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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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.