Dynamique d'un plasma non collisionnel interagissant avec une impulsion laser ultra-intense

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Submitted on 9 Mar 2015

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Dynamique d’un plasma non collisionnel interagissant

avec une impulsion laser ultra-intense

Rémi Capdessus

To cite this version:

Rémi Capdessus. Dynamique d’un plasma non collisionnel interagissant avec une impulsion laser ultra-intense. Autre [cond-mat.other]. Université Sciences et Technologies - Bordeaux I, 2013. Français. �NNT : 2013BOR15268�. �tel-01127949�

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22 2 22 2

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χe∼1

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t t t t a0 ne Ex

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℄ ℄ 1025 2

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℄ ℄ 22 2

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℄ ℄ ILaser>1024 2

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×

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1023 2 22 2 ℄ ℄

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℄ F =τr ˙ F ℄ ℄ ℄ ℄ ℄ ℄ ℄ ℄ ℄ ℄ ℄ ℄

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℄ ℄ ℄ ℄ ℄ ℄ ℄ ℄ ℄

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1022 2 10 22 2 t

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χe∼ 1

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xµ _(ct_,x) ˙ xµ _γ e(c,v) aµ _γ4 ev.˙cv,γe4v.˙c2v.v+γ_e2dv_dt Pray≡(FL.v,FL) pµ _(γ emec;pe) jµ _(cρ_,ρv) Aµ_=(A0_,A) A µ Fµν₌∂Aν ∂xµ− ∂Aµ ∂xν Fµν ∂Fµν ∂xl +∂F νl ∂xµ +F lµ ∂xν=0

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Fµν E B Fµν₌    

0 −Ex −Ey −Ez

Ex 0 −Bz By Ey Bz 0 −Bx Ez −By Bx 0     ∂Fµl ∂xµ =− 1_cǫ₀jl, ǫ0 ǫ0≃8.85×10 −12 ∇×E=−∂B dt, ∇.B=0, ∇.E=ρ/ǫ0, ∇×B=µ0j+_c1₂∂E_dt, m F µν maµ₌₋e cFµνx˙ν ≡ ≃1.6×10 −19 m0≡ 9.11×10 −31 me≡ m0 λL≡ ωL≡

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γe≡(1−ve2/c2)−1/2 FLe≡ (E+v×B) ne ni ωpe≡ _mne_ee_ǫ2₀ 1/2 τr≡_6πǫ₀e_m2_e_c3 τr≃6.2×10 −24_s ǫ0=8.85×10−12 µ0=1/(ǫ0c2) ℄ A0_(r,t_{)= 1} 4πǫ0 ρ(r′_,t′ r) r−r′d3r′ A(r,t)=_4πµ0 J(r′,t′r) r−r′d3r′ tr=t+1 cr−r′

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tr t r−r′_/c A(r,t)E B E=−∂A_∂t−∇r.A0,B=∇r×A, ρ(r,t)=−eδ(r) J(r,t)= −eδ(r)ve E B E(r,t)=−_4πǫe 0 (n−β_e) γ2 er2κ3 t=tr −_c4πǫe 0 n κ3_r× (n−βe)× ˙βe _t=t_r, B(r,t)=n×E, κ=(1−n.βe),n= r−r ′ r−r′ = R_R . E r−r ′2 r−r′ A = E×B µ0 R Rn n Θ r v˙e =E×B_µ 0 = 1 cµ0   −e c4πǫ0 n× n× ˙βe R   2 t=tr n. dPray dΩ = 38πτrmev˙2e 2Θ.

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Θ Ω Pray=τrmev˙e2. Pray=meτraµaµ. ve 0 v µ_=(γ ec;γev)=(c;0) γe=1 Pray=_6πǫ1 0c3m2ee 4 _Fi0_F i0−FikFik= τ_mr ee 2_E′2_,k∈[1_,3_], E′ Pray E′_{= γ} e(E+v×B), P=τr meγ 2 ee2(E+v×B)2−(βe.E)βe (E,B) ω mec 2 E=

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0 B PT= τ_mr eγ 2 e(eE)2. σT= 8π 3( e 2 4πǫ0mec2) 2₌8π 3r02 r0 PT=cǫ0σTγe2E2. (v×B) 2_=s_in2_(ψ)v2_B2 P = τr mee 2_v2 eB2γe2 2(ψ), UB= B2 2µ0 ψ=(v;B) ψ [0,2π] P =2₃σTUBcγe2β2. (ve → c) Pv→c=2₃τrc 2 mee 2_B2_γ2 e. dε dt=−P ǫ

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ψ=π/2 dγe dt=− τre 2_B2_v2 m2 ec2(1−v 2 e c2) dε dt=−τrω 2 B mec2 ε 2_−(m ec2)2 ωB=eB_m e. mec 2 dε dt=−τrω 2 B mec2ǫ 2 dε ε2=−τr ω2 B mec2dt

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ε(t)= 1 (1 mec2 t t0+α) +mec2 t−1 0 =ωB2τr α t=0 t→ ∞ ε→ mec2 εc=(γe−1)mec 2 α t=0 ε(t=0)=γemec 2_| t=0 ε(t) α= 1 (γe−1)mec2|t=0 = 1 εc|t=0 ε(t)= (γ0−1) (γ0−1)(_tt₀)+1+1 mec 2 εray εray(t)=ε(t=0)−ε(t). ε(t=0) = γ0mec2 ε(t) εray(t)= (γ0−1) 2_(t/t 0) (γ0−1)(t/t0)+1 mec 2_. t = t0 (γ0−1) t1/2

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0 1 2 3 4 5 6 7 8 9 10 temps (ns) 0 10 20 30 40 50 En er gi e r ay on ne e ( Me V)

Energie cinetique (sans pertes) Energie rayonnee

Energie cinetique (avec pertes)

Energie = f(t)

1 electron. gamma=100 et B=100MG

t1/2 E= E0 √ 2cos(ωLt−

kx)ey+E√0₂sin(ωLt−kx)ez

ve= veex a4 0ωLτr∼1 dγe dt=−ω2Eτrγe2 ωE= eE_2m0 ec=a 0ωL 2 γe(t)=_1+γγ0 0ωE2tτr

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εray(t)=(γ0−1) τ_1+τrγ0a0/4ωL rγ0a0/4ωL t = (τrγ0a0/4ωL)−1 γ0= 1+a20 dP′ dΩ′= 3 2meτra′2sin2(Θ′) Θ′_≡−_a→′_;−→_n θ′ θ µ ≡ cos(θ) =µ′_+β_e 1+βeµ′ dµ= dµ′ γ2 e(1+βeµ′)2 dW=

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γe(dW′+cdp′x)=γe(1+βeµ′)dW′ dP dΩ=dP ′ dΩ′( dt′ dt)γe3(1+βeµ′)3. dt=γedt′. dt=γe(1−βeµ)dt′.

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dP dΩ= dP ′ dΩ′(dt ′ dt)γe3(1+βeµ′)3=3meτrγ 4 e(a2⊥+γe2a2//) 8π sin2(Θ′)γ2e_γ16 e(1−βeµ) −3 (1+βeµ ′_{) =} 1 γ2 e(1−βeµ) sin 2_(Θ′_{) =s}_in2_(θ′_{) =} sin2_(θ) γ2 e(1−βeµ)2 a ′₌ γ2 e(a⊥+γea//) dP// dΩ = 3meτra2_// 8π(1−βeµ)5sin 2_(θ) γe≫ 1 (1−β e.n/c) (1−βecos(θ)) θ→ 0 1−βecos(θ)=1₂1−βe2+θ2 +O θ2 θ∼ 1−β 2 e 1/γe2 ℄ θ sin(θ) µ sinθ∼θ µ∼1−θ2 2 βe=1−2γ12 e+O 1 γ2 e γe(1−βeµ)≃(1+(γ_2γe_eθ)2) dP// dΩ = 12meτra2_//(γeθ)2 π(1+(γeθ)2)5γ 8 e dP⊥ dΩ = 3me τra2⊥ 8π(1−βeµ)3(1−sin 2_(θ)cos2_(φ) γ2 e(1−βeµ)2) φ= ωB γet dP⊥ dΩ = 3me τra2⊥ π(1+(γeθ)2)5γ 6 e(1−2(γeθ)2cos(2φ)+(γeθ)4) cos(2φ)≃ 1 φ= ωB γet≃ ωB γe△t≃ 1 γ3 e ≪ 1 △t

