Fresnel-Kirchhoff Diffraction Theory Research Articles

A software platform for phase contrast x-ray breast imaging research

PurposeTo present and validate a computer-based simulation platform dedicated for phase contrast x-ray breast imaging research. MethodsThe software platform, developed at the Technical University of Varna on the basis of a previously validated x-ray imaging software simulator, comprises modules for object creation and for x-ray image formation. These modules were updated to take into account the refractive index for phase contrast imaging as well as implementation of the Fresnel–Kirchhoff diffraction theory of the propagating x-ray waves. Projection images are generated in an in-line acquisition geometry. To test and validate the platform, several phantoms differing in their complexity were constructed and imaged at 25keV and 60keV at the beamline ID17 of the European Synchrotron Radiation Facility. The software platform was used to design computational phantoms that mimic those used in the experimental study and to generate x-ray images in absorption and phase contrast modes. ResultsThe visual and quantitative results of the validation process showed an overall good correlation between simulated and experimental images and show the potential of this platform for research in phase contrast x-ray imaging of the breast. The application of the platform is demonstrated in a feasibility study for phase contrast images of complex inhomogeneous and anthropomorphic breast phantoms, compared to x-ray images generated in absorption mode. ConclusionsThe improved visibility of mammographic structures suggests further investigation and optimisation of phase contrast x-ray breast imaging, especially when abnormalities are present. The software platform can be exploited also for educational purposes.

Fast one-dimensional wave-front propagation for x-ray differential phase-contrast imaging.

Numerical wave-optical simulations of X-ray differential phase-contrast imaging using grating interferometry require the oversampling of gratings and object structures in the range of few micrometers. Consequently, fields of view of few millimeters already use large amounts of a computer's main memory to store the propagating wave front, limiting the scope of the investigations to only small-scale problems. In this study, we apply an approximation to the Fresnel-Kirchhoff diffraction theory to overcome these restrictions by dividing the two-dimensional wave front up into 1D lines, which are processed separately. The approach enables simulations with samples of clinically relevant dimensions by significantly reducing the memory footprint and the execution time and, thus, allows the qualitative comparison of different setup configurations. We analyze advantages as well as limitations and present the simulation of a virtual mammography phantom of several centimeters of size.

Using holography to measure extinction.

This work presents a new concept to measure the extinction cross section for a single particle in situ. The concept involves recording the hologram produced by the interference of a particle's forward-scattered light with the incident light. This interference pattern is fundamentally connected to the energy flow that gives rise to extinction, and, by integrating this measured pattern, one obtains an approximation for the cross section. Mie theory is used to show that this approximation can be as little as 1% in error of the true value for many cases of practical interest. Moreover, since an image of the particle can be computationally reconstructed from a measured hologram using the Fresnel-Kirchhoff diffraction theory, one can obtain the cross section simultaneously with the particle shape and size.

Holographic interferometry for aerosol particle characterization

Using simulations based on Mie theory, this work shows how double-exposure digital holography can be used to measure the change in size of an expanding, or contracting, spherical particle. Here, a single particle is illuminated by a plane wave twice during its expansion: once when the particle is 27λ in radius, and again when it is 47λ. A hologram is formed from each illumination stage from the interference of the scattered and unscattered, i.e., incident, light. The two holograms are then superposed to form a double exposure. By applying the Fresnel–Kirchhoff diffraction theory to the double-exposed hologram, a silhouette-like image of the particle is computationally reconstructed that is superposed with interference fringes. These fringes are a direct result of the change in particle size occurring between the two illumination stages. The study finds that expansion on the scale of ~6λ is readily discerned from the reconstructed particle image. This work could be important for improved characterization of single and multiple aerosol particles in situ. For example, by illuminating an aerosol particle with infrared light, it may be possible to measure photothermally induced particle expansion, thus providing insight into a particle׳s material properties simultaneous with an image of the particle.

Spectral purification and infrared light recycling in extreme ultraviolet lithography sources

We present the design of a novel collector mirror for laser produced plasma (LPP) light sources to be used in extreme ultraviolet (EUV) lithography. The design prevents undesired infrared (IR) drive laser light, reflected from the plasma, from reaching the exit of the light source. This results in a strong purification of the EUV light, while the reflected IR light becomes refocused into the plasma for enhancing the IR-to-EUV conversion. The dual advantage of EUV purification and conversion enhancement is achieved by incorporating an IR Fresnel zone plate pattern into the EUV reflective multilayer coating of the collector mirror. Calculations using Fresnel-Kirchhoff's diffraction theory for a typical collector design show that the IR light at the EUV exit is suppressed by four orders of magnitude. Simultaneously, 37% of the reflected IR light is refocused back the plasma.

