Diffraction Intensity Distribution Research Articles

Diffraction intensity distribution of an axicon illuminated by polychromatic light

A theoretical treatment is given for the diffraction intensity distribution of an axicon in polychromatic light illumination. Based on the dispersing theories of polychromatic light and the diffraction pattern of an axicon in monochromatic light illumination, the diffraction patterns of an axicon illuminated by a red high-brightness LED are derived. Experimental results show that the diffraction intensity in polychromatic light is a superposition of the intensities in individual monochromatic patterns. Furthermore, the central spot radius decreases with an increase in the distance l between the point source and the lens L, and the intensity of the central spot increases with an increase in l.

Diffraction intensity and symmetry of single-wall carbon nanotubes

We present a thorough full symmetry-based analysis of the diffraction space of single-wallcarbon nanotubes. This enables direct insight into various effects of the tube’ssymmetry on intensity distribution, among which are the pattern’s symmetry andorientational dependence. Also, some particular features of diffraction intensityare singled out. These are directly linked to a tube’s symmetry parameters andtherefore may give new valuable information about the tube itself. Finally, we discusshow these parameters can be determined from diffraction intensity distribution.

Influence of the size of the grating on Talbot effect

The Fresnel theory of the Talbot effect of the grating is reviewed in this paper, and the general formula of the Fresnel diffraction of the grating with finite aperture size is presented. The aperture effect of the grating on the Talbot image is discussed and a number of numerical calculations are performed. The variation rule of the diffraction intensity distribution at the Talbot distance with the aperture size of the grating and the case at different Talbot distances are obtained through the calculations, and the corresponding explanations of these phenomena are given in this paper. We think that these discussions provide the theoretic basis for the practical application of the Talbot effect of the actual grating.

Birefringence-induced bifocusing for selection of radially or azimuthally polarized laser modes

We develop a round-trip matrix diagonalization method for quantitative description of selection of radially or azimuthally polarized beams by birefringence-induced bifocusing in a simple laser resonator. We employ different focusing between radially and tangentially polarized light in thermally stressed laser rods to obtain low-loss stable oscillation in a radially polarized Laguerre-Gaussian, LG(0,1)*, mode. We derive a free-space propagator for the radially and azimuthally polarized LG(0,1)* modes and explain basic principles of mode selection by use of a round-trip matrix diagonalization method. Within this method we calculate round-trip diffraction losses and intensity distributions for the lowest-loss transverse modes. We show that, for the considered laser configuration, the round-trip loss obtained for the radially polarized LG(0,1)* mode is significantly smaller than that of the azimuthally polarized mode. Our experimental results, obtained with a diode side-pumped Nd:YAG rod in a flat-convex resonator, confirm the theoretical predictions. We achieved a pure radially polarized LG(0,1)* beam with M(2)=2.5 and tens of watts of output power.

Structural Changes of Polyacrylonitrile Film by Catalytic Effects of Nickel Particles under Carbonization Process

Structural changes of polyacrylonitrile (PAN) film, prepared by gelation/crystallization from solutions, were studied by the effect of nickel particles on the carbonization at various temperatures under argon atmosphere. The carbonized PAN at 800 °C showed thread-like structures extended from nickel particles. At 1000 °C, the thread-like structures changed to rod-like structures whose one end side was closed by a nickel particle or the aggregation. The disruption of rod-like structures occurred at 1200 °C, and the carbonization at 1400 °C provided new growth of carbon layers on the surface of spherulitic nickel particles. At 1600 °C, fibrous textures were observed as residual traces of disrupted carbon layers by the overflow of melted nickel due to the thermal expansion. The graphitization degrees for G- and T-components in the films carbonized at 1600 °C were investigated on the basis of X-ray diffraction intensity distribution from the (002) plane. The analysis was done in terms of the comparison between the experimental and theoretical diffraction intensity curves. The theoretical calculation was carried out by using a concept concerning the para-crystalline theory proposed by Hoseman and Bagchi from the viewpoint of the lattice fluctuation of the c-axis.

