A new approach to vector scattering: The 3S boundary source method

© 2019 OSA - The Optical Society. All rights reserved. This paper describes a novel Boundary Source Method (BSM) applied to the vector calculation of electromagnetic fields from a surface defined by the interface between homogenous, isotropic media. In this way, the reflected and transmitted fields are represented as an expansion of the electric fields generated by a basis of orthogonal electric and magnetic dipole sources that are tangential to, and evenly distributed over the surface of interest. The dipole moments required to generate these fields are then calculated according to the extinction theorem of Ewald and Oseen applied at control points situated at either side of the boundary. It is shown that the sources are essentially vector-equivalent Huygens’ wavelets applied at discrete points at the boundary and special attention is given to their placement and the corresponding placement of control points according to the Nyquist sampling criteria. The central result of this paper is that the extinction theorem should be applied at control points situated at a distance d = 3s (where s is the separation of the sources) and consequently we refer to the method as 3sBSM. The method is applied to reflection at a plane dielectric surface and a spherical dielectric sphere and good agreement is demonstrated in comparison with the Fresnel equations and Mie series expansion respectively (even at resonance). We conclude that 3sBSM provides an accurate solution to electromagnetic scattering from a bandlimited surface and efficiently avoids the singular surface integrals and special basis functions proposed by others

Holography, tomography and 3D microscopy as linear filtering operations

In this paper we characterise 3D optical imaging techniques as 3D linear shift invariant filtering operations. From the Helmholtz equation that is the basis of scalar diffraction theory we show that the scattered field, or indeed a holographic reconstruction of this field, can be considered to be the result of a linear filtering operation applied to a source distribution. We note that if the scattering is weak, the source distribution is independent of the scattered field and a holographic reconstruction (or in fact any far-field optical imaging system) behaves as a 3D linear shift invariant filter applied to the refractive index contrast (which effectively defines the object). We go on to consider tomographic techniques that synthesise images from recordings of the scattered field using different illumination conditions. In our analysis we compare the 3D response of monochromatic optical tomography with the 3D imagery offered by confocal microscopy and scanning white light interferometry (using with quassi-monochromatic illumination) and explain the circumstances in which these approaches are equivalent. Finally, we consider the 3D response of polychromatic optical tomography and in particular the response of spectral optical coherence tomography and scanning white light interferometry

Optical diffraction tomography in fluid velocimetry: the use of a priori information

Holographic Particle Image Velocimetry (HPIV) has been used successfully to make threedimensional, three-component flow measurements from holographic recordings of seeded fluid. It is clear that measurements can only be made in regions that contain particles, but simply adding more seeding results in poor quality images that suffer from the effects of multiple scattering. Optical Diffraction Tomography provides a means to reconstruct a 3D map of refractive index from coherent recordings of scattered fields with different illumination conditions. Although the Born Approximation limits the applicability of the technique to weakscattering problems, this approach has been used to create three-dimensional images using a Digital Holographic Microscope (DHM). A non-linear optimization technique, the Conjugated Gradient optimisation Method (CGM) has been previously proposed in microwave imaging for strong scattering problems. In this paper we propose a modification of the CGM which uses apriori information to reduce the number of unknown variables that characterize the object to the position of the seeders. Some 2D numerical experiments have been computed, showing promising results and the value of these is fluid velocimetry is discussed

Effects of defocus on the transfer function of coherence scanning interferometry

Coherence scanning interferometry (CSI) offers three dimensional (3D) measurement of surface topography with high precision and accuracy. Defocus within the interferometric objective lens, however, is commonly present in CSI measurements, and reduces both the resolving power of the imaging system and the ability to measure tilted surfaces. This paper extends the linear theory of CSI to consider the effects of defocus on the 3D transfer function and the point spread function in an otherwise ideal CSI instrument. The results are compared with measurements of these functions in a real instrument. This work provides further evidence for the validity of the linear systems theory of CSI

Optical measurements of cavitation in tribological contacts

The paper describes the use of a white light interformeter to measure the cavitation bubble and oil film thickness in a tribological contact and compares the results to theory. It is found that oil film thickness is best predicted by the theory proposed by Coyne and Elrod.

