM-LWFA is appearance of peaks at • 1995: Modena et al, observed waves driven to breaking point for first time — self-injection of electrons into a wakefield accelerator

Modena Nature 1995

! = !0 ± n!p

Propagation instabilities: Hosing

– Consider a laser pulse with spatio-temporal coupling issue »back of pulse sits in plasma wave created by front of pulse

»but off-axis so feels focusing “force” due to transverse density gradient »back of pulse will overshoot and oscillate - laser “hoses” as it propagates.

laser propagation direction

�hosing ⇡ 4p2�2

L

a0w0

✓ncr

ne

◆3/2

Kaluza PRL 2010

Propagation instabilities: Hosing

• Hosing has been observed by imaging the side-scatter/ self emission • Hosing wavelength matches the theory

Kaluza PRL 2010

Evolution of plasma waves due to non-linear plasma optics

• Combined effects of non-linear plasma optics lead to co-evolution of laser pulse and plasma wave • This is crucial for injection in many LWFAs

Evolution of plasma waves due to non-linear plasma optics

– ponderomotive + relativistic self-focusing at back of pulse

– pulse front etching/compression

– power amplification leading to injection (see Streeter PRL 2018, Sävert PRL 2015)

Summary

• Introduced concepts needed to understand how lasers propagate inside a LWFA

– self-focussing / guiding – pulse compression – photon “deceleration”

• Introduced propagation instabilities – self-modulation – hosing

• Discussed role of pulse evolution in self-injection

stuart.mangles@imperial.ac.uk @plasmaStu