Talbot effect in X-Ray Waveguides

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Talbot effect in X-Ray Waveguides
I. Bukreeva1,2, A. Cedola1, A. Sorrentino1, F.
Scarinci1, M. Ilie1, M. Fratini1, and S.
Lagomarsino1 1Istituto di Fotonica e
Nanotecnologie CNR, Roma Italy 2Russian
Academy of Science, P. N. Lebedev Physics
Institute, Moscow, Russia
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Waveguides
Resonant beam coupling
Front-coupling

• Advantages of front-coupling WG
• Core layer vacuum
• FC WG as optics for X-ray tubes

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Computer simulation
Optics Waveguide with front-coupling
Source incoherent radiation coherent radiation

• Method
• Numerical solution of wave equation
• Analytical solution of wave equation

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Talbot effect in periodical structures
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Talbot effect. Phase gratings (phase ?) vs plane
FC WGs
In general self image fenomenon occur in wave
field composed of discrete modes
Plane wave
Waveguide with vacuum gap d100 nm, ?0.154 nm,
Rfr1
Phase grating (?) Period D2d200 nm
The interference pattern has a maximum modulation
at distances XmmD2/(8?) fractional Talbot
distance, for WG D2d
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Talbot effect. Phase gratings (phase ?) vs plane
FC WGs
Diffraction pattern has a maximum modulation at
fractional Talbot distances XT mD2/8?
Au, l 0.154 nm d 100 nm
1) Phase grating. Period D, 0.5 duty
cycle and p phase shift. Diffraction
pattern has half the period of the
grating dD/2 2) FC waveguide with vacuum
gap dD/2.
The width of the energy distribution at odd
fractional Talbot distances XT mDeff2/8? is one
half that at even distances
deffd2z, where z1/k(qc-qm)1/2 is the
penetration depth for m-th mode
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Talbot effect. Front coupling WGs
Diffraction pattern in the far field zone
D2
D2
The width of the energy distribution in far field
zone for even fractional Talbot distances is one
half that for odd distances D2 2 D1
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Montgomery self-imaging
Talbot self-imaging (sufficient condition)
Wave field with
a lateral periodicity Py is
periodic in longitudinal direction with period
Montgomery self-imaging (necessary condition)
Wave field has a longitudinal periodicity Px in
direction x if the lateral components of the
k vector obeys the condition
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Montgomery self-imaging
WG with the longitudinal periodicity
1. Wave field with lateral periodicity
2. The longitudinally periodic wave field

• 3. Lateral Py and longitudinal period Px
• are varied independently

Ewald sphere
Paraxial domain
Jürgen Jahns, and Adolf W. Lohmann Appl. Opt.
, Vol. 48, No. 18 / 20 June 2009
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Multimodal WGs with longitudinal periodicity
WG with longitudinal periodicity. Period
PLdeff2/? , duty 2/3
Multimodal WG, Au, l0.1nm,d100 nm
The energy distribution at far field zone. WG
(red line) and WG with grating (blue line)
The energy distribution at exit apertue of the
WG (red line) and WG with grating (blue line)
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Multimodal WGs with longitudinal periodicity.
Incoherent source.

• WG with grating.
• Period Pdeff 2/?119.9 ?m
• (39.97 ?m vacuum, 79.94 ?m Cr)
• deff 109.5 nm

Propagation of the incoherent wave in the WG
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Conclusion

• 1. The field in multimode x-ray waveguides with
  longitudinal periodicity are studied.
• 2. Modal structure of WGs depends on the ratio
  of the lateral and longitudinal periods
• 3. Montgomery condition for the wave vector
  includes the wavelength of the x-ray radiation
  and therefore x-ray waveguides with longitudinal
  periodicity can influence on temporal frequencies
  of the field

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• Thank you