Preparation and characterization of carbon films prepared from poly(vinyl alcohol) containing metal oxide and nano fibers with iodine pretreatment

This research focused on the combination of catalyst effect of metal oxide, thermal conductive effect of carbon nano fibers and iodine pretreatment during carbonization of polymer precursor in order to prepare tough carbon films. Poly(vinyl alcohol) (PVA) composites containing metal oxide (Fe 3O 4) and vapor-grown carbon fibers (VGCFs) were prepared by gelation/crystallization method with the freezing/thawing technique. The dry gel films as precursor were pretreated in the atmosphere of vapor iodine, and then heat-treated at 600–1200 °C. The combined effect of iodine, Fe 3O 4 and VGCF on the carbonization of PVA was analyzed with thermo-gravimetric analysis, X-ray diffraction, and scanning electron microscope in details. Iodine pretreatment for 24 h significantly promoted the dehydration of PVA, and resulted the carbon film with a high crystallinity. Fe 3O 4 as catalyst facilitated the carbonization of PVA at a low temperature of 800–900 °C. The addition of VGCFs was found to play an important role to prepare tough films by mild carbonization due to its high thermal conductivity. The degree of graphitization in the carbon film depended on the filler contents, pretreatment conditions and carbonization conditions. The graphitization degrees for G- and T-components in the film were investigated on the basis of X-ray diffraction intensity distribution from the (0 0 2) plane.

Determination of the chiral indices (n,m) of carbon nanotubes by electron diffraction

The atomic structure of a carbon nanotube can be defined by the chiral indices, (n,m), that specify its perimeter vector (chiral vector), with which the diameter and helicity are also determined. The fine electron beam available in a modern Transmission Electron Microscope (TEM) offers a unique and powerful probe to reveal the atomic structure of individual nanotubes. This article covers two aspects related to the use of the electron probe in the TEM for the study of carbon nanotubes: (i) to express the electron diffraction intensity distribution in the electron diffraction patterns of carbon nanotubes and (ii) to obtain the chiral indices (n,m) of carbon nanotubes from their electron diffraction patterns. For a nanotube of given chiral indices (n,m), the electron scattering amplitude from the carbon nanotube can be expressed analytically in closed form using the helical diffraction theory, from which its electron diffraction pattern can be calculated and understood. The reverse problem, i.e., assignment of the chiral indices (n,m) of a carbon nanotube from its electron diffraction pattern, is approached from the relationship between the electron diffraction intensity distribution and the chiral indices (n,m). The first method is to obtain indiscriminately the chiral indices (n,m) by reading directly the intensity distribution on the three principal layer lines, l(1), l(2), and l(3), which have intensities proportional to the square of the Bessel functions of orders m, n, and n + m: I(l1) proportional, variant |J(m) (pidR)|(2), I(l2) proportional, variant |J(n) (pidR)|(2), and I(l3) proportional, variant |J(n+m) (pidR)|(2). The second method is to obtain and use the ratio of the indices n/m = (2D(1)-D(2))/(2D(2)-D(1)) in which D(1) and D(2) are the spacings of principal layer lines l(1) and l(2), respectively. Examples of using these methods are also illustrated in the determination of chiral indices of isolated individual single-walled carbon nanotubes, a bundle of single-walled carbon nanotubes, and multi-walled carbon nanotubes.

Analysis of one-dimensional disordered structures by simulation technique

A study on one-dimensional disordered structures (ODDS) in close packed (cp) crystals by Monte Carlo computer simulation is reported. Calculations of diffraction intensity distributions along the 10.L reciprocal lattice row from: 5T, 8H(44) and 12R(31)3 structures with deformation stacking faults (SFs) are presented. In particular, using the simple frequency function of fault-to-fault distances, both random and non-random distributions of SFs are considered. Distinctive features of the diffraction patterns corresponding to the chosen examples of transformations from the parent structures 5T, 8H(44) and 12R(31)3, into other polytypes are discussed in detail.

Diffraction of an ultrashort pulsed beam with arbitrary polarization state from a volume holographic grating in LiNbO 3 crystals

Based on a modified coupled wave theory of Kogelnik, we have studied the diffraction of an ultrashort pulsed beam with an arbitrary polarization state from a volume holographic grating in photorefractive LiNbO3 crystals. The results indicate that the diffracted intensity distributions in the spectral and temporal domains and the diffraction efficiency of the grating are both changed by the polarization state and spectral bandwidth of the input pulsed beam. A method is given of choosing the grating parameters and input conditions to obtain a large variation range of the spectral bandwidth of the diffracted pulsed beam with an appropriate diffraction efficiency. Our study presents a possibility of using a volume holographic grating recorded in anisotropic materials to shape a broadband ultrashort pulsed beam by modulating its polarization state.

The Phase Transitions of KNbO3 Under High Pressure

A ferroelectric KNbO 3 has been investigated to 11.4 GPa at room temperature by angle-dispersive powder diffraction with diamond anvil cell using synchrotron radiation, which was performed by a high-resolution evaluation setting. KNbO 3 transforms from orthorhombic to tetragonal lattice at around 6 GPa, and further transforms to a cubic lattice at about 9.2 GPa. Single-crystal X-ray diffraction studies under pressures clarified the space groups of the three phases and their lattice constants. The extinction rule of the observed reflections and diffraction intensity distribution proved Cm2m, P4mm and Pm3m for orthorhombic, tetragonal and cubic polymorphs, respectively.

