Talbot effect in X-Ray Waveguides I. Bukreeva 1,2, A. Cedola 1, A. Sorrentino 1 F. Scarinci 1, M. Ilie 1, M. Fratini 1, and S. Lagomarsino 1 1 Istituto

Talbot effect in X-Ray Talbot effect in X-Ray WaveguidesWaveguides

I. Bukreeva1,2, A. Cedola1, A. Sorrentino1F. Scarinci1,M. Ilie1, M. Fratini1, and S. Lagomarsino1

1Istituto di Fotonica e Nanotecnologie – CNR, Roma – Italy2Russian Academy of Science, P. N. Lebedev Physics Institute, Moscow, Russia

Poland,Krakow 2010

____________________________________________________________________________ I. Bukreeva, S. LagomarsinoI. Bukreeva, S. Lagomarsino

WaveguidesWaveguides

Advantages of front-coupling WGAdvantages of front-coupling WG

Core layer: vacuum

FC WG as optics for X-ray tubes

Resonant beam couplingResonant beam coupling Front-couplingFront-coupling


Computer simulationComputer simulation

Method

- Numerical solution of wave equation- Analytical solution of wave equation

Optics Waveguide with front-

coupling

Source

incoherent radiation coherent radiation


Talbot effect in periodical Talbot effect in periodical structuresstructures


Talbot effect. Phase gratingsTalbot effect. Phase gratings (phase (phase ) vs) vs plane FC WGs plane FC WGs

Phase grating ()) Period D=2d=200 nm

Waveguide with vacuum gapWaveguide with vacuum gap

d=100 nm, d=100 nm, =0.154 nm, R=0.154 nm, Rfrfr=1=1

Pla

ne w

ave

Pla

ne w

ave

The interference pattern has a maximum modulation at distancesThe interference pattern has a maximum modulation at distances

XXmm=mD=mD2/(8/(8) – ) – fractional Talbot distance, forfractional Talbot distance, for WG D=2d WG D=2d

In general self image fenomenon occur in wave field composed of discrete modesIn general self image fenomenon occur in wave field composed of discrete modes

WG

WGPla

ne w

ave

Pla

ne w

ave


Talbot effect. Phase gratingsTalbot effect. Phase gratings (phase (phase ) vs) vs plane FC WGs plane FC WGs

Au,nm d = 100 nm

1) Phase grating. Period D, 0.5 duty cycle and π phase shift. Diffraction pattern has half the period of the grating d=D/2

2) FC waveguide with vacuum gap d=D/2.

The width of the energy distribution at odd fractional

Talbot distances XT= mDeff2/8λis

one half that at even distances

Diffraction pattern has a maximum modulation Diffraction pattern has a maximum modulation at at fractional Talbot distances XT= mD2/8λ

deff≈d+2where=1/k(c-m)1/2

is the penetration depth for m-th mode

WG

U0(

y)


Talbot effect. Talbot effect. Front coupling WGsFront coupling WGs

Diffraction pattern in the far field zone

The width of the energy distribution in far field

zone for even fractional Talbot distancesis one half that for odd distances

d d ddd

I, a

.u.

d

WG

U0(

y)U

0(y)


Montgomery self-imagingMontgomery 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

)exp()()()( , yikAyUyUyUm

mym0Pxxx

2

x

2

22my P

m12k )(,

)()()()( yUyUPyUyUxPxxy00

2y

x

P2Px

y

U0(

y)

the paraxial case


1P

nd2

m

2

x

22

2

3. Lateral P3. Lateral Pyy and longitudinal period P and longitudinal period Pxx

are varied independentlyare varied independently

2

2Tx

2My

2kk

Montgomery self-imagingMontgomery self-imaging

Ewald sphere

...,,,,, 210mP

m12

P

m12k

y

2

y

2mxT

1. Wave field with lateral periodicity1. Wave field with lateral periodicity

2. The longitudinally periodic wave field2. The longitudinally periodic wave field

WG with the longitudinal periodicityWG with the longitudinal periodicity

y

x

Paraxial domain /~,max xpNnmmJürgen Jahns, and Adolf W. Lohmann Appl. Opt. , Vol. 48, No. 18 / 20 June 2009

...,,,, 21nP

n12k

2

x

2

nyM

U0(

y)

2

2y

2x

2kk


Multimodal WGs with longitudinal Multimodal WGs with longitudinal periodicityperiodicity

WG with longitudinal periodicity. Period PL=deff

2/ , duty 2/3 Multimodal WG, Au, nm,d=100 nm

The energy distribution at exit apertue of the WG (red line) and WG with grating (blue line)

The energy distribution at far field zone. WG (red line) and WG with grating (blue line)

U0(

y)

U0(

y)


Multimodal WGs with longitudinal Multimodal WGs with longitudinal periodicity. Incoherent source.periodicity. Incoherent source.

WG with WG with gratinggrating..Period P=deff Period P=deff 22//=119.9=119.9 mm(39.97(39.97 m vacuum, 79.94m vacuum, 79.94 m Cr)m Cr)ddeffeff =109.5 nm =109.5 nm

WG

WGRan

dom

sou

rce

Propagation of the incoherent wave in the WG

Ran

dom

sou

rce


ConclusionConclusion

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


Poland,Krakow 2010

Documents

Talbot effect in X-Ray Waveguides I. Bukreeva 1,2, A. Cedola 1, A. Sorrentino 1 F. Scarinci 1, M. Ilie 1, M. Fratini 1, and S. Lagomarsino 1 1 Istituto