Radiation by Charged Particles: a Review Contents F.Sannibale Contents

RadiationRadiation byby ChargedCharged Particles:Particles: aa ReviewReview

Fernando Sannibale

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

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.

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 ττ generates a R(τ ) Coulomb potential that will contribute t = τ + to theϕ potential at the point P at a later c ϕ τ δ time t given by (cgs units): τ time t givenδ by (cgs units): τ τ τ τ q d ()r,t = []− t + R()τ c dt R( )= R( ) = r − r′(τ ) τ R () τ δ τ So the total δpotential at the point P at the time t is given by: τ τ ∞ τ ∞ []− t + R c () [ − t + r − rτ′()c] Lienard-Wiechert ()r,t = q τ d = q dτ ∫ R() τ ∫ r − r′() Potentials −∞ −∞ δ τ And analogously for the vector potential: τ q ∞ []− t + R( ) c q ∞ [ − t + r − rτ′()c] A()r,t = ∫ v d = ∫ v dτ c −∞ R() c −∞ r − r′() 4

Accelerator-Based Sources of Coherent Terahertz Radiation – UCSC, Santa Rosa CA, January 21-25, 2008 γ 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 γ q q  dβ  v −1 2 E = ()n − β + n × ()n − β × with β = , = ()1− β 2 2 2 3 3   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 ω ElectronElectronγ inin FreeFree SpaceSpace The radiated electric field can be expressed in frequency domain: +∞ ω q n×[]()n−β ×dβ dt +ωcR−1 −2(n−β) E = exp[]i ()τ +R c dτ ω ∫ ()2 ω L. D. Landau c −∞ R⋅ 1−n⋅β τ iq +∞ E = R −1[β − [1+ ic ( R)] n] exp[i ( + 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:ω ~ ic ~ ~ iω exp ikR 2π E()=ω k × [Aω()× k] A()ω = q ()exp[]i()ω t − kr′ dr′ where k = ω cR ∫ λ ~ ~ 0 B()= 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: By applying theω Heisenberg uncertainty principle for the photon case we obtain: σ σ τ σ h E hν hω 2π σ ≥ p = = = = = k σ z = cσ τ pz z σ ω h h 2 σ c c τ c λ σ 1 σλ λ = h c ⇒ σ σ ≥ or λ ≥ pz z c 2 z 4π σ c we can define the longitudinal coherence length as zc = 2σ ω σ ω ω 2π σ ≥ h h h pw w pw = psinθ wσ= sin()θ w ≅ θ w = h θ w w = x, y 2 λ θ σc c λ p σ 2π λ p σ w θ pwσ w = h w ⇒ θ σ w ≥ 4π λ and the transverse coherence length as σ wc = 4πσ θw By using the previous results, we can define the λ 3 VC = ( 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 θ that the magnetic field must have a component ω parallel to k due to diffraction. θ ∂B 1 λ E d ≈ d 2 = B d 2 = 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 pwλ≥ h 2 and ε nw = σ wσ pw m0c = βγ σ wσ w ⇒ ε nw ≥ λCompton 4π w = x, y, z

Compton ≡ Compton wavelength = h m0c = 2.426 pm for electrons,

ε nw ≡ normalized emittance, w′ = dw ds

λ 3   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.

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 δδ is defined as the number of particles

(photons, electrons, ... ) in the volume of coherence VC

The limit value of δδ is infinity for bosons, and 2 for non polarized-fermions because of the Pauli exclusion principle.

