Characterization of statistical properties of x-ray FEL radiation by means of few-photon processes...
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Transcript of Characterization of statistical properties of x-ray FEL radiation by means of few-photon processes...
Characterization of statistical properties of x-ray FEL radiation
by means of few-photon processes
Nina Rohringer and Robin Santra
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
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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 FELat Low Energy Undulator Testline (APS)
Random phases and amplitudes ! Statistical description necessary
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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 Tb Gain bandwidthcohT
1∝ωσ
Single-shot spectrum Average over shots
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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 ?
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Classical field-correlation functions
)'()(:)',(1 tEtEttG = 1st order time correlation function(Michelson interferometer)
)','(),(:)',',,(1 tzEtzEtztzG = 1st order time-space correlation functionYoung’s double slit experiment
2nd order correlation function(Hanbury-Brown and Twiss experiment)
222 )','(),(:)',',,( tzEtzEtztzG =
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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).
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Statistical description by density matrix formalism
Initial state:
atomMulti-mode density-matrixin Fock representation { } ,....),(
21 kknnn rr=
Final state:
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Perturbative Quantum Electrodynamics Approach
H=Hatomic+HField+HI
1st and 2nd order perturbation theory in A and A2 termsto calculate transition matrix elements
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One-photon absorption
kr
if
Generalized cross correlation function of 1st order
×
Atomic part
Field
Correlations of different anglesof incident radiation
{ }fffTrfP ρ̂)( ΨΨ=
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Generalized cross correlation function of 1st order
Restriction to single propagation direction: rkkz/||$=
kn=
Average number of photons with frequency
Spectral intensity distribution
kω
kn=
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One-photon single ionizationkr
ifqAfr,*+=
×
Ai =
×
Atomic part
Field
),,()(',
kkqAtomicqPkk
rrrrrr∑= ×
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Elastic X-ray scattering
krSk
r
l ii
{ }fkkk sssnnTrnP ρ̂)( 00 ⊗ΨΨ=
Negligible if far from resonanceii
kr
Skr
Field
Atomic part
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Two-photon absorption
ii
kr
'kr
negligible ?
kr'k
r
l if
Field
Atomic part
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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:
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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 1st order correlation functions(Generalized Siegert Relations)
Intensity distribution:
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In principle,
only first order correlation function G1 is needed!
But
Experimental verification needed.
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Feasibility Study for Helium and NeonRate-equations
Neon: Auger-decay and valence-shell ionization included
Gaussian pulse envelopeHelium:
)()()()()()()( 22 HePtjAHePtjAHeP σσ −−=&
),()()(~),()()(~)()()(),(
22ω
ωω
σσσ
eHePtjA
eHePtjAHePtjAeHeP+
++
−
−=&
),()()(~),()()(~)()()(),(
222
222
2
ω
ωω
σ
σσ
eHePtjA
eHePtjAHePtjAeHeP+
++
−
−=&
),()()(~)2,( ωω σ eHePtjAeHeP +++ =&
),()()(~),()()(~),( 2222 ωωωω σσ eHePtjAeHePtjAeeHeP ++++ +=+&
),()()(~)2,( 222
2 ωω σ eHePtjAeHeP +++ =&
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Helium transition probabilities in dependence of intensityRate equations for Gaussian-shaped pulse
ii
kr
'kr
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Expected experimental event rates
pulse-duration 100 fsenergy 1.4 keVrepetition rate 120 Hzphotons/pulse 5. 1012
gas density 1014 cm-3
Helium not suitable,…
1m 0.5 m 0.25 mHe+ 1.7d7 3.9d6 6.9d5He++
sequ. 3.2d3 2.8d3 1.7d3He++
corr. 4.4d5 1.d5 1.8d4
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Neon transition probabilities in dependence of intensity
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Expected experimental event rates
pulse-duration 100 energy 1.4 keVrepetition rate 120 Hzphotons/pulse 5. 1012
gas density 1014 cm-3
2.5m 2 m 1 m Ne2+ 3.8d9 5.2d8 5.4d2Ne4+ 5.0d9 1.1d9 5.4d3Ne6+ 4.8d9 1.9d9 1.1d5Ne8+ 3.1d9 2.1d9 3.6d7Ne9+ 1.2d9 1.5d9 1.3d7Ne10+ 2.1d8 8.3d8 4.9d8
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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