Coherence in Beams
Jörg Rossbach University of Hamburg & DESY
CERN Accelerator School (CAS) on Free-Electron Laser and Energy Recovery Linac 31 May – 10 June, 2016, Hamburg Contents This lecture will address some basic issues -- details and solutions will be given in later lectures --- or (even better) in your own studies
• Why coherence ? • Basics on wave propagation • Interference • Transverse coherence • Longitudinal coherence • Correlation functions • Coherent matter waves
0 p o s itio n / m m
CAS on FELs and ERLs, Hamburg 2016 Jörg Rossbach, Univ HH 2 Why coherence? Electro-magnetic radiation comes in waves: amplitudes and phases -- and we have to cope with it !
Structure of biological macromolecule reconstructed from diffraction pattern of protein crystal:
LYSOZYME , MW=19,806 Images courtesy Janos Hajdu
CAS on FELs and ERLs, Hamburg 2016 Jörg Rossbach, Univ HH 3 Coherence of a single photon pulse from an FEL
courtesy Janos Hajdu
SINGLE MACROMOLECULE simulated image
Needs very high radiation power @ λ ≈1Å
CAS on FELs and ERLs, Hamburg 2016 Jörg Rossbach, Univ HH 4 CAS on FELs and ERLs, Hamburg 2016 Jörg Rossbach, Univ HH 4 Why Coherence ?
Diffraction from LCLS X-ray FEL/Stanford: D. Li, et al., DOI: 10.1038/ncomms10140 (2016)
CAS on FELs and ERLs, Hamburg 2016 Jörg Rossbach, Univ HH 5 Basics on wave propagation 1. At some distance from the source, a wave front propagates indepentently ‰ Huygens Principle: Each point on the wave front can be considered the origin of a spherical wave, the new wave front being the envelope of these wavelets ‰ applicable for e.m., water, acoustic, matter waves ‰ Basics for diffraction
2. The e.m. wave equation is linear ‰ If two waves , are solutions, then will be a solution as well !
‰ Basics for interference
CAS on FELs and ERLs, Hamburg 2016 Jörg Rossbach, Univ HH 6 Diffraction from a slit ˆ Consider a plane wave (complex amplitude E 0 ) to screen of arriving from the left at slit of width a observation
We use Huygens principle and consider the radiation field consisting of wavelets from tiny areas of size ∆y . We observe under an angle θ from a “very far distance”. We call the complex electric wave vector of the first wavelet ˆ . E1 ∆δ ∆y sin θ The next wavelet is: ˆ ⋅ i with ∆ πδ = 2 E1 e λ
δ a ⋅sin θ And the last wavelet: ˆ ⋅ i N with πδ = 2 E1 e N λ
2 ∆δ δ 2 ∝ˆˆˆ =++in ++ ˆ i N The intensity at the screen is IEsum E1KK Ee 1 Ee 1
∆y λ The amplitude of each wavelet is E ˆ = E ˆ = E ˆ ∆ δ ‰ 1a 0 0 2 θπ a sin CAS on FELs and ERLs, Hamburg 2016 Jörg Rossbach, Univ HH 7 Diffraction from a slit Intensity at the screen: 2 2 λ 2 ∆δ iδ IE∝ˆ = E ˆ ∆++++δ ()1 KK eein N sum 2 θπ a sin 0
δ δ= 2 2 N 2 λ λ δ δ 2 ⇒ Eedˆi δ = − Eie ˆ ()i N − 1 θ π θ π 0 ∫ 0 2sinaδ =0 2sin a λ λ 2 2 δ iδ ∗ − i I∝ EeEeˆ()()N −−=1 ˆ N 1 21cos E 2 () −=δ θ π 2 θ π a sin0 0 2 a sin 0 N θπ 2 asin 2 sin λ δ λ θπ =2 2 N = 2 ∝ 2 asin 4 E0 sin E0 sinc 2sin θπ a 2 θπ asin 2 λ λ Can be interpreted as a Fourier transform of the transmission function of the slit (note δ ≈ 2πyθ/λ is a linear function of y):
δ δ= = N iδ y πθ y 2 2πθ +∞ 2 e iy iy τ τ dδ ∝ λ eλ dy =() ye dyFT = () () y ∫θ ∫ ∫ slit δ = = −∞ 0 y y 1 CAS on FELs and ERLs, Hamburg 2016 Jörg Rossbach, Univ HH 8 Diffraction from two narrow slits Makes it simple to calculate more complicated slit systems: e.g. Young’s double slit experiment (1801)
τ Fourier transform of two delta functions at ±a/2: 2δ +∞ 2πθ a ai y π θ λ − π θ a δδ y++− y edyeλ =+ia/ e ia / = 2cos ∫ λ −∞ 1444244432 2 τ 2δ τθ ∝ ()2 I() FT2δ () y 2 π θπ θ a 2 a ∝cos =1 1 + cos : λ λ 2
CAS on FELs and ERLs, Hamburg 2016 Jörg Rossbach, Univ HH 9 Diffraction from two real slits We can even take into account the finite width of the slits: τ τ τ =′ − ⋅ ′ ′ realistic()y∫ 2δ ( yy ) slit () ydy The Fourier transform of such a convolution integral is just the product of the Fourier transforms: τ τ τ θ ∝()2 = ⋅ 2 I() FTrealistic () y FT ()():2δ FT slit Note 1: “diffraction from two slits” (or of a finite number of separated waves) is mostly called “ interference ”. Interference from a continuum of waves is called “diffraction ”. There is no fundamental difference !