The relation between brightness B and δδ is:

N δ 3 B = ε λ ε  C  nx nyε nz = B   4π  N ≡ number of particles

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 δ e kT −1 α 3 for synchrotron sources of radiation in the visible range ≈ 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 δ (maximum allowed for unpolarized electrons) 3 ≈ N D ≈ 2×10−12 e ε ε for electrons from RF photo guns x 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 σ ε β ⇒ x x′ = 0 ⇒ x 2 = x 2 + z 2 x′2 = x 2 (1+ z 2 x′2 x 2 ) 0 0 w w w σ w w σ σ 2 2 2 2 ε 1 2 For particles x = = and x′ = ′ = β 2 2 x x x x x x x = w (1+ z β x ) σ σ 2 2 2 2 2 λ σ 2 2 1 2 For photons x x′ = x ′x = ( 4π ) = σ (1+ z z ) πσ x w 0 2 π 2 Where we have defined 4 λ w w0 and the photon z = = w0 = 2σ w the Rayleigh range as 0 λ beam size as

Note that the z0 in optics plays the same role of ββ 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 >> Ph B 4π σ 4π σ ε ε ε 2 β 2 ε 2 = z σ ′2 = Ph = σ ′ = B wPh Ph 0 wR σ wB B w wB z0 β w σ σ  z 2   z 2  2 (z) = σ 2 1+  2 (z) = 2 1+  Ph wPh  2  B wB  2   z0   β w  ϕ Gouy phase Betatron phase z z ζ ()z = arctan ()z = arctan z 0 β w z z

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. Transverse Electro-Magnetic or TEM modes are of particular interest. These can present cylindrical simmetry (Laguerre-Gaussian modes radially polarized) or rectangular (Hermite-Gaussian modes linearly polarized):

LG modes pq HGpq modes

Gaussian mode: the fundamental mode for both LG and HG modes

The emittance of the higher order modes is λ m ε ≈ m = proportional to the number m of transverse spots 4π 2k 14 Accelerator-Based Sources of Coherent Terahertz Radiation – UCSC, Santa Rosa CA, January 21-25, 2008 Radiation by Charged WeizsäckerWeizsäcker--WilliamsWilliams MethodMethod Particles: a Review F.Sannibale ofof VirtualVirtual PhotonsPhotons The method exploits the fact that the field of a relativistic particle is very similar to the one of a plane wave. Because of this, the particle can be replaced by virtual photons (plane wave) that with their field represent the field of the particle. y q In the particle rest frame (cgs units): E ′ = r ′ r′3 z(t) z qb qb qct′ E′ = = ′ ′ x 3 2 3 2 Ez = 3 2 E y = 0 b []b 2 + z′2 (t′) []b 2 + (ct′) 2 []b 2 + (ct′) 2 x q v ≈ c ω and in the laboratory frame: γ qct γ qb v≈c ϕ E = Ez ω= − 3 2 x 3 2 E y = 0 []bπ2 + (γωct) 2 []b 2 + (γ ct) 2 ω 2 2 ~ 1 γ FWHM for E 2 By Fourier transforming, the spectrum of the energy W x K1 (x) x π ω 1 per ϕunitω area due to the two terms is obtained: 2 γ 2 0.8 K :modified Bessel function dW (),b cπ ω 2 q 1  γ b  ω b  1 z = E (),b =   K 2   of the 2nd kind z 2 2 2   0   0.6 b db d d 2 cb ω c γ c π     0.4 2 2 ω dWx (),b c 2 q  γ b  2  b  0.2 = Ex (),b =   K1   b db d d 2 2cb 2  c   γ c  0 15     1 2 3 4 x 5 Accelerator-Based Sources of Coherent Terahertz Radiation – UCSC, Santa Rosa CA, January 21-25, 2008 Radiation by Charged TheThe PowerPower SpectrumSpectrum Particles: a Review F.Sannibale ofof thethe VirtualVirtual PhotonsPhotons ω ω π The total energy spectrum is obtained by integrating the dW ( ) ∞ dIω( ,b) = 2 b db previous spectrum over the possible values of b: d ∫ dω bmin The complete analytical ωsolution can be derived ω but the following approximationsω are very useful: ω dW ( ) q 2  2b  for ω >>πγ c/b min for ω >>γ c/bmin γ ≈ exp− γ ω  ω d 2c  c  and for ωω <<γγ c/bmin dW ()2 q 2  1.123 c  1 π2 q 2   γ c  1 ≈ ln  −  ≅ ln  −  d ωc bmin 2 c ω bmin 2 ω        ω 1 dW (ω) The number of virtualα photonsγ per mode is given by: n()()d ω = dω π ω dω ω h ω 2   c  1 dω n()d ω≈ α ln  −  Low frequency regime   bmin  2 ω e 2 1 with α = ≅ h c 137  2γbminω  dω n()d ≈ exp−  High frequency regime 2  c  ω The spectrum of the virtual photons associated with a γ c ωC = particle extends up to about the critical wavelength ωC bmin 16 Accelerator-Based Sources of Coherent Terahertz Radiation – UCSC, Santa Rosa CA, January 21-25, 2008 Radiation by Charged Particles: a Review The Calculation of b F.Sannibale The Calculation of bmin The quantity b is the distance between the observation point and the particle trajectory (the impact parameter in collision terminology) λ βγσ Comptonπ h h We already derived that for a particle wσ w′ ≥ = = w = x, y 4 4π m0c 2m0c σ λ in our case ββ ~ 1 and σσ' ~ 1/2γγ ≥ σ ~ Compton = h ωx w w min ω 2π m0c α γ λ The position of the particle cannot beπ defined ∗ Compton h bmin ~ σ w min ~ = within σσw min, the coherence length. It is 2π m c ω ω 0 natural than to assume o b∗ ~ 4 ×10 −3 Α for electrons minω α γ π that used in a 2   m c 2  1 d 2  m c 2  dω n()d ≈ ln 0  − ≅ ln ω0  previous result for e-       previous result for e   h  2  h  ω This expression shows how many virtual photons per mode are readily "available" for radiation! ω 2 The virtual photon spectrum is limited to ~ h C ~ γ m0c