Analysis on diffraction properties of the transmission phase grating

Diffraction properties of the transmission phase grating are analyzed in terms of the Fourier optics theory. The analysis results show that the diffraction intensity distribution of the transmission phase grating is closely related to the optical thickness of the grating and the wavelength of the incident light, and not only determined by the grating's period and slit width. For monochromatic incident light, more than 80% of the diffraction energy will concentrate to the first diffraction order under certain conditions. For white incident light, the energy of the first order of diffraction spectrum may be much higher than other orders. Based on these results, some possible applications of the transmission phase grating are discussed.

Internal strains and stresses measured in cortical bone via high-energy X-ray diffraction

High-energy synchrotron X-ray diffraction was used to study internal stresses in bone under in situ compressive loading. A transverse cross-section of a 12–14 year old beagle fibula was studied with 80.7 keV radiation, and the transmission geometry was used to quantify internal strains and corresponding stresses in the mineral phase, carbonated hydroxyapatite. The diffraction patterns agreed with tabulated patterns, and the distribution of diffracted intensity around 00.2/00.4 and 22.2 diffraction rings was consistent with the imperfect 00.1 fiber texture expected along the axis of a long bone. Residual compressive stress along the bone’s longitudinal axis was observed in the specimen prior to testing: for 22.2 this stress equaled −95 MPa and for 00.2/00.4 was between −160 and −240 MPa. Diffraction patterns were collected for applied compressive stresses up to −110 MPa, and, up to about −100 MPa, internal stresses rose proportionally with applied stress but at a higher rate, corresponding to stress concentration in the mineral of 2.8 times the stress applied. The widths of the 00.2 and 00.4 diffraction peaks indicated that crystallite size perpendicular to the 00.1 planes increased from t = 41 nm before stress was applied to t = 44 nm at −118 MPa applied stress and that rms strain ε rms rose from 2200 με before loading to 4600 με at the maximum applied stress. Small angle X-ray scattering of the unloaded sample, recorded after deformation was complete, showed a collagen D-period of 66.4 nm (along the bone axis).

Crystal Structure of Natural Non-metamict Ti- and Fe2+-rich Chevkinite-(Ce)

The crystal structure of non-metamict Ti- and Fe2+-rich chevkinite-(Ce) has been redetermined with the single-crystal sample collected from Bayan Obo, Inner Mongolia, China. The chemical formula of the sample is Ce4Fe2Ti3Si4O22. The crystals are monoclinic with the unit cell parameters a = 13.4656(15) angstrom, b = 5.7356(6) angstrom, c = 11.0977(12) angstrom, beta= 100.636(2)degrees, V = 842.39 (16) angstrom(3) and Z = 2. The structures of Ti- and Fe2+-rich chevkinite-(Ce) were refined with space groups P2(I)/a and C2/m. Least-squares refinement results show that both structural models of Ti- and Fe2+-rich chevkinite-(Ce) are very good, R[F-2> 2 sigma(F-2)] =0.027 with P2(1)/a and R[F-2> 2 sigma(F-2)] =0.021 with C2/m. In order to illustrate the relationship between the two space groups P2(1)/a and C2/m, the distribution of diffraction intensities was inspected. Pseudo extinction was found, i.e., reflections with h+k=2n are systematically strong, while those with h+k=2n+1 are weak. By neglecting the systematically weak (h+k=2n+1) reflections the space group becomes C2/m. There is a mirror plane in the C2/m perpendicular to the b axis. However, oxygen atoms in the P2(1)/a model are of a symmetrical relationship with the corresponding pseudo mirror plane. It is concluded that the crystal structure of non-metamict Ti- and Fe2+-rich chevkinite-(Ce) is a superstructure with the space group of P2(1)/a, which is of pseudo symmetry corresponding to the space group C2/m.

Catalytic effect of nickel under carbonization of polyimide films

In attempt to produce graphite films with high crystallinity by heat-treatment at temperatures lower than 3000 °C, the catalytic effect of nickel was investigated for carbonization of polyimide films prepared by mixing polyamic acid with nickel particles. The three-layered polyimide film was prepared to obtain graphite films with thickness beyond 100 μm. The middle layer composed of polyimide with nickel particles. The thickness of each layer was 50 μm and the film thickness became 150 μm. The carbonization was done at ca. 1600 °C for 5 h. The morphology of the carbonized films was observed by scanning electron microscopy. The graphitization degree was investigated on the basis of X-ray diffraction intensity distribution from the (0 0 2) plane. The analysis was done in terms of the comparison between the experimental and theoretical diffraction intensity curves. The theoretical calculation was carried out by using a concept for the para-crystalline theory proposed by Hoseman and Bagchi from the viewpoint of the lattice fluctuation of the c-axis.

