Transport Intensity Equation Research Articles

X-ray volumetric quantitative phase imaging by Foucault differential filtering with linear scanning

Non-interferometric X-ray quantitative phase imaging (XQPI), much simpler than the interferometric scheme, has provided high-resolution and reliable phase-contrast images. We report on implementing the volumetric XQPI images using concurrent-bidirectional scanning of the orthogonal plane on the optical axis of the Foucault differential filter; we then extracted data in conjunction with the transport-intensity equation. The volumetric image of the laminate microstructure of the gills of a fish was successfully reconstructed to demonstrate our XQPI method. The method can perform 3D rendering without any rotational motion for laterally extended objects by manipulating incoherent X-rays using the pinhole array.

A suggested optical setup for duplicated-image transport intensity equation technique to accurate determine the optical anisotropy in fibers

In this paper, an optical setup is suggested to simultaneously record a duplicated-image transport intensity equation technique (TIE) for an anisotropic material. The suggested method is characterized by its ability to simultaneously determine the refractive indices of anisotropic fibers at the different polarization directions using only single set captured micrographs. The advantage of suggested optical setup, for determining the birefringence, is minimizing the number of captured micrographs used in computations compared to the traditional TIE technique. The optical anisotropy of the High-density polyethylene (HDPE) fiber was determined using the traditional and the suggested method to verify our suggested optical setup. The suggested method is used to determine the refractive indices and birefringence profiles of highly oriented Poly (ethylene terephthalate) fiber (PET), partially oriented poly (ether ether ketone) fiber (PEEK) and medium anisotropic polyamide-6 fibers. The obtained results are compared with the published data for these fibers and found to be in good agreement.

An immersion microscopy method for determining the optical anisotropy in fibres using transport intensity equation technique

A method is proposed for investigating the refractive indices and birefringence of anisotropic fibres using transport intensity equation (TIE) technique. The method depends on utilizing the matching and mismatching between refractive indices of fibres and their immersion liquids. In which, when an anisotropic fibre immersed in a liquid of refractive index matches with its parallel or perpendicular refractive index, the focus position is obtained at a definite angle of the polarizer. This angle gives the polarization state in the selected direction. The second polarization state position can be obtained by rotating the polarizer 90° from the first position. The Canny edge detector software was used to detect the best focus position at each direction of the polarized light. The optical system of the TIE technique was modified and calibrated, with the aid of the proposed method, to differentiate between the polarizations directions of anisotropic fibre from recording intensity image. Poly (ethylene terephthalate) highly oriented fibre was used as a reference sample to calibrate the optical system. The proposed method was applied to investigate refractive indices and birefringence profiles of isotropic polypropylene (PP) fibre, low birefringent PP fibres and anisotropic nylon-6 fibre. The results of the method for different types of fibres show a good agreement with the published data for the same fibres.

Phase retrieval exact solution based on structured window modulation without direct reference waves

Arising in coherent diffraction imaging (CDI), the phase retrieval problem is to reconstruct a complex object by measuring only diffraction intensity patterns. Existing approaches, such as iterative Fourier algorithm (IFA), transport intensity equation (TIE) methods and matrix optimisation, result in near-optimal solutions rather than exact solutions. This paper proposes an analytic method to get the exact phase retrieval solution without relying on direct reference waves. Based on structured template windows, the object far-field phase is solved analytically from the recorded Fourier intensities, from which the exact object reconstruction can be achieved. Sufficient conditions for obtaining exact phase retrieval solution are analysed rigorously and verified by numerical simulations, which also show the robustness of the method through simulations under noises, dynamic range, nonlinear response and bad alignment. A simple experimental result is provided to demonstrate the practical feasibility of the method. It achieves analytic phase retrieval for arbitrary complex-valued objects and may be applicable to quick phase imaging.

A New Method to Retrieve the Three-Dimensional Refractive Index and Specimen Size Using the Transport Intensity Equation, Taking Diffraction into Account

Refractive index retrieval is possible using the transport intensity equation (TIE), which presents advantages over interferometric techniques. The TIE method is valid only for paraxial ray assumptions. However, diffraction can nullify these TIE model assumptions. Therefore, the refractive index is problematic for reconstruction in three-dimensions (3D) using a set of defocused images, as diffraction effects become prominent. We propose a method to recover the 3D refractive index by combining TIE and deconvolution. A brightfield (BF) microscope was then constructed to apply the proposed technique. A microsphere was used as a sample with well-known properties. The deconvolution of the BF-images of the sample using the microscope’s 3D point spread function led to significantly reduced diffraction effects. TIE was then applied for each set of three images. Applying TIE without taking into account diffraction failed to reconstruct the 3D refractive index. Taking diffraction into account, the refractive index of the sample was clearly recovered, and the sectioning effect of the microsphere was highlighted, leading to a determination of its size. This work is of great significance in improving the 3D reconstruction of the refractive index using the TIE method.

Direct observation of curvature of the wave surface in transmission electron microscope using transport intensity equation

In optics it is well known that the image surface is curved, even when an illumination is an ideal plane wave. However, in transmission electron microscopy (TEM) the curvature of field, or wave surface, has not been discussed seriously. We have observed the curvature of field in TEM using the transport of intensity equation (TIE). The TIE describes the relation between the phase and intensity distributions of the wave that propagates in vacuum. The common way to obtain the phase distribution using the TIE requires solving the Poisson equation, which needs the information on the boundary (boundary condition). In this report, we observe the images that pass through a selected area (SA) aperture on the intermediate image plane in order to resolve the boundary value problem. Then, we have developed a new iterative scheme to solve the TIE based on the discrete cosine transform (DCT) valid for the Neumann boundary condition. Using the iterative DCT solver we have observed the phase modulation that shows curvature of field within the SA aperture. To the authors' knowledge this is the first direct observation of the additional phase term corresponding to the curvature of field on the image plane in transmission electron microscope. We have also explained why electron holography does not detect this spherical wave on the image plane.

