undulator equation and rediated power - peopleattwood/sxr2009/lecnotes/10...ch05_f24_left_feb07.ai...
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Undulator Equationand 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
Ch05_LG189_Jan07.ai
Pcen =
cen =
πeγ 2I
0λuK2
2
eB0λu2πm0c
∆λλ
1γ ∗ N
1N
γ ∗ = γ / 1 + K2
2
λx = (1 + + γ 2θ2)λu
2γ 2K2
2
K =
θcen =
N periods
1 γ∗1
Nγ∗θcen =
e–
λu
Photon energy (eV)
Pce
n (W
)Tuningcurve
λ∆λ
= N
ALS, 1.9 GeV400 mAλu = 8 cmN = 55K = 0.5-4.0
0 100 200 300 4000
0.50
1.00
1.50
2.00
(1 + )2
K2f(K)
Professor David AttwoodUniv. California, Berkeley Undulator Equation and Radiated Power, EE290F, 15 Feb 2007
Ch05_F24_left_Feb07.ai
Spectral Brightness of Synchrotron Radiation
1-2 GeVUndulators
6-8 GeVUndulators
Wigglers
Bendingmagnets
10 eV 100 eV 1 keV 10 keV 100 keVPhoton energy
Spe
ctra
l brig
htne
ss
12 nm 1.2 nm 0.12 nm1020
1019
1018
1017
1016
1015
1014
Professor David AttwoodUniv. California, Berkeley Undulator Equation and Radiated Power, EE290F, 15 Feb 2007
The Equation of Motion in an Undulator
Ch05_F15_Eq16_19.top.ai
(5.16)
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
where p = γmv is the momentum. The radiated fields are relatively weak so that
Taking to first order v vz, motion in the x-direction is
By = Bo cos 2πzλu
y
z
x
ve–
= –e(E + v × B)dpdt
–e(v × B)dpdt
Professor David AttwoodUniv. California, Berkeley Undulator Equation and Radiated Power, EE290F, 15 Feb 2007
The Equation of Motion in an Undulator (cont.)
Ch05_F15_Eq16_19.bot.ai
(5.17)
(5.19)
(5.18)
integrating both sides
is the non-dimensional “magnetic deflection parameter.”The “deflection angle”, θ, is
θ = = sinkuzvxvz
vxc
Kγ
Professor David AttwoodUniv. California, Berkeley Undulator Equation and Radiated Power, EE290F, 15 Feb 2007
The Axial Velocity Depends on K
Ch05_Eq22_23VG.ai
(5.22)
(5.23a)
In a magnetic field γ is a constant; to first order the electron neither gains nor looses energy.
thus
Taking the square root, to first order in the small parameter K/γ
Using the double angle formula sin 2kuz = (1 – cos 2kuz)/2, where ku = 2π/λu,
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.
Reducedaxial velocity
A double frequencycomponent of the motion
Professor David AttwoodUniv. California, Berkeley Undulator Equation and Radiated Power, EE290F, 15 Feb 2007
K-Dependent Axial VelocityAffects the Undulator Equation
Ch05_Eq25_29VG.ai
(5.25)
(5.26)
(5.29a)
Averaging the z-component of velocity over a full cycle (or N full cycles) gives
We can use this to define an effective Lorentz factor γ* in the axial direction
As a consequence, the observed wavelength in the laboratory frame of reference is modified fromEq. (5.12), taking the form
that is, the Lorentz contraction and relativistic Doppler shift now involve γ* rather than γ
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
(5.28)
Professor David AttwoodUniv. California, Berkeley Undulator Equation and Radiated Power, EE290F, 15 Feb 2007
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
Ch05_T4_topVG.ai
e–
γ*e–
z
N periods
x′
z′
Lorentz transformation
Lorentztransformation
Θ′θ′
sin2Θ′
= Nλ′∆λ′
λu
λu
x
z= N
θcen =
λ∆λ
1γ* N
′ =λuγ*
Determine x, z, t motion:
x, z, t laboratory frame of reference x′, z′, t′ frame of reference moving with theaverage velocity of the electron
Dipole radiation:
=
x′, z′, t′ motiona′(t′) acceleration
= –e (E + v × B)
mγ = e B0 cos
vx(t); ax(t) = . . .
vz(t); az(t) = . . .
dpdt
dvxdt
dzdt
2πzλu
dP′dΩ′
e2 a′2 sin2 Θ′16π20c3
= (1–sin2 θ′ cos2 φ′) cos2 ω′ut′dP′dΩ′
e2 cγ2
40λu
K2
(1 + K2/2) 22
Professor David AttwoodUniv. California, Berkeley Undulator Equation and Radiated Power, EE290F, 15 Feb 2007
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.)
