A combined diffraction and geometrical optics approach for lidar overlap function computation

This book treats optics at the level of students in the later stage of their bachelor or the beginning of their master. It is assumed that the student is familiar with Maxwell’s equations. Although the book takes account of the fact that optics is part of electromagnetism, special emphasis is put on the usefulness of approximate models of optics, their hierarchy and limits of validity. Approximate models such as geometrical optics and paraxial geometrical optics are treated extensively and applied to image formation by the human eye, the microscope and the telescope. Polarisation states and how to manipulate them are studied using Jones vectors and Jones matrices. In the context of interference, the coherence of light is explained thoroughly. To understand fundamental limits of resolution which cannot be explained by geometrical optics, diffraction theory is applied to imaging. The angular spectrum method and evanescent waves are used to understand the inherent loss of information about subwavelength features during the propagation of light. The book ends with a study of the working principle of the laser.

In this paper changes of size functions under morphological transformations are studied. Some inequalities concerning size functions of a subset of the Euclidean plane, its dilation by an open disk and its skeleton are proved. Such inequalities prove the stability of size functions with respect to these morphological transformations and give a new approach for computation in Size Theory.

The sub-diffraction-limit spatially structured light patterns have attracted more and more attention for their important applications in many frontier scientific fields. The present paper aims at developing sub-diffraction-limit spatially structured beam patterns which might have great potential to improve the light performance in fields such as super resolution imagery, optical tweezer, micro/nano lithography, etc. Here, a variety of spatially structured beam patterns are obtained by the phase modulation of polarized beams and studied in detail experimentally and numerically. Firstly, a new kind of phase plate, which combines the merits of circular and vortex 2 phase plates, is proposed based on the wave front design; it is composed of two spiral-shaped phase plates with their phases changing from 0 to 2 and - to , respectively. Later, the phase plate is applied to the circularly polarized Gaussian beam modulation in a high NA system. By combining a self-made circular with a commercial vortex 2 phase plate, the designed new phase plate is implemented in the experiment. The morphology of the spatially structured light pattern, which is generated on the focal plane, is observed by a CCD camera in the experiment. The beam pattern presents a donut shape on the focal plane, while the dimension of the donut-shaped pattern becomes smaller as the imaging plane axially deviates from the focal plane. It is found that the beam patterns captured in experiment highly consist with the numerical simulation results carried out by the vectorial diffraction integral theory. It can be deduced that the spatially structured beam is capillary-shaped. Meanwhile, at the two ends of the capillary-shaped beam, the inner diameter is smaller than the diffraction limitation. Furthermore, the structured beam pattern presents a spatial voxel distribution with center and axis symmetry. Finally, the characteristics of the spatially structured beam patterns, which are generated by modulating circular, linear, radial and azimuthal polarized beams with the new designed phase plate, are analyzed and discussed in detail. It is found that for circular, linear, radial and azimuthal polarization, the full widths at half maximum (FWHMs) of the minimum dark spots in the horizontal direction are 0.31, 0.32, 0.24 and 0.36, respectively. On the optical axis, the FWHMs of the dark spots created by linearly, radially and azimuthally polarized light, are 0.8, 0.78 and 0.76 , respectively, and no axial intensity is found with circularly polarized beam incidence.

In this paper, we extend the paraxial conical refraction model to the case of the partially coherent light using the unified optical coherence theory. We demonstrate the decomposition of conical refraction correlation functions into well-known conical refraction coherent modes for a Gaussian Schell-model source. Assuming randomness of the electrical field phase of the input beam, we reformulated and significantly simplified the rigorous conical refraction theory. This approach allows us to consider the propagation of light through a conical refraction crystal in exactly the same way as in the classical case of coherent radiation. Having this in hand, we derive analytically the conical refraction intensity both in the focal plane and in the far field, which allows us to explain and rigorously justify earlier experimental findings and predict new phenomena. The last include the counterintuitive effect of narrowing of the conical refraction ring width, disappearance of the dark Poggendorff’s ring in the Lloyd’s plane, and shift of Raman spots for the low-coherent conical refraction light. We also demonstrate a universal power-law dependence of conical refraction cones coherence degree on the input correlation length and diffraction-free propagation of the low-coherent conical refraction light in the far field.

