Spatial Temporal Coherence; Coherent Undulator Radiation

Spatial and Temporal Coherence;Coherent Undulator Radiation

David Attwood

University of California, Berkeley

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

Spatial and Temporal Coherence; Coherent Undulator Radiation, EE290F, 22 Feb 2007

d

Ch08_F00VG.ai

λθ

λ

e–

Pinhole

d

λ

λθ

lcoh = λ2/2∆λ {temporal (longitudinal) coherence}

d θ = λ/2π {spatial (transverse) coherence}

or d 2θFWHM = 0.44 λ

(8.3)

(8.5)

(8.5*)

Pcoh,λ/∆λ = 1– ƒ(K)

(8.6)

(8.9)

Pcoh = Plaser (8.11)

(λ/2π)2

(dxθx)(dyθy)

(λ/2π)2

(dxθx)(dyθy)

ωω0

eλuIη(∆λ/λ)N2

8π0dxdy

Pcoh,N = Pcen

λu

COHERENCE AT SHORT WAVELENGTHSChapter 8

Professor David AttwoodUniv. California, Berkeley Spatial and Temporal Coherence; Coherent Undulator Radiation, EE290F, 22 Feb 2007

d θ

λλ

CH08_YoungsExprmt_v3.ai

Young’s Double Slit Experiment: Spatial Coherence and the Persistence of Fringes

Persistence of fringes as the source grows from a point source to finite size.


d 2θ|FWHM λ/2

λcoh = λ2/2∆λ = Ncohλ12

Spatial and Spectral Filteringto Produce Coherent Radiation

Ch08_F08VG.ai

Courtesy of A. Schawlow, Stanford.Professor David AttwoodUniv. California, Berkeley Spatial and Temporal Coherence; Coherent Undulator Radiation, EE290F, 22 Feb 2007

Ch08_F01VG.ai

Coherence, Partial Coherence, and Incoherence

θd

λ λ, ∆λ

Point source oscillator– < t <

Source of finite size,divergence, and duration


Spatial and Temporal Coherence

Ch08_Eq1_12_F2VG.ai

(8.1)

(8.12)

(8.3)

(8.5)

Mutual coherence factor

Longitudinal (temporal) coherence length

Full spatial (transverse) coherence

Normalize degree of spatial coherence(complex coherence factor)

A high degree of coherence (µ → 1)implies an ability to form a high contrastinterference (fringe) pattern. A low degreeof coherence (µ → 0) implies an absenceof interference, except with great care.In general radiation is partially coherent.

Transverse (spatial)coherence

Longitudinalcoherence length

Point source,harmonic oscillator

P1

P2

P2

coh = λ2/2∆λ

P1


Spectral Bandwidth andLongitudinal Coherence Length

Ch08_F03VG.ai

(8.3)

Define a coherence length coh as the distance of propagation over which radiation of spectral width ∆λ becomes 180° out of phase. For a wavelength λ propagating through N cycles

and for a wavelength λ + ∆λ, a half cycle less (N – )

Equating the two

so that

λ ∆λ

∆λ

λcoh

180 phase shift1.00

coh = Nλ

coh = (N– ) (λ + ∆λ)

N = λ/2∆λ

12

12


• Associate spatial coherence with a spherical wavefront.• A spherical wavefront implies a point source.• How small is a “point source”?

Ch08_XBL 915-6740AVG.ai

From Heisenberg’s UncertaintyPrinciple (∆x · ∆p ≥ ), the smallest source size “d” you can resolve, withwavelength λ and half angle θ, is

d · θ = λ2π

d

λ

θ

2

A Practical Interpretationof Spatial Coherence


Ch08_XBL883-8849.modf.ai

Partially Coherent Radiation ApproachesUncertainty Principle Limits

∆x ∆p ≥ /2

quantities1e

∆x k∆θ ≥ 1/2

∆x ∆k ≥ /2

2∆x ∆θ ≥ λ/2π

Standard deviations of Gaussian distributed functions(Tipler, 1978, pp. 174-189)

