2.2 Synchrotron Radiation SourcesCharacteristics of Bending Magnet RadiationThe bending magnet radiation is described as a sweeping “searchlight”. This effect is a general manifestation associated with radiation from relativistic particles undergoing acceleration.

Non-relativistic radiation Relativistic radiation

For highly relativistic electrons β approaches unity, and γ >>1. The large emission angle Θ in non-relativistic radiation is compressed, and the radiation is folded into a narrow radiation cone of half angle;

.

1957 and, 2

energyelectronisEwhere

[GeV]Ecm

E

e

ee

e ==≈ γ2γ1θ

( ) 2/122 /-11/ ,/ −== cvcv γβ

a

v

2.2 Synchrotron Radiation SourcesFrom the solution of Maxwell’s equations for relativistic electron in a uniform magnetic field, the photon flux FB for bending magnet radiation is given by

)%1.0(sec/)/( ][ ][1033.1

/ 22213

0

3

BWmradphotonsEEHAIGeVE

dddFd

ceB

⋅×=

=ψωωψθ

θ

( )

requency agular f:critical ω nergy photon e:critical E

ω/ω y onel functiified Bess); K(y/KyyH

% BW..is where dω

c

c

c

/

== mod :2

10 /

3/22

322

2

ω

mBeE cc 2

3 2γω hh ==

(E/Ec) is the photon energy normalized with respect to critical photon energy;

2.2 Synchrotron Radiation SourcesThe critical photon energy is that for which half the radiated power is in higher energy photons and half in lower energy photons.

][ ][665.0)( 2 TBGeVEkevE ec =

The corresponding wavelength is

[GeV] B[T]E.or

eBcm

ec

ec 2

0

2

6418A 34

=⎟⎠⎞

⎜⎝⎛= λ

γπλ

keV(APS) keV(ALS), . keV(PLS), .: EExample, c 201355

It is convenient to know the bending magnet photon flux per unit horizontalangle θ, integrating out the vertical plane -dependence.φ

( ) ( ) functionBesselifiedisKdyyKyyGwhere

BWmradphotonsEEGAIGeVE

ddFd

ceB

mod :''

)%1.0(sec/)/( ][ ][1046.2

/

3/50 3/51

113

2

∫∞

=

⋅×=

ωωθ

2.2 Synchrotron Radiation Sources

The functions H2(y), and G1(y) are tabled and are shown graphically below

2.2 Synchrotron Radiation Sources

The functions H2(y), representingon-axis photon flux from bendingmagnet, and G1(y), representing the vertically integrated photon flux, as function of photon energynormalized to the critical photonenergy. Half the radiated poweris in photons of energy greaterthan Ec, and half in photons of energy less than Ec.

Note that for a photon energy of 4Ec the photon flux is reduced a factor of about 10 from its value at Ec. Practically usable photon energy is the energy up to 4Ec.

2.2 Synchrotron Radiation Sources

electron

Polarization of radiationi) σ-component: horizontally polarizedii) π -component: vertically polarized

ψ=0; 100% σ-componentψ≠0; π-component exists

)( component−σ

)( component−π

Characteristics of Undulator Radiation2.2 Synchrotron Radiation Sources

An electron traversing a periodic magnet structure will undergo a small amplitude of oscillation and therefore radiate. If the electron’s angular excursions are small compared with the natural radiation, θc<1/2γ, the device is referred to as an undulator.

2.2 Synchrotron Radiation Sources

Since the speed of electron is close to c, Lorentz contraction and relativisticDoppler shift of λu lead to a reduction in the radiated wavelength by a factorof . 22γ

Moving electron sees a periodic magnetic structure (Lorentz contraction);

λλλ u='

Due to the periodic magnet, the electron experiences an oscillation and consequently radiates. In the frame moving with the electron this problem is that of the classical dipole, a point charge oscillating with an amplitudemuch smaller than the radiated wavelength. The frequency of this emittedradiation, in the reference frame of the electron, is

u

ccfλγ

λ==

''

2.2 Synchrotron Radiation SourcesTo the observer in the fixed laboratory reference frame, the radiationwavelength is further reduced by Doppler shift. The relativistic form of theDoppler frequency formula is

( ) ( )θβλθβγ cos1cos1'

−=

−=

u

cff

Radiation from an oscillating charge moving at (i) a non-relativist and (ii) a relativistic speed.