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dP⊥ dΩ =3meτra 2 ⊥(1−(γeθ)2)2 π(1+(γeθ)2)5 γ 6 e. a//=0 a⊥= ωB γev⊥ dP dΩ= 2e 2_ω2 Bv⊥2 πc3_(1+(γ_e_θ)2₎5(1−(γeθ)2)2γe4. a//= −e mecγeE// a⊥ = −e mecγeE⊥ dP dΩ= e2_τ r 2πme(1+(γeθ)2)5{8γ 2 eE//2(γeθ)2+2E⊥2[1−(γeθ)2]2}γe4. dP// dΩ = 24meτr2a2_//(γeθ)2γe10 (1+(γeθ)2)6 dP⊥ dΩ =6τrmea 2 ⊥{1−(γeθ)2}2 (1+(γeθ))6 γ 8 e dP dΩ= 6τrmeω 2 Bv2⊥{1−(γeθ)2}2 (1+(γeθ)2)6 γ 6 e

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dP dΩ= 4τee2{4γe2E//2(γeθ)2+E⊥2(1−(γeθ)2)2} πme(1+(γeθ)2)6 γ 6 e 1 γe κ ∆s=κ∆θ≃ 2κ γe. ∆ve = ve∆θ ∆t′=△s_v γemv2∆θ △s =−e×v×Bsc in(ψ) κ=∆s ∆θ κ= ωBvγsin(ψ)e ∆t ′₌ 2 ωBsin(ψ) ωB = eB me ∆t=γe(1−βe)∆t′≈ △t′ 2γe △t=_γ₂ 1 eωBsin(ψ) ψ B ∆s γ2 e 1 γe 1 γ_e2 (γe→ 1)

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ω= ωB −7 _ω B≃104 γe≈10 4 △t 10−10_s ℄ dW dΩdω= e 2 16π3_ǫ₀_c| (n×(n−β)× ˙β) 1 (1−n.β)3 t=tr eiωt_dt|2_. tr dt=dtr(1−n.β). t tr R0≫ n.r(tr) d dtr n×(n×β) 1−n.β = n× (n−β)× ˙β (1−n.β)2 dW dΩdω= e 2_ω2 16π3_ǫ₀_c| eiω(tr−r(tr).n/c)n×(n×β)dtr|2. n×(β×n) n,e⊥,e n×(β×n)=e⊥ vt_κr −e vtr κ θ. β(tr) n tr e n

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e⊥ (n,e⊥) ≃2/γ 1 γ2 βe=1− 1_2γ₂ e+O 1γ2e cosθ=1−θ2 2+O 1γ2 e tr−n.r(tr)/c=tr−κ_c θ vtr κ = 12γ2 e 1+(γeθ) 2 _t r+c 2_γ2 et3r 3κ2 +O 1_γ2 e θ<1 γe ctr<κ/γe d2_W dΩdω= d 2_W ⊥ dΩdω+ d2_W dΩdω, d2_W ⊥ dΩdω= e 2_ω2 16π3_ǫ₀_c| ctr κ 2γiω2 θγ2tr+c2 γ2_t3 r 3κ2 dtr|2, d2_W dΩdω= e 2_ω2_θ2 16π3_ǫ₀_c| iω 2γ2 θγ2tr+c2 γ2_t3 r 3κ2 dtr|2

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θ 2 γ=(1+(θγ)2) d2_W ⊥ dΩdω= 9e 2_ω2 32π3_ǫ₀_c κθ2 γ cγ2 e K 2 2/3(η), d2_W dΩdω=9e 2_ω2_θ2 32π3_ǫ₀_c κθ2 γ cγe K 2 5/3(η) η=ωκθ_3cγ₃γ3 e γe≫ 1 d dΩ → δ Ω− pp dP⊥ dΩdω=3 √ 3τrωB (α)βmec2 8π δ Ω− pp [F(x)+G(x)], dP dΩdω=3 √ 3τrωBsin(α)βmec2 8π δ Ω− pp [F(x)−G(x)], F(x)=x ∞ x dξK5/3(ξ), G(x)=xK2/3(x), x=_ωω cr S(x) 35/2 8πF(x) ∞ −∞S(x)dx=1 Prad=γ 2 eτr me (eBvesin(α)) 2 d2_P dωdΩ =Pωcrδ(Ω− p p)S(ωωcr) ωcr γ 3 e ωB

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ωcr= ncrωB ncr= 3 2γe3sinα △t △t pe ∼2/γe pe/pe S(x) ω S(x) ω≪ ωcr d2_P dωdΩ∝(ωωcr) 1/3 ω≫ ωcr d2_P dωdΩ∝ ωωcre −ω/ωcr ℄ ωcr=3₂γe3FLep×p2_e e FLe=−e(E+ve×B) FLe a0 pe=γemeve ωcr∼ 3₂γe2a0ωL a0 γe ωcr≃3×106ωL ωL ωcr

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P =F .ve

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F =meτr da dt, medv µ dτ=− emeF µν_v ν+meτrd 2_vµ dτ2 e (t/τr) a µ _Fµ ext vµ ℄ Fµ =P µν_X ν=(ηµν+v µ_vν c2 )Xν Pµν v µ Xν Xν= avν+b˙vν+z¨vν PµνXν=(ηµν+vµ_c2vν)(avν+b˙vν+z¨vν) Pµν_X ν= b˙vµ+z¨vµ v µ_v_˙ µ=0 Pµνvν=(ηµν+vµ_cv2ν)vν=0 b= −δm me=m+δm Pµν_v_¨ ν=(¨vµ+vµ_c2vνv¨ν) vνv¨ν=_dτdvνv˙ν−˙vαv˙α=−˙vαv˙α z=τr mev˙µe=Fextµ +Fµ Fµ =meτr[¨v µ_{− ˙}vαv˙α c2 vµ]

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Fµ =F µ sch+Frayµ F µ sch=meτrv¨µ Frayµ =−meτrv˙ α_v_˙ α c2 vµ Fµ ℄ τr re− 3 2 τr λc h mec λc≃ 2.43×10−12 ℄ ℄

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λc Es=mec 2 λce, χe Es fµ L γe(FLe.βe,FLe) χe χe=(−f µ LefLe,µ)1/2 eEs χe=γeF2Le−(βe.FLe)2 1/2/eEs γe χe γe Es≃1.3×10 18 Icr≃ 2.3×10 29 2 χe χe=γe 1−βe a0ωL/mec2 χe=γeeB0cβe2sin(α)/Es B0 χe γe∼a0 a0

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≃3.3×10 23 2 χe mec 2_/ecτ r≃ mec2/ere re ℄ Es λc≫ re τr dvµ dτ =m−eeF µν_v ν, d2_vµ dτ =mee ∂Fµν ∂xµvνvle+me22_e_cFµνFνlvl medv_dτµ=−eFµνvν−eτr∂F_∂xµνlvkv_el−τre 2 meF µl_F νlvν+e_m2τ_er Fνlvl(Fνmvm)vµ γe≫ γ2 e≫ ℄ dpe dt=FLe+Fsf,dxcdte=βe Fsf=−eτ_mr e[FLe×B−eE(βe.E)]− τ r mecγ 2 e F2Le−(FLe.βe)2 βe.