Spherical wavefronts occurin the limiting case

or(d 2θ)FWHM λ/2 FWHM quantities

Note: ∆p = ∆k ∆k = k∆θ

d = 2∆x

(spatially coherent)d θ = λ/2π

θ

∆k∆θk

(8.4)


Ch08_F05VG.ai

Propagation of a Gaussian Beam

r0

Waist

λ

z Gaussian intensitydistribution, sphericallypropagating wave

r(z) θ

In

with waist diameter d = 2r0, we have TEM00 radiation with d θ = λ/2π

(Siegman, Lasers)


Ch08_HeisenbergVG.aiProfessor David AttwoodUniv. California, Berkeley Spatial and Temporal Coherence; Coherent Undulator Radiation, EE290F, 22 Feb 2007

Spatially Coherent Undulator Radiation

λ = 11.2 nm λ = 13.4 nm

1 µmD pinhole

25 mm wide CCDat 410 mm

Ch08_Coh_SXR_Sci .PPTProfessor David AttwoodUniv. California, Berkeley

Courtesy of Patrick Naulleau, LBNL.

Spatial and Temporal Coherence; Coherent Undulator Radiation, EE290F, 22 Feb 2007

Ch08_SpatialFiltrng.ai

Spatial Filtering of Undulator Radiation

Undulator central radiation cone ( = N ; θ = ):

With spatial filtering (a pinhole and an angular aperture):

(5.41a)

(8.6)

(8.9)

With eq.(5.28), . Convert to photon

energies where for θ = 0, , and where

corresponds to K = 0. For small electron beamdiverence, σ′ << θcen, the spatially coherent power is

λ∆λ

1γ* N

≤ 1 ≤ 1

2 2x,y


500 100 150 200 250 300 350 400 450

5

0

10

15

20

25

30

35

Photon energy (eV)

Tuningcurve

Coh

eren

t pow

er (

mW

)

λ∆λ = N

Pcoh = · Pcen

coh = Nλ/2

(λ/2π)2

(dxθx)(dyθy)

Spatially Filtered Undulator Radiation

Ch08_F09VG.ai

(8.6)

(8.9)

Using a pinhole-aperture spatial filter, passing only radiation that satisfies d θ = λ/2π

for dx = 2σx, dy = 2σy, θTx → θx, θTy → θy,and σ′2 << θcen.2

Photon energy (eV)

Pce

n (W

)

Tuningcurve

λ∆λ = N

Pcen = ·πeγ 2I0λu

K2f(K)

(1 + K2/2)2

λu = 8 cmN = 551.9 GeV0.4 ampn = 1(only)

0 100 200 300 4000

0.50

1.00

1.50

2.00Undulatorradiation

d

Spatialfilter

Pinhole

N periods

1 γ∗1

Nγ∗

1

Nγ∗

θcen =

θcen = ; = N; γ* = γ/ 1 + K2/2

λ = (1 + + γ2θ2)λu

2γ2

λ∆λ cen

θ

Angularaperture

K2

2

e–

λu


Ch08_SpatialSpectral.ai

Spatial and Spectral Filtering of Undulater Radiation

In addition to the pinhole – angular aperture for spatialfiltering and spatial coherence, add a monochromator for narrowed bandwidth and increased temporal coherence:

which for σ′ << θcen (the undulator condition) gives thespatially and temporally coherent power ( ; )

which we note scales as N2.

(8.10a)

(8.10c)

2 2x,y


Spatially and Spectrally FilteredUndulator Radiation

Ch08_F11VG.ai

(8.10a)

(8.10c)

• Pinhole filtering for full spatial coherence• Monochromator for spectral filtering to λ/∆λ > N

M1planemirror

M2spherical

mirror

Water-cooledbeam definingapertures

8.0 cm periodundulator

EUVinterferometer

166°

Variedline-

spacegrating

Exitslit

166°

M6bendable

EntrancepinholeM5

planeM4

bendable

M3retractable

planemirror

M0retract-

ableplanemirror

EUV/soft x-rayphotoemission

microscope

500 100 150 200 250 300 350 400 4500

100

50

200

150

Photon energy [eV]