(i) (ii)

2.2 Synchrotron Radiation Sources

In general, since β≈1 and θ <<1,

( ) ( )22222

2

12

,12 θγ

γλλ

θγλγ

+=+

= u

u

andcf

22γWe see the contraction on axis, but with the off-axis radiation having a wavelength in creased by a factor . ( )22θγ1+

Undulator Radiation Pattern

( ) 2/122 /-11/ ,/ −== cvcv γβ

a

v

2.2 Synchrotron Radiation Sources

Undulator Radiation Pattern

a

v

For undulating electron motion in the frame of reference moving electron, electron oscillated simple harmonic motion. However, the radiation pattern observed in the laboratory frame is relativistically contracted into a narrowradiation cone, so-called search-light effect as

Radiation pattern in laboratory frame

Radiation pattern in electron frame

An oscillating charge and the resultant radiation pattern

2.2 Synchrotron Radiation SourcesThe central Radiation Cone

(a) Radiation spectrum as seen in the frame of reference moving withthe electron is narrow with a relative spectral bandwidth of order 1/N(Δω’/ω’=1/n is derivation form frequency –time transformation) , where N is the number of oscillation periods. (b) In the laboratory frame ofreference, the wavelength a shorter, but spectrum is broader due to off-axis Doppler effects.

Recall

( )222 1

2 θγ

γλλ += u

2.2 Synchrotron Radiation SourcesThe radiation is contained in a narrow cone of half angle θ=1/2γ, then thecorresponding spectral width within cone can be estimated by taking thedifference for two angles.

41 22 ≈=

Δ θγλλ

Use of aperture spectral filtering, or use of grating monochromatorspectral bandwidth can be obtained up to

N1 =

Δλλ

The central radiation cone of the half width becomes

Ncen γθ 1 =

2.2 Synchrotron Radiation Sources

Spectrum of undulator radiation before and after selecting a angular cone near axis

Illustration of grating monochromator as used filter undulator radiation to annatural spectral width 1/N, and the concomitant cone narrowing to .N1/γ

2.2 Synchrotron Radiation Sources

Undulator Radiation: Calculation of Radiated Power, Brightness, and HarmonicsUndulator Equation

uy

zBBλπ2cos0=

)( BvEp×+= q

dtd

parameter) deflection (magnetic

][ ][9337.02eBK where, 2sin u0

0 cmTBcm

zKcve

u

ux λ

πλ

λπ

γ==⎟⎟

⎠

⎞⎜⎜⎝

⎛=

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛ +−=

uz

zcKKcvλπ

γγ22cos

422/11 2

2

2

2

2.2 Synchrotron Radiation Sources

The angle of the electron motion is bounded by ±K/λ, i.e.

⎟⎟⎠

⎞⎜⎜⎝

⎛≈=

uz

x zKvv

λπ

γθ 2sintan e

Maximum excursion angle

γθ K

e ≈max,

Recall that the characteristic half angle for emission of radiation is θrad≈1/2γ.For K<1, the electron angular excursions lie within the radiation cone. For K>>1, we refer to wiggler radiation. In this case, interference opportunitiesare lost because the radiation from various segments of an oscillation arewidely separated in the angle and therefore do not overlap in space after somepropagation distance. Wiggler radiation provides a 2N increase in radiated power and a broad shift to higher photon energies.

2.2 Synchrotron Radiation Sources

After the Lorentz contraction and relativistic Doppler shift, we can get the undulator equation: fundamental wavelength

⎟⎟⎠

⎞⎜⎜⎝

⎛++

=⎟⎟⎠

⎞⎜⎜⎝

⎛++=

222

222

2

21

21 ][

][9496.0)( 2

12

θγλ

θγγλλ

Kcm

GeVEkeVEandK

u

eu

Undulator Harmonics

In addition to fundamental wavelength. The effect of transverse oscillationsintroduces higher harmonics into the motion. Let us consider as a 2-nd harmonic motion, vz.

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛ +−=

uz

zcKKcvλπ

γγ22cos

422/11 2

2

2

2

2.2 Synchrotron Radiation SourcesThis expression displays both the decreases axial velocity and an axial modulation at twice the fundamental frequency.

If the first order (fundamental) motion leads to radiation at ω’1 in the electron frame, then the axial harmonic will radiate at ω’2=2 ω’1. The second harmonic oscillations of the electron are at right angles to the fundamental oscillations. As a result, the polarization is different.