A new technique for generating quasi non-diffracting beam with long propagation distance by using simple optical elements including the convergent lens and axicons is proposed. The theory of generating such a beam is studied with geometrical optics and diffraction theory. The formation process of the beam is simulated, and the transverse intensity distributions at various distances are obtained. The simulation results show that the transverse intensity distribution at long-distance accords with Bessel distribution. A comparison between the quasi non-diffracting beam, which is obtained by our experiment, and that in the literature (Belyi et al. 2010 Opt. Exp. 18 1966) shows that its propagation distance is more than 50 m longer, and the beam divergence angle is compressed by 22 times. In the experiment, the beam patterns are captured at different propagation positions, and the obtained results are in good agreement with the theoretical analyses.

In his landmark paper, dated February 1962, Prof. Joseph Keller detailed the notion of Keller's cone. He stated, “Geometrical optics, the oldest and most widely used theory of light propagation, fails to account for certain optical phenomena called diffraction....” I had a personal encounter with Keller's cone at a hotel in Florida! In the early morning on October 14, 2007, I witnessed Keller's cone on the door of my hotel room, resulting from edge diffraction from a TV stand due to the sun's rays coming through an opening of a window curtain. By now we know how to construct the local plane-wave behavior of the diffracted field along the diffracted rays, by invoking the fact that an edge forms one of the caustics of the diffracted wavefront. We also know that the diffracted field is of the order of k-1/2, in comparison to the geometrical-optics field, which is of the order of k0. In many antenna and scattering problems, this added diffracted term immensely enhances the accuracy of the total field. Due to the boundary-layer properties of the diffracted ray field along the incident and reflected shadow boundaries and also caustics, the original form of Keller's construction fails. These shadow-boundary shortcomings have been overcome through construction of the Uniform Theory of Diffraction (UTD, by Kouyoumjian and Pathak), the Uniform Asymptotic Theory (UAT, by Ahluwalia, Boersma, Lewis, Lee, and Deschamps), and the Spectral Theory of Diffraction (STD, by Rahmat-Samii, Ko, and Mittra). An overview and the salient features of these theories are revisited in a novel and unified manner, and a representative reflector-antenna example is highlighted.

Diffraction, Fourier Optics and Imaging

For the electromagnetic modeling of scattered fields from large and complex objects, a very efficient simulation tool based on the well-known Shooting-and-Bouncing-Rays (SBR) technique has been developed [1]. This algorithm uses field-based high-frequency approaches for finding the relevant propagation paths, i.e. rays are traced through the scenario according to the laws of Geometrical Optics (GO) and diffracted rays are generated according to the laws of the Uniform Theory of Diffraction (UTD). This GO/UTD part is combined with a source-based approach, where each ray generates an equivalent surface current at each interaction with the object. These currents are calculated using Physical Optics (PO) and Physical Theory of Diffraction (PTD), and the superposition of all these contributions leads to the total scattered field strengths. Further technical details about the ray tracing approach can be found in [1, 2, 4]. With this simulation tool, large and complex objects can be modeled, which are still too costly for numerically exact methods. Generally, both metallic and dielectric objects [2], and also combined metallic-dielectric objects, can be modeled.

The moderate resolution imaging spectroradiometer (MODIS) will be one of the primary instruments observing the earth on the Earth Observing System (EOS) scheduled for launch at the end of this decade. Its 36 spectral bands ranging from 0.4 to 14.4 microns will image the entire earth. The radiometric requirements are such that the contributions to the optical point spread function from optical ghosts, scattered light and cross-talk in the very small spectral filters, located near the focal planes, must be characterized very well and kept to a minimum. This paper describes a series of Monte Carlo simulations of the propagation of light through the optics in order to determine the extent of the ghosting and scattered light. These results are also compared with experimental data.