Ch05_T4_bot.VG.ai
x′
z′
Lorentz transformation Θ′θ′
sin2Θ′
= Nλ′∆λ′
x
z= N
θcen =
λ∆λ
1γ* N
= (1–sin2 θ′ cos2 φ′) cos2 ω′ut′dP′dΩ′
e2 cγ2
40λu
K2
(1 + K2/2) 22
Ne uncorrelated electrons:
Ne = IL /ec, L = Nλu
= 8γ*2dP
dΩ
=
∆Ωcen = πθ2 = π/γ*2 N
Pcen =
dP
dΩ
dP′dΩ′
e2 cγ4
0λu
K2
(1 + K2/2)
K2
(1 + K2/2)
K2
(1 + K2/2)
K ≤ 1θ ≤ θcen
2 3
πe2 cγ2
0λu N2 2
Pcen = πe γ2I
0λu2
cen
Professor David AttwoodUniv. California, Berkeley Undulator Equation and Radiated Power, EE290F, 15 Feb 2007
Power Radiated in the Central Radiation Cone
Ch05_Eq32_34.top.ai
=dP′dΩ′
Kkuγ
z′c
e2 a′2 sin2 Θ′16π20c3
To use the “dipole radiation” formula
Take z ct , and ωu = kuc
Integrating once
We can use the Lorentz transformation to the primed frame of reference
Thus in the primed frame of reference
, ku = 2π/λu
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
x = x′
x′ = – cos ωuγ*(t′ + )
ωu′ small: z′ is the small axialmotion about the mean
Professor David AttwoodUniv. California, Berkeley Undulator Equation and Radiated Power, EE290F, 15 Feb 2007
Power Radiated in the Central Radiation Cone (cont.)
Ch05_Eq32_34.bot.ai
Kkuγ
z′c
(5.33)
(5.34)
Taking the second derivative
since
Then
x′ = – cos ωuγ*(t′ + )
ωu′ small: z′ is the small axialmotion about the mean
Professor David AttwoodUniv. California, Berkeley Undulator Equation and Radiated Power, EE290F, 15 Feb 2007
Power in the Central Radiation Cone (continued)
Ch05_Eq37_41b.top.ai
(5.37)
(5.38)
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 laboratoryframe of reference gives
so that
For the central radiation cone (1/N relative spectral bandwidth)
So that for a single electron, the power radiated into the central cone is
∆Ωcen = πθ2 = π/(γ* N )2cen
Professor David AttwoodUniv. California, Berkeley Undulator Equation and Radiated Power, EE290F, 15 Feb 2007
Power in the Central Radiation Cone (continued)
Ch05_Eq37_41b.bot_Oct05.ai
(5.38)
For Ne electrons radiating independently within the undulator
The power radiated into the central cone is then
or
Ne = IL/ec, where L = Nλu
(K ≤ 1) (5.39)
(5.41b)
Professor David AttwoodUniv. California, Berkeley Undulator Equation and Radiated Power, EE290F, 15 Feb 2007
Corrections to Pcen for Finite K
Ch05_Eq39_41a_T5_Oct05.ai
(5.39)
(5.41a)
(5.40a)
(5.40b)
Our formula for calculated power in the central radiation cone (θcen = 1/γ* N , ∆λ/λ = 1/N)
where
* 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).