Coh

eren

t pow

er (W

)

λ∆λ = 103

8.0 cm period, N = 551.9 GeV, 400 mAd θ = λ/2πcoh = 103λ/2η = 10%

Tuningcurve


Ch08_CohOptBLwGraf_Feb07.ai

Tuningcurves

8.0 cm period, N = 551.9 GeV, 400 mAd θ = λ/2πcoh = 1000 λ/2ηeuv = 10%, ηsxr = 10%

Coherent Power with a Monochromator

M2spherical2 mirror

Water-cooledbeam definingapertures

8.0 cm periodundulator

Variedline-space

grating

Exitslit172°

M4bendable2 mirror

M3bendable2 mirror

M0retractable

plane2 mirror

Entrancepinhole

CoherentSoft X-ray

End Station

K-Bfocus

system

Coherent Soft X-Ray Beamline: Use of a Higher Harmonic (n = 3) to AccessShorter Wavelengths

0

100

200

300

102 103

Photon Energy (eV)

Pco

h, λ

/∆λ (

µW)

n = 1

n = 3

λ∆λ

= 103


Coherent Power at the ALS

Ch08_U8_Feb07.ai

1.9 GeV, 400 mAλu = 80 mm, N = 550.5 ≤ K ≤ 4.0σx = 260 µm, σx′ = 23 µrσy = 16 µm, σy′ = 3.9 µr

0 200 400 600 800 10000

0.5

1.0

1.5

0

10

20

30

40

50

0

100

200

300

102 103102 103

Photon Energy (eV)

Photon Energy (eV)

Photon Energy (eV)

Pco

h, N

(mW

)

Pco

h, λ

/∆λ

(µW

)

Pce

n (W

)

U8

n = 1

n = 3

n = 1

n = 3

n = 1

n = 3

λ∆λ

= 55

λ∆λ = 55

λ∆λ = 165

λ∆λ

= 103

λ∆λ

= 103

ηeuv = 10%ηsxr = 10%


Coherent Power at the ALS

Ch08_U5_Feb07.ai

1.9 GeV, 400 mAλu = 50 mm, N = 890.5 ≤ K ≤ 4.0σx = 260 µm, σx′ = 23 µrσy = 16 µm, σy′ = 3.9 µrη = 10%

U5

0 500

n = 1

n = 3

n = 1

n = 3

n = 1

n = 3

1000 1500 20000

0.5

1.0

1.5

2.0

2.5

0

10

20

30

40

50

0

100

200

300

400

102 103102 103

Photon Energy (eV)Photon Energy (eV)

Photon Energy (eV)

Pco

h, N

(mW

)

Pco

h, λ

/∆λ

(µW

)

Pce

n (W

)

λ∆λ

= 89 λ∆λ

= 103


Ch08_ALS_U5epu_Feb07.ai

1.9 GeV, 400 mAλu = 50 mm, N = 270.5 ≤ K ≤ 4.0σx = 260 µm, σx′ = 23 µrσy = 16 µm, σy′ = 3.9 µrθcen = 61µr @ K = 0.87 (500 eV)

U5 EPU

0 500

n = 1

n = 3

n = 1

n = 3

n = 1

η = 1.3%

n = 3

1000 1500 20000

0.5

1.0

1.5

2.0

2.5

0

5

10

15

20

0

2

4

6

102 103102 103


Photon Energy (eV)

Pco

h, N

(mW

)

Pco

h, λ

/∆λ

(µW

)

Pce

n (W

)

λ∆λ

= 27 λ∆λ = 103


Coherent Power for an EPU at the ALS

0 200 400 600 800 10000

0.2

0.4

0.6

0.8

1.0

0

2

4

6

8

102 103102 103


Photon Energy (eV)

Pco

h, N

(mW

)

Pco

h, λ

/∆λ

(µW

)

Pce

n (W

)

