RadiationRadiation byby ChargedCharged Particles:Particles: aa ReviewReview Fernando Sannibale 1 Accelerator-Based Sources of Coherent Terahertz Radiation – UCSC, Santa Rosa CA, January 21-25, 2008 Radiation by Charged Particles: a Review Contents F.Sannibale Contents • Introduction • The Lienard-Wiechert Potentials • Photon and Particle Optics • The Weizsäcker-Williams Approach Applied to Radiation from Charged Particles • Incoherent and Coherent Radiation 2 Accelerator-Based Sources of Coherent Terahertz Radiation – UCSC, Santa Rosa CA, January 21-25, 2008 Radiation by Charged Particles: a Review Introduction F.Sannibale Introduction The scope of this lecture is to give a quick review of the physics of radiation from charged particles. A basic knowledge of electromagnetism laws is assumed. The classical approach is briefly described, main formulas are given but generally not derived. The detailed derivation can be found in any classical electrodynamics book and it is beyond the scope of this course. A semi-classical approach by Max Zolotorev is also presented that gives an "intuitive" view of the radiation process. 3 Accelerator-Based Sources of Coherent Terahertz Radiation – UCSC, Santa Rosa CA, January 21-25, 2008 Radiation by Charged TheThe FieldField ofof aa MovingMoving Particles: a Review F.Sannibale ChargedCharged ParticleParticle A particle with charge q is moving along the trajectory r' (t), the vector r defines the observation point P. R = r - r' is the vector with magnitude equal to the distance between the particle and the observation point. The particle at the time τ generates a R(τ ) Coulomb potential that will contribute t = τ + to the potential at the point P at a later c time t given by (cgs units): q dϕ()r,t = δ []τ − t + R()τ c dt R(τ ) = R(τ ) = r − r′(τ ) R()τ So the total potential at the point P at the time t is given by: ∞ δ []τ − t + R()τ c ∞ δ [τ − t + r − r′()τ c] Lienard-Wiechert ϕ()r,t = q ∫ dτ = q ∫ dτ −∞ R()τ −∞ r − r′()τ Potentials And analogously for the vector potential: q ∞ δ []τ − t + R(τ ) c q ∞ δ [τ − t + r − r′()τ c] A()r,t = ∫ v dτ = ∫ v dτ c −∞ R()τ c −∞ r − r′()τ 4 Accelerator-Based Sources of Coherent Terahertz Radiation – UCSC, Santa Rosa CA, January 21-25, 2008 Radiation by Charged AcceleratedAccelerated Particles: a Review F.Sannibale ParticlesParticles RadiateRadiate The field components can be calculated from the Lienard-Wiechert potentials and the relations: 1 ∂A E = − − ∇ϕ B = ∇ × A R = R n with R = R c ∂t q q dβ v −1 2 E = ()n − β + n × ()n − β × with β = , γ = 1− β 2 2 2 3 3 () γ R ()1− n ⋅ β cR()1− n ⋅ β dt c B = n × E ⇒ B is perpendicular to E where the quantities on the RHS of the expressions are calculated at τ = t - R(τ )/c. The first term of the electric field depends on the particle speed and converges to the Coulomb field when v goes to zero. The second term is non zero only if the particle is accelerated. Charged particles when accelerated radiate electromagnetic waves. When the observation direction n is parallel to the particle trajectory β and the acceleration dβ/dt is perpendicular to β, the resulting electric field is parallel to the acceleration. If dβ/dt is parallel to R there is no radiation. 