and
is strictly valid for K << 1. This restriction is due to our neglect of K2 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:
x = K2/4(1 + K2/2)
Professor David AttwoodUniv. California, Berkeley Undulator Equation and Radiated Power, EE290F, 15 Feb 2007
Pcen in Terms of Photon Energy
Ch05_Eq41c_eVG.ai
(5.41c)
(5.41e)
(5.41d)
From the undulator equation
On axis, θ = 0, and with f λ = c
In terms of photon energy (on-axis)
We can now replace K 2/(1 + K2/2)2 in Pcen by an expression involving ω. Introducing the limiting photon energy ωo, corresponding to K = 0,
then
or
where
For λu = 5.00 cm and γ = 3720, ωo = 686 eV. For λu = 3.30 cm and γ = 13,700, ωo = 14.1 keV
λ = (1 + + γ 2θ2)λu2γ2
K 2
2
ω = 4πγ2c /λu (1 + )
ωo = 4πγ 2c /λu
K 2
2
f = 2γ2c /λu(1 + )K 2
2
Professor David AttwoodUniv. California, Berkeley Undulator Equation and Radiated Power, EE290F, 15 Feb 2007
0 500 1000 1500 20000
0.5
1
1.5
2
2.5
0 200 400 600Photon energy (eV)
800 10000
0.5
10 200 400 600
γ = 3720λu = 80 mmN = 55Ι = 400 mAθcen = 45 µr
θcen =
n = 1
1γ* Nn = 1
n = 1
n = 3
n = 3
n = 3
800 10000
0.5
1
Pcen (W)
1.5
Power in the Central Radiation ConeFor Three Soft X-Ray Undulators
Ch05_PwrCenRadCone1rev.ai
ALS
γ = 3720λu = 50 mmN = 89Ι = 400 mAθcen = 35 µr
ALS
γ = 2940λu = 52 mmN = 49Ι = 250 mAθcen = 59 µr
MAX II
Pcen (W)
Pcen (W)
1N
13N
=∆λλ 1
=∆λλ 3
λ∆λ
= 89
λ∆λ
= 270
Professor David AttwoodUniv. California, Berkeley Undulator Equation and Radiated Power, EE290F, 15 Feb 2007
θcen =
1γ* N
Power in the Central Radiation ConeFor Three Soft X-Ray Undulators
Ch05_PwrCenRadCone2.ai
1N
13N
=∆λλ 1
=∆λλ 3
γ = 3330λu = 49 mmN = 84Ι = 200 mAθcen = 35 µr
0 500 1000 1500 2000
n = 1
n = 30
0.2
0.4
0.6
0.8
1.0
λ∆λ
= 84
λ∆λ
BESSY II
γ = 3910λu = 56 mmN = 81Ι = 300 mAθcen = 35 µr
ELETTRA
γ = 5870λu = 22 mmN = 80Ι = 200 mAθcen = 23 µr
Australian Synchrotron
= 250
0
0.5
1.5
2.0
1.0
102 103
n = 1
n = 3
Photon Energy (keV)
Photon Energy (eV)
Photon Energy (eV)
0 2 4 6 8 10
8
6
2
0
n = 1
n = 3
4
Pcen (W)
Pcen (W)
Pcen (W)
Professor David AttwoodUniv. California, Berkeley Undulator Equation and Radiated Power, EE290F, 15 Feb 2007
θcen =
1γ* N
Power in the Central Radiation ConeFor Three X-Ray Undulators
Ch05_PwrCenRadCone3.ai
1N
13N
=∆λλ 1
=∆λλ 3
0 10 20 30 400
5
10
15
n = 1APS
n = 3
Photon Energy (keV)
0 5 10 15 20 25
15
10
5
0
n = 1
n = 3
Pcen (W)
γ = 11,800λu = 42 mmN = 38Ι = 200 mAθcen = 17 µr
γ = 13,700λu = 33 mmN = 72Ι = 100 mAθcen = 11 µr
SPring-8γ = 15,700λu = 32 mmN = 140Ι = 100 mAθcen = 6.6 µr
ESRF
Pcen (W)
Pcen (W)
λ∆λ = 38
= 120λ∆λ
n = 1
n = 3
0 10 20 30 40 50 600
5
10
15
20
Professor David AttwoodUniv. California, Berkeley Undulator Equation and Radiated Power, EE290F, 15 Feb 2007
Ch05_F20VG.ai
Power Radiated into the Central Radiation Cone
0
200 300 400 500
Photon energy (eV)600 700 800
21.5
0.51
K0 4
6 8 10
Photon energy (keV)12 14 162
1.5
0.51
K
0
0.5
1
1.5Pcen(watt)
Pcen(watt)
2
2.5
ALS
02468
10
12
14
APS
λu = 5.0 cm
N = 89
γ = 3720
I = 400 mA
λu = 3.3 cm
N = 72
γ = 13,700
I = 100 mA
(θcen = 1/γ* N , λ/∆λ = N, n = 1)
Professor David AttwoodUniv. California, Berkeley Undulator Equation and Radiated Power, EE290F, 15 Feb 2007
Ch05_F21VG.ai
Spectral Bandwidth of UndulatorRadiation from a Single Electron
(On-axis radiation, θ = 0)
2πω0
N cycles
Radiated Wavetrain Spectral Distribution
E
t
Sin2 Nπu(Nπu)2
Ι(ω)Ι0
1.0
0.5
00 ω0 ω
∆ωω0
0.8895N=
Professor David AttwoodUniv. California, Berkeley Undulator Equation and Radiated Power, EE290F, 15 Feb 2007