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ve |ve|ex γ2 e Fsf ℄ ℄ Fsf,x=− e 2 mecτrγ 2 e (Ey−cBz)2+(Ez+cBy)2 ℄ χe χe≪ ǫ Ees=1₂e 2 4πǫ0 ∞ ǫ dr r2

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x˙ µ pµ em= ˙x µ c2Ees= e 2 8πǫ0 ˙ xµ c2 ∞ ǫ dr r2 pµ0=m0x˙µ me=m0+ e 2 8πǫ0ǫc2 x˙ µ ℄ pµ_=pµ em+m0x˙µ=mex˙µ−meτrx¨µ, ˙ xµ_=vµ_+(˙_x)µ rad (˙x)µ_rad≈τrx¨µ ˙ pµ_=−e cFµνx˙ν−(˙p)µrad,˙xµ= p µ me+(˙x) µ rad. T µν ˙ pµ_+(˙_p)µ rad+_dτd Tµ0dV=0. (˙p)µ_rad d dτ Tµ0dV= ecFµνx˙ν

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(˙x)µ rad e cFµν(˙x)νrad ˙ x2_=˙_xµ_x_˙ µ=c2 p2_=pµ_p µ=m2ec2 (˙x) µ rad (˙p)µrad ℄ (˙x)µ

radx˙µ=0, me(˙x)µradx¨µ=−(˙p)µradx˙µ

(˙x)µ_rad (˙p)µ_rad p2_=pµ_p µ=m2e c2+τr2x¨2 ≤m2ec2 E2_/c2_=pµ_p µ+p2=m2ec2+p2, τ 2 r mec 2 gi ℄ d2_u ds2

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−epµ_F µν(˙x)νrad=pµ(˙pµ)rad (˙pµ₎ rad=_mPrad_e_c2pµ, (˙xµ₎

rad=_mτr_e_PP_{rad, e}rad fLµ

Prad Prad Prad,e τrfL2/me fµ L=−eF µν_p ν me Prad Prad,e ˙ pµ_=−eFµν_x_˙ ν−τrf 2 L m2 ec2p µ _,˙_xµ₌pµ me+ τr mef µ L p µ v 2_{= c}2_{− ˙}_x2 rad= c2 v 2 c2 x˙ ˙ x2_=˙_xµ_x_˙ µ=c2 1+ _mτr ec 2 fµ LfL,µ

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χe ˙ x2_≃c2 _1−1._05×χ2 e v2_{≃ c}2 χe∼ Prad F µν vµ _x_˙µ _F˜µν eτr meF µν mevµ=pµ+ ˜Fµνpν ˜ Fµν pµ_=m evµ+O(0) (τreE/mec) pµ_=m evµ− ˜Fµνvν+O(˜Fµν) ℄ p=me n 0 (−1)n _F_˜n_.˙_x F.˙˜x ˜F ik_x_˙ k ˜ F2_{.v ˜}_F.˜_F.˙_{x ˜}_Fik_F˜ klx˙l

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p µ _m ex˙µ τr mecFLe ≪ 1 mec 2_/cτ r FLe ≪ eEs pµ_=m e n 0 (−1)n F˜n.˙x=₁₊_τm_rex˙µ mec2(ve.FLe) dxe dt=1+τrve mec2(ve.FLe) +_mτr e Fµν_v ν 1+τr mec2(ve.FLe) ve 1+ τr mec2(ve.FLe) =ve−_mτr ec2(ve.FLe) ve 1+τr mec2(ve.FLe) , −eFµν_v ν FLe dxe dt =c(βe+δβe) δβe=mτrec FLe−(FLe.βe)βe 1+_mec2τr (ve.FLe)= (τr/c)γewe 1+_mec2τr (ve.FLe) cδβe t cβ/¨xe ¨ xe ¨ xe=_γ1 eme p˙e− ˙ pe.pe γ2 em2ec2pe +τrγew˙e+τrwe ˙ pe.pe γe

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we≡dv_dte[0]FLe−(F_γ_eLe_m_e.βe)βe (τreE /mec) cδβe ∼4×1025 2 δβe we δβe dpe dt=FLe−eδve×B−γe2(δβe.FLe)βe Frad ,dxe dt=c(βe+δβe) ve δve ve τr Ee dxe dt

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dEe dt =−eE.(ve+δve)−γe2(δve.FLe) δve δ(x)δve δ(x) neδve d2_I dΩdω=δ Ω− pe pe γ 2 e(δve.FLe)S _ωω cr , d2_I dΩdωdΩdω=γe2(δve.FLe) (χe≪ 1) ℄

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p x ˙ pµ_=−eFµν_x_˙ ν− τrf 2 L m2 ec2p µ_, ˙ xµ_{= p}µ me+ τ r mef µ L φ φ=nµ_x µ=nx=ωt−kx nµ₌ ω c,k =k(1,n) k=kn n=ex ∂µAµ(φ) Aµ_{(φ)= 0,0,a} 1m_eecψ1(φ),a2m_eecψ2(φ) a0≃0.85 I[1018_W/cm2_]λ2_L[_µm2_] ψ1(φ) ψ2(φ) ψ1(φ)=sin(φ),ψ2(φ)=cos(φ) a1=a2=a0/√2 a1=a0 a2=0 ˙ pµ mex¨µ−τrf˙Lµ=−eFµνx˙ν−τrf 2 L m2 ec2mep µ nµ n µ_f L,µ =0 n µ_p µ≡ menµvµ= menµx˙µ τr/mefLµ n µ_f L,µ=0 xµ x˙µ x¨µ

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nµ x µ φ d2_φ dτ2− τrfL2 m2 ec2 dφ dτ=0 φ τ dτ dφ= dφ dτ −1 =_(n_µ_v(φ)1 µ) f 2 L d2_τ dφ2= τre2 m2 ec2 ψ ′ 1(φ) 2 a2 1+ ψ ′ 2(φ) 2 a2 2 ρ≡ µ µ(φ) (φ) φ ρφ=_h(φ)ρ0 =dφ_dτ ρ0/ω=(nv0)/ω=γ0(1−β.n), γ φ0 h(φ) [φ0;φ] (a0ωLτr) h(φ)=1+ρ0τr φ φ0 dϕ ψ₁′(ϕ)2a2 1+ ψ ′ 2(ϕ) 2 a2 2 d dφ=dτddφdτ=dτdh(φ)ρ0

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Fj(φ)≡ _φφ₀dϕh(ϕ)ψj′(ϕ) vµ_(φ)= 1 h(φ) v0µ− c 2 2ρ0 h 2_(φ)−1nµ − c h(φ)[F1(φ)ζ1µν+F2(φ)ζ2µν]v0,ν +_2h(φ)ρc2 0 a 2 1F12(φ)+a22F22(φ)nµ ζ_jµν≡nµ_aν j−nνaµj v µ γ(φ)= 1_h(φ)γ0+_2γ 1 0(1−β.n)h 2_{(φ)−1 +} +_h(φ)1 _2γ 1 0(1−β.n)a 2 1F1(φ)2+a22F2(φ)2 + F1(φ)ζ10,ν+F2(φ)ζ20,νp0,ν γβx(φ)= 1_h(φ)ǫ±γ0β0+_2γ 1 0(1−βx)h 2_{(φ)−1 +} + 1 h(φ) 1 γ0(1−β.n)(a1F1γy,0βy,0+a2F2γz,0βz,0)+ 1 2γ0(1−βx)a 2 1F12+a22F22 ǫ± ǫ±=cos(k,ve) γβy(φ)=_h(φ)1 [γ0βy,0+a1F1(φ)] γβz(φ)= 1_h(φ)[γ0βz,0+a2F2(φ)] h(φ) h(φ) γ(φ)px(φ) γ0−px0 h(φ0) h(φ) γ(φ)−p_mx(φ) ec =h(φ0) γ0− px0 mec

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py pz h(φ)py(φ)−meca1F1(φ)=h(φ0)py0(φ)−meca1F1(φ0), h(φ)pz(φ)−meca2F2(φ)=h(φ0)pz0(φ)−meca2F2(φ0) ω ′ (φ) ω′₌ 1 h(φ)ρ0 (φ) ℄ xµ_(φ)=xµ 0+ρ10   φ φ0dϕh(ϕ)v µ_(ϕ)+τ r[F1(φ)ζ1µν+F2(φ)ζ2µν]vν   x µ_(φ) τ=τ(φ0)+1_ρ 0 φ φ0 [h(ϕ)γ(φ)+h(φ)ωτr(F1a1γβy+F2a2γβz)]dϕ, x(φ)=x(φ0)+c_ρ 0 φ φ0 [h(ϕ)γβx(ϕ)+h(ϕ)τrω(F1a1γβy(ϕ)+F2a2γβz(ϕ))]dϕ, y(φ)=y(φ0)+_ρc 0 φ φ0 [h(ϕ)γβy(ϕ)−ρ0τrF1a1]dϕ, z(φ)=z(φ0)+_ρc 0 φ φ0 [h(ϕ)γβz(ϕ)−ρ0τrF2a2]dϕ.