Ch08_MAXII_Feb07.ai

1.5 GeV, 250 mAλu = 52 mm, N = 490.1 ≤ K ≤ 2.7σx = 300 µm, σx′ = 26 µrσy = 45 µm, σy′ = 20 µrη = 10%

n = 1

n = 3

n = 1

n = 3

n = 1

n = 3

λ∆λ

= 49 λ∆λ

= 103

0

10

20

30

40

Coherent Power at MAX II


Coherent Power at Elettra

Ch08_Elettra_Feb07.ai


0

0.5

1.5

2.0

1.0

0

5

10

15

20

0

50

100

150

102 103102 103

102 103

Photon Energy (eV)

Photon Energy (eV)

Pco

h, N

(m

W)

Pco

h, λ

/∆λ

(µW

)

Pce

n (W

)

n = 1

n = 3

n = 1

n = 3

n = 1

n = 3

λ∆λ

= 81

λ∆λ

= 81

λ∆λ

= 103


PohangPowerCurves.ai

2.5 GeV, 180 mAλu = 70 mm, N = 590 ≤ K ≤ 6.5σx = 433 µm, σx′ = 43 µrσy = 27 µm, σy′ = 6.8 µrθcen = 33 µr @ K = 1η = 10%

0 500

n = 1

n = 3

n = 1

n = 3

n = 1

n = 3

1000 1500 2000 25000

0.5

1

1.5

0

2

4

6

8

0

1

2

3

4

5

101 102 103101 102 103


Photon Energy (eV)

Pco

h, N

(mW

)

Pco

h, λ

/∆λ

(µW

)

Pce

n (W

)

λ∆λ

= 59

λ∆λ

= 59

λ∆λ

= 1000

Coherent Power at the Pohang Light Source


Coherent Power at New Subaru

Ch08_NewSubaru_Feb07.ai


n = 1

n = 3

n = 1

n = 3

n = 1

n = 3

0 100 200 300 400 500 6000

0.05

0.10

0.15

0.20

0

0.2

0.4

0.6

0.8

1.0

0

5

10

15

20

Pco

h, N

(mW

)

Pco

h, λ

/∆λ

(µW

)

Pce

n (W

)

101 103102102101 103


Photon Energy (eV)

λ∆λ

= 103λ∆λ

= 200


CohPwr_CLS_Feb07.ai

2.9 GeV, 500 mAλu = 75 mm, N = 21.50 ≤ K ≤ 5.2σx = 440 µm, σx′ = 47 µrσy = 88 µm, σy′ = 21 µrθcen = 47 µr @ K = 1η = 10%0 1000 20001500 2500 3000500

1020

4

6

2

8

10

Photon Energy (eV)

Pco

h, λ

/∆λ

(µW

)

Pce

n (W

)

n = 1

n = 3

n = 3

n = 1

n = 3λ

∆λ = 103

0

1

2

4

3

5

0

1

2

3

4

102 103 103 3 1033 103

Photon Energy (eV)

CLS, 75 mm EPU

Photon Energy (eV)

Pco

h, N

(mW

)

n = 1

λ∆λ = 22

λ∆λ = 65

λ∆λ = 22

Coherent Power at the Canadian Light Source


Coherent Power Predictedwith SPEAR 3 at SSRL

Ch08_SPEAR3_SSRL_Feb07.ai

3.0 GeV, 500 mAλu = 3.3 cm, N = 1050 ≤ K ≤ 2.2σx = 436 µm, σx′ = 43 µradσy = 30 µm, σy′ = 6.3 µradθcen = 17 µrad @ K = 1η = 10%

Photon Energy (eV)

n = 1

n = 3λ∆λ

= 103

Photon Energy (eV)

Photon Energy (keV)

n = 1

n = 3λ

∆λ= 105

0 2 4 6 8

104103104103

P coh

, N (m

W)

Pco

h, λ

/∆λ

(µW

)

Pce

n (W

)

