Wigner distribution measurement of the spatial coherence properties of FLASH

Transcription

1 Wigner distribution measurement of the spatial coherence properties of FLASH Tobias Mey Laser-Laboratorium Göttingen e.v. Hans-Adolf-Krebs Weg 1 D Göttingen

2 EUV wavefront sensor Experimental setup at BL2 = nm Hartmann plate Adjustment of beam line optics by online wavefront monitoring 0.5 Yaw [mrad] w rms =15.5nm w rms =3.5nm w rms =2.6nm w rms =2.6nm w rms =5.9nm 0.0 w rms =13nm [1] B. Flöter et al, Beam parameters of FLASH beamline BL1 from Hartmann wavefront measurements, Nucl. Instrum. Meth. A 635 (2011) Pitch [mrad]

3 EUV wavefront sensor Experimental setup at BL2 = nm Hartmann plate before adjustment after adjustment Wavefront w rms ~ 10nm w rms ~ 2.5nm ~ /10 Intensity [2] B. Flöter et al, EUV Hartmann sensor for wavefront measurements at the Free-electron LASer in Hamburg, New J. Phys. 12 (2010)

4 EUV wavefront sensor High pulse-to-pulse stability! Adjustment of active KB-system 10µm x 10µm focal size Online optics alignment at MLS synchrotron at 13.5 nm 4

5 Motivation Coherent diffractive imaging [4] [3] 200nm 1µm 2µm [5] Decreasing coherence [3] H. N. Chapman et al., Femtosecond diffractive imaging with a soft-x-ray free-electron laser, Nature Phys. 2, (2006) [4] M. M. Seibert et al., Single mimivirus particles intercepted and imaged with an X-ray laser, Nature 470, (2011) [5] B. Chen et al., Diffraction imaging: The limits of partial coherence, Phys. Rev. B 86, (2012) 5

6 Coherence x x 1 x 2 Mutual coherence function Γ x, s = E x 1, t E (x 2, t) = E x s/2, t E (x + s/2, t) s = x 2 x 1 Local degree of coherence Γ x, s γ x, s = I x s/2 I x + s/2 Global degree of coherence K = Γ x, s 2 dxds Γ x, 0 dx 2 required for interference effects [6] M. Born and B. Wolf, Principles of Optics, Cambridge University Press (1980) 6

7 Coherence Interference of elementary waves γ(x, s) 2a x + s/2 x s/2 s I x, y = I 0 J 1 2πar λd 2πar λd γ x, s cos 2πs λd x [6] r = x² + y² d γ x, s = 0.7 [6] M. Born and B. Wolf, Principles of Optics, Cambridge University Press (1980) 7

8 Coherence γ 0, 0, s x, 0 = exp s x 2 2 l c 2 Gaussian Schell model l c Coherence length l c 8

9 Coherence Gaussian Schell model K = 0.42 ± 0.09 γ x, y, s x, s y 4D-distribution γ(0,0, s x, 0) γ( l h = 6.2µm l v = 8.7µm [7] A. Singer et al., Spatial and temporal coherence properties of single free-electron laser pulses, Opt. Expr. 20, (2012) 9

10 Wigner distribution function y Spatial coordinate x = x y Mutual coherence function v z x h x, u = k 2π 2 Γ x, s e iku s d 2 s Wigner distribution Radiation angle u = u v Eugene Paul Wigner Nobel price 1963 (with J. H. D. Jensen and Generalized radiance : emitted M. Goeppert-Mayer) radiation from position x in direction u Wilhelm-Weber-Straße 22, Göttingen h = W m 2 sr [8] M. J. Bastiaans, Application of the Wigner distribution function to partially coherent light, J. Opt. Soc. Am. A 3, (1986) 10

11 Wigner distribution function y Spatial coordinate x = x y Mutual coherence function v z x h x, u = k 2π 2 Γ x, s e iku s d 2 s Wigner distribution Radiation angle u = u v Irradiance Radiance Global degree of coherence I x = h x, u dudv Near field I u = 2π 2 h x, u dxdy Far field h x, u ²dx²du² K = λ2 h x, u dx²du² [8] M. J. Bastiaans, Application of the Wigner distribution function to partially coherent light, J. Opt. Soc. Am. A 3, (1986) 11

12 Gaussian Schell-model y Separability h x, u = h x x, u h y (y, v) θ x d 0,x Wigner distribution at waist position h x x, u = h 0 exp 8 x2 2 exp 8 u2 2 d 0,x θ x Global degree of coherence K = 4 π λ d 0,x θ x Phase space volume (constant after Liouville) 12

13 Gaussian Schell-model y Separability h x, u = h x x, u h y (y, v) Wigner distribution at waist position h x x, u = h 0 exp 8 x2 2 exp 8 u2 2 d 0,x θ x Propagation of the Wigner distribution h x x, u z = h x (x z u, u) 13

14 Gaussian Schell-model y Separability h x, u = h x x, u h y (y, v) Wigner distribution at waist position h x x, u = h 0 exp 8 x2 2 exp 8 u2 2 d 0,x θ x Propagation of the Wigner distribution h x x, u z = h x (x z u, u) 14

15 Caustic scan Phosphor screen Microscope objective Manipulator CCD Camera FLASH Wavelength 24.7 nm Pulse energy 35 µj Repetition rate 10 Hz Camera Eff. pixel size 0.645µm Exposure time 1.5s 15

16 Caustic scan Phosphor screen Mikroscope objective Manipulator CCD Camera 16

17 Wigner distribution Projection-slice theorem [9] h q x, z q x = I z q x I(x, y)dy [9] A. Torre, Linear ray and wave optics in phase space, Elsevier B.V. Netherlands (2005) 17

