Undulator Equation and Rediated Power - Peopleattwood/sxr2009/lecnotes/10...Ch05_F24_left_Feb07.ai...

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


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


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


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


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)


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


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


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


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


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


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)


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)


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


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


ALS


MAX II

Pcen (W)

Pcen (W)

1N

13N

=∆λλ 1

=∆λλ 3

λ∆λ

= 89

λ∆λ

= 270


θcen =

1γ* N

Power in the Central Radiation ConeFor Three Soft X-Ray Undulators

Ch05_PwrCenRadCone2.ai

1N

13N

=∆λλ 1

=∆λλ 3


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


ELETTRA


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)


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


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)


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=


Undulator Equation and Rediated Power - Peopleattwood/sxr2009/lecnotes/10...Ch05_F24_left_Feb07.ai...

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