15

10

5

0

0

5

10

n = 1

n = 3

0

50

100


Coherent Power at the Australian Synchrotron

CohPwr_AustralSynch_Feb07.ai

Photon Energy (eV)

n = 1

n = 3λ

∆λ= 103

Photon Energy (eV)

Photon Energy (eV)

n = 1

n = 3

λ∆λ

= 80

λ∆λ

= 80

λ∆λ= 240

0 2,000 4,000 6,000 8,000 10,000

104103104103

P coh

, N (m

W)

Pco

h, λ

/∆λ

(µW

)

Pce

n (W

)

8

6

2

0

0

2

4

6

n = 1

n = 3

4

0

10

20

30

40

50

3.0 GeV, 200 mAλu = 22 mm, N = 800 ≤ K ≤ 1.8σx = 320 µm, σx′ = 34 µradσy = 16 µm, σy′ = 6 µradθcen = 23 µrad @ K = 1η = 10%


Coherent Power at the UK’s Diamond Synchrotron Facility

CohPwr_DiamondUK_Feb07.ai

3.0 GeV, 300 mAλu = 2.4 cm, N = 820 ≤ K ≤ 1.4σx = 123 µm, σx′ = 24 µrσy = 6.4 µm, σy′ = 4.2 µrθcen = 23 µr @ K = 1η = 10%1000 5000 900070003000

103 1040

200300

100

400500600

Photon Energy (eV)

Pco

h, λ

/∆λ

(µW

)

Pce

n (W

)

n = 1

n = 3

n = 3

n = 1

n = 3λ

∆λ= 103

0

2

4

8

6

10

0102030405060

103 104

Photon Energy (eV)

Photon Energy (eV)

Pco

h, N

(mW

) n = 1

λ∆λ

= 82

Courtesy of Brian Kennedy (King’s College London),Susan Smith (Daresbury), and Yanwei Liu (LBNL)

Professor David AttwoodUniv. California, Berkeley Spatial and Temporal Coherernce; Coherent Undulator Radiation, EE290F, 22 Feb 2007

Ch08_ESRF_highβ_Feb07.ai

6.0 GeV, 200 mAλu = 42 mm, N = 380 ≤ K ≤ 2.1σx = 395 µm, σx′ = 11 µrσy = 9.9 µm, σy′ = 3.9 µrη + 10%(high beta)

Photon Energy (eV)

n = 1

n = 3λ

∆λ= 103

Photon Energy (eV)

Photon Energy (keV)

n = 1

n = 3

λ∆λ

= 38

0 5 10 15 20 25

104103104103

P coh

, N (m

W)

Pco

h, λ

/∆λ

(µW

)

Pce

n (W

)

15

10

5

0

0

5

10

15

0

20

40

60

n = 1

n = 3

Coherent Power at ESRF


Ch08_APS_Feb07.ai


0 10 20 30 400

5

10

15

0

0.2

0.4

0.6

0.8

0

2

4

6

104103 105 103 104 105


Photon Energy (eV)

Pco

h, N

(mW

)

Pco

h, λ

/∆λ

(µW

)

Pce

n (W

)

APS

n = 1

n = 3

n = 1

n = 1

n = 3

n = 3

λ∆λ

= 72 λ∆λ

= 103

Coherent Power at the APS


Ch08_SPring8_Feb07.ai

8 GeV, 100 mAλu = 32 mm, N = 1400 ≤ K ≤ 2.46σx = 393 µm, σx′ = 15.7 µrσy = 4.98 µm, σy′ = 1.24 µrη = 10%

n = 1

n = 3

n = 1

n = 3

n = 1

n = 3

λ∆λ

= 103λ∆λ

= 140

0 10 20 30 40 50 600

5

10

15

20

0

5

10

15

20

0

100

200

300

Photon Energy (eV)

Photon Energy (keV)

103 104 105

Photon Energy (eV)103 104 105

Pco

h, N

(mW

)

Pco

h, λ

/∆λ

(µW

)

Pce

n (W

)Coherent Power at SPring-8


Spatial Temporal Coherence; Coherent Undulator Radiation

Documents