Undulator Equation and Radiated Power

David Attwood

University of California, Berkeley

(http://www.coe.berkeley.edu/AST/srms)

Undulator Equation and Radiated Power, EE290F, 15 Feb 2007 Undulator Radiated Power in the Central Cone

N periods λ 2 – u K 2 2 e λx = (1 + + γ θ ) 2γ 2 2

πeγ 2I K2 Pcen = 2 f(K) K 2 0λu (1 + ) 2 λu 1 1 γ∗ 1 θcen = ∗ θcen = γ N γ ∗ N 2.00 ALS, 1.9 GeV ∆λ  1 = 400 mA Tuning  λ cen N 1.50 λu = 8 cm curve N = 55 K = 0.5-4.0 (W) eB0λu 1.00 K = cen 2πm0c P λ = N 0.50 ∆λ ∗ K2 γ = γ / 1 + 2 0 0 100 200 300 400 Photon energy (eV)

Professor David Attwood Univ. California, Berkeley Undulator Equation and Radiated Power, EE290F, 15 Feb 2007 Ch05_LG189_Jan07.ai Spectral Brightness of Synchrotron Radiation

12 nm 1.2 nm 0.12 nm 1020

6-8 GeV 1019 Undulators 1-2 GeV 1018 Undulators

1017

1016 Wigglers Spectral brightness 15 10 Bending magnets 1014 10 eV 100 eV 1 keV 10 keV 100 keV Photon energy

Professor David Attwood Univ. California, Berkeley Undulator Equation and Radiated Power, EE290F, 15 Feb 2007 Ch05_F24_left_Feb07.ai The Equation of Motion in an Undulator

Magnetic fields in the periodic undulator cause the electrons to oscillate and thus radiate. These magnetic fields also slow the electrons axial (z) velocity somewhat, reducing both the Lorentz contraction and the Doppler shift, so that the observed radiation wavelength is not quite so short. The force equation for an electron is dp y = –e(E + v × B) (5.16) 2πz dt By = Bo cos λ where p = γmv is the momentum. The u radiated fields are relatively weak so that x dp  –e(v × B) dt v e– z

Taking to first order v  vz, motion in the x-direction is

Professor David Attwood Univ. California, Berkeley Undulator Equation and Radiated Power, EE290F, 15 Feb 2007 Ch05_F15_Eq16_19.top.ai The Equation of Motion in an Undulator (cont.)

integrating both sides

(5.17)

(5.19)

(5.18)

is the non-dimensional “magnetic deflection parameter.” The “deflection angle”, θ, is

vx vx K θ =  = sinkuz vz c γ

Professor David Attwood Univ. California, Berkeley Undulator Equation and Radiated Power, EE290F, 15 Feb 2007 Ch05_F15_Eq16_19.bot.ai The Axial Velocity Depends on K

In a magnetic field γ is a constant; to first order the electron neither gains nor looses energy.

thus (5.22)

Taking the square root, to first order in the small parameter K/γ

(5.23a)

2 Using the double angle formula sin kuz = (1 – cos 2kuz)/2, where ku = 2π/λu,

Reduced A double frequency axial velocity component of the motion The first two terms show the reduced axial velocity due to the finite magnetic field (K). The last term indicates the presence of harmonic motion, and thus harmonic frequencies of radiation.

Professor David Attwood Univ. California, Berkeley Undulator Equation and Radiated Power, EE290F, 15 Feb 2007 Ch05_Eq22_23VG.ai K-Dependent Axial Velocity Affects the Undulator Equation

Averaging the z-component of velocity over a full cycle (or N full cycles) gives

(5.25)

We can use this to define an effective Lorentz factor γ* in the axial direction

(5.26)

As a consequence, the observed wavelength in the laboratory frame of reference is modified from Eq. (5.12), taking the form

that is, the Lorentz contraction and relativistic Doppler shift now involve γ* rather than γ