5 Accelerator-Based Sources of Coherent Terahertz Radiation – UCSC, Santa Rosa CA, January 21-25, 2008 Radiation by Charged EmissionEmission byby RelativisticRelativistic Particles: a Review F.Sannibale ElectronElectron inin FreeFree SpaceSpace The radiated electric field can be expressed in frequency domain: +∞ −1 −2 q n×[]()n−β ×dβ dt +cR γ (n−β) E = expiω()τ +R c dτ ω ∫ 2 [] L. D. Landau c −∞ R⋅()1−n⋅β iqω +∞ E = R −1[β − [1+ ic (ωR)] n] exp[iω(τ + R c)]dτ ω ∫ I.M.Ternov c −∞ The equivalence of the two expressions can be shown by integration by parts and the quantities on the RHS of the expressions are again calculated at τ = t - R(τ )/c. Landau also showed that when r >> r' and R ~ R0 = r then the vector potential in frequency domain can be written as: ~ ic ~ ~ iω exp()ikR 2π E()ω = k × [A()ω × k] A()ω = q exp[]i()ω t − kr′ dr′ where k = ω cR ∫ λ ~ ~ 0 B()ω = i k × A ()ω The last integral is calculated on the particle trajectory and shows that for r >> r' , the net radiation is the result of the interference between plane waves emitted by the particle during its motion. For a relativistic particle in rectilinear motion in a uniform media the interference is fully destructive and no radiation is emitted. 6 Accelerator-Based Sources of Coherent Terahertz Radiation – UCSC, Santa Rosa CA, January 21-25, 2008 Radiation by Charged CoherenceCoherence LengthsLengths Particles: a Review F.Sannibale andand CoherenceCoherence VolumeVolume By applying the Heisenberg uncertainty principle for the photon case we obtain: h E hν hω 2π σ = cσ σ pzσ z ≥ p = = = = h = hk z τ 2 c c c λ 1 σ λ σ σ = h σ cσ ⇒ σ σ ≥ or λ σ ≥ pz z c ω τ ω τ 2 λ z 4π c we can define the longitudinal coherence length as σ zc = 2σ ω ω ω 2π σ σ ≥ h h h pw w pw = psinθ w = sin()θ w ≅ θ w = h θ w w = x, y 2 c c λ p 2π λ p w θ σ pwσ w = h σ θ σ w ⇒ σ θ σ w ≥ λ 4π λ and the transverse coherence length as σ wc = 4πσ θw By using the previous results, we can define the 3 VC = (λ 4π ) volume of coherence VC in the 6-D phase space Two photons inside VC are indistinguishable, or in other words are in the same coherent state or mode. 7 Accelerator-Based Sources of Coherent Terahertz Radiation – UCSC, Santa Rosa CA, January 21-25, 2008 Radiation by Charged AlternativeAlternative DerivationDerivation Particles: a Review F.Sannibale ofof thethe CoherenceCoherence LengthsLengths Let us consider a wave focused into a waist of diameter d. Field components and wave vector as in the figure. From Stokes theorem and Faraday law (SI units): ∂ E E ⋅dl = ()∇ × E ⋅n dS = B ⋅n dS B d ∫ ∫S ∂t ∫S k E If we Integrate over the dotted path, we notice that B the integral on the left is not vanishing. This implies that the magnetic field must have a component parallel to k due to diffraction. ∂B 1 λ E d ≈ d 2θ = Bω d 2θ = Bckd 2θ But E = Bc ⇒ θ ≈ = dif ∂t dif dif dif kd 2πd One can say that the waist diameter is diffraction limited and d λ represents the transverse coherence length when θ is the d ⊥ ≈ radiation angular aperture 2πθ c The transform limited length of a pulse with bandwidth ∆ω is l|| ≈ τ = 1/∆ω, so the longitudinal coherence length is defined as ∆ω C 8 Accelerator-Based Sources of Coherent Terahertz Radiation – UCSC, Santa Rosa CA, January 21-25, 2008 Radiation by Charged TheThe CoherenceCoherence VolumeVolume Particles: a Review F.Sannibale forfor ParticlesParticles By applying the Heisenberg uncertainty principle to emittance: σ wσ pw ≥ h 2 and ε nw = σ wσ pw m0c = βγ σ wσ w′ ⇒ ε nw ≥ λCompton 4π w = x, y, z λCompton ≡ Compton wavelength = h m0c = 2.426 pm for electrons, ε nw ≡ normalized emittance, w′ = dw ds 3 λ This allows to define a 6-D phase space volume V Compton C VC = 4π Two particles inside VC are indistinguishable, or in other words are in the same coherent state. By analogy with the photon case we can say that VC is the coherence volume for the particle. 