Takashi TANAKA (RIKEN SPring-8 Center) Cheiron2008: Light Source II Characteristics of SR (2) • Electron Trajectory in the Undulator • Qualitative Description of Undulator Radiation
Cheiron2008: Light Source II

2 SR emitted by an electron moving at r = (x,y,z) Observation of SR at R = (X, Y,Z) lf the far-field approximation (/r)<<Z) is applicable, the radiation pattern depends only on the observation angle θ = (θ x,θ x).
Cheiron2008: Light Source II

Equation of motion of an electron moving in a magnetic field B myvx
ma UzAt I 1 , I 2 : 1st and 2nd field integrals of the undulator
Cheiron2008: Light Source II

Bx(z)= 0 By(z) ～ Bo sin magnetic field velocity position eBoヘッ93.37 Bo(T)入u(cm) 2πmc K value, Deflection parameter
Cheiron2008. Light Source

velocity longitudinal velocity :average velocity oscillating term Undulator field induces: • transverse(x) oscillation • longitudinal (z) oscillation • effective deceleration(Δβ¸=Κ²/4γ²)
Cheiron2008: Light Source II

• Horizontal oscillation with a period of λ μ • Major contribution to radiation
• Longitudinal oscillation with a period of λ ψ 2 • Amplitude 1/y times lower than βχ. • Minor contribution, but source of vertical polarization observed vertically off-axis.
Cheiron2008: Light Source II

+ (θx − βx)² + (θy − βy)² Time squeezing takes place most significantly when the direction of the electron motion coincides with that of observation ( β = θ ).
Cheiron2008: Light Source II
Characteristics of SR (2) • Electron Trajectory in the Undulator • Qualitative Description of Undulator Radiation
Cheiron2008: Light Source II

period of electron motion   time squeezing = period of emitted light T' = T(1 − βz cos θ) period of observed light H. Kitamura et al., J. Appl. Phys. 21 (1982) 1728 Fundamental Wavelength λ₁ λ u ( 1 − β z COS θ ) 1 + γ²θ² + K² (2) 31
Cheiron2008: Light Source II

• If the undulator length is infinite, the pulse duration is infinitely long, and thus the radiation is completely monochromatic with line spectrum. d2 F 4πcγ= | λυ dx'duの1 + K²/2 + γ2θ2 •  In practice, the undulator length is finite, so the line spectrum is broadened.
Cheiron2008: Light Source II

にもEn sin ω t ; -T/2 < t < T/2 , ω 1 = 2πc/λ 1 ; t < −T/2,T/2 < t 0 Fourier Transform d2 F π ν = ω 1 (θ) x | E(w)|² × sinc² dx'dy' ω1 ( θ ) Square of “sinc” function dominates the UR
Cheiron2008: Light Source II

• In the previous derivations of UR spectral function, no knowledge on electro- dynamics is required. •  In practice, Eo is a complicated function of θ and K, and needs to be calculated by Fourier transforming the electric field derived from the Lienard-Wiecherd potential. •  However, the simple derivation gives us a clear understanding on UR properties.
Cheiron2008: Light Source II
Qualitative Descriptions of UR
d2F(w,θ) dΩdω / ω π N − ω 1 ( θ ) Fosinc w1(8) Energy Profile at θ = 0 Fosinc² ( Ν πε ) ;ε = [ω − ω1(0)]/ω1(0) Angular Profile at ω=αω₁(0) Fosinc²[ Νπ(αΘ² + α − 1)] ; Θ = γθ/√1 + K²/2
Cheiron2008: Light Source II

ω=0.9ω₁(0) lower energy ω=ω₁(0) fundamental energy ω=1.05ω₁(0) higher energy Cheiron2008: Light Source II
Angular Profile at ω=ω₁(0) πN(γθ)² Ia(θ) = Fosinc² + K2/2 approximation Gaussian Profile with σ I G (θ) = F 0 exp( −θ 2 / 2σ) Angular Divergence of UR (L=Nλ,,) Diffraction Limit (UR is Spatially Coherent)入1 Beam Size of UR 4 π
• In addition to the fundamental radiation at ω₁, higher-energy radiation at nω₁, called higher harmonics, is observed. The integer n is referred to as a harmonic number. • This is a consequence of the fact that the time-squeezing factor depends on the longitudinal electron position and thus the electric field in the time domain is distorted.
Cheiron2008: Light Source II

d T 1 + K² cos² (2π2 / 2π on-axis observation: n =(0,0,1) Large K value brings a modulation in the time squeezing factor Distortions of the electric field takes place due to the nonuniform time squeezing. Due to symmetry, even harmonics do not appear. Cheiron2008: Light Source II

Cheiron2008: Light Source II

For the n-th harmonic radiation, d2 F Hosinc dx'dy nw」( θ
Cheiron2008: Light Source II

• As in the wigglers, no circular polarized radiation (CPR) is observed due to cancellation of CPR components. • The direction of the linear polarization observed off axis is tilted due to the longitudinal oscillation of electron motion.
Examples of the direction of linear polarization for various observation angles. H. Kitamura, JJAP 19 (1980) L185
Cheiron2008: Light Source II

Cheiron2008: Light Source II
• Effects due to Finite Emittance of the Electron Beam – Injection to the undulator with angular and positional offset SR with Position and Slope Offset
Cheiron2008: Light Source II

Off-axis observation at θeff Peak shift to lower energy 1 + γ2θ² + K² / 2 noton Flu Energy
Cheiron2008: Light Source II

Electron with an offset of Δγ 31 Energy shift of ω₁ 1 + K²/2 noton Flu Energy

Cheiron2008: Light Source II

Optical Emittance of UR:ｽ/4π Bandwidth of UR: ~1/nN Effects due to the e- beam are larger for higher harmonics
Under Gaussian approximation Ox',y' *effective beam size T x,y *effective divergence *Note: Valid only near the resonance energy (nω₁). Cheiron2008: Light Source II
Simple scheme to estimate the on-axis flux density and brilliance. on-axis flux density with zero-emittance beam Effective Brilliance d- F
Cheiron2008: Light Source II

•  SR emitted from the light source is processed by several optical elements before irradiation to the sample, such as the focusing mirror, monochromator. • These elements can be easily damaged by the heat load brought by the SR. •  It is thus important to reduce the heat load as much as possible without sacrificing the flux, which is actually done by the XY slit at the front-end section.
Cheiron2008: Light Source II

The power profile is much broader than the flux. Extraction of SR with an appropriate slit significantly reduces the heat load.
Cheiron2008: Light Source II

Cheiron2008: Light Source II

• Wigglers are identical to undulator from the point of view of magnetic circuit. • It is generally said that the K value distinguishes between the two, however, this is not exactly correct. • What we should take care is the region of photon energy to be utilized for application.
Cheiron2008: Light Source II

•   Quantitative descriptions of SR • Light sources for circular polarization and schemes for fast helicity switching –  helical undulator & elliptic wiggler –  chicanes&choppers, kicker magnets •   Effects on the electron beam –  natural focusing –  beam-axis fluctuation due to COD variation o R&Ds toward shorter magnetic period –  superconducting undulators –  cryogenic permanent magnet undulators • Coherent SR for intense THz light •   Undulators for SASE-based X-ray FEL