(The "log" term for typical cases ranges from few units to few tens) 17

Accelerator-Based Sources of Coherent Terahertz Radiation – UCSC, Santa Rosa CA, January 21-25, 2008 Radiation by Charged Particles: a Review The Radiation Divergence F.Sannibale The Radiation Divergence

We just showed that the quantity bmin represents the transverse coherence of the radiation at the critical wavelength. γ c σ c ~ bmin ωC ~ bmin We will see later in the talk that each radiation process is characterized by its own value of b (always > b* ). But before going into that, we can still min σ min λ extract some additional information common to all cases. πσ λ π We previously found that: σ cσ θ c ~ λ 4π ω

C C c c bγmin 1 so at the critical wavelength: θ c ~ ~ = ~ = 4 c 4 bmin 2bmin C 2bmin c 2γ σ σ So independently from the radiating process, the angular θ 1 ~ θ c ~ width of the radiation at the critical wavelength is always: 2γ 18

Accelerator-Based Sources of Coherent Terahertz Radiation – UCSC, Santa Rosa CA, January 21-25, 2008 Radiation by Charged Particles: a Review Virtual Photons Became Real: F.Sannibale Virtual Photons Became Real: We now know that a drifting particle can be considered as surrounded by a cloud of virtual photons responsible for the particle field. Such photons cannot be distinguished from the particle itself but... - If the charged particle receives a kick that delays it from its virtual photons the photons can be separated and become real In vacuum when γ >> 1 the only practical way is by a transverse kick: Synchrotron radiation v Edge Radiation Synchrotron Bremsstrahlung, Beamstrahlung Radiation - If in a media the speed of light at a given wavelength is smaller than the particle speed the photons lag behind the particle and separate.

v > c n()λ Cerenkov radiation v Radially polarized and hollow due to symmetry - If a particle goes through an aperture with diameter 2b smaller than or comparable with the transverse coherence length of some of its virtual photons those photons will be diffracted and reflected. Diffraction Radially polarized and πσλ λ 2σ wc = ~ γ > b Transition radiation hollow due to symmetry 2 2π θw (not Smith-Purcell)19 (Smith-Purcell) metal (not Smith-Purcell) Accelerator-Based Sources of Coherent Terahertz Radiation – UCSC, Santa Rosa CA, January 21-25, 2008 Radiation by Charged Particles: a Review The Formation Length F.Sannibale The Formation Length

The formation length LF is the trajectory length that a particle has to travel in order that the radiated wavefront advances one λ/2π (one radian) ahead of the particle trajectory projection along the observation direction.

