1. Characterization of statistical properties of x-ray FEL radiation by means of few-photon processes Nina Rohringer and Robin Santra

2. 2 Outline Motivation –SASE FEL –Amplification starts from “Shot noise” Theoretical tools –Classical and quantum mechanical field-correlation functions –Density matrix formalism –Quantum electrodynamics Atomic physics –1-photon absorption –Elastic scattering –2-photon absorption Characterization of FEL radiation – Feasibility study –Rate equations for Helium and Neon Outlook

3. 3 Single-shot measurements of SASE FEL Yuelin Li et al. Phys. Rev. Lett. 91, 243602 (2003). Field intensities and phases of the 530 nm chaotic output of a SASE FEL at Low Energy Undulator Testline (APS) Random phases and amplitudes ! Statistical description necessary

4. 4 Theoretical methods to predict statistical properties of SASE FEL amplification starts from “Shot Noise” Gaussian random process: random arrival times of electrons at the entrance of the undulator E. L. Saldin, E. A. Schneidmiller, and M. V. Yurkov, The Physics of Free Electron Lasers, (Springer-Verlag, Berlin 2000). Krinsky, Gluckstern Phys. Rev. ST Accel. 6, 50701 (2003). Simulations in the non-saturated regime: Electron bunch-duration T b Gain bandwidth Single-shot spectrum Average over shots

5. 5 Questions we have to ask: (1) Which statistical information of the radiation field is necessary to interpret a given experiment ? (2) Which experiments would allow to determine those relevant statistical properties of the radiation field ?

6. 6 Classical field-correlation functions 1 st order time correlation function (Michelson interferometer) 1 st order time-space correlation function Young’s double slit experiment 2 nd order correlation function (Hanbury-Brown and Twiss experiment)

7. 7 Quantum mechanical field-correlation functions - quantum mechanical concept of coherence -R. J. Glauber, The quantum theory of optical coherence, Phys. Rev. 130, 2529 (1963). -P. Lambropoulos, C. Kikuchi, and R.K. Osborn, Coherence and two-photon absorption, Phys. Rev. 144, 1081 (1966). -G.S. Agarwal, Field-correlation effects in multiphoton absorption processes, Phys. Rev. A 1, 1445 (1970).

8. 8 Statistical description by density matrix formalism Initial state: atom Multi-mode density-matrix in Fock representation Final state:

9. 9 Perturbative Quantum Electrodynamics Approach H=H atomic +H Field +H I 1 st and 2 nd order perturbation theory in A and A 2 terms to calculate transition matrix elements

10. 10 One-photon absorption Generalized cross correlation function of 1 st order Atomic part Field Correlations of different angles of incident radiation

11. 11 Generalized cross correlation function of 1st order Restriction to single propagation direction: Average number of photons with frequency Spectral intensity distribution

12. 12 One-photon single ionization Atomic part Field

13. 13 Elastic X-ray scattering Negligible if far from resonance Field Atomic part

14. 14 Two-photon absorption negligible ? Field Atomic part

15. 15 Correlation Functions of coherent and chaotic single-mode radiation field Coherent-state representation of density matrix: (Glauber’s quasi-probability p-representation) Coherent field: (pure coherent state) Chaotic field:

16. 16 1-dimensional classical models of SASE FEL Predictions E. L. Saldin, E. A. Schneidmiller, and M. V. Yurkov, The Physics of Free Electron Lasers, (Springer-Verlag, Berlin 2000). S. Krinsky and R.L. Gluckstern,Phys. Rev. ST Accel. 6, 50701 (2003). C. B. Schroeder, C. Pellegrini, and P. Chen, Phys. Rev. E 64, 56502 (2001). S. O. Rice, Bell Syst. Tech. J 24, 46 (1945). Relation of higher-order to 1 st order correlation functions (Generalized Siegert Relations) Intensity distribution:

17. 17 In principle, only first order correlation function G 1 is needed ! But Experimental verification needed.

18. 18 Feasibility Study for Helium and Neon Rate-equations Neon:Auger-decay and valence-shell ionization included Gaussian pulse envelope Helium:

19. 19 Helium transition probabilities in dependence of intensity Rate equations for Gaussian-shaped pulse

20. 20 Expected experimental event rates pulse-duration100 fs energy1.4 keV repetition rate 120 Hz photons/pulse 5. 10 12 gas density 10 14 cm -3 Helium not suitable,… 1  m 0.5  m 0.25  m He + 1.7d7 3.9d6 6.9d5 He ++ sequ. 3.2d3 2.8d3 1.7d3 He ++ corr. 4.4d5 1.d5 1.8d4

21. 21 Neon transition probabilities in dependence of intensity

22. 22 Expected experimental event rates pulse-duration100 energy1.4 keV repetition rate 120 Hz photons/pulse 5. 10 12 gas density 10 14 cm -3 2.5  m 2  m 1  m Ne 2+ 3.8d9 5.2d8 5.4d2 Ne 4+ 5.0d9 1.1d9 5.4d3 Ne 6+ 4.8d9 1.9d9 1.1d5 Ne 8+ 3.1d9 2.1d9 3.6d7 Ne 9+ 1.2d9 1.5d9 1.3d7 Ne 10+ 2.1d8 8.3d8 4.9d8

23. 23 Conclusions and Outlook Density matrix approach for statistical treatment of radiation field Perturbative quantum electrodynamics approach For few photon processes: –Shot to shot characterization of radiation field not necessary –Necessary information: generalized correlation functions of the radiation field Low order correlation functions could in principle be determined by means of single- and double ionization of well-studied atomic systems Theoretical Challenges: –Accurate atomic matrix-elements for elementary processes needed –Inversion problem