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℄ Fj E = Ecos(φ)ey φ=ωt−kx β0,y=0 β0,z=0 ρ ω 1−βx 1+βx h(φ) F1 h(φ)=1+ωτra20 2 1−β1+βxx,0,0[(φ−φ0)−(cos(2φ)−cos(2φ0))] F1=cos(φ)h(φ)−cos(φ0)+ωτra 2 0 4 1−βx,0 1+βx,0[cos(φ)−cos(φ0)] −ωτra20

2 1−β1+βxx,0,0 sin(φ)−sin(φ0)+56[cos(φ)cos(2φ)−cos(φ0)cos(2φ0)]

+ωτra20

12

1−βx,0

1+βx,0[sin(φ)sin(2φ)−sin(φ0)sin(2φ0)]

(60)

γβx φ a0=100 γ0βx a0=200 γ0βx a0=300 γ0βx γβx βxd= a2 0 a2 0+4 x a0 γ0βx,0 a0γ0βx,0 a0 γβx,0 a0≥γβx,0 a0 γβx,0 a0 γβx,0

(61)

h(φ) γ0a 2 0 γβx,0≥ 0 h(φ)−1 βxkx≥ 0 βx∼ 1 E γ2 γβx (v.E=0) v.E ψ2(φ) sin(φ) a µ 1 aµ2 cos(φ)2 sin(φ)2 h(φ)F1 F2 h(φ)=1+ 1−β_1+βx,0 x,0ωτra 2 0(φ−φ0) F1= (φ)h(φ)− (φ0)+ 1−βx,0 1+βx,0ωτra 2 0[ (φ)− (φ0)] F2= (φ)h(φ)− (φ0)+ 1−β_1+βx,0 x,0ωτra 2 0[ (φ)− (φ0)] Ey Ez γ0≫ a0

(62)

γ φ a0 γβx,0 a0 γβx,0 a0 γβx,0 γ0≫ a0 ℄ a0γβx,0 a0>γ

(63)

τr q(χe) χe a0 γe γe a0 ve×B A µ _(A0_,A) Aµ E=−∂A ∂t−∇r(A0) B=∇r×A p mec=γe(φ)− 1h(φ)γ0−p,0 p⊥= 1_h(φ) φ φ0 ∂A ∂ϕh(ϕ)dϕ γe(φ)= 1+ p 2 ⊥ (mec)2+ p2 (mec)2

(64)

γe(φ) γe(φ)= 1+_(mp_e⊥_c)2 h(φ) 2 γ0−_mecp + γ0−_(mec)p h(φ) fp=−mec2∇rΓe Γ= 1+(pe/mec)2+a20 1022 2 δfi fi=fi0+δfi fi δfi≪ fi γeme∼a0me

(65)

≃1/γ2 0ω2Bτr=τB/γ0 TB 2πγ0 ωB γ 2 0ωBτr∼ 1 ℄ e2 4πǫ0cγ0ωBτr e2 4πǫ0c ωcr= (−eve×B)×pe/p2e≃γ02ωB γ0mec 2 γ0>4πǫ0c/e2 ℄ γ0>137 γ0 B =Bez p=p⊥+p p⊥=px+py,p =pz γ0≫ 1 dp2 ⊥ dt=−τB(m2ec)2 p4 ⊥ γe, dp dt=− 1 τB(mec) pp⊥ γe τB≡(τrωB2)−1 ωB≡ eB me γe γ0≫ 1)γe(t) γ(t)=γ0 1+tγ_τ0 B −1 γ γ ℄

(66)

u=p2 ⊥ du u2=− 2 (mec)2τBγ0 1+ tγ0 τB dt p⊥= p⊥,0 1+2p2⊥,0/(mec)2 γ0τB t1+2τBtγ0 1/2 γ2 e px+ipy dX dt=iωγe(t)B X px py px= ωB γ0t1+ γ0t 2τB p⊥,0 1+2p2⊥,0/(mec)2 γ0τB t1+ tγ0 2τB 1/2 py= ω_γBt 0 1+t 2_γ 0 2τB p⊥,0 1+2p2⊥,0/(mec)2 γ0τB t1+ tγ0 2τB 1/2 p⊥ p p⊥ p⊥ γ(t)dt p⊥ γ(t)dt γ(t) p⊥ p⊥ γ(t)dt= pγ⊥00 1+γ0t τB 1+2p2⊥,0/(mec)2 γ0τB t1+ tγ0 2τB 1/2dt = τB(m_pec)2 ⊥,0 1+ 2p2 ⊥,0/(mec)2 γ0τB t 1+tγ 0 2τB 1/2

(67)

p p =p,0 _pm_⊥,0ec−m_p_⊥,0ec 1+2p 2 ⊥,0/(mec)2 γ0τB t1+ tγ0 2τB 1/2 p(t=0)=p,0 p⊥/p⊥,0 ωBt B 9 _γ _{B 10}5 γ=1000 p⊥ p⊥ p Q=γ2 0τrωB p⊥ p

(68)

p/p,0 ωBt B 9 _γ _{B 10}5 γ=1000 B B 9 _γ _{B 10}5 _γ=1000

(69)

Tp Tp= 1+4πQ−1τ_γB 0, τB= ωB2τr−1 τB Q≪ 1 Tp= 2πγ0 ωB γ(t) 1/(1+γ0t/τB) γ(t)/γ0 (1+4πQ) −1/2 ℄ θi θi=   v v2_+v2 ⊥   θi ωt B 9 _γ _B 5 _γ ≃ 1fs t≃10fs

(70)

t <1fs θi≪ 1/γ ≃ 2/γ ℄ t≃4ps ℄ fe

(71)

dN dt=0 N = _Ωfe(r,pe,t)drdpe fe dΩ rdpe Ω {r,pe} ∇ ={∇r,∇pe}= ∂ ∂r,∂p∂e U={Ur,Upe}= dR dt={˙r,˙pe}={c(βe+δβe),FLe+F } R dN dt= Ω ∂fe ∂t+∇.(feU)dΩ=0 Ω ∇fe(r,pe,t).UdΩ= S(Ω) fe(n.U)dS n= Ω Ω S(Ω) Ω fe S(Ω) ∇ U ∂fe ∂t+∇r.(fec(βe+δβe))+∂p∂e.(fe(FLe+F ))=0 ℄ ℄ ∇.(fU)=∇f.U+f∇.U,

(72)

∂ ∂tfe+c(βe+δβe). ∂ ∂rfe+(FLe+F ). ∂ ∂pefe+ fe _∂r∂.(cβe+cδβe)+ ∂_∂p e.(FLe+F ) =0 ℄ dVp dVp dt= Ω ∇pe.dp_dtedΩ ∂ ∂pe.FLe ∂ ∂pe.F ∂ ∂tfe+c(βe+δβe).∂r∂fe+(FLe+F ).∂p∂efe=−fe ∂ ∂r.(cδβe)+∂p∂e.(F ) −fe∂F_∂p_e fγ    pγ ωΩ_c εγ _R3dpγ ωfγ dpγ _cω 2d _cω dΩ

(73)

χe≪ 1 ∂fγ ∂t+cΩ·∇rfγ= τrc3 4_ω3 R3dpefe(pe)γ 2 eFLe 2 meωcr 1− (ψ) 2_β2 e δ Ω−p_pe e S ωωcr ψ=(ve;E) Ω ω χe≪ λcr= 2πc ωcr ∼n −1/3 e

(74)

∼ 10 22_× (mi/me)2 2 _∼3×1028 2 Ex χe≪ 1 fe fγ fi (a0ωLτL) ∂fe ∂t+∂r∂.(fec(βe+δβe))+_∂p∂ e.(fe(FLe+F ))=0, ∂fγ ∂t+cΩ·∇rfγ= τrc3 4_ω3 R3dpefe(pe)γ 2 eFLe 2 meωcr 1− (ψ) 2_β2 e δ Ω−p_pe e S ωωcr , ∂fi ∂t+∇r·(ficβi)+_∂p∂ i·(FLifi)=0. χe∼ τr (χe) χe