Virtual photonsβ become real after the parent particle travels for one L βVirtual photons become real after the parent particle travels for one LF θ θ Example: formation length for diffraction or transition radiation wavefront D emitted during βtransition fromβ mediaθ to vacuum: θ particle γ LF = β ct F θ  1  β D D = ct F − ct F cos()= LF  − cos() ⇒ θLF =   1 − cos()θ

2 For ~ 1, << 1 ⇒ 1 ≅ 1+1 2 2 and cos()≅ 1− 2 2 L ≅ γ D F 1 2 +θ 2

2 If we observe the radiation at λ ~ λC at the peak for θ ~ 1/γ : LF ~ γ D 20

Accelerator-Based Sources of Coherent Terahertz Radiation – UCSC, Santa Rosa CA, January 21-25, 2008 Radiation by Charged SynchrotronSynchrotron RadiationRadiation Particles: a Review F.Sannibale FormationFormation LengthLength

λ/2π P With reference to the figure and using the definition of L : ρθ F wavefrontθ L O T F =ρ ct − OP and LF = F or t F = β ρ D F θ β c θ particle ρ ρ ρ Lβ ρ L θF/2  F   F   F  ρ F F OP = OT cos  = 2 sin cos  = sinθ F ⇒ D = − sin θ  2   2  β 2  ρ ρ ρ L λ L 1 L3   1 1 L2  L  1 1 L2  ⇒ ≅ γ F −  F − F  = L  −1+ F  ≅ F  γ + F  D  3  F  2   2 2  π 6   6  2  3 ρ  ρ 1 L2 2 1  1 L3  If F = F >> ⇒ ~  F  ⇒ L ~ λ1 3 ρ 2 3 Low frequency regime 2 2  2  F 3 3 2  6  γ 2λ High frequency regime or LF ~ π

ϑ 1 3 The angle θ = L /ρ also indicates θ λ  Low frequency F F = ~   the radiation angular width: F   angular width  ρ  21

Accelerator-Based Sources of Coherent Terahertz Radiation – UCSC, Santa Rosa CA, January 21-25, 2008 Radiation by Charged TheThe CalculationCalculation ofof bb Particles: a Review min F.Sannibale forfor SynchrotronSynchrotron RadiationRadiation

As it was ρshown before, the parameter bmin represents the transverse coherenceθ length for the radiation. For synchrotron λ/2π P radiation bmin is given byρ the segment AH when θF/2~1/γ θ wavefront 2 2 A ρθ  1  ρθ T () F  F O AH = []1− cos F 2 ≅ 1−1+  = H ρ  2 4  8 ω 2 2 ρ particle F min  γ2  b ≅ ω ~   = And using previous results: ρ θ /2 min 8 α8   2 F π   γ 2γ

γ c γ 3 c λω ρ Synchrotronω radiation critical ω = ~ 2 ρ~ π C C 3 frequency and wavelength bmin ρ γ ω α π ω 3 ω 2   c  1 d 2   ωC  1 dω ω dω n()d ≈ ln2  −  = ln  −  ⇒ n()d ~ α ω ω    2    2 ω ω ω In the rest of the lecture, we will neglect the log and the -1/2 terms and the 2/π factor ω ω because for all radiation processes they are together of the order of the unit. γ dP c e 2 1 3 ω = n()~ 2 3 for << ω Low frequency h C 2 C power spectrum d LF c power spectrum 22 Accelerator-Based Sources of Coherent Terahertz Radiation – UCSC, Santa Rosa CA, January 21-25, 2008 Radiation by Charged Particles: a Review Diffraction Radiation F.Sannibale Diffraction Radiation