Developments in laser Doppler accelerometry (LDAc) and comparison with laser Doppler velocimetry

This paper outlines the principles and early development of an interferometric technique for remote measurement of vibration acceleration — laser Doppler accelerometry (LDAc). One of the key advantages of LDAc over laser Doppler velocimetry (LDV) is its ability to measure extremely high vibration accelerations and shocks, effectively without limit, and this point is expanded upon in the paper. Early LDAc development showed how unwanted, velocity-dependent optical beats could occur on the photodetector but novel use of a frequency shifting device, whose primary purpose was for direction discrimination, was successful in isolating the required acceleration-dependent beat. A problem remained in the rate at which the velocity-dependent and acceleration-dependent beats broadened during target motion. In a further development, it was possible to 'select’ a back reflection to produce a velocity-dependent beat that was NOT modulated in the presence of target motion. The acceleration-dependent beat could then be demodulated and preliminary results are given to demonstrate this outcome

Focus variation microscope: linear theory and surface tilt sensitivity

In a recent publication [3rd International Conference on Surface Metrology, Annecy, France, 2012, p. 1] it was shown that surface roughness measurements made using a focus variation microscope (FVM) are influenced by surface tilt. The effect appears to be most significant when the surface has microscale roughness (Ra ≈ 50 nm) that is sufficient to provide a diffusely scattered signal that is comparable in magnitude to the specular component. This paper explores, from first principles, image formation using the focus variation method. With the assumption of incoherent scattering, it is shown that the process is linear and the 3D point spread characteristics and transfer characteristics of the instrument are well defined. It is argued that, for the case of micro-scale roughness and through the objective illumination, the assumption of incoherence cannot be justified and more rigorous analysis is required. Using a foil model of surface scattering the images that are recorded by a FVM have been calculated. It is shown that for the case of through the objective illumination at small tilt angles, the signal quality is degraded in a systematic manner. This is attributed to the mixing of specular and diffusely reflected components and leads to an asymmetry in the k-space representation of the output signals. It is shown that by using extra-aperture illumination or at tilt angles greater than the acceptance angle of aperture (such that the specular component is lost), the incoherent assumption can be justified once again. The work highlights the importance of using ring-light illumination and/or polarizing optics, which are often available as options on commercial instruments, as a means to mitigate or prevent these effects

Depth-resolved imaging and displacement measurement techniques viewed as linear filtering operations

The last 5 years have seen the emergence of a family of optical interferometric techniques that provide deformation measurements throughout three-dimensional (3-D) weakly scattering materials. They include wavelength scanning interferometry (WSI), tilt scanning interferometry (TSI), phase contrast spectral optical coherence tomography (PC SOCT) and hyperspectral interferometry (HSI) and can be thought of as a marriage between the phase sensing capabilities of Phase Shifting Interferometry and the depth-sensing capabilities of Optical Coherence Tomography. It was recently shown that some closely related 3-D optical imaging techniques can be treated as shift-invariant linear filtering operations. In this paper, we extend that work to include WSI, TSI, PC SOCT and HSI as spatial filtering operations and also relate the properties of their transfer functions in the spatial frequency domain to their spatial resolution and phase sensitivity, for depth-resolved displacement measurements

Microstructural and mechanical characterisation of laser-welded high-carbon and stainless steel

Laser welding is becoming an important joining technique for welding of stainless steel to carbon steel and is extensively used across various sectors, including aerospace, transportation, power plants, electronics and other industries. However, welding of stainless steel to high-carbon steel is still at its early stage, predominantly due to the formation of hard brittle phases, which undermine the mechanical strength of the joint. This study reports a scientific investigation on controlling the brittle phase formation during laser dissimilar welding of high-carbon steel to stainless steel. Attempts have been made to tailor the microstructure and phase composition of the fusion zone through influencing the alloying composition and the cooling rate. Results show that the heat-affected zone (HAZ) within the high-carbon steel has significantly higher hardness than the weld area, which severely undermines the weld quality. To reduce the hardness of the HAZ, a new heat treatment strategy was proposed and evaluated using a finite element analysis-based numerical simulation model. A series of experiments has been performed to verify the developed thermo-metallurgical finite element analysis (FEA) model, and a qualitative agreement of predicted martensitic phase distribution is shown to exist