(75)

εγ= _R3dpγ ωfγ pγ Fγ Fγ= R3 dpγcωfγΩ γ 2 e F F =γ2 eA(α,t)1− 2_(ψ)β2 e ve+O(a0ωLτL) A(α,t)=τrωLG(α,t)a20mec2ωL, α=Ex/Ey, G(α,t)= +α2 2 +α2 g(t) S(x)dx=1 ∂ ∂tεγ+∇r.Fγ=Wγ

Wγ=A(α,t)[I1−I2]

   I1= _Rγe2fedpe I2= _R 2_(ψ)(γ2 e−1)fedpe ψ≡(E,ve)

(76)

γe fe=_θKne 2(1/θ) −γ e−1 θ θ= Te mec2 K2(1/θ)≃2θ2 θ≫ Wγ Wγ≃6neA(α,t)θ2 ℄ Te= 1+a20−1 mec2 a0≫ 1) a0mec 2 Wγ a0 Iγ Wγ l Iγ= lrad Wγdx=lradWγ l ne

(77)

     w(t,r)=mec2 (γe−1)fe(t,r,pe)dpe : u(t,r)=1 2 ǫ0E2+µ10 B 2 _: Wγ : w ∂w ∂t=mec2 R3 (γe−1)∂f_∂tedpe ∂w ∂t=−∇r. R3 mec2(γe−1)vefedpe−∇r. R3 mec2(γe−1)δvefedpe − R3 mec2(γe−1)∇pe.(fe(F +FLe))dpe R3 (γe−1)∇pe.(fe(F +FLe))dpe=

(78)

[fe(γe−1)]_R3− R3 ∇pe(γe).(FLe+F )fedpe ∇peγe= ve/mec2 ∂w ∂t=−mec2∇r. R3 (γe−1)vefedpe−mec2∇r. R3 (γe−1)δvefedpe + R3 ve.FLefedpe+ R3 ve.F fedpe ∂u ∂t=−∇r. −j∗e.E ≡ 1 µ0(E×B) j∗e=−e R fe(ve+δve)dpe ∂E ∂t+∇r.σ= −δje.E (1) −A(α,t)   R3 γ2 e−1 1− (ψ) 2_β2 e fedpe   (2) −mec2∇r. R3 (γe−1)γeδvefedpe (3) +τrωpe2 c2_n_e .je (4)        E=w+u σ=mec2 (γe−1)vefedpe+ je=−e_R3vefedpe δje=−e_R3δvefedpe

(79)

σ Wγ εγ ∼γ2 e t t ,1e− t ,1e− =(γ e−1)mec2 γ2 e(δve.FLe)∼ 1 2πγeωLτra20TL

(80)

Wγ nemec 2 r t ∼ Wγ nemec2 −1 = mec2 6A(α,t)θ2 A(α,t)=τrωLG(α,t)a20mec2ωL t ≃ m ec2 6A(α,t)θ2= 1 12πτrωL2G(α,t)a20θ2TL TL pe/(mec)∼ γe θ∼ a0 G(α,t) a0 t ≃ 1 12πτrωLG(α,t)a40TL fe

(81)

1021 1022 1023 1024 I_laser (W/cm2) 10-6 10-4 10-2 100 102 tcooli ng /TL I ∇p e.F ℄ S=− Ω felnfedΩ fe d dt=∂t∂+c(βe+δβe)._∂r∂+(FLe+F )._∂p∂ e d dtlnfe=−∇pe.F −∇r.cβe dS dt= Ω [∇pe.F +∇r.δβe]fedΩ

(82)

∇pe.F =− p2 γem2ec2τrsin 2_(ϕ)ω2 B 3−_γ2₂ e ≤0 ∇r.δβe=−eτ_mr e ∂2 ∂r2(ve.A) ϕ≡(ve;B),ωB=eB_m e ∇r.A ∇r.δβe A 0 A E B ∇r.δβe= eτ_mr e ∂2 ∂r2A0+ ∂ 2 ∂r2(ve.A)+ eτ r mec∇r. E. veve c2 +O(a0ωLτr) δne ∂2 ∂r2A0= −eδne ǫ0 ∇r.δβe=−τrωpe2δn_ne e+eτ r me ∂2 ∂r2(ve.A)+eτ_m_er_c∇r E.veve c2 +O(a0ωLτr) ∇r.δβe ∇pe.F γ2e ∇r.δβe

(83)

(84)

I≥Icr >Icr ℄ ℄ ≃

(85)

A+e−_{→ A}′_+2e−_+e+ Z+γ→ Z+e−_+e+ ℄ η χ µ Nγ ℄

(86)

γL e+nγL→ e′+γ ωL ≃ λL ω′ L ℄ ℄ ωL γemec 2 εγ εγ=4γ2ωL ε′ γ=4γ∗2ωL γ∗=γ 2_/(1+a2 0)

(87)

m∗≃me 1+a 2 0 ℄ I 2.2×1017 2 <γe>≃105 a0<1 ℄ ℄ dσ dΩ=r 2 e 2 ω ′ ω 2 _ω ω′+ ω′ ω− 2θ ω′₌ ω 1+ ω mec2(1− θ) ω ω ′ re=_4πǫ₀e_m2_e_c2 Ω σ≈σT 1−2x+26x 2 5 +... , x≪ 1 σ≈ 8₃σT1_x ln(2x)+1₂ , x≫ 1 x=_mω ec2,σT:

(88)

10-4 10-3 10-2 10-1 100 101 102 103 104 105 106 x= hν/m_ec2 10-6 10-5 10-4 10-3 10-2 10-1 100 101 σ/ σ T x= ω/mec 2

χ

e

∼ 1

χe χe∼1 k

(89)

dpµ dτ=−ecFµνvν+ImQEDec2 kµ (k.p)mec−pµ IQED=Icl/(1+1.04χe)4/3 Icl=_mτr eF 2 Le χe τr ℄ τr→ τrIQED_I cl =τrq(χe) q(χe)=_(1+1._04χ1 e)4/3 0 1 2 3 4 5 χ_e 0 0,2 0,4 0,6 0,8 1 q( χ e ) (χe) χe (χe) χe∼

(90)

τr (χe) χe τr ℄ χ 2 e χe∼1 dIQED dωdΩ=Icl 35/2 8πr   ∞ r_χe K5/3(r′)dr′+χ2errχeK2/3(rχe)  , IQED=Iclq(χe),r= ω_ω cr rχe=r/(1−χer) × 22 2 ωc0≃ λL µ

(91)

me→ me 1+a20 ωcr∝m−1e ω≫ ωc0 rχe = _1−χr_e_r χe∼ rχ e≥r= ω ωcr ω≤102_ω c0 τr q(χe) τr

(92)

(93)

(94)

(95)

fe(r,pe,t) λD≡(Teǫ0/(nee2))1/2 ≡ n −1/3 e neλD neλ3D≫ 1 1010 neλ3Dω−1pe ∂fα ∂t+ ∂∂r.(feUα)+ ∂∂pα.(feFα)=0 α α→ e α→ Fe FLe F −e(E+ve×B)−ecδβe×B−γe2(δβe.FLe)βe Ue=ve+δve Fi FLi Ze(E+vi×B) Ui=vi ρ= α qα fαdp j= α qα fαvαdp.

