Course 2: Basic Technologies

Transcription

1 Course 2: Basic Technologies Part II: X-ray optics

2 What do you see here? Seite 2

3 wavefront distortion erration.asp Seite 3

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5 Zernike Polynominals The Zernike polynomials are a set of orthogonal polynomials that arise in the expansion of a wavefront function for optical systems with circular pupils. The odd and even Zernike polynomials are given by ou n m ρ, φ = R n m ρ sin mm eu m n ρ, φ = R m n ρ cos mm where the radial function R n m ρ is defined for n and m integers with n m 0 by R n m ρ = (n m)/2 l=0 1 l n l! l! 1 2 n + m l! 1 2 n m l! ρ n 2l 0 for n m even for n m odd Seite 5

6 Zernike polynominals II Sometimes the Zernike polynominals are also denoted as Z m n ρ, φ = R m n ρ sin mm Z m n ρ, φ = R m n ρ cos mm φ is the azimuthal angle with 0 φ 2π ρ is the radial distance with 0 ρ 1 Remember e.g. the gaussian beam: Seite 6

7 Some specific terms Examples of Zernike polynomials(top raw) and their corresponding far-field pattern (bottom raw). From left to right, you see defocus, 2 astigmatisms, 2 comas and spherical aberration. Ji-Ping Zou and Benoit Wattellier (2012). Adaptive Optics for High-Peak-Power Lasers in Adaptive Optics, Dr. Bob Tyson (Ed.), DOI: / Seite 7

8 wavefront sensing: Shack Hartman Wavefront Sensor Seite 8

9 Shack Hartmann LIVE Seite 9

10 Zernike Modes LIVE! Seite 10

11 If the orthonormal set of Zernike is used, the Zernike representation leads quickly to the wave front RMS, by the formula: Seite 11

12 X-ray Optics Seite 12

13 Refractive from the IR to the x-ray region of the electromagnetic spectrum Attwood, D.; Soft X-Ray and Extreme Ultraviolett Radiation, Cambridge University Press, 1999 Seite 13

14 Reflection and refraction of EUV/soft x-ray radiation Attwood, D.; Soft X-Ray and Extreme Ultraviolett Radiation, Cambridge University Press, 1999 Seite 14

15 Refractive Index from the IR to X-ray Spectral Region n = 1 δ + ii = 1 r 0λ 2 2π n a(f 1 0 i f 2 0 ) δ β f 0 0 1, f 2 n a r 0 deviation of the refractive index from one absorption index real and imaginary part of the atomic scattering factor number of atoms per volume classical electron radius Seite 15

16 Phase shift relative to vacuum propagation Attwood, D.; Soft X-Ray and Extreme Ultraviolett Radiation, Cambridge University Press, 1999 Seite

17 Spatial continuity along the interface Attwood, D.; Soft X-Ray and Extreme Ultraviolett Radiation, Cambridge University Press, 1999 Seite 17

18 Total external reflection of soft x-ray and EUV radiation Attwood, D.; Soft X-Ray and Extreme Ultraviolett Radiation, Cambridge University Press, 1999 Seite 18

19 Total external reflection Attwood, D.; Soft X-Ray and Extreme Ultraviolett Radiation, Cambridge University Press, 1999 Seite 19

20 Galcing incidence reflection (s-polarization) Attwood, D.; Soft X-Ray and Extreme Ultraviolett Radiation, Cambridge University Press, 1999 Seite 20

21 The attosecond filter Combines a glacing incidence mirror and a filter Modest resolution E/ E = 3-5 Commonly used Seite 21

22 Determining f 1 2 and f 2 0 Attwood, D.; Soft X-Ray and Extreme Ultraviolett Radiation, Cambridge University Press, 1999 Seite 22

23 Is there a Brewster angle in the x-ray and EUV? Attwood, D.; Soft X-Ray and Extreme Ultraviolett Radiation, Cambridge University Press, 1999 Seite 23

24 grazing incidence optics P. Kirkpatrick, A. V. Baez, Journal Opt. Soc. Am., vol. 38, pp , 1948 Seite 24

25 Wolter optics Type I DOI: /andp Seite 25

26 Wolter Optics Type II DOI: /andp Seite 26

27 Wolter Optics Type III DOI: /andp Seite 27

28 Schwarzschild optics K. Schwarzschild, Untersuchungen zur geometrischen Optik, II. Astronomische Mitteilungen der Königlichen Sternwarte zu Göttingen, vol. 10, pp. 4-28, 1905 Seite 28

29 Zone plates Seite 29

30 Zone plates II Attwood, D.; Soft X-Ray and Extreme Ultraviolett Radiation, Cambridge University Press, 1999 Seite 30

31 Zone Plate III Seite 31

32 Fig. 1 Optical design of the U125/1-PGM. This monochromator was the first collimated PGM installed at BESSY II. The versatility of collimated plane grating monochromators Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment Volumes 467/468, Part Seite 32