Note 2: Due to interference of two waves, there are suddenly locations of permanently ZERO intensity (energy density) where each individual wave did generate intensity. ‰ Energy density is re-distributed in space due to interference ! CAS on FELs and ERLs, Hamburg 2016 Jörg Rossbach, Univ HH 10 Far field condition We have assumed a plane wave arriving at the slits. ‰ Typically realized by a point source in very far distance L. How far is “far”? ‰ There should be no significant phase difference of the wave between the two slits (or the illuminated object) ‰ Compare the optical path lengths ! a2 L− L = << λ 1 2 8L ‰ Point source a2 L >> distance should be 8λ
The same condition holds for the distance from object to observation screen !
‰ Fraunhofer diffraction
Otherwise: Fresnel diffraction (no fun) 11
CAS on FELs and ERLs, Hamburg 2016 Jörg Rossbach, Univ HH transverse coherence What matters for interference is the superposition of (two) e.m. waves. So far we assumed they both come from the same source. This is not realistic. Let’s now drop this assumption but let’s still consider monochromatic waves.
δ iδ i =+=1 + 2 (Let’s assume A 1 = A 2) E E121 E Ae Ae 1 Intensity: 2 2 ? ∝ =+()δ δ − I E2 A 1 1 cos( 2 1 ) period =() + ∆ δ ⇒ I I 0 1 cos asin θ ∆= πδ 2 +const ∝⋅+ a x const λ ? ? ?
‰ Such radiation from slits is called transversely coherent
CAS on FELs and ERLs, Hamburg 2016 Jörg Rossbachsely, Univ HH 12 transverse coherence =( + ∆ δ ) I I 0 1 cos Moving along the screen, the cos ∆δ - term oscillates between +1 and -1 (see page 9) ‰ The interference contrast (=“visibility”) is then V=1: I− I V ≡max min →1 (perfect coherence and A =A ) + 1 2 Imax I min 2π π θ a 2 c Phase difference ∆δ can be expressed as time delay, e.g.: τ ω τ∆=δ = = λλ 0
Imagine, the two fields E1 and E2 do NOT stem from the same point source. We can still calculate the intensity on the screen according to p.9, but the phase difference on the screen does NOT only stem from the time delay τ (due to the observation angle), but there may be a further time difference τ’ between E1 and E2 : ∆δ ω τω τ τ ′+ ′ =i i0 = i 0 ( ) E2 Eee 1 Ee 1
E() tE⋅∗ ( t − τ ) O,1 O ,2 T ≡ τγ c 12 (a , ) normalized complex correlation function 2⋅ 2 EO,1 E O ,2 We consider intensities ‰ need to average over osc. period CAS on FELs and ERLs, Hamburg 2016 Jörg Rossbach, Univ HH 13 transverse coherence E() tE⋅∗ ( t − τ ) O,1 O ,2 ≡ τγ c 12 (a , ) 2 2 normalized complex correlation function E⋅ E 2π θ θ a a O,1 O 2 - depends on observation angle through τ = = λω 0 c In our simple case A 1 = A 2: =( + γ γ ∆δ ) +ℜc =( + ) II01 cos⇒ I 0( 1 12) I 0 1 12 ω τ τω ∗ − +′ − it0 it 0 ( ) Ae1 Ae 1 −ω τ τ ′ − Indeed: τγ c = = i 0 ( ) 12 (a , ) 2 e A1 ω τ ∆δ τγ c 0 = 12 (a , ) ⇒ e e δ γ γ =ℜc =ℜ∆δ = ∆ 12 12 e cos γ c The visibility of interference is determined by: 12 =γ c = In this case (A 1 = A 2): V 12 1 q.e.d.