(96)

℄ ℄ ∆x<3λDe. dpα dt =Fα dx dt=U ℄ pn+1/2 α =pnα+q₂αEn∆t

(97)

pn+1/2 α =pn+1/2α +qαvnα×Bn∆t,vn+1/2α =p n+1/2 α mαγα pn+1 α =pn+1/2α +q₂αEn∆t ℄ ℄ ℄

(98)

(99)

∆t pn+1 e = ˜pen+1−δpne δpn e=_m1 ecγ˜e n+1_(˜_δβn+1 e ·Fn+1Le )˜pen+1∆t+ ˜δβen+1×Bn+1∆t ˜ pn+1 e δpn e ∇pe.F ≤0, F = −ecδβ e×B−γe2(δβe.FLe)βe xn+1/2 e −xn−1/2e =(βne+δβne) ℄

(100)

℄ ωpe∆t<2 ∆x ω<c/∆x λL

(101)

λmax 2πc/ωmin ≃n−1/3e 2πc ωmin ≪ n −1/3 e nc nc 1.1×1021 −3_λ2 [µm] ωmin≫ 2.75×10−3 n_ne c 1/3 λ−2/3_L_,[_µm] ne nc λL µm 10−2 a0≫ 1 γe≫ 1 ωmin ωs= 2π/∆t ωre= pe×F_p₂Le e ∼ a0 γeωL γe ωre∆t γe

(102)

∆t ∆t≤TL a0 ωcr,j nj ∆˜Prad,j j j j=ne0(∆x) α Nmpm ne0 ∆x N α ω2 p, = je 2 jmeǫ0× N (∆x)α= j N (∆x)α =ne e2 meǫ0=ω 2 p, j d2_P rad dΩdω≡ _j j∆˜Prad,j

(103)

∆˜Prad,j n dΩ dω 2/γe δ Ω− pe pe kL θ φ θ= px (p2 x+p2y+p2z)1/2    φ= py p2 y+p2z py≥ φ=2π− py p2 y+p2z py<0 θ φ θ φ ω ω θ φ 0≤θ≤π 0≤φ≤2π ω θφ

(104)

θ φ θ φ [∆E] ∆E= |E −E | E ∆E I ×10 23 2 _a y=az=200 ne nc Max[∆E] ω <1/∆t

(105)

[∆E] l=λL l= λL l λL ne nc ne nc 10−4 [∆E] a0 λL µ ne nc l 100λL lλL ∼ ℄ ℄ ℄ l= λL

(106)

ne nc Temec 2 ₁₀−2 8×1022 2 TL t=0 t=100TL ℄ mec mec px,i≤ mic mic t=100TL

(107)

-100 -75 -50 -25 0 25 50 75 100 x/λ_L -20 0 20 40 60 eE x/ m e cω L t=100TL Ex t=100TL ≤ Ex γe 1−βe,

(108)

χe

(109)

(110)

(111)

℄ ℄ ℄ ℄ ℄ ℄ ℄ 10 22 2 ℄ ℄

(112)

εk(t)= t 0 dt′_I k(t′) ηk=εk/εL, γ εγ εe εi ζk=dηk/dt/TL, ζγ ζe ζi TL nc=ωL2meǫ0/e2 ω2_=ω2 pe/γe +k2c2, ωpe= _mne_ee_ǫ2₀ γe N N 1−ne nc nc= meǫ0ω 2 L/e2

(113)

ne>nc γe ≫ 1 γe <γe>≃(1+a 2 0)1/2 ℄ nc nc(1+a 2 0)1/2 ℄ ℄ a0≪ 1 l=1µ ℄ ℄ cβp

(114)

℄ ℄ ℄ 2ωL ℄ βp ω ′ L

(115)

2I′ c I I′ I βp I′_=I(1−βp) (1+βp) 2I′_/c x ℄ 2n′ ivpmaγpvp=2ρc2γp2βp2, ρ=(mi+Zme)noi noi βp ℄ ℄ I ρc3 1−βp 1+βp=γ 2 pβp2 B =(I/ρc3₎1/2 ℄ βp= B_1+B ℄ ay∝a0/√ne I=8×1022 2 l=100λL ne0 nc λL

(116)

t=100TL TL ℄ a0∼n_ne c l λL

(117)

℄ ℄ I l d dt(βLSγLS)=2I(t−XLS /c) ρlc2 R(ω′)1−β_1+βLS_LS, γLS (1−βLS2)−1/2 dXLS/dt=VLS I ρ=(mini+mene) l R(ω ′₎

(118)

ω′ _{ω (1−β} LS)/(1+βLS) βLS=(1+K) 2₋₁ (1+K)2₊₁, K=_ρ2F_lc2=2π Z Ammep a2 0τ ξ, Idt=a2 0τ τ ξ πne nc l λL ¯ ξ=πn_ne c l λL× 1a0= ξa0 ξ ∼ ¯ ξ≥1 η ℄ η=2βLS/(1+βLS), β=VLS/c VLS (β→ 1) η ω N ω N ω ′ ωr ωr 1−βLS 1+βLSω 2 βLS 1+βLSN ω N ω η ne0=100nc l=0.5λL

(119)

TL ne=100nc l= λL ×1022 2 ×1022 2 _a 0=136 ξ≃ a0 ¯ ξ > Ex 10 22 2 _3._3×1023 2 ne nc l=100λL λL

(120)

TL λL λL mi/me 0 10 20 30 40

t

/T

_L 0 10 20 30 40

t

/T

_L 1 10 100 1000 10000 1e+05 1e+06 1e+07 1e+08 1e+09 1e+10 Ra di at ed e ne rg y [ J/ c m²]

with self-force without self-force

a) b) λL 10 21 2 1022 2 _×1022 2 _×1023 2 10 22 2 ∼ 2 5×108 2 3.3×1023 2

(121)

× 1022 2 _×1023 2 t=50TL 8× 1022 2 _3._3×1023 2 χe≤ δve

(122)

℄ TL I 1.1×1023 2 ne0=10nc ne0/nc≪ a0 l= λL l= λL (px,py) t=0 τr(γe2/me)FLe 2 ω2 Lτrγe2a20mec2 px,e γ 2 e py pe,x≫ pe,y γ 2 e γ2 e[FLe−(FLe.βe)βe] γ 2 eFLe 2 e,y≫ e,x≫ mec γ 2 e(1−βe2)eELe 2 eELe 2 ℄

(123)

-200 0 200 400 600 γβ_x -300 -200 -100 0 100 200 300 γβy -400 -200 0 200 400 600 800 -300 -200 -100 0 100 200 300 γβy Electrons a) c) Circulaire Lineaire -0,2 -0,1 0 0,1 0,2 0,3 0,4 γβ_x -0,05 0 0,05 γβy -0,4 -0,2 0 0,2 0,4 0,6 -0,06 -0,04 -0,02 0 0,02 0,04 0,06 γβy Ions b) d) Circulaire Lineaire (px,py) l=1λL ne nc t=40TL I ×10 23 2 t≃ TL t≃ TL Ex Ex∼ene0l/ǫ0≪ vxBz∼a0Ec=EL, Ec=mecωL/e

(124)

py,i=vx,iBz∝eAy(x−ct)/mi x,i x,i Ex ωL (px,py) l=100λL ne nc t=100TL I ×10 23 2

(125)

l TL TL ℄ Ex FLe pe,x> 0 pe,x<0 py 2ωL ∼

(126)

≤ 0 Ex Ay pi,y∝Ay

(127)

TL l λL ne=10nc pi,x>0.5 ic

(128)

λL l= λL ne 100nc a0 a0,y a0,z ℄

(129)

ne,RR ne,sans RR∼ εe−_{,sans RR} εe−_,RR ∼ 1 1− 1 t ωL Ey 2TL r∼ 1 2TL

(130)

βLS ω ′ L ω′ L=ωL 1−β_1+βLS LS λ ′ L

(131)

82 83 84 85 86 87 88 89 90 x/λ_L 0 1 2 3 4 5 6 7 8 9 10 82 83 84 85 86 87 88 89 90 x/λ_L 0 1 2 3 4 5 6 7 8 9 10 with RR _w_i_thou_t_RR a) b) Ex/Ec ni/ni0×50 ne/ne0×50 γ2 e γe(1−βecos(θ)) θ R ξ a0

(132)

t= TL ℄ R≃ ξ2/(1+ξ2) a0≤(1+ξ2)1/2 ξ2_/a2 0 a0≥(1+ξ2)1/2 ∆max

∆max≡ pi,rad.−_p pi,norad. i,norad. , pi,rad pi,norad ∆max

(133)

82 83 84 85 x/λ_L 0 0,05 0,1 0,15 0,2 n i /n 0 -60-40-20 0 20 40 60 80 x/λ_L -2 0 2 4 6 8 γβ i, x TL l= 1/8λL × 22 2

(134)

1022 1023 1024

I

(W

/cm

2

)

0 50 100 150 200 250 300 350

∆

ma x

(

%)

l/λ_L= 0.3 = 0.5 = 1 ∆max ne nc ∆max 1022 2 1023 2

(135)