(5.28)

where K  e Bo λu/2πmc. This is the undulator equation, which describes the generation of short (x-ray) wavelength radiation by relativistic electrons traversing a periodic magnet structure, accounting for magnetic tuning (K) and off-axis (γθ) radiation. In practical units

(5.29a)

Professor David Attwood Univ. California, Berkeley Undulator Equation and Radiated Power, EE290F, 15 Feb 2007 Ch05_Eq25_29VG.ai Calculating Power in the Central Radiation Cone: Using the well known “dipole radiation” formula by transforming to the frame of reference moving with the electrons

x, z, t laboratory frame of reference x′, z′, t′ frame of reference moving with the average velocity of the electron

e– e– γ* z λu λ′ = u γ* λu N periods Lorentz transformation x′, z′, t′ motion Determine x, z, t motion: a′(t′) acceleration dp = –e (E + v × B) Dipole radiation: dt dP′ e2 a′2 sin2 Θ′ dvx dz 2πz = mγ = e B0 cos dΩ′ 16π2 c3 dt dt λu 0 2 2 2 vx(t); ax(t) = . . . dP′ e cγ K 2 2 2 = 2 2 2 (1–sin θ′ cos φ′) cos ω′ut′ dΩ′ 40λu (1 + K /2) vz(t); az(t) = . . .

x x′ 1 sin2Θ′ θcen = Lorentz transformation γ* N Θ′ θ′ λ z = N z′ ∆λ λ′ = N ∆λ′

Professor David Attwood Univ. California, Berkeley Undulator Equation and Radiated Power, EE290F, 15 Feb 2007 Ch05_T4_topVG.ai Calculating Power in the Central Radiation Cone: Using the well known “dipole radiation” formula by transforming to the frame of reference moving with the electrons (cont.)

x x′ 1 sin2Θ′ θcen = Lorentz transformation γ* N Θ′ θ′ λ z = N z′ ∆λ λ′ = N ∆λ′ dP *2 dP′ = 8γ dΩ dΩ′ dP e2 cγ4 K2  K ≤ 1 dP′ e2 cγ2 K2 = = (1–sin2 cos2 ) cos2 t 2 2 3  2 2 2 θ′ φ′ ω′u ′ dΩ 0λu (1 + K /2)  θ ≤ θcen dΩ′ 40λu (1 + K /2)

2 *2 ∆Ωcen = πθ cen = π/γ N πe2 cγ2 K2 Pcen = 2 2 2 0λu N (1 + K /2)

Ne uncorrelated electrons:

Ne = IL /ec, L = Nλu

πe γ2I K2 Pcen = 2 2 0λu (1 + K /2)

Professor David Attwood Univ. California, Berkeley Undulator Equation and Radiated Power, EE290F, 15 Feb 2007 Ch05_T4_bot.VG.ai Power Radiated in the Central Radiation Cone

To use the “dipole radiation” formula 2 2 2 dP′ = e a′ sin Θ′ 2 3 dΩ′ 16π 0c we need the acceleration a′(t′) in the frame of reference moving with the electron. We already know the velocity in the laboratory frame of reference

, ku = 2π/λu Take z  ct , and ωu = kuc

Integrating once

We can use the Lorentz transformation to the primed frame of reference

x = x′ Thus in the primed frame of reference K z′ x′ = – cos ωuγ*(t′ + ) kuγ c ωu′ small: z′ is the small axial motion about the mean Professor David Attwood Univ. California, Berkeley Undulator Equation and Radiated Power, EE290F, 15 Feb 2007 Ch05_Eq32_34.top.ai Power Radiated in the Central Radiation Cone (cont.)

