X-ray optical Fourier transformation based on refractive x-ray - - PowerPoint PPT Presentation

The work was supported by the Ministry of Education and Science of the Russian Federation, unique project identification number RFMEFI57817X0252.

Petr Ershov,

Sergey Kuznetsov, Irina Snigireva, Vyacheslav Yunkin and Anatoly Snigirev

X-ray optical Fourier transformation based

• n refractive x-ray optics for the study
• f crystalline nanostructures

Immanuel Kant Baltic Federal University

Laboratory of x-ray Optics and Physical materials Science

“One of the most remarkable and useful properties of a converging lens is its inherent ability to perform two-dimensional Fourier

• transforms. This complicated analog operation

can be performed with extreme simplicity in a coherent optical system, taking advantage of the basic laws of propagation and diffraction

• f light. “



  

             dx xu f j x t f j f ku j A u U    2 exp ) ( 2 exp ) (

2

(a) Object placed against the Lens



  

              dx xu f j x t f j f d f ku j A u U    2 exp ) ( ) 1 ( 2 exp ) (

2



  

             dx xu d j x t d f d j d ku j A u U    2 exp ) ( 2 exp ) (

2

(b) Object placed in front of the Lens (c) Object placed behind the Lens

The scale of the Fourier transform depends on the distance d from the focal plane. The Fourier transform is not a complete one, due to the presence of the quadratic phase factor that precedes the integral.

Introduction. Concept of the Fourier transform

Introduction. X-ray Fourier imaging by refractive optics In addition to focusing x-ray beams by refractive

• ptics, it is possible to perform a Fourier

transformation of the wave front as per visible light with conventional lenses. Theoretical calculations Experimental observations Fourier images obtained from x- rays transmitted through a photonic crystal.

Experimental observations

• n colloidal crystals

How it will work for Bragg diffraction of periodic structures?

Periodic structure of a crystal Bragg condition Fourier transform by lenses Introduction. X-ray Fourier imaging by refractive optics

Experiment ESRF beamline BM5 Distance to source 39 m X-ray wavelength (energy) 0,75 Å (16,5kEv) Lens effective aperture 130 μm Intensity transmission 17,5% Source-size projection 2,7 μm Silicon Bragg reflection (111) Δω = ±9’’ Sample: profiled Si Sample: SiO2 grating

Object – 2D grating Period 360 nm Si substrate and Si pillars By Parabolic Compound Refractive Lenses with point focusing we performed 2D Fourier transforms of crystal reciprocal space

• P. Zaumseil and et. al.

Object – 2D grating Period 360 nm Si substrate and Si pillars By Parabolic Compound Refractive Lenses with point focusing we performed 2D Fourier transforms of crystal reciprocal space

• P. Zaumseil and et. al.

CRL Fourier transform vs CDI

• 7.8
• 5.2
• 2.6

0.0 2.6 5.2 7.8 0.01 0.1 1

Normilized int. q (um

• 1)
• 7.8
• 5.2
• 2.6

0.0 2.6 5.2 7.8 0.01 0.1 1

Normilized int. q (um

• 1)

First order maxima contrast ~ 93 % First order maxima contrast ~ 64 % To analyze diffraction from Si-Ge nano- heterostructure is more efficient to use CRL, because:

• CRL concentrate more x-ray intensity
• More area could be analyzed by one shot
• CRL perform higher resolution in

reciprocal space Exposure time = 30 sec Exposure time = 1 sec Contrast is higher Resolution is higher Exposure is lower Resolution ~0.05 um-1 Resolution ~1.5 um-1

• 100
• 50

50 100 1E-3 0.01 0.1 1

Normilized int. qz(um

• 1)

qz By thetha-2theta scan we recorded a reciprocal space volume near Si 004 reflection.

[010] [100] [001]

Scanning in reciprocal space.

Reciprocal space

[001] [100] [010]

[010] [100] [001] [100] [001] [100] [010]

2.7 um-1 2.7 um-1 200 um-1

The surface represent reciprocal space near Si 004

• reflection. These techniques gives a lot of

information about crystals orientation, form factors, periodicity and perfection in three dimensions.

!Whole recording time is 1 min!

Si 004 reciprocal space volume

19.2 um-1 57 um-1 38 um-1 365 um-1

Si 004 Ge 004 Due to the Germanium dots lattice tilt there are no periodical structure in the reciprocal space. Germanium peak 004 is broad in all directions.

[010] [100] [001]

Ge 004 reciprocal space

Conclusions

• X-ray refractive optics is a powerful wave front analyzer with high angular resolution
• As well as studying the sample using the transmitted beam, we could also use the

diffracted beam.

• Due to Bragg diffraction, we could perform a crystallographical analysis of an object
• Fourier transform techniques could be used to design an X-ray diffraction microscope

similar to transmission electron microscope where both electron microscope images (information in real space) and diffraction patterns (information in reciprocal space) for the same region can be observed. Part of the x-ray microscope. Experimental hutch, ID06, ESRF, Grenoble, France

Thank for you attention!

The work was supported by the Ministry of Education and Science of the Russian Federation, unique project identification number RFMEFI57817X0252.