µ (lene)2 l λL ∼ ξ a0 ℄ a0 ∼

(136)

-50 0 50 100 x/λ_L -2 0 2 4 6 8 -50 0 50 100 x/λ_L -2 0 2 4 6 8

with radiationlosses without radiationlosses

a) b) -100 -50 0 50 100 x/λ_L -5 0 5 10 15 -50 0 50 100 x/λ_L -5 0 5 10 15

with radiationlosses without radiationlosses

c) _d) l= λL ne nc I ×10 22 2 I ×10 23 2

(137)

t= TL 10nc λL t= TL l= λL ne nc

(138)

l=100λL ne0 nc 0 20 40 60 80 100

t

/T

_L 100 101 102

η

e

(

%)

-8 -6 -4 -2 0 2 4 6

log[

ε

_γ

/m

_e

c

2

]

10-3 10-2 10-1 100 101 102

ζ

γ

∗m

e

c

2

(

%)

0 20 40 60 80 100

t

/T

_L 100 101 102 103

ζ

γ

(

%)

-30-20-10 0 10 20 30

x

/λ

_L 10-5 10-4 10-3 10-2 10-1 100

ζ

γ

*l

m

(

%)

(a) (b) (c) (d) t=40TL

(139)

χe

(140)

Ey t=80TL

(141)

t=20TL βpc vb≈1₂a0c menc/mini ℄ t=40TL ω≫ ωcr t=50TL 10 4 ₁₀2 pe,x 10 4_m ec 102mec pe,x TL ∆max ∆max ∆max

(142)

l=100λL × 22 2 ne nc

(143)

I=Ir+Iγ+Ie Ir Iγ Ie Iγ Wγ εγ Iγ Wγlrad nel α≡Ex/Ey Iγ I

(144)

0 20 40 60 80 100 t/T_L 0 10 20 30 40 50 60 70 η e , η γ ( %) 0 20 40 60 80 100 t/T_L 0 10 20 30 40 50 60 70 a) b) ηγ ηe l=100λL ne0=10nc It=I−Iγ It c+ Ir c 1−βp 1+βp=2γ 2 pβp2ρc2 R(ω′₎ Ir=R(ω′)It (R(ω ′_{) =1)} It ρc3 1−βp 1+βp=γ 2 pβp2 βp= Γ_1+Γ

(145)

Γ= I_ρc₃ 1/2 1−I_Iγ 1/2≃   a20nc ne 1+_mm_ei   1/2 1−12πa2 0n_ne c l λL(ωLτr) 1/2 Iγ I=a 2 0mec3nc l 1022 1023

I

_laser

(W

/cm

2

)

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 n_e/n_c= 10 = 50 = 100 1022 1023

I

_laser

(W

/cm

2

)

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7

β

p n_e/n_c= 10 = 50 = 100 a) b) βp 1− Iγ I l l

(146)

γeme∼a0me l ne nc Γ I ρc3 ℄ R=1−β_1+βp p= 1 1+2Γ (1−R)=_1+2Γ2Γ , ηi ηi=1−R ℄ ǫi=mic2(γi−1)=2mic2 Γ 2 1+2Γ ηi (1−R)

(147)

1022 1023

I

_laser

(W

/cm

2

)

0 20 40 60 80 ne/nc= 10 = 50 = 100 1022 1023

I

_laser

(W

/cm

2

)

0 20 40 60 80

η

i

(

%)

n_e/n_c= 10 = 50 = 100 a) b) ηi 10 23 2 Iγ lrad λL N x

(148)

-25 -20 -15 -10 -5 0 5 10 x/λ_L -1 -0,5 0 0,5 1 n_e/n_c E_y/E_c*a₀-1 x TL TL TL ℄ ℄

A(x,t)=a(t)Re yei(ωt−kx)

(149)

1.4×1022 _5._5×1022 2 _λ L µ 2.5×10 9 2 _t=0 -14 -13 -12 -11 -10

x

/λ

_L -100 -80 -60 -40 120 130 140 150 160 170 180

θ (°)

10-4 10-3 10-2 10-1 100 101 ζ γ /st r ( %) 120 130 140 150 160 170 180

θ (°)

-4,6-4,4-4,2 -4 -3,8

x

/λ

_L -100 -80 -60 p x /( m e c) t= 5 T_L _t=₁₅_T L t= 5 T_L t= 15 T_L (1) b) a) c) d) γ0 a0 θ 2/γe

(150)

-8 -6 -4 -2 0

x

/λ

_L -200 -100 0 100 200 300 0 30 60 90 120 150 180

θ (°)

10-4 10-3 10-2 10-1 100 101 ζ γ /st r ( %) 0 30 60 90 120 150 180

θ (°)

-4 -3,8-3,6-3,4-3,2 -3

x

/λ

_L -100 -50 0 p x /( m e c) t= 5 T_L _t=₁₅_T L t= 5 T_L t= 15 T_L (2) a) b) c) d) γ0 a0 θ a0 >γe t=15TL θ θ t=25TL ≃ 150

(151)

0 100 200 300 400 500 ε_e-/m_ec2 t= 5 T_L t= 25 T_L t= 45 T_L 0 100 200 300 400 500 ε_e-/m_ec2 101 102 103 104 105 A. U (a) (b) a0 γe ∼ ≃ TL ≃39TL ∆ǫγ≃ 0.5 τs p ≡ px_N1_e N1epx Ne

(152)

250 300 350 400 ε_e-/m_ec2 101 102 103 104 105 ( A. U) 20 25 30 35 40 θ (°) 10-3 10-2 10-1 100 ζ γ /st r ( %) -2 -1 0 1 2 log[ε_γ/m_ec2] 10-3 10-2 10-1 100 ζγ *( ε γ / m e c 2 ) ( %) a) _b) c) a0 γ0 τs a0>γe τs a0 a0>γe γe a0 τs τs ηe ηγ ×109 2

(153)

0 2 4 6 8 10

(Temps de montee)/T_L

120 140 160 180 200 a₀ 0 10 20 30 40 τ s /T L 0 2 4 6 8 10 (FWMH)/T_L 0 2 4 6 8 10 12 14 16 (Plateau )/T_L 0 10 20 30 40 τ s /T L a) b) c) b) τs a0 γ0

(154)

0 10 20 30 t/T_L 0 5 10 15 20 0 10 20 30 40 50 60 70 t/T_L 0 5 10 15 20 η γ , η e ( %)

Avecla reaction du rayonnement Sansla reaction du rayonnement

17.5% 5.3% 3.6% 0.68% 3.7% 3.6% a) b) ηe ηγ 1022 2

(155)

l1λL ne0 100nc a0 ne/nc TL l=100λL ne0 10nc ℄ ξ¯

(156)

¯ ξ ℄ τs

(157)

(158)

t t t t a0 ne Ex

(159)

℄ ℄ Ilaser≤1022 2 22 2 ℄

aL(x,t)=a0(t)Re(y−iz)eiωL(t−x/c)

l= λL TL 10 −3 _n c a0 TL nc a0 a0 2

(160)

ηγ≃ ≃ TL TL ℄ 0 20 40 60 80 100

t

/T

_L 10-2 10-1 100 101 102

η

γ

(

%)

1 10 100

t

/T

_L 10-5 10-4 10-3 10-2 10-1 100 101 102

ζ

γ ( %) (a) (b)

(161)

Ti π/ωpi 2π mi Zmeω −1 pe ωpe nee 2_/γ_e_m_e_ǫ₀ ne nc a0=200 ωpi ωpe/γe γe≫ 2πne ne l λLEc Ec Ti≤t t t t ≃ TL t ≃ TL tmax≃ TL Ti t t + + + 2+ e ∼ Ex/Ec≃a0

(162)

0 2 4 6 8 10

x

/λ

_L 0 50 100 150 200 E x /E c 0 10 20 30 40

x

/λ

_L -20 0 20 40 60 E x /E c 0 5 10 15 20

x

/λ

_L 0 50 100 E x /E c a) b) c) Ex t=t Ec mecωL/e pe,x

(163)

100 200 _a 300 400 0 100 150 200 250 300 Ma x[ Ex /Ec ] protons deuterons ionsimmobiles

