Propagation of an X-ray beam modified by a photonic crystal

Kohn, V. G.; Snigirev, A.

doi:10.1107/s160057751401056x

Cited by 7 publications

(7 citation statements)

References 17 publications

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“…Theory of Ptychographic Experiment : Propagation of X‐rays through the colloidal crystal can be described by the paraxial wave equation, which in Cartesian coordinates can be presented as

- 2 i k \frac{\partial E (r)}{\partial z} = {normalΔ}_{⊥} E (r) + k^{2} χ (r) E (r)

where

E (r)

is the complex‐valued wave amplitude, k = 2π/λ is the wavenumber, λ is the wavelength of X‐rays, Δ ⊥ is the transverse part of the Laplace operator and

χ (r)

is the complex‐valued susceptibility of the sample. Here the following coordinate system was adopted with the z ‐axis along the optical axis of a narrowly collimated coherent X‐ray beam and r = { x , y } is a 2D vector in transverse direction along the surface of the colloidal crystal film, as shown in Figure .…”

Section: Theory and Experimental Sectionmentioning

confidence: 99%

“…In a thick colloidal film, the ESW may be determined by a multislice approach . In a thin colloidal film, the free‐space propagation, which is represented by the propagation operator Δ ⊥ in Equation , can be neglected, and the differential equation gives relatively simple solution for the object function

O (x, y) = \exp ([], i \frac{k}{2} true \int_{0}^{normald false(x, y false)} χ false(x, y, z false) normald z)

The complex‐valued susceptibility can be represented as a sum of real and imaginary parts

χ (x, y, z) = χ' (x, y, z) + i χ'' (x, y, z)

and after substitution of Equation in Equation , the complex valued object function can be written as

O (x, y) = (||, O false(x, y false)) {normale}^{i ϕ (x, y)} = \exp ([], - \frac{k}{2} true \int_{0}^{normald false(x, y false)} χ'' false(x, y, z false) normald z) \exp ([], i \frac{k}{2} true \int_{0}^{normald false(x, y false)} χ' false(x, y, z false) normald z)

Here the amplitude of the object function is defined as

| O (x, y) | = \exp ([], - \frac{k}{2} true \int_{0}^{normald false(x, y})

…”

Section: Theory and Experimental Sectionmentioning

confidence: 99%

“…Further access to local structure of colloidal crystals can be gained by using microscopy with Fresnel optics and soft X‐rays, but unfortunately the use of soft X‐rays significantly limits the penetration depth. Alternatively, compound refractive lenses (CRLs) can be used as an objective lens for hard X‐rays to study thicker samples . This technique provides full field of view, however, resolution is limited by the resolving power of the optical elements and contrast is limited by using hard X‐rays …”

Section: Introductionmentioning

confidence: 99%

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Ptychographic X‐Ray Imaging of Colloidal Crystals

Lazarev

Besedin

Zozulya

et al. 2017

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dipolar interactions. [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15] The flexibility of colloidal crystals and their response to external stimuli [16][17][18] along with their unique optical properties such as the strongly pronounced structural color and photonic band gap makes them attractive for many applications. [19][20][21][22][23][24][25] Moreover, defects, which determine the mechanical properties of many engineering materials such as metals, [26] can be studied with "atomic" resolution using colloidal crystals. [27][28][29] The high penetration depth of X-rays makes them an ideal tool to access important statistically averaged spatial information about the colloidal crystal structure and disorder over macroscopic sample volume. [7,30,31] Further access to local structure of colloidal crystals can be gained by using microscopy with Fresnel optics and soft X-rays, [32,33] but unfortunately the use of soft X-rays significantly limits the penetration depth. Alternatively, compound refractive lenses (CRLs) can be used as an objective lens for hard X-rays to study thicker samples. [34][35][36] This technique provides full field of view, however, resolution is limited by the resolving power of the optical elements and contrast is limited by using hard X-rays. [37] Ptychographic coherent X-ray imaging is applied to obtain a projection of the electron density of colloidal crystals, which are promising nanoscale materials for optoelectronic applications and important model systems. Using the incident X-ray wavefield reconstructed by mixed states approach, a high resolution and high contrast image of the colloidal crystal structure is obtained by ptychography. The reconstructed colloidal crystal reveals domain structure with an average domain size of about 2 µm. Comparison of the domains formed by the basic close-packed structures, allows us to conclude on the absence of pure hexagonal close-packed domains and confirms the presence of random hexagonal close-packed layers with predominantly face-centered cubic structure within the analyzed part of the colloidal crystal film. The ptychography reconstruction shows that the final structure is complicated and may contain partial dislocations leading to a variation of the stacking sequence in the lateral direction. As such in this work, X-ray ptychography is extended to high resolution imaging of crystalline samples. Ptychography [+]