18 Wigner distribution Projection-slice theorem [9] h q x, z q x = I z q x [9] A. Torre, Linear ray and wave optics in phase space, Elsevier B.V. Netherlands (2005) 18

19 Wigner distribution Projection-slice theorem [9] Log h h q x, z q x = I z q x 4D reconstruction h q, z q = I z q [9] A. Torre, Linear ray and wave optics in phase space, Elsevier B.V. Netherlands (2005) 19

20 Wigner distribution h x x, u = h x, y, u, v dydv h y y, v = h x, y, u, v dxdu [10] T. Mey et al., Wigner distribution measurements of the spatial coherence properties of the free-electron laser FLASH, Opt. Expr. 22, (2014) 20

21 Wigner distribution Measurement Reconstruction 21

22 Coherence properties h x, y, u, v Coherent area FFT Γ x, y, s x, s y Normalization Beam area γ x, y, s x, s y Wavelength Beam diameter Coherence length Global degree of λ [nm] d x / d y [µm] l x / l y [µm] coherence K Wigner [10] / / Double pinhole [7] / / [7] A. Singer et al., Spatial and temporal coherence properties of single free-electron laser pulses, Opt. Expr. 20, (2012) [10] T. Mey et al., Wigner distribution measurements of the spatial coherence properties of the free-electron laser FLASH, Opt. Expr. 22, (2014) 22

23 Coherence properties Hanbury Brown-Twiss [12] (Singer, 2013) Double pinhole [7] (Singer, 2012) Michelson interferometer [13] (Hilbert, 2014) Double slit [11] (Singer, 2008) Wigner [10] (Mey, 2014) [7] A. Singer et al., Opt. Expr. 20, (2012) [10] T. Mey et al., Opt. Expr. 22, (2014) [11] A. Singer et al., Phys. Rev. Lett. 101, (2008) [12] A. Singer et al., Phys. Rev. Lett. 111, (2013) [13] V. Hilbert et al., Appl. Phys. Lett. 105, (2014) 23

24 Thanks to Optics/Short Wavelengths Dr. Klaus Mann Dr. Bernd Schäfer Dr. Barbara Keitel Dr. Marion Kuhlmann Dr. Elke Plönjes-Palm and to you for your kind attention!

25 Saturation effects Short wavelength Raise electron energy γ Divergence θ 1 γ Saturation after 22 gain lengths (L g 2.5m) L g γ z/l g 25

26 4D - Wigner distribution 26

27 4D - Wigner distribution TEM 10 TEM 02 TEM 03 +TEM 10 TEM 01 27

28 4D - Wigner distribution TEM 10 TEM 02 TEM 03 TEM 01 +TEM 10 h 1 h 2 h 3 h 1 h 0 + h 0 h 1 dxdu h dydv h dxdu h dxdu h dxdu h n x, u = 1 n π exp 8x2 2 d 8u2 2 L 0,x θ n 2 8x2 2 x d + 8u2 2 0,x θ x L n Laguerre Polynom vom Grad n [7] A. Torre, Linear ray and wave optics in phase space, Elsevier B.V. Netherlands (2005) [12] T. Mey, Measurement of the Wigner distribution function of non-separable laser beams employing a toroidal mirror, New J. Phys. 16, (2014) 28

29 4D - Wigner distribution TEM 10 TEM 02 TEM 03 TEM 01 +TEM 10 Global degree of coherence K Theory Experiment h 3 h 1 h 0 + h 0 h 1 dxdu TEM TEM TEM h dxdu h dxdu TEM TEM 01 +TEM [7] A. Torre, Linear ray and wave optics in phase space, Elsevier B.V. Netherlands (2005) [12] T. Mey, Measurement of the Wigner distribution function of non-separable laser beams employing a toroidal mirror, New J. Phys. 16, (2014) 29

30 Brillanz 30

31 Funktionsprinzip FEL Photonen Wellenlänge λ p = λ u 2γ K 2 u 2 Undulator Parameter K u = eλ ub 0 2πm e c 31

32 Streifen durch Spiegel 32

33 FLASH - Wigner-Verteilung Messung Rekonstruktion separierbar 33

34 FLASH - Wigner-Verteilung Messung Rekonstruktion nicht-separierbar 34

35 FLASH - Wigner-Verteilung Messung 2D Rekonstruktion 4D Rekonstruktion 35

36 Fluktuationen FLASH - Schwerpunkt Nahfeld d 0 = 50/40 µm Fernfeld θ = 5.1/3.7 mrad 36

37 Fluktuationen FLASH - Durchmesser Nahfeld d 0 = 50/40 µm Fernfeld θ = 5.1/3.7 mrad 37

38 Fluktuationen FLASH - Kohärenz K = 16λ2 π 2 1 d 0,x d 0,y θ x θ y ΔK = Δd 0,x d 0,x 2 + Δd 0,y d 0,y 2 + Δθ x θ x 2 + Δθ y θ y 2 K Durchmesser/Divergenz Kohärenz-Fluktuation K 1.5 K ΔK = 0.08 K K = ±

39 Wigner-Verteilung und CDI 39

40 FLASH - Wigner-Verteilung System-Matrix: Propagation von Strahltaille zu Kamera-Position S(z, φ) = A B C D Rekonstruktions-Vorschrift Freie Propagation: h q x, z q x = I z q x Allgemein: h A q x, B q x = I z q x 40

41 Systemmatrix 4D Messung S z, φ = Sprop z S tilt,α Srot φ S toroid Srot φ 1 1 S tilt,α Sprop( z 0 ) Sprop z = z z S tilt,α = cosα 0 0 1/ cosα / cosα 0 0 cosα S toroid = cosφ sinφ sinφ cosφ 0 0 Srot(φ) = 2/R t cosφ sinφ 0 2/R 0 0 sinφ cosφ s