Profil gaussien

Ex Ec a0 t= t

(164)

0 30 60 90 120 150 180 θ (°) 10-3 10-2 10-1 100 dζ /d θ ( %) a) t=t t= t l= λL

(165)

l= λL t ≃35TL t ≃22TL t=t 100 150 200 250 300

(γ

_e

-1)

103 104 105 106

N

e -

/c

m

100 200 300 400 500

(γ

_e

-1)

103 104 105 a) b) t=t ne=10nc l= λL t=t γemec 2_≃ε e εe Te t=t ωL

(166)

0 5 10 15 20 -300 -200 -100 0 100 200 300

γ

e

β

x,e 0 5 10 15 20 -200 -100 0 100 200 300 400 a) b) (x(t),px(t)) ∈[t −20TL;t +20TL] ne=10nc l= λL [Ey] [Ex] t=t 2ωL

(167)

∆t [t −δt;t +δt] δt≪ t δpe,x δpe,x pe,x t t t=t t=t ℄ rc ωre ωcr λL

(168)

300 600 900 1200

(γ

_e

-1)

103 104 105 106 100 200 300 400 500 600

(γ

_e

-1)

103 104 105 106

N

e -

/c

m

a) _b) t=t ne=10nc l= λL 0 10 20 30 40

x

/λ

_L -200 0 200 400 600 0 10 20 30 40

x

/λ

_L -200 0 200 400 600

γ

e

β

x,e c) _d) (x(t),px(t)) ∈[t −20TL;t +20TL] ne=10nc l= λL [Ey] [Ex] t=t

(169)

t

l t t TL l=100λL t Ex t l=1λL ηi t≃t t Ex ne nc l λL ≪ a0 Ex

(170)

T 0 10 20 30 40 50 60 t/T_L 0 10 20 30 40 50 60 ζγ ( %) a) *(1/2)

Impulsionlaser

0 5 10 15 20 25 30 35 40 45 50 55 t/T_L 0 0,5 1 1,5 2 2,5 3 ζγ ( %) b) *(1/40)

Impulsionlaser

0 10 20 30 40 50

t

/T

_L 5 10 15 20 25

E

x

/E

y

(

%)

0 10 20 30 40 50

t

/T

_L 0 10 20 30 40 50 60 70

E

x

/E

y

(

%)

c) d) l= λL l= λL Ex t

(171)

t t

t

t ne nc 1 10 100 l/λ_L 15 20 25 30 35 1 10 100 l/λ_L 15 20 25 30 35 t max /T L a) b) t t t ξ¯ λL λL tmax

(172)

T Ex ζγ t=t Ex ζγ l=1λL l=100λL l ζ l l

(173)

t

t a0 ay az t t 0 5 10 15 20 25 30 (FWHM)/T_L 0 20 40 60 80 t max /T L t t

(174)

T TL t Ti t t t TF WHM a0 t t ≥TF WHM a0 t Ti ωpe ne/ncπ/TL Ti= nc Zne mi γeme TL t ≃ t t ≃ 35TL t t a0 t TF WHM    mi<∞ FWHM≥3TL t ∼8₃FWHM −3TL mi→ ∞ FWHM≥3TL t ∼5₃FWHM ≤3TL t ≃5TL∀mi

(175)

t

t εγ t Iγ t ∂Iγ ∂t=0 g(t) ∂g ∂t+2α∂α∂t=0 α=Ex/Ey t fe α≪ 1 t g(t) α≃ g(t) g(t)∝ (t−t_2σ₂ ) σ≃FWHM/2.3548 t =t t t t t tmax ωLl c∼ω 2 L ω2_pea0 ωLtmax∼ ωLl c∼20010∼

(176)

t TL a0 a0 ne Ex ξ¯ Ex ξ¯ l=100λL l=1λL 20λL H+ λL

(177)

0 20 40 60 80 100 η γ ( %) 1 10 100 l/λ_L 1 10 100 η e -, η i ( %) 1 10 100 l/λ_L 0 20 40 60 80 100 η γ ( %) 1 10 100 η e -, η i ( %) a) b) c) d) ηγ ηe ηi l ne nc ℄ εL ηe ωL

a

0

n

e ηγ ηe ηi a0 ne ηtot

(178)

ηγ ηe ηi a0≥ χe 0 100 200 300 400 500 600 a₀ 40 45 50 55 60 65 70 η i ( %) 0 20 40 60 80 100 η γ ( %) 0 5 10 15 20 25 30 η e - ( %) 0 100 200 300 400 500 600 a₀ 40 50 60 70 80 90 100 η tot. ( %) a) b) c) d) ηγ ηe ηi ηtot a0 ne nc TL a0≥ a0 χe

(179)

10-3 10-2 10-1 100 101 102 n_e/n_c 10-3 10-2 10-1 100 101 102 η tot. ( %) 10-4 10-2 100 102 η e - ( %) 10-4 10-2 100 102 η γ ( %) 10-3 10-2 10-1 100 101 102 n_e/n_c 10-5 10-4 10-3 10-2 10-1 100 101 102 η i ( %) a) b) c) d) ηγ ηe ηi ηtot ne a0 a0≥ a0 a0 a0≥ ℄

(180)

℄ ne/nc a0 nc ℄ ℄ ne/nc≃ 3 ne/nc≃ a0 ne/nc

(181)

a0mec 2 a0 a0Ec >250 l= λL l= λL ωcr a0Ec FLe∼√2a0mecωL

(182)

EX -3 -2 -1 0 1 2 3 10-4 10-3 10-2

ζ

γ

*

m

e

c

2 -3 -2 -1 0 1 2 3

log[

ε

_γ

/m

_e

c

2

]

10-4 10-3 10-2 10-1 -3 -2 -1 0 1 2 3 10-4 10-3 10-2 -3 -2 -1 0 1 2 3

log[

ε

_γ

/m

_e

c

2

]

10-4 10-3 10-2 10-1

ζ

γ

*

m

e

c

2 a) _b) c) _d)

circulaire _l_inea_ire

circulaire lineaire

l= 1λ_L l= 1λ_L l= 100λ_L l= 100λ_L t=t ωcr ∼ 3_2m ecγ 3 e pe_p×a₂ 0 e ωL∼10 7_ω L pe,x ∆θ pz,e∼ py,e pe, ≤√2pz θ

(183)

θ t=t l= λL l= λL (θ) = px,e √ p2 e,y+p2e,z ∆θ ωcr>107ωL

(184)

EX Ex ∼ λL 10 ≤ ωcr≤100 θ≥ ∼ θ ≤ ωcr≤ ≤ ωcr≤

(185)

θ t=t l= λL l= λL Ex <γe>∼ E_Ex c lrad λL ∼ 10a0

(186)

EX Ex pe,x≫pe,y pe,x Ex l Ex θ θ≃

(187)

x ηi ℄ 1λL 100λL 1.1×10 23 2 TL χe χe χe l∼1λL χe χe

(188)

EX 20 40 60 80 100120140 10-5 10-4 10-3 10-2 10-1 100 101

N

e

(

%)

20 40 60 80 100 120 140

t

/T

_L 10-5 10-4 10-3 10-2 10-1 100 20 30 40 50 60 10-5 10-4 10-3 10-2 10-1 100 101 20 40 60 80 100

t

/T

_L 10-5 10-4 10-3 10-2 10-1 100

N

e

(

%)

a) _b) c) d) χ_e> 0.2 χ_e > 0.1 χ_e > 0.2 χ_e> 0.1 χ_e > 0.1 χ_e > 0.2 χ_e > 0.1 χ_e > 0.2 χe> 0.1 χe > 0.2 ℄ l=1λL l=100λL l∼100λL χe χe χe ∼10 23 2 ℄ ℄ χe

(189)

0 10 20 30 40 50 60 70 80

t

/T

_L 10 20 30 40 50 60 70 80 90 100

η

γ

(

%)

~80%

~50%

~30%

ηγ a0=200 ne0=10nc TL TL

(190)

ηγ mi mi

(191)

Z/mi

(192)

χe∼1

(193)

A µ_(ω_,k) δfi χe

(194)

22 2 ∼

(195)

ne0=10nc ξ≪ 1¯ ξ∼1¯ ¯ ξ≫ 1