Structure modifications in silicon irradiated by ultra-short pulses of XUV free electron laser

Structural modifications of solid Si (001) targets exposed to the XUV TESLA free electron laser radiation were studied. The samples were irradiated with the photon energy centered at 14eV, in short pulses of only 80fs and of peak power up to 1 GW. The FEL beam was focused on sample surfaces to microspots of size 10–100μm. The energy density in the spots varied from below the ablation threshold up to far above this threshold. The structural modifications induced with the irradiation were studied by AFM, Nomarski contrast microscopy and by X-ray diffraction methods. A variety of morphological structures created in the damaged areas was found. The maps of the X-ray diffracted intensity distribution recorded around chosen spots on the Si surface made it possible to probe the damage distribution range around the spots. The observed features are related to the FEL irradiation fluencies applied.

Simulation of optical projection with polarization- dependent stray light to explore the difference between dry and immersion lithography

When imaging at high incident angles to the resist, both the reflected light from the resist surface and the image forming process of the transmitted light into the resist are polarization dependent. The transmitted TM component and the reflected TE and TM components tend to induce stray light as a function of the incident angle. In this work, these components are analyzed and evaluated quantitatively. The polarization-dependent stray light (PDS) and the system stray light found in regular optical imaging systems are incorporated into a simulation program for diffracted intensity distribution to construct exposure-defocus (E-D) trees and windows to evaluate the exposure latitude and depth of focus (DOF) of typical circuit elements. The DOF of line-space pairs and contact holes with and without PDS are compared in the cases of two- and three-beam interference, first- and second-order beams, binary intensity masks, and phase shifting masks. Even though the 193-nm immersion system is shown to be better than 157- and 193-nm dry systems, using polarized illumination can improve DOF, particularly in low k1 situations, and in other situations when the stray-light-free image contrast is low.

Monte Carlo Technique Investigation of One-Dimensional Disorder in Polytypes

To investigate one-dimensionally disordered (ODD) structures in close packed (cp) crystals, the Monte Carlo simulation technique has been applied. Calculations of diffraction intensity distributions along the 10.L reciprocal lattice row from 6H and 9R structure with both random and non-random deformation stacking faults (SFs) are presented.

Far-field diffraction characteristics of a time dependent Gaussian-shaped pulsed beam from a rectangular mask with a circular aperture array

The diffraction characteristics of a broad-band light source from a rectangular mask with a circular aperture array in the far field are theoretically investigated with a rapid and precise mathematical derivation. We provide a rigorous treatment of a time dependent Gaussian-shaped pulsed beam passing through a rectangular mask with a circular aperture array and give more accurate and concrete criteria judging the timing for the spectral red shift, blue shift, and splitting of light distributions as well. In addition, the distortion of the space-time diffraction intensity distribution is also presented in detail. This study shows the mathematically analytic form for a pulsed beam incident on an array of small apertures, when the usage of lenses is not practical. Also, the energy distribution of the pulse in the frequency domain is presented to benefit the understanding of its applications in optics information systems.

Far-field diffraction characteristics of a time-dependent Gaussian-shaped pulsed beam incident on the reflection grating

The diffraction characteristics of a broadband light source from a reflection grating in the far field are theoretically investigated with a prompt and precise mathematical derivation. We provide a rigorous treatment of a time-dependent Gaussian-shaped pulsed beam reflecting from a grating and give more accurate and concrete criteria judging the timing for the spectral red and blue shifts and the splitting of light distributions. In addition, the distortion of the space-time diffraction intensity distribution is also presented in detail.

Near-field diffraction characteristics of a time-dependent Gaussian-shaped pulsed beam from a circular aperture

The diffraction characteristics of a broadband light source from a circular aperture in the near field are theoretically clarified via a prompt and precise mathematical derivation. We provide a rigorous way of treating in the case of a time-dependent Gaussian-shaped pulsed beam passing through a circular aperture and give a more accurate and concrete criteria for judging the timing for the spectral red and blue shifts and the splitting of light distributions. In addition, the distortion of the space-time diffraction intensity distribution is also presented in detail. Hence, the results offer a clear understanding about the Gaussian beam quality of the output of optical resonator in the near-field diffraction.