J.T. Mendonça

Instituto Superior Técnico, Lisboa

Photon accelerationas a scattering process

Outline

• Historical background;• Photon acceleration models;• Acceleration in a laser wakefields;• Ionization fronts and particle beams;• Scattering by relativistic plasma bubbles;• Photon acceleration in gravitational fields;• Generalized Sachs-Wolfe effect;• Conclusions.

Historical background

- Relativistic mirror (Einstein, 1905)

- Moving ionization fronts:(Semenova, 1967; Lampe et al., 1978).

- Moving nonlinear perturbation: adiabatic frequency shift (Mendonça, 1979).

- Photon acceleration in a laser wakefield (Wilks et al., 1989).

-Time refraction (Mendonça, 2000).

- Laser self blue shift: flash ionization (Yablonovich, 1974, Wood et al., 1993).

- Microwave experiments: ionization fronts (Savage et al., 1992).

-2D optical experiments (Dias et al., 1997).

- Photon trapping in a wakefield (Murphy et al., 2006).

Related optical phenomena: self and cross-phase modulation, super-continuum (70’s); Unruh radiation!! (2010)

Experimental evidence

Theoretical models

- Classical model:Geometric optics of space and time varying media

- Kinetic modelPhoton kinetic theory (laser as a photon gas, photon Landau damping);

- Quantum model Full wave theory (similar to scattering theory in Quantum Mechanics);

- Second-Quantization modelQuantum optics approach (theory of time refraction and temporal beam splitters);

-Extension to other types of interaction neutrino-plasma physics, gravitational waves.

n x

1

1

n2

y

n x

1

1

n2

ct

Refraction Time refraction

Photons cannot travel back in the past

W = Inv

€

n1ω1 = n2ω2

€

n1 sinθ1 = n2 sinθ2

1

c t

xn2n1

2

Space-time refraction(moving boundary)

€

n1 sinθ1 = ω2ω1

⎛ ⎝ ⎜ ⎞

⎠ ⎟n2 sinθ2

Generalized Snell’s law

€

ω j (1− βn j cosθ j ) = Inv.

Invariant

• Photons relativistic particles with effective mass • Photon Dynamics canonical equations of motion

Photon ray equations

€

dr

dt=

∂ω

∂k=

k

ωc 2

dk

dt= −

∂ω

∂r= −

1

2ω∇ωp

2

€

ω = k2c 2 + ωp2

Force acting on the photon: refraction (inhomogeneous plasma) and photon acceleration (non-stationary plasma).

Hamiltonian

Photon interraction wit ionization

fronts

Reflection in counter-propagation (relativistic mirror)

Δωωω

ββ

≈±

po2

02 1

ωi,ki

ωf,kf

v=cβ

ωi,ki

ωf,kf

v=cβ

v=cβv=cβ

Co-propagation (-)

Counter-propagation (+)

Plasma formation (flash ionization) β

ωf,kfplasma

gasgas

plasma

€

Δω ≈ω po

2

2ω0

€

Δω =ω0

2β

1− β

Shadow images of a relativistic front

t0

t1

Intense laser pulse in a gas jetRelativistic ionization fronts

(Experiments done in collaboration with LOA, France)

Counter-PropagationCounter-Propagation Co-PropagationCo-Propagation

Laser pulse 65 fs, 2.5 mJ @ 620 nm

Simultaneous measurement: β=0.942, n=4.26 x 1019 cm-3

Dias et al. PRL (1997)

Photon acceleration

Photon dynamics in a wake field

Phase space plot

frequency shift

Laser probe diagnostic for laser accelerators

Wakefield = relativistic electron plasma wave produced by a pump laser pulse

Photon trapping in laser wakefield

R. Trines (simulations)

C. Murphy (experiments)

Confirmed by PIC code simulations

Laser wake field experiments at RAL

C. Murphy et al. PoP (2006)

Photon scattering by a ionization front

Beta0.997

nAr

1x1019 cm-3

€

ωr = ωiγ mirror2 (1+ β mirror )

2

Photon Scattering by a relativistic plasma bubble

x

z

€

Inv. = k 2c 2 + ωp2 (

r η ) −

r v ⋅

r k

€

ωs = ω0

1+ β 1+ ωp 02 /ω0

2

1− β cosθ 1+ ωp02 /ωs

2

Scattered frequency

€

r =

rr −

r v t

X=ωs/ωi

β=0.99

0.95

Photon-bubble scattering problem

(k0,ω0)

(ks,ωs)

v

€

ω p2 = ωp 0

2 1−εf (r r −

r v t)[ ]

€

f (r η ) = exp −

η 2α

2a2α

⎛

⎝ ⎜

⎞

⎠ ⎟, α =1,2

€

S(ω0,ω,θ) =| E(ω,θ) |2

| E0(ω0) |2

Cosmic Temperature map

WMAP_2008

Photon acceleration in a gravitational field scenario

Sachs-Wolfe effect is photon acceleration

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Low frequency shift limit

Low plasma density limit

Mendonça, Bingham and Wang, CQG (2008)

Moving perturbations QuickTime™ and aTIFF (Uncompressed) decompressor

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Dynamical invariant

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QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.Purely vacuum perturbations

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This is not a Doppler correction!

Generalized Sachs-Wolfe

Perturbed flat space time

ijijij hg +=

Weak gravitational wave

ahh =−= 33

Photons in a Gravitational wave

€

ω =kc 1+a

2

ky

k

⎛

⎝ ⎜

⎞

⎠ ⎟

2

−kz

k

⎛

⎝ ⎜

⎞

⎠ ⎟2 ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

⎧ ⎨ ⎪

⎩ ⎪

⎫ ⎬ ⎪

⎭ ⎪

Photon dispersion relation

02460246810-1-0.500.510246

Nearly parallel photon propagation

x

y∫+

=t

y dttk

tackty '

)'(

)'(1)(

k

kc

dt

dx x= t

a

k

k

dt

dk yx

∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

1

Parallel photon motion

Perpendicular photon motion

Typical photon trajectories

Photon acceleration by gravitational waves

Gravitational wave frequency: =104 s-1

and amplitude A =10-4

=Act/2

dt

dkc

dt

d x=ω

Conclusions

• Photon acceleration is a first order effect;

• Experimental evidence: relativistic fronts, wakefields (plasmas), self and cross phase modulations (optics);

• It can be seen as space-time refraction;

• It can also be seen as a scattering process;

• Interaction with Gfields and Gwaves ( ray bursts, Sachs-Wolfe effect);

• Applications: plasma diagnostics, ultra-short laser pulses, sub-cycle and attosecond optics, tunable radiation sources.