33 dynamic adaption to optimise focus/wavefront major classes: piezoelectric bimorph systems mechanically actuated systems Extension of these technologies Increased number of actuators to correct local figure errors Active optics (adaptive optics) Seite 33

34 Multilayer Interference Coatings Attwood, D.; Soft X-Ray and Extreme Ultraviolett Radiation, Cambridge University Press,

35 Attwood, D.; Soft X-Ray and Extreme Ultraviolett Radiation, Cambridge University Press, 1999 Seite 35

36 High Refelectivity Mirror Coatings Require Refractive index contrast at the interfaces Minimal Absorption in the low-z material Thin high-z layer where possible Γ τ H /( τ H + τ L ) Interfaces wich are chemically stable with time Minimal interdiffusion at the interfaces Minimal interfacial roughness (no crystallite formation with the layers, no shadowing in the coating process, surface mobility) Thermal stability during illumination Chemically stable vacuum interface (e. g. capping layer) Uniform coating thickness Seite 36

37 an angular scan of a multilayer mirror performs a fourier-transform of the density profile Seite 37

38 Multilayer Interface Mirrors Computational Modelling Seite 38

39 Fresnel equations s-polarized light p-polarized light Seite 39

40 Example Seite 40

41 Methods to simulate multilayer mirrors Transfer Matrix Method Abeles matrix formalism recursive Rouard method simulation resources Seite 41

42 Boundary Conditions at interfaces The boundary conditions require that the electric and magnetic fields be continous at each interface. This can be expressed as follows: first interface n 0 E 0 n 0 E 0 = n 1 E 1 n 1 E 1 second interface n 1 E 1 e iii n 1 E 1 e iii = n T E T where l is the length of the medium In matrix form this can be expressed as E 0 = n 0 n 0 E 0 cos kk in 1 sin kk i sin kk n 1 cos kk 1 n T E T E 0 Fowles, G. A.; Introduction to modern Optics; Dover Publications, 2009 Seite 42

43 Multiple optical interfaces Using the following abbreviations r = E 0 E 0 t = E T E 0 with r the reflection coefficient and t the transmission coefficient we can write r = M 1 t n 0 n 0 n T The matrix M is known as the transfer matrix M = cos kk in 1 sin kk i sin kk n 1 cos kk Seite 43

44 Assuming N interfaces For a number of N layers the total transfer matrix is derived by N M i = M 1 M 2 M 3 M N = M i=1 where M = A C B D thus the total reflection and transmission of a system with N interfaces is written as r = An 0 + Bn 0 n T C Dn T An 0 + Bn T n 0 + C + Dn T t = 2n 0 An 0 + Bn T n 0 + C + Dn T Seite

45 Transfer Matrix for periodic high reflectance films Example: quarter wavelength film of alternating layer of high and low refractive index materials Transfer matrix for a double layer: i 0 n L in L 0 i 0 n H in H 0 = n H n L 0 0 n L n H Transfer matrix for 2N layers: M = n H n L 0 0 n L n H N = n H n L 0 N 0 n L n H N Seite 45

46 Total reflectivity for a 2N layer film Assuming n 0 = 1 for the first interface R = r 2 = n H nl n H nl N N n L n H + n L n H N N 2 = n H nl n H nl 2N 2N Seite 46

47 Smooth Multilayer Cotings are Required to Minimize Wavefront Errors in EUV Optical Systems Attwood, D.; Soft X-Ray and Extreme Ultraviolett Radiation, Cambridge University Press, 1999 Seite 47

48 Example: Mo/Si Multilayer Seite 48

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50 Optimized Reflectivity with Thin High-Z Layers Attwood, D.; Soft X-Ray and Extreme Ultraviolett Radiation, Cambridge University Press, 1999 Seite 50

51 Influence on Reflectivity R = R 0 e 4 ππ sin θ g λ 2 where R 0 R θ g λ σ smooth surface reflectivity attenuated reflectivity grazing angle of incidence wavelength of incidence rms surface roughness Minghong Yang et al, Thin Solid Films, Volume 517, Issue 2, (2008) Seite 51

52 Optic Simulation Software Seite 52

53 HHG generating focus (mirror, f= mm, Ausl 8 mm, inc_angl 6 deg) Aussenspiegel (mirror in 2 m Abstand von HHG spot, r=180 mm, Ausl 1 Zoll, inc_angl 5 deg, Ausl incl. Kern) Innenspiegel (Ausleuchtung 5 mm) Seite 53

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56 Additional Sources from the WWW: 2nd EIROforum School on Instrumentation: X-ray Optics for Synchrotron Radiation Beamlines David Attwood; Soft X-Rays and Extreme Ultraviolet Radiation Franz Kärtner: Ultrafast Optics Michette, A. G. (1986). Optical Systems for Soft X Rays. Boston, MA: Springer US. doi: / Seite 56

57 Thank you! Seite 57