K z′ x′ = – cos ωuγ*(t′ + ) kuγ c ωu′ small: z′ is the small axial motion about the mean Taking the second derivative

since

(5.33)

Then

(5.34)

Professor David Attwood Univ. California, Berkeley Undulator Equation and Radiated Power, EE290F, 15 Feb 2007 Ch05_Eq32_34.bot.ai Power in the Central Radiation Cone (continued)

The central radiation cone, , corresponds to only a small part of

the sin2 Θ′ radiation pattern, near Θ′  π/2 , where

Within this small angular cone, a Lorentz transformation back to the laboratory frame of reference gives

so that (5.37)

For the central radiation cone (1/N relative spectral bandwidth) 2 2 ∆Ωcen = πθcen = π/(γ* N ) So that for a single electron, the power radiated into the central cone is

(5.38)

Professor David Attwood Univ. California, Berkeley Undulator Equation and Radiated Power, EE290F, 15 Feb 2007 Ch05_Eq37_41b.top.ai Power in the Central Radiation Cone (continued)

(5.38)

For Ne electrons radiating independently within the undulator

Ne = IL/ec, where L = Nλu The power radiated into the central cone is then

(K ≤ 1) (5.39)

or

(5.41b)

Professor David Attwood Univ. California, Berkeley Undulator Equation and Radiated Power, EE290F, 15 Feb 2007 Ch05_Eq37_41b.bot_Oct05.ai Corrections to Pcen for Finite K

Our formula for calculated power in the central radiation cone (θcen = 1/γ* N , ∆λ/λ = 1/N)

(5.39)

2 is strictly valid for K << 1. This restriction is due to our neglect of K terms in the axial velocity vz. The Pcen formula, however, indicates a peak power at K = 2 , suggesting that we explore extension of this very useful analytic result to somewhat higher K values. Kim* has studied undulator radiation for arbitrary K and finds an additional multiplicative factor, f(K), which accounts for energy transfer to higher harmonics:

(5.41a)

where (5.40a) and x = K2/4(1 + K2/2)

(5.40b)

* K.-J. Kim, “Characteristics of Synchrotron Radiation”, pp. 565-632 in Physics of Particle Accelerators (AIP, New York, 1989), M. Month and M. Dienes, Editors. Also see: P.J. Duke, Synchrotron Radiation (Oxford Univ. Press, UK, 2000). A. Hofmann, “The Physics of Synchrotron Radiation” (Cambridge Univ. Press, 2004).

Professor David Attwood Univ. California, Berkeley Undulator Equation and Radiated Power, EE290F, 15 Feb 2007 Ch05_Eq39_41a_T5_Oct05.ai Pcen in Terms of Photon Energy

From the undulator equation λ K 2 λ = u (1 + + γ 2θ2) 2γ2 2 On axis, θ = 0, and with f λ = c K 2 f = 2γ2c /λ (1 + ) u 2 In terms of photon energy (on-axis) K 2 ω = 4πγ2c /λ (1 + ) u 2 2 2 2 We can now replace K /(1 + K /2) in Pcen by an expression involving ω. Introducing the limiting photon energy ωo, corresponding to K = 0, 2 ωo = 4πγ c /λu then (5.41c)

or (5.41e) where (5.41d)

For λu = 5.00 cm and γ = 3720, ωo = 686 eV. For λu = 3.30 cm and γ = 13,700, ωo = 14.1 keV

Professor David Attwood Univ. California, Berkeley Undulator Equation and Radiated Power, EE290F, 15 Feb 2007 Ch05_Eq41c_eVG.ai Power in the Central Radiation Cone For Three Soft X-Ray Undulators

2.5 ALS 1 2 γ = 3720 θcen = n = 1 γ* N λu = 50 mm 1.5 N = 89 ∆λ   = 1 P (W) λ Ι = 400 mA cen = 89  λ 1 N 1 ∆λ n = 3 θcen = 35 µr ∆λ   = 1 λ λ 0.5 = 270  3 3N ∆λ 0 0 500 1000 1500 2000

1.5 ALS n = 1 γ = 3720 λu = 80 mm 1 N = 55 Ι = 400 mA Pcen (W) n = 3 θcen = 45 µr 0.5

0 0 200 400 600 800 1000 1 MAX II n = 1 γ = 2940 λu = 52 mm N = 49 Pcen (W) 0.5 Ι = 250 mA n = 3 θcen = 59 µr