Error analysis of ellipsoidal mirrors for soft X-ray focusing by wave-optical simulation

The ellipsoidal mirror is an ideal soft X-ray focusing optic that enables achromatic and highly efficient focusing to a nanometer spot size; however, a high-quality surface is necessary for ideal focusing. Knowledge of the required figure accuracy is important for fabrication. In this paper, we analyze the effects of figure errors on the focusing performance through wave-optical calculations based on the Fresnel–Kirchhoff diffraction theory, assuming coherent soft X-rays. Figure errors are classified into three types from the viewpoint of manufacturing. The effect of the alignment error is also investigated. The analytical results quantitatively indicate criteria regarding figure accuracy, which are expected to be essential for the development of high-performance ellipsoidal soft X-ray focusing mirrors.

光栅刻划刀架系统的运行精度对光栅光谱性能的影响

A grating diffractive spectral imaging model with ruling line bend and ruling line position errors was established based on the Fresnel-Kirchhoff diffraction theory, and the effect of these errors mentioned above on grating spectral properties was analyzed. For the instability of ruling carriage system, an optical measuring system was designed, and a mechanical improvement for the ruling carriage system was proposed.It was implemented by adopting a bilateral flexible hinge structure to replace the fixed linking method of original saddle slider pattern and ruling carriage system. Finally, The experiments on the stability test of ruling carriage system and the grating ruling test were performed. The experiment on the stability test of ruling carriage system shows that running stability of the improved score tool holder system is improved significantly as compared to that before improvement, and the displacement curve repeatability error (PV value) has reduced from 127 nm to 13 nm, decreased about 89%. Moreover, the grating ruling experiment shows that the grating spectrum performance after improving ruling carriage system has improved greatly, and the stray light has been effectively suppressed. These results obtained are in agreement with that of simulation, and provides theory and technological bases for improving grating ruling quality.

Numerical correction of distorted images in full-field optical coherence tomography

We propose a numerical method which can numerically correct the distorted en face images obtained with a full field optical coherence tomography (FF-OCT) system. It is shown that the FF-OCT image of the deep region of a biological sample is easily blurred or degraded because the sample has a refractive index (RI) much higher than its surrounding medium in general. It is analyzed that the focal plane of the imaging system is segregated from the imaging plane of the coherence-gated system due to the RI mismatch. This image-blurring phenomenon is experimentally confirmed by imaging the chrome pattern of a resolution test target through its glass substrate in water. Moreover, we demonstrate that the blurred image can be appreciably corrected by using the numerical correction process based on the Fresnel–Kirchhoff diffraction theory. The proposed correction method is applied to enhance the image of a human hair, which permits the distinct identification of the melanin granules inside the cortex layer of the hair shaft.

Applications of Fresnel-Kirchhoff diffraction theory in the analysis of human-motion Doppler sonar grams

Observed human-gait features in Doppler sonar grams are explained by using the Boulic-Thalmann (BT) model to predict joint angle time histories and the temporal displacements of the body center of mass. Body segments are represented as ellipsoids. Temporally dependent velocities at the proximal and distal end of key body segments are determined from BT. Doppler sonar grams are computed by mapping velocity-time dependent spectral acoustic-cross sections for the body segments onto time-velocity space, mimicking the Short Time Fourier Transform used in the Doppler sonar processing. Comparisons to measured data indicate that dominant returns come from trunk, thigh and lower leg.

Higher‐order phase shift reconstruction approach

Biological soft tissues encountered in clinical and preclinical imaging mainly consists of atoms of light elements with low atomic numbers and their elemental composition is nearly uniform with little density variation. Hence, x-ray attenuation contrast is relatively poor and cannot achieve satisfactory sensitivity and specificity. In contrast, x-ray phase-contrast provides a new mechanism for soft tissue imaging. The x-ray phase shift of soft tissues is about a thousand times greater than the x-ray absorption over the diagnostic x-ray energy range, yielding a higher signal-to-noise ratio than the attenuation contrast counterpart. Thus, phase-contrast imaging is a promising technique to reveal detailed structural variation in soft tissues, offering a high contrast resolution between healthy and malignant tissues. Here the authors develop a novel phase retrieval method to reconstruct the phase image on the object plane from the intensity measurements. The reconstructed phase image is a projection of the phase shift induced by an object and serves as input to reconstruct the 3D refractive index distribution inside the object using a tomographic reconstruction algorithm. Such x-ray refractive index images can reveal structural features in soft tissues, with excellent resolution differentiating healthy and malignant tissues. A novel phase retrieval approach is proposed to reconstruct an x-ray phase image of an object based on the paraxial Fresnel-Kirchhoff diffraction theory. A primary advantage of the authors' approach is higher-order accuracy over that with the conventional linear approximation models, relaxing the current restriction of slow phase variation. The nonlinear terms in the autocorrelation equation of the Fresnel diffraction pattern are eliminated using intensity images measured at different distances in the Fresnel diffraction region, simplifying the phase reconstruction to a linear inverse problem. Numerical experiments are performed to demonstrate the accuracy and stability of the proposed approach. The proposed reconstruction formula is a generalization of the transport of intensity equation (TIE). It has the second-order accuracy compared to the linear model used in the conventional phase retrieval approach. The numerical experiments demonstrate that the accuracy and stability of the proposed phase reconstruction method outperforms the TIE-based reconstruction method. A novel approach has been proposed to retrieve an x-ray phase shift image induced by an object from intensity images measured at different distances in the Fresnel diffraction region. The authors' approach has the second-order accuracy and is able to retrieve the phase shift of an object stably, overcoming the restriction of slow phase variation assumed by the conventional phase retrieval techniques.