The point-spread function (PSF) is used in optics for design and assessment of the imaging capabilities of an optical system. It is therefore of vital importance that this PSF can be calculated fast and accurately. In the past 12 years, the Extended Nijboer-Zernike (ENZ) approach has been developed for the purpose of semi-analytic evaluation of the PSF, for circularly symmetric optical systems, in the focal region. In the earliest ENZ-years, the Debye approximation of the diffraction integral, by which the PSF is given, was considered for the very basic situation of a low-NA optical system and relatively small defocus values, so that a scalar treatment was allowed with a focal factor comprising a quadratic function in the exponential. At present, the ENZ-method allows calculation of the PSF in low- and high-NA cases, in scalar form and for vector fields (including polarization), for large wave-front aberrations, including amplitude non-uniformities, using a quasi-spherical phase focal factor in a virtually unlimited focal range around the focal plane, and no limitations in the off-axis direction. Additionally, the application range of the method has been broadened and generalized to the calculation of aerial images of extended objects by including the finite distance of the object to the entrance pupil. Also imaging into a multi-layer is now possible by accounting for both forward and backward propagation in the layers. In the advanced ENZ-approach, the generalized, complex-valued pupil function is developed into a series of Zernike circle polynomials, with exponential azimuthal dependence (having cosine/sine azimuthal dependence as special cases). For each Zernike term, the diffraction integral reduces after azimuthal integration to an integral that can be expressed as an infinite double series involving spherical Bessel functions, accounting for the parameters of the optical system and the defocus value, and Jinc functions comprising the radial off-axis value. The contribution of the present paper is the formulation of truncation rules for these double series expressions, with a general rule valid for all circle polynomials at the same time, and a dedicated rule that takes into account the degree and the azimuthal order of the involved circle polynomials to significantly reduce computational cost in specific cases. The truncation rules are based on effective bounds and asymptotics (of the Debye type) for the mentioned spherical Bessel functions and Jinc functions, and show feasibility of computation of practically all diffraction integrals that one encounters in the ENZ-practice. Thus it can be said that the advanced ENZ-theory is more or less completed from the computational point of view by the achievements of the present paper.

Calculations in technical optics are often done with a technique called raytracing. This is a treatment of optical systems in the framework of ray optics. It provides a particularly clear, if incomplete, view of the properties of optical systems. Strictly speaking, light propagation needs to be treated by taking the wave nature of light into account. The difference is that waves give rise to diffraction and interference phenomena which are disregarded in ray optics. Whenever the geometrical dimensions of the problem are so small as to become comparable to the wavelength of light, the ray optic treatment breaks down. This is the case in single-mode fibers.

This paper describes a digital method that is capable of automatically focusing optical coherence tomography (OCT) en face images without prior knowledge of the point spread function of the imaging system. The method utilizes a scalar diffraction model to simulate wave propagation from out-of-focus scatter to the focal plane, from which the propagation distance between the out-of-focus plane and the focal plane is determined automatically via an image-definition-evaluation criterion based on information entropy theory. By use of the proposed approach, we demonstrate that the lateral resolution close to that at the focal plane can be recovered from the imaging planes outside the depth of field region with minimal loss of resolution. Fresh onion tissues and mouse fat tissues are used in the experiments to show the performance of the proposed method.