9 Accelerator-Based Sources of Coherent Terahertz Radiation – UCSC, Santa Rosa CA, January 21-25, 2008 Radiation by Charged Particles: a Review The Degeneracy Parameter F.Sannibale The Degeneracy Parameter The degeneracy parameter δ is defined as the number of particles (photons, electrons, ... ) in the volume of coherence VC The limit value of δ is infinity for bosons, and 2 for non polarized-fermions because of the Pauli exclusion principle. The relation between brightness B and δ is: N 3 B = λC ε nxε nyε nz δ = B 4π N ≡ number of particles 10 Accelerator-Based Sources of Coherent Terahertz Radiation – UCSC, Santa Rosa CA, January 21-25, 2008 Radiation by Charged TypicalTypical DegeneracyDegeneracy Particles: a Review F.Sannibale ParameterParameter ValuesValues Photons (spin 1) 1 δ = hν << 1 for thermal sources of radiation in the visible range e kT −1 3 for synchrotron sources of radiation in the visible range δ ≈ α N e ωτ b ≈ 10 15 -1 9 (ω ~ 10 s , τb ~ 10 ps, Ne ~ 10 , α ~ 1/137) 18 δ ≈ N ph ≈ 3×10 for a 1 Joule laser in the visible range Electrons (spin 1/2) for electrons in a metal at T = 0 oK δ = 2 (maximum allowed for unpolarized electrons) 3 D −12 δ ≈ N e ≈ 2×10 for electrons from RF photo guns ε xε yε z δ ≈ 10−6 for electrons from needle (field emission) cathodes 11 Accelerator-Based Sources of Coherent Terahertz Radiation – UCSC, Santa Rosa CA, January 21-25, 2008 Radiation by Charged RayleighRayleigh RangeRange Particles: a Review F.Sannibale andand BetaBeta FunctionFunction For a beam (particles or photons in paraxial approximation) drifting in a free space of length z: x 1 z x = 0 2 2 2 2 2 x = (x0 + zx0′ ) = x0 + z x0′ + 2z x0 x0′ x′ 0 1 x0′ Let's assume that the beam for z = 0 is in a waist 2 2 2 2 2 2 2 2 ⇒ x0 x0′ = 0 ⇒ x = xw + z x′w = xw (1+ z x′w xw ) For particles x 2 = σ 2 = ε β and x′2 = σ ′2 = ε β 2 2 1 2 x x x x x x σ x = σ w (1+ z β x ) 2 For photons x 2 x′2 = σ 2σ ′2 = (λ 4π ) 2 2 1 2 x x σ x = σ w (1+ z z0 ) 2 2 Where we have defined 4πσ w πw0 and the photon z = = w0 = 2σ w the Rayleigh range as 0 λ λ beam size as Note that the z0 in optics plays the same role of β in particle physics12 Accelerator-Based Sources of Coherent Terahertz Radiation – UCSC, Santa Rosa CA, January 21-25, 2008 Radiation by Charged Particles: a Review A complete Analogy F.Sannibale A complete Analogy Light optics Accelerator optics (paraxial approximation) λ λC ε = ε B >> Ph 4π 4π ε 2 2 ε 2 2 Ph ′ B σ wPh = ε Ph z0 σ w′ R = σ wB = ε B β w σ wB = z0 β w z 2 z 2 σ 2 (z) = σ 2 1+ σ 2 (z) = σ 2 1+ Ph wPh 2 B wB 2 z0 β w Gouy phase Betatron phase z z ζ ()z = arctan ϕ ()z = arctan z 0 β w z z 13 Accelerator-Based Sources of Coherent Terahertz Radiation – UCSC, Santa Rosa CA, January 21-25, 2008 Radiation by Charged Particles: a Review Transverse Modes F.Sannibale Transverse Modes Transverse modes define the intensity profile of photon beams.