Closed aperture z-scan technique using the Fresnel–Kirchhoff diffraction theory for materials with high nonlinear refractions

The close aperture z-scan experiment was carried out using Fresnel–Kirchhoff diffraction theory, and the special shape of close z-scan curves is determined based on the properties of the diffraction ring pattern. As an experimental case, the nonlinear refractive index of a dye (acid blue 9) is calculated using the mentioned method for the first time. Some new applications of the Fresnel–Kirchhoff diffraction theory are also explained which may be useful to characterize the nonlinear optical properties of materials. Finally, a numerical relation is evaluated to simply calculate the nonlinear refraction index in cases with high nonlinearity orders.

Aspheric silicon lenses for terahertz photoconductive antennas

We report on the design and characterization of aspheric focusing silicon lenses for terahertz photoconductive antennas. The lenses are engineered using ray-tracing software and characterized using an optical fiber terahertz time-domain spectroscopy system. We find that using aspheric lenses improves terahertz radiation coupling from the emitter and enables improved collection by the detector. The signal-to-noise ratio and the cutoff frequency of measured terahertz spectra are improved. Minimized aberrations also reduce the focal spot size. Simulations based on Fresnel–Kirchhoff diffraction theory, taking into account the radiation pattern of the emitter and aberrations of the lenses, show good agreement with our measurements.

Point-source holographic imaging of nanostructures and interfaces with low energy electrons

A lensless holographic in-line point source microscope was envisioned more than half a century ago, but its realization with electron waves has come short due to not only difficulties inherent in Fresnel-type reconstruction methods, but also to the lack of an adequate (spatially and temporally coherent) point source. With the recent creation of ultrasharp nanotips, which can field emit electrons from a single atom at their apex, an extremely coherent electron source is available that provides a great boost to the holographic method. The spatial coherence of such nanotips is a few Å, while their temporal coherence is characterized by a value of energy dispersion (FWHM) as low as 0.1 eV. In this work we ascertain the use of such a microscope in the imaging of nanoscale structures and interfaces. The method is suitable for two- and three- dimensional imaging of solid nanoparticles, thin crystals, and surfaces, but also for biological entities. We show how improvements in the reconstruction method can be made by applying the rigorous Fresnel-Kirchhoff diffraction theory adapted to Electron Optics. Sub-nanometer resolution is achievable for beam energy between 100–200 eV.

Progress in extreme ultraviolet interferometric and holographic lithography

The Center for Nanotechnology has developed an advanced beamline dedicated to nanopatterning using the radiation from a new undulator on the Aladdin storage ring at the Synchrotron Radiation Center of the University of Wisconsin-Madison. Computer generated holograms and transmission interferometric gratings were fabricated and tested on the new extreme ultraviolet (EUV) exposure system. The authors have developed an accurate model, based on Fresnel-Kirchhoff integral diffraction theory, to analyze performance of real EUV interferometric and holographic lithography systems.

Structure-change-dependent transmittance of PtOx thin film in super-resolution near-field structure: Diffraction effect analysis

To study working mechanism of super-resolution near-field structure (super-RENS) optical disk from a far-field optics view is very necessary because of the actual far-field writing/readout process in the optical disk system. A Gaussian diffraction model based on Fresnel-Kirchhoff diffraction theory of PtOx-type super-RENS has been set up in this Letter. The relationship between micro-structural deformation (change of bubble structure and refractive index profile) with far-field optical response of PtOx thin film has been studied with it in detail. The simulation results are in good agreement with the experimental results reported in literatures with a designed configuration. These results may provide more quantitative information for better understanding of the working mechanism of metal-oxide-type super-RENS. (c) 2007 Elsevier B.V. All rights reserved.

Numerical Analysis and Comparison of Three Metal-Oxide-TypeSuper-Resolution Near Field Structures

Based on the Fresnel–Kirchhoff diffraction theory, webuild up a Gaussian diffraction model of metal-oxide-type super-resolution nearfield structure (super-RENS), which can describe far field optical properties. Thespectral contrast induced by refractive index and the structural changes inAgOx, PtOx and PdOx thin films, which are the key functional layersin super-RENS, are studied by using this model. Comparison results indicate that thespectral contrast depends intensively on the laser-induced distribution and changeof the refractive index in the metal-oxide films. The readout mechanism of themetal-oxide-type super-RENS optical disc is further clarified. This Gaussiandiffraction model can be used as a simple and effective method for choosing properactive materials in super-RENS.