Non-interferometric determination of optical anisotropy in highly-oriented fibres using transport intensity equation technique

The optical setup of the transport intensity equation (TIE) technique is developed to be valid for measuring the optical properties of the highly-oriented anisotropic fibres. This development is based on the microstructure models of the highly-oriented anisotropic fibres and the principle of anisotropy. We provide the setup of TIE technique with polarizer which is controlled via stepper motor. This developed technique is used to investigate the refractive indices in the parallel and perpendicular polarization directions of light for the highly-oriented poly (ethylene terephthalate) (PET) fibres and hence its birefringence. The obtained results through the developed TIE technique for PET fibre are compared with that determined experimentally using the Mach-Zehnder interferometer under the same conditions. The comparison shows a good agreement between the obtained results from the developed technique and that obtained from the Mach-Zehnder interferometer technique.

Noninterferometric topography measurements of fast moving surfaces

The topography of moving surfaces is recovered by noninterferometric measurements. The phase reconstruction is derived by measuring the intensities of a backscattered pulsed laser light and solving the transport intensity equation (TIE). The TIE is solved by expanding the phase into a series of Zernike polynomials, leading to a set of appropriate algebraic equations. This technique, which enables us to make a direct connection between experiments and the TIE, has been successfully tested in gas gun experiments. In particular, the topographies of a moving projectile and the free surface of a shocked target were recovered.

Topography retrieval using different solutions of the transport intensity equation

The topography of a phase plate is recovered from the phase reconstruction by solving the transport intensity equation (TIE). The TIE is solved using two different approaches: (a) the classical solution of solving the Poisson differential equation and (b) an algebraic approach with Zernike functions. In this paper we present and compare the topography reconstruction of a phase plate with these solution methods and justify why one solution is preferable over the other.

Fast optical computerized topography

The topography of samples is recovered from the phase reconstruction by solving the Transport Intensity Equation (TIE). The TIE is solved by expanding the equation into a series of Zernike polynomials, leading to a set of appropriate algebraic equations. In the experiments laser light was used and the illuminated region defined the boundary conditions on the target. The phase was uniquely reconstructed and the geometry of the target was calculated. The novel technique has been successfully tested on a transparent phase plate as well as on a gold coated one. Using this technique with illumination of a short laser pulse makes it well suited for reconstructing surfaces of moving objects.

Quantitative phase retrieval in transmission hard x-ray microscope

Quantitative phase retrieval with a sub-100-nm resolution is achieved from micrographs of a zone plate based transmission x-ray microscope. A plastic zone plate containing objects of sizes from micrometers down to tens of nanometers is used as a test sample to quantify the retrieved phase. Utilizing the focal serial images in the image plane, the phase information is retrieved quantitatively across the entire range of sizes by combining the transport intensity equation and self-consistent wave propagation methods in this partial coherence system. The study demonstrates a solution to overcome the deficiency encountered in the two phase retrieval approaches.

Phase measurement of atomic resolution image using transport of intensity equation

Since the Transport Intensity Equation (TIE) has been applied to electron microscopy only recently, there are controversial discussions in the literature regarding the theoretical concepts underlying the equation and the practical techniques to solve the equation. In this report we explored some of the issues regarding the TIE, especially bearing electron microscopy in mind, and clarified that: (i) the TIE for electrons exactly corresponds to the Schrödinger equation for high-energy electrons in free space, and thus the TIE does not assume weak scattering; (ii) the TIE can give phase information at any distance from the specimen, not limited to a new field; (iii) information transfer in the TIE for each spatial frequency g will be multiplied by g2 and thus low frequency components will be dumped more with respect to high frequency components; (vi) the intensity derivative with respect to the direction of wave propagation is well approximated by using a set of three symmetric images; and (v) a substantially larger defocus distance than expected before can be used for high-resolution electron microscopy. In the second part of this report we applied the TIE down to atomic resolution images to obtain phase information and verified the following points experimentally: (i) although low frequency components are attenuated in the TIE, all frequencies will be recovered satisfactorily except the very low frequencies; and (ii) using a reconstructed phase and the measured image intensity we can correct effectively the defects of imaging, such as spherical aberrations as well as partial coherence.

Imaging a dense nanodot assembly by phase retrieval from TEM images

Phase retrieval is a classical inverse problem in many fields dealing with waves that is becoming of increasing interest in transmission electron microscopy (TEM). A non-interferometric approach is here applied to TEM images. Phase retrieval possibilities given by the transport intensity equation are compared to the ones deriving from the weak phase object approximation. In the limit of small angles, both methods lead to a similar equation between the phase and a set of defocus images. This equation can be solved by an image processing equivalent to using a specific filter in Fourier space. This processing leads to phase images with a spatial resolution here essentially limited by the defocus amount between images. A dense assembly of silicon nanodots is used as a model case to illustrate the interest of this approximate phase retrieval method which can be carried out on standard equipment. The dot heights estimated using the phase images are found to be in good agreement with ones measured by atomic force microscopy. Since image noise and large defocus values may strongly affect the solution given by the approximate method, an iterative phase retrieval method is also used as a test for working conditions.

Estimation of Optical Phase Aberrations of Micro-Optics Components by the Irradiance Transport Equation

A mathematical method for the recovery of the optical phase by using the transport intensity equation is proposed. The method, based on phase decomposition of in orthogonal series expansion, allows us to calculate the least-squares estimation of each mode coefficient of the phase as a weighted integral over intensity measurements. Experimental results of focal length and the third and fifth order spherical aberration of planar microlenses are presented. These results are compared with those of other experimental methods.