0 0 200 400 600 800 1000 Photon energy (eV)

Professor David Attwood Univ. California, Berkeley Undulator Equation and Radiated Power, EE290F, 15 Feb 2007 Ch05_PwrCenRadCone1rev.ai Power in the Central Radiation Cone For Three Soft X-Ray Undulators

1.0 BESSY II θ = 1 γ = 3330 cen γ* N 0.8 n = 1 λu = 49 mm N = 84 ∆λ 0.6   = 1 Pcen (W) λ Ι = 200 mA = 84  λ 1 N 0.4 ∆λ θcen = 35 µr ∆λ 1 λ = 0.2 = 250  λ  3N ∆λ n = 3 3 0 0 500 1000 1500 2000 Photon Energy (eV)

2.0 ELETTRA n = 1 γ = 3910 1.5 λu = 56 mm N = 81 Ι = 300 mA Pcen (W) 1.0 θcen = 35 µr 0.5 n = 3 0 102 103 Photon Energy (eV)

8 Australian Synchrotron 6 γ = 5870 λu = 22 mm n = 1 N = 80 Pcen (W) 4 Ι = 200 mA θcen = 23 µr 2 n = 3

0 0 2 4 6 8 10 Photon Energy (keV)

Professor David Attwood Univ. California, Berkeley Undulator Equation and Radiated Power, EE290F, 15 Feb 2007 Ch05_PwrCenRadCone2.ai Power in the Central Radiation Cone For Three X-Ray Undulators

15 ESRF 1 θcen = n = 1 γ = 11,800 γ* N 10 λu = 42 mm ∆λ N = 38   = 1 P (W) λ = 38  λ  N cen ∆λ Ι = 200 mA 1 5 θcen = 17 µr ∆λ   = 1 λ = 120 n = 3  λ 3 3N ∆λ 0 0 5 10 15 20 25

15 APS n = 1 γ = 13,700 10 λu = 33 mm N = 72 Pcen (W) Ι = 100 mA n = 3 5 θcen = 11 µr

0 0 10 20 30 40

20 SPring-8 γ = 15,700 15 λu = 32 mm N = 140 n = 1 Pcen (W) 10 Ι = 100 mA θcen = 6.6 µr 5 n = 3

0 0 10 20 30 40 50 60 Photon Energy (keV)

Professor David Attwood Univ. California, Berkeley Undulator Equation and Radiated Power, EE290F, 15 Feb 2007 Ch05_PwrCenRadCone3.ai Power Radiated into the Central Radiation Cone

(θcen = 1/γ* N , λ/∆λ = N, n = 1)

ALS APS

2.5 14 12 2 10 P 1.5 P cen = 5.0 cm cen 8 (watt) λu (watt) λu = 3.3 cm 6 1 N = 89 N = 72 = 3720 4 0.5 γ γ = 13,700 I = 400 mA 2 I = 100 mA 0 0 2 2 1.5 1.5 800 16 1 700 1 14 600 12 K 0.5 500 K 0.5 10 400 8 300 6 0 0 4 200 Photon energy (eV) Photon energy (keV)

Professor David Attwood Univ. California, Berkeley Undulator Equation and Radiated Power, EE290F, 15 Feb 2007 Ch05_F20VG.ai Spectral Bandwidth of Undulator Radiation from a Single Electron

(On-axis radiation, θ = 0)

Radiated Wavetrain Spectral Distribution

1.0 2 Ι(ω) Sin Nπu N cycles 2 Ι0 (Nπu) E ∆ω 0.8895 t 0.5 = ω0 N

2π ω0 0 0 ω0 ω

Professor David Attwood Univ. California, Berkeley Undulator Equation and Radiated Power, EE290F, 15 Feb 2007 Ch05_F21VG.ai