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

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 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