We already calculated the formation length for this case: We already calculatedβ the formation length for this case: θ 2 wavefront L = D ~ D F γ 2 2 D 1 − cos()1 +θ θ particle

bmin ~ a ⇒ ωC = γ c bmin ~ γ c a

So for a = 1 mm and a 1 GeV electron, the diffraction

radiation spectrum extends to up ~ ωC/2π ~ 100 THz (λC ~ 3 µm). The intensityω ω peaks at θ ~ 1/γ where L ~ λγ2/2π and the power spectrum is: ω F 2 dP c e 1 Low frequency power = h n()~ 2 ω d L c γ spectrum @ θ ~ 1/γ ω F ω dP e 2 γ1  ω  ~ exp− 2  High frequency power d c 2  ω  spectrum @ θ ~ 1/γ  C  23

Accelerator-Based Sources of Coherent Terahertz Radiation – UCSC, Santa Rosa CA, January 21-25, 2008 Radiation by Charged Particles: a Review Transition Radiation F.Sannibale Transition Radiation

The fields of a relativistic particle crossing a media interact with ω 2 the electrons of the media itself . Such electrons move under the e P ~ 4π ne action of the time varying electric field up to frequencies of the m0 order of the plasma frequency. Above this frequency the n ≡ e − density electrons in the media cannot respond to the too fast excitation e anymore and the media becomes transparent at these high cgs units frequency components. Transition radiation can be viewed as diffraction radiation through a hole of the size of ~ a plasma wavelength!

λP n2 n bmin ~ ⇒ ωC = γ c bmin ~ 2γ c λP = γωP π 1 2 16 For a unity density material, ωP ~ 3 x 10 s-1 and with a 1 GeV electron, the transition radiation spectrum extends 18 to up ~ ωC/2π ~ 3 x 10 Hz (λC ~ 0.1 nm - hard x-rays)! The intensity peaks at θ ~ 1/γ where L ~ λγ2 ω ω F Transition radiation and the power spectrum becomes: γ = 30 ω dP c e 2 1 = n()~ ω Low frequency power h γ2 d LF c spectrum @ θ ~ 1/γ ∼1/γ 24

Accelerator-Based Sources of Coherent Terahertz Radiation – UCSC, Santa Rosa CA, January 21-25, 2008 Radiation by Charged Particles: a Review Cerenkov Radiation F.Sannibale Cerenkov Radiation

For the emissionβ of Cerenkov radiation: β cω c c >= = ε n() r ()ω θ θ (cβ n)t 1 Fromβ θthe figure: cos ( = = C ct β n θ

c  1  2 2 D ( ( ( ( ⇒ L = D = ctF − t F cos C = LF 1− cos C  = LF ()1− cos C = LF sin θC F 2   ( n  n  sin θ C As for the transition radiation case, in principle also for the Cerenkov b ~λ . ω ω min P Nevertheless, the requirementω β c > c/n(ω) imposes limitations to the bandwidth. Additionally, in order to extractω the radiation from the media the latter must be transparentθ at that wavelength. 2 2 ω dP c e 2 e 2 Low frequency = n()~ sin ( ~ sin θ ( h C C power spectrum d LF c c 25

Accelerator-Based Sources of Coherent Terahertz Radiation – UCSC, Santa Rosa CA, January 21-25, 2008 Radiation by Charged RadiationRadiation fromfrom aa BeamBeam Particles: a Review F.Sannibale ofof ChargedCharged ParticlesParticles We now want to investigate the case where many particles radiate together in a beam. We will show that for whatever radiation process (synchrotron radiation, Cerenkov radiation, transition radiation, etc.) the incoherent component of the radiation is due to the random distribution of the particles along the beam. Example: "Ideal" coasting beam moving on a circular trajectory with the particles equally separated by a longitudinal distance d : No synchrotron radiation emission for frequencies with λλ < ~ d. The interference between the radiation emitted by the evenly distributed electrons produces a vanishing net electric field. In a more realistic coasting beam, the particles are randomly distributed causing a small modulation of the beam current. The interference is not fully destructive anymore and the beam radiates also at longer wavelengths. 26 Accelerator-Based Sources of Coherent Terahertz Radiation – UCSC, Santa Rosa CA, January 21-25, 2008 Radiation by Charged Particles: a Review Radiation Fluctuations F.Sannibale Radiation Fluctuations