CAS on FELs and ERLs, Hamburg 2016 Jörg Rossbach, Univ HH 14 transverse coherence E() tE⋅∗ ( t − τ ) O,1 O ,2 ≡ τγ c 12 (a , ) 2 normalized complex correlation function ⋅ 2 EO,1 E O 2
≠ If I 1 I 2 ‰ ω ωτ τ τ2 τ ωτ τ ω +′ − 2 ′− − ′ − ∝=+i0 t i 0 () t =++2 2 i0 ()() + i 0 Itotal E sum Ae1 Ae 2 AAAAe 1212 ( e ) =++ γ ()ω τ τ ′ − + + ℜ c ⇒ IIIIItotal 12 12cos 0 ( ) ⇒ IIII 12 1212
I− I 2 I I V ≡max min =1 2 γγ c ≡ eff + + 12 12 Imax I min II 1 2
CAS on FELs and ERLs, Hamburg 2016 Jörg Rossbach, Univ HH 15 transverse coherence
∗ 2 E( y ,0)⋅ E ( y ,0) dy dy This is a single number describing the entire beam ς ≡ ∫ 11 22 12 2 from slit (object), independent of angle of observation. ()E( y ,0) dy ∫ 1 1 1 ς →1 for perfectly coherent radiation.
Example from FLASH FEL: From: A. Singer, et. al: Opt.Express 20, No.16 (2012)
ς = ± x 0.59 0.1 ς = ± y 0.72 0.08
CAS on FELs and ERLs, Hamburg 2016 Jörg Rossbach, Univ HH 16 Transverse coherence S. Reiche Z=25 m Z=37.5 m Z=50 m
Z=62.5 m Z=75 m Z=87.5 m m
Single mode dominates Ï close to 100% transverse coherence CAS on FELs and ERLs, Hamburg Jörg Rossbach, Univ HH 17 2016 impact of focusing (needed for most experiments)
CAS on FELs and ERLs, Hamburg 2016 Jörg Rossbach, Univ HH 18 transverse coherence
Why are the phases in 1 and 2 uncorrelated? Probably because the source is extended !
Consider a ray arriving at double slit at an angle ϕ and with intensity I(ϕ ) It contributes a phase difference π ϕ to double slit of: ∆δ = 2 a ϕ λ The averaged phase difference between the ωτ ω ∗ − +′ − it0 it 0 ( ) Ae1 Ae 1 −ω τ ′ − τγ c = = i 0 ( ) extended source to the double slit is: 12 (a , ) 2 e A1
+∞ 2π ϕa 1−i I (ϕ ) a a Van Zitter-Zernicke-Theorem: IedFT ϕ(ϕ ) λ = ≡ I⇒ VI= Visibility of interference pattern corresponds to ∫ λλ Fourier Transform of source distribution function Itotal −∞ I total
CAS on FELs and ERLs, Hamburg 2016 Jörg Rossbach, Univ HH 19 temporal coherence
2 2 −π t − π t ()∆τ 2 πν ()∆ τ 2 =i2 0 t ℜ = πν EtEe( )0 e ; EtEe ( )0 cos(2 0 t ) ν ν− 2 −π (0 ) ν −∆−π ντ ν ()()22 ∆ 2 FT( E ( t )) ∝ e(0 ) = e
Phase information will be lost after longitudinal coherence time: τ ≈ 1 c ∆ν CAS on FELs and ERLs, Hamburg 2016 Jörg Rossbach, Univ HH 20 . temporal coherence
∗ ⋅∗ − τ Et()⋅ E ( t − τ ) Et1() E 1 ( t ) 1 2 ≡ τγ c ≡ γ τc ( ) 12 (a , ) 2⋅ − τ 2 ‰ Et()2⋅ Et ( − τ ) 2 Et1() Et 2 ( ) 1 1 normalized complex autocorrelation function And again: γ τ γ τ τ = +ℜc =( + ) I() I0( 1 ()) I 0 1()
2 FT(γ τc ())= () FT () E = Power spectral density e.g. our single Gaussian wave packet: π τ 2 π τ 2 − − 2 2 2()∆τ πν τ 2 ()∆ τ πν γ τ τ γ τ γ τ c= i2 0 ℜ c = = ( )e e⇒ ( ) ( ) e cos(20 )
CAS on FELs and ERLs, Hamburg 2016 Jörg Rossbach, Univ HH 21 . temporal coherence Possible way of measurement: Split, delay and overlap at a detector
From: A. Singer, et. al: Opt.Express 20, No.16 (2012)
CAS on FELs and ERLs, Hamburg 2016 Jörg Rossbach, Univ HH 22 temporal coherence Add up many wave packets at
15 arbitrary time (phase) differences: 12.852
5 10
0
5
5 500 400 300 200 100 0 100 200 300 400 500 xi Fourier 3 0 T 20 30 40 50 60 70 80 20 i 80 ω If we want a single characteristic number for the average duration of coherence in the beam, we can calculate the +∞ coherence time : 2 γ τ τ τ = c c ∫ ( ) d −∞ Typical values for FELs: FLASH in SASE mode @ 8 nm: τc = 6 fs FERMI in seeded mode @ ca. 20 nm: τc = 100 fs
CAS on FELs and ERLs, Hamburg 2016 Jörg Rossbach, Univ HH 2323 . Full optical phase control
If the transverse correlation is perfect, one may think of manipulating the optical phase in a perfectly controlled way.