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“…Theory of Ptychographic Experiment : Propagation of X‐rays through the colloidal crystal can be described by the paraxial wave equation, which in Cartesian coordinates can be presented as

- 2 i k \frac{\partial E (r)}{\partial z} = {normalΔ}_{⊥} E (r) + k^{2} χ (r) E (r)

where

E (r)

is the complex‐valued wave amplitude, k = 2π/λ is the wavenumber, λ is the wavelength of X‐rays, Δ ⊥ is the transverse part of the Laplace operator and

χ (r)

Section: Theory and Experimental Sectionmentioning

confidence: 99%

O (x, y) = \exp ([], i \frac{k}{2} true \int_{0}^{normald false(x, y false)} χ false(x, y, z false) normald z)

The complex‐valued susceptibility can be represented as a sum of real and imaginary parts

χ (x, y, z) = χ' (x, y, z) + i χ'' (x, y, z)

and after substitution of Equation in Equation , the complex valued object function can be written as

O (x, y) = (||, O false(x, y false)) {normale}^{i ϕ (x, y)} = \exp ([], - \frac{k}{2} true \int_{0}^{normald false(x, y false)} χ'' false(x, y, z false) normald z) \exp ([], i \frac{k}{2} true \int_{0}^{normald false(x, y false)} χ' false(x, y, z false) normald z)

Here the amplitude of the object function is defined as

| O (x, y) | = \exp ([], - \frac{k}{2} true \int_{0}^{normald false(x, y})

…”

Section: Theory and Experimental Sectionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Ptychographic X‐Ray Imaging of Colloidal Crystals

Lazarev

Besedin

Zozulya

et al. 2017

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“…That is why the accuracy of the approximation (1) must be analyzed. Kohn et al (2018) carried out a detailed analysis of the results of two methods: phase-contrast imaging and a more accurate iterative method developed for studying photonic crystals (Kohn & Tsvigun, 2014;Kohn, Snigireva & Snigirev, 2014). It has been shown that the phase-contrast method based on the transmission function (1) is fairly accurate when applied to specimens containing the number of tubules N 100 along the beam.…”

Section: Computer Simulations Of Phase-contrast Images Of a Model Objmentioning

confidence: 99%

Computer simulations of X-ray phase-contrast images and microtomographic observation of tubules in dentin

Argunova

Kohn

Lim³

et al. 2020

J Synchrotron Radiat

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An investigation of the problems of X‐ray imaging of dentinal tubules is presented. Two main points are addressed. In the first part of this paper, the problem of computer simulating tubule images recorded in a coherent synchrotron radiation (SR) beam has been discussed. A phantom material which involved a two‐dimensional lattice of the tubules with parameters similar to those of dentin was considered. By a comparative examination of two approximations, it was found that the method of phase‐contrast imaging is valid if the number of tubules along the beam is less than 100. Calculated images from a lattice of 50 × 50 tubules are periodic in free space but depend strongly on the distance between the specimen and the detector. In the second part, SR microtomographic experiments with millimetre‐sized dentin samples in a partially coherent beam have been described. Tomograms were reconstructed from experimental projections using a technique for incoherent radiation. The main result of this part is the three‐dimensional rendering of the directions of the tubules in a volume of the samples. Generation of the directions is possible because a tomogram shows the positions of the tubules. However, a detailed tubule cross‐section structure cannot be restored.

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“…The field of applications of refractive optics is not limited to beam conditioning, but can be extended into the area of Fourier optics, as well as coherent diffraction and imaging techniques [12][13][14][15]. Using the intrinsic property of the refractive lens as a Fourier transformer, the coherent diffraction microscopy and high-resolution diffraction methods have been proposed to study 3-D structures of photonic crystals and mesoscopic materials [16][17][18].…”

Section: Future Gamma Systems Roundmentioning

confidence: 99%