2.2 Synchrotron Radiation Sources

(a) Figure 8 electron motion in the frame of reference moving with theaverage electron velocity.

(b) The resultant radiation patterns at the fundamental and second harmonic frequencies in the frame of reference moving with theelectron.

(c) The resultant radiation patterns at the fundamental and second harmonic frequencies in the laboratory frame of reference.

2.2 Synchrotron Radiation SourcesFurther analysis of the electron motion, we can fine a general undualtorequation

2

12

222

2 ⎟⎟⎠

⎞⎜⎜⎝

⎛++= θγ

γλλ K

nu

n

.

# ,1

periodsmagnetic e # of N is th

andharmonictheisnwherenN

=Δλλ

Due to the increased number of cycles, the relative spectral bandwidth is also improved

Even harmonics (n=2, 4, 6, …) radiate a pattern that peaks off axis and has zero intensity on axis. Odd harmonics (n=1, 3, 5, …) radiate on axis with narrow spectrum and into a narrow forward cone. (Note that within a fullelectron, this will be modified due to random individual motions slightlyoff axis, i.e., to finite phase space effects.) As consequence, the even harmonics tend to be relatively weak on the axis, upon transformation to laboratory frame, radiate in to hollow cone of radiation.

2.2 Synchrotron Radiation SourcesPower Radiated in the Central Radiation One

Considering electron motion in an undulator of finite K, the undulatorparameter can be written by

2

12

222

2 ⎟⎟⎠

⎞⎜⎜⎝

⎛++= θγ

γλλ Ku

We can obtains a corrected formula for central radiation cone

2/1 2

cen NK

γθ +

=

For the fundamental frequency with a relative spectral bandwidth Δλ/λ≈1/N,the average power radiated into the central cone is given by

( ) . ,21

22

2

0

2

cen currentelctronisIwhere/K K

λεIπeγP

u +≈

2.2 Synchrotron Radiation Sources

( )22

226

cen21][

][)1069.5( /K K

cmλAIγWP

u +×= −

Total Radiated Power

( )

( )22

226

1,

22

2

0

2

1,

2/1][][)1090.1(

,2/13

KK

cmANIWP

orKKINeP

uT

uT

+

×=

+=

−

λγ

λεγπ

2.2 Synchrotron Radiation Sources

Special Type of Undulator

Polarization control undulator: linearly, circularly, or elliptically polarizedradiation.

2.2 Synchrotron Radiation SourcesCharacteristics Wiggler RadiationTransition from Undulator to Wiggler RadiationAs K increases toward unity the radiated power in higher harmonics growsrapidly. For K>>1, analysis shows the emergence of large number of everstronger harmonics, extending to ever higher photon energies.

2.2 Synchrotron Radiation Sources

It can be seen the development of this comblike harmonic structure for increasing values the magnetic deflection parameter K. The comblikespectrum eventually merges to continuum for finite emittance and acceptance angle in the wiggler limit (K>>1).

2.2 Synchrotron Radiation SourcesWiggler Power and Flux

For wiggler radiation with K>>1, although the electron motion become very complex, the shape of the spectrum irradiated from wiggler is basically the same shape of H2(ω/ωc).

Photon flux is 2N times higherthan that of bending magnet.

Critical energy of wiggler radiationis proportional to magnetic fieldstrength.

2.2 Synchrotron Radiation Sources

Total radiated power in all harmonics, integrated over all angles and wavelengths, for arbitrary K, is given by

][][)1090.1(

,3

2260

22

cmANIKWP

orINKeP

uT

uT

λγ

λεγπ

−×=

=

)%1.0(sec/)/( ][ ][1065.2

/ 22213

0

2

BWmradphotonsEEHAIGeVNE

ddFd

ceB

⋅×=

Ω=ψωω

Photon flux per unit solid angle, per unit relative spectral bandwidth, Is given by

2.2 Synchrotron Radiation Sources

)%1.0(sec/)/( ][ ][1092.4

/ 113

2

BWmradphotonsEEGAIGeVNE

ddFd

ceB

⋅×=

ωωθ

The photon flux per unit horizontal angle θ, integrating out the vertical planeψ-dependence, is given by

2.2 Synchrotron Radiation SourcesHomework

Due date: October-18, 20071. Solve the problems 2.1, 2.3, 2.9, and 2.10