‰ Use a micro-electrical mechanical system (MEMS) filter
‰ See http://www.physik.uni-wuerzburg.de/femto-welt/formerframe.html
CAS on FELs and ERLs, Hamburg 2016 Jörg Rossbach, Univ HH 24 . General case coherence ∗ E() tE⋅ ( t − τ ) Et()⋅ E∗ ( t − τ ) O,1 O ,2 ≡ τγ c 1 1 c 12 (a , ) ≡ γ τ ( ) 2 ⋅ 2 2⋅ − τ 2 EO,1 E O 2 Et1() Et 1 ( ) Maybe, the autocorrelation function depends on the source point? Maybe the transverse correlation function depends on wavelength? ‰ All this will happen and is characteristic for an FEL ‰ See talks by K.-J. Kim and M. Yurkov
CAS on FELs and ERLs, Hamburg 2016 Jörg Rossbach, Univ HH 25 Coherence combined For a full description of the coherence properties, we need to combine transverse and longitudinal coherence properties.
Example from A. Singer, et. al: Opt.Express 20, No.16 (2012) Note: This is only vertical/longitudinal, the horizontal/longitudinal similar
26 CAS on FELs and ERLs, Hamburg 2016 Jörg Rossbach, Univ HH . Electron diffraction According to quantum mechanics, each electron carries wave properties at the λ =h = h de Broglie wavelength: e γβ p m0 c
According to p. 19 each (incoherently - remember the star!) radiating source with opening angle ∆ϕ will act like a point source and generate perfect interference as long as the slit separation a is λ λ < ≈ e a ‰ transverse coherence length of electron beam: Lcoh ∆ϕ 2πσϕ
ε n σ σ ε = = x xϕ x γβ
CAS on FELs and ERLs, Hamburg 2016 Jörg Rossbach, Univ HH 27 . Electron diffraction λ ε n h h ≈ e σ σ ε = = x Combine λ = = , L , and x ϕ x e γβ coh πσ γβ ‰ p m0 c 2 ϕ βhσ h Transv. coherence length of electron beam: L ≈x = x coh γβεε n n mc0x mc 0
‰ can perform interference experiment on nanocrystals of 24nm diameter
From: C. C. Chang, et. al: Nanotechnology 20 (2009) 115401 (89 eV, carbon nanotubes)
CAS on FELs and ERLs, Hamburg 2016 Jörg Rossbach, Univ HH 28 Left open….
1. I restricted myself to 1D; extension to 2D needed. 2. I restricted myself to linear polarization case. 3. What happens with coherence when focusing? (Nothing in certain cases, Fermat’s principle helps.) 4. I discussed only 1st order correlations. 5. Most things can be done also in frequency domain. Further reading
Text books: 1. Many illustrations are from: D. C. Giancoli: Physics (Pearson, 2006) 2. A. Singer, et. al: Spatial and temporal coherence properties of single free-electron laser pulses, Optics Express 20, No.16 (2012) 2. M.V. Klein, T.E. Furtak: Optics (Wiley, New York, 1986) 3. J. W. Goodman: Statistical Optics (Wiley, New York, 2000) 4. L. Mandel and E. Wolf: Optical Coherence and Quantum Optics (Cambridge University Press, 1995) 5. M. Born and E. Wolf: Principles of Optics(Cambridge University Press, 2002) 6. E.L. Saldin, E.A. Schneidmiller, M.V. Yurkov: The Physics of Free Electron Lasers (Springer 1999)