The late 1800s saw a great effort on the part of microscope manufacturers to understand the basis of image quality, resolution, and contrast in their products. There was also a perceived commercial advantage in having a product that could be marketed as “scientifically designed.” Both of these aims were achieved through the work of Ernst Abbe, a physicist who worked at the microscope factory of Carl Zeiss. This chapter explains the seminal contributions of Abbe to the construction of microscope objectives and his diffraction theory and its role in image formation in the light microscope. The diffraction theory of Abbe and his experimental sets of gratings, apertures, and lenses used to observe the diffraction pattern in the back focal plane of the microscope objective provided evidence that there is an upper limit to the ability of a lens to resolve very fine spatial details. An optical imaging system uses surfaces that refract (lenses) and/or reflect (mirrors) the light from an object to form its image. This chapter also explains how a lens forms the image of an object; describes the role of diffraction in the process of image formation; and shows how the collection angle of the objective, the refractive index of the medium between the specimen and the objective, and the wavelength of the light affect the limiting optical resolution. Finally, it discusses how optical aberrations confound this limit, derived on the basis of diffraction theory in the absence of aberrations. 3.1 The Contributions of Abbe In the beginning of the 19th century, there were attempts to provide a scientific basis to imaging in the optical microscope. Both Fraunhofer and Airy attempted to use the theory of diffraction and interference to understand image formation. Nevertheless, it was in Jena, Germany, where Ernst Abbe made his important contribution to this problem. Abbe studied physics and mathematics, first two years at University of Jena, and then three additional years at University of GA¶ttingen, where he concentrated on the theory and practice of precision measurements. Abbe took courses that included individual practice on the construction of precision measuring instruments.

A numerical synthesis algorithm is presented to design inhomogeneous 3-D-printed lenses for any aperture field and, hence, beam pattern. The algorithm is based on geometrical optics (GO), a numerical method for wave propagation in inhomogeneous media, particle swarm optimization, and a global stochastic optimization technique. The approach allows the synthesis of either optimal geometry with known constant permittivity (shape only), optimal nonconstant permittivity with known geometry (material only), or optimal geometry and nonconstant permittivity (shape and material). The presented approach is validated against the analytic test case of the Maxwell fish eye and is shown to provide excellent agreement. A spherical lens is optimized for nonconstant permittivity function and is shown to recover the permittivity of the Luneburg lens. Instructions on how to fabricate the resultant inhomogeneous lenses via 3-D-printing techniques is given. Examples of optimum inhomogeneous lenses synthesized with our approach are provided along with the full-wave simulation verification. A 30 cm inhomogeneous lens is fabricated via 3-D-printing and measured in the UCLA's plane bipolar near-field measurement range and compared with the results of the GO algorithm.

We have studied the impact of nanowire alignment and measurement direction at the percolation
threshold on the effective resistance (R) of two-dimensional (2D) films. This helps us to analyze the
effect of anisotropy on the conductivity and transmittance of the nanowire-based network characterized by the disorder parameter (s). These optoelectronic properties are determined for systems
with monodisperse as well as bimodal distribution of length. The 2D systems that are simulated using our computational approach are assumed to be transparent and conductive in which percolative transport is the primary conduction mechanism.
We obtain our results numerically using a computational and geometrical approach, i.e., a Discrete
(grid) method, advantageous in algorithm speed. For a particular disorder parameter s,
the conductivity and transmittance increase as the length fraction increases for the bimodal
distribution of the length of nanowires in networks. We have observed the maximum conductivity
when the nanowires are highly aligned along the measurement direction of percolation, in contrast
to the isotropic arrangement of nanowires. Significantly, alignment introduced in nanowires leads
to a higher percolation threshold which leads to a decrease in the transmittance of the network. We
show that the resistivity of the monodisperse network in the direction parallel (perpendicular) to the
alignment decreases (increases) with the disorder parameter and scales as s (s2). This scaling holds
true for the bimodal distribution of nanowires as well. For a particular length fraction, the electrical
anisotropy increases with s. The anisotropy is maximum for nearly aligned nanowires in a bimodal
network with the highest proportion of the longest wire considered. For the maximally aligned wires
and highest length fraction, we obtained an approximately 50% enhancement in the figure of merit,
denoted by ϕ. Hence, incorporating longer length wires and increasing the alignment in nanowire
networks increases the conductivity, anisotropy, and figure of merit.