If the particle turn by turn position along the beam changes (longitudinal dispersion, path length dependence on transverse position), the current modulation changes and the radiated energy and its spectrum fluctuate turn by turn.

Coherent Single passage component spectrum By averaging over multiple Average passages, the measured spectrum Spectrum converges to the characteristic incoherent spectrum of the radiation process under observation.

(synchrotron radiation in the Log Brightness example).

Log Frequency In the case of bunched beams, a strong coherent component at those wavelengths comparable or longer than the bunch length shows up But the higher frequency part of the spectrum remains essentially unmodi27fied. Accelerator-Based Sources of Coherent Terahertz Radiation – UCSC, Santa Rosa CA, January 21-25, 2008 Radiation by Charged Particles: a Review More Quantitatively... F.Sannibale More Quantitatively... The electric field associated with the radiation N E()t = e t − (t ) emitted by the beam at the time t is: ∑ k k =1 where e is the electric field of the electromagnetic pulse radiated by a ω single particle and tk is the randomly distributed arrivaω l time of the particle ω (Poisson process). ∞ N In the frequencyω domain: Eˆ ()= E t e ()i t dt = eˆ ω eiω ()tk ω ∫−∞ ∑ k =1 ω N N And for the radiated power 2 2 P ()∝ Eˆ ()= eˆ()ω eiω()tk −tl per passage: ∑∑ k ==11l The previous quantity fluctuates passage to passageω , and the average radiated power from a beam with normalized distribution f (t) is: N ∞ ∞ 2 2 2 P ()∝ eˆ() dt dt f ()()t f t eiω()tk −tl = eˆ()N + N()()N −1 fˆ ω  ∑ ∫∫ k l k l   k,l=1 −∞ −∞   ∞ Incoherent term Coherent term where fˆ()ω = ∫ f t e ()iωt dt −∞ 28 Accelerator-Based Sources of Coherent Terahertz Radiation – UCSC, Santa Rosa CA, January 21-25, 2008 Radiation by Charged Particles: a Review References F.Sannibale References

Max Zolotorev

Oleg Chubar

Gennady Stupakov

L. D. Landau, E. M. Lifshitz "The Classical Theory of Fields", Vol.2, Editori Riuniti J. D. Jackson "Classical Electrodynamics" 3rd Edition, Wiley G. R. Fowles "Introduction to modern Optics" 2nd Edition, Dover

The web

Accelerator-Based Sources of Coherent Terahertz Radiation – UCSC, Santa Rosa CA, January 21-25, 2008 Radiation by Charged Physical Constants Particles: a Review Physical Constants F.Sannibale ((SISI Units)Units)

From: http://physics.nist.gov

Accelerator-Based Sources of Coherent Terahertz Radiation – UCSC, Santa Rosa CA, January 21-25, 2008 Radiation by Charged Particles: a Review Homework F.Sannibale Homework

Using the expression for the electric field derived from the Lienard- Wiechert potentials describe the polarization (direction of the electric field) when the acceleration is parallel to the velocity but the observation direction is not.

Explain what happens when a charged particle goes through a periodic iris structure.

Derive the formula for the coherent synchrotron radiation.

In the case of the ideal coasting beam, explain what happens when

λλ > = d. 31

Accelerator-Based Sources of Coherent Terahertz Radiation – UCSC, Santa Rosa CA, January 21-25, 2008