Formation and evolution of far-field diffraction patterns of divergent and convergent Gaussian beams passing through self-focusing and self-defocusing media

The formation and the evolution of the far-field patterns of the Gaussian beams passing through a thin self-focusing medium and a thin self-defocusing medium are studied using the Fresnel–Kirchhoff diffraction theory, and the effects of the different Kerr media and the different Gaussian beams on the far-field pattern formation and evolution are analysed by taking into consideration both the change in the additional phase shift induced by the refractive index and the change in the sign of the radius of wavefront curvature of the laser beam passing through the nonlinear medium. Our results show that, when either the divergent Gaussian beam passes through the self-defocusing medium or the convergent Gaussian beam passes through the self-focusing medium, the far-field intensity distribution pattern is a series of thick diffraction rings with a central dark spot; while, when either the divergent Gaussian beam traverses the self-focusing medium or the convergent Gaussian beam traverses the self-defocusing medium, the far-field intensity distribution pattern is a series of thin diffraction rings with a central bright spot. Our results also show that whether the central dark spot or the central bright spot occurs depends mainly on the sign of the product of the additional phase shift and the radius of the wavefront curvature. When the sign is negative, that is, when the divergent Gaussian beam passes through the self-defocusing media, or the convergent Gaussian beam passes through the self-focusing media, the central dark spot surrounded by the thick diffraction rings will emerge in the far field; while, when the sign is positive, that is, when the divergent Gaussian beam traverses the self-focusing media, or the convergent Gaussian beam passes through the self-defocusing media, the central bright spot with the thin diffraction rings will occur in the far field. The difference between the diffraction patterns is attributed to the interplay of the strong spatial self-phase modulation caused by the refractive index change of the medium and the changes in sign of the radius of the wavefront curvature of the incident Gaussian beam. The results obtained in this paper are of significance to the design of practical nonlinear optical limiters for the eye or sensor protection and many other applications.

Experimental validation of the Wigner distributions theory of phase‐contrast imaging

Recently, a new theory of phase-contrast imaging has been proposed by Wu and Liu [Med. Phys. 31, 2378-2384 (2004)]. This theory, based upon Wigner distributions, provides a much stronger foundation for the evaluation of phase-contrast imaging systems than did the prior theories based upon Fresnel-Kirchhoff diffraction theory. In this paper, we compare results of measurements made in our laboratory of phase contrast for different geometries and tube voltages to the predictions of the Wu and Liu model. In our previous publications, we have used an empirical measurement (the edge enhancement index) to parametrize the degree of phase-contrast effects in an image. While the Wu and Liu model itself does not predict image contrast, it does measure the degree of phase contrast that the system can image for a given spatial frequency. We have found that our previously published experimental results relating phase-contrast effects to geometry and x-ray tube voltage are consistent with the predictions of the Wu and Liu model.

Reproduction in free space of fields confined by 3-dimensional optical waveguides

It is shown by using the Fresnel-Kirchhoff diffraction theory and the method of images that a scalar field confined by a 3-dimensional optical waveguide can be generated in free space by a suitable light source. In the method the boundaries of a waveguide are replaced by virtual sources. This allows one to present the wave guiding as the interference and diffraction of multiple light beams produced in free space by the guide equivalent source (Fresnel waveguide). Thus, the scalar optics of a 3-dimensional waveguide is presented as the free-space beam optics. The method is illustrated by construction of the Fresnel sources for the triangular, rectangular and hexagonal waveguides. The numerical examples demonstrate the diffraction-free and self-imaging propagation in the free-space of the eigenmodes of the Fresnel rectangular-waveguide.

Z-scan theory based on a diffraction model

Based on Fresnel–Kirchhoff diffraction theory, a diffraction model of nonlinear optical media interacting with a Gaussian beam has been set up that can interpret the Z-scan phenomenon in a new way. This theory not only is consistent with the conventional Z-scan theory for a small nonlinear phase shift but also can be used for larger nonlinear phase shifts. Numerical computations indicate that the shape of the Z-scan curve is greatly affected by the value of the nonlinear phase shift. The symmetric dispersionlike Z-scan curve is valid only for small nonlinear phase shifts (|Δϕ0| <π), but, with increasingly larger nonlinear phase shifts, the valley of the transmittance is severely suppressed and the peak is greatly enhanced. The power output through the aperture will oscillate with increasing nonlinear phase shift caused by the input laser power. The aperture transmittance will attenuate and saturate with increasing Kerr constant.