Transfer function and near-field detection of evanescent waves

We consider characterization of a near-field optical probe in terms of detection efficiency of different spatial frequencies associated with propagating and evanescent field components. The former are both detected with and radiated from an etched single-mode fiber tip, showing reciprocity of collection and illumination modes. Making use of a collection near-field microscope with a similar fiber tip illuminated by an evanescent field, we measure the collected power as a function of the field spatial frequency in different polarization configurations. Considering a two-dimensional probe configuration, numerical simulations of detection efficiency based on the eigenmode expansion technique are carried out for different tip apex angles. The detection roll-off for high spatial frequencies observed in the experiment and obtained during the simulations is fitted using a simple expression for the transfer function, which is derived by introducing an effective point of (dipolelike) detection inside the probe tip. It is found to be possible to fit reasonably well both the experimental and the simulation data for evanescent field components, implying that the developed approximation of the near-field transfer function can serve as a simple, rational, and sufficiently reliable means of fiber probe characterization.     

A closed-form solution for wave propagation in optical waveguides of longitudinally invariant structures without using beam propagation algorithm

In this paper, we propose a different method to study wave propagation in longitudinally invariant waveguides with arbitrary index profile. In our method, both the electric field and the refractive index profile are expanded into two Fourier cosine series. With these series substituted into the wave equation, a differential matrix equation can then be obtained. We show here that such a matrix equation can be solved and an explicit expression for the wave field at any longitudinal position along an optical waveguide can be obtained. The solution proposed in this method can thus exclude the use of the beam propagation algorithm. This study demonstrates that our approach yields the same results as those obtained by using commercial softwares in which a beam propagation method with the Padé approximation is used.     

Fraunhofer and Fresnel field distributions from the LP01 mode in a micro-sized hollow optical fibre

Abstract

The electric field distribution of a LP₀₁ mode and properties of the evanescent-wave field in a micro-sized hollow optical fibre under the weakly guiding approximation are analysed. The far- and near-field distributions of the output beam from the LP₀₁ mode and its propagation characteristics in free space are calculated numerically from Fraunhofer and Fresnel diffraction theory. We also derive an analytical expression of the far-field distribution of the LP₀₁ mode and discuss its applicable conditions. Our study shows that the output beam of the LP₀₁ mode in the hollow fibre is a dark hollow divergent beam whose near-field divergent angle is slightly smaller than the far-field divergent angle. The dark spot size (DSS) of the beam in the near field is about equal to the beam radius r ₀, whereas the DSS in the far field is smaller than its beam radius. We analyse the dependences of the far-field divergent angle of the output beam from the LP₀₁ mode on various fibre parameters and briefly discuss potential applications of the dark hollow beam in atomic physics and atomic optics.     

Silicon planar-apertured probe array for high-density near-field optical storage

We propose a novel, to our knowledge, silicon planar-apertured probe array as an optical head for high-density near-field optical storage. In comparison with a conventional fiber probe employed for near-field optical storage the apertured probe array has a higher readout data-transmission rate and better mechanical durability. A probe array with an aperture size of 100 nm was fabricated by use of photolithography and wet etching of a silicon wafer. Subwavelength-readout capability was demonstrated by use of one aperture of the probe array. Furthermore, we achieved a 16 times increase in the light-transmission efficiency of the probe array by installing glass-sphere microlenses on each aperture. The increase was confirmed by measurement of the near-field optical intensity.     

Acoustic wave propagation along cladded fibers of cubic crystalsymmetry

An approximate method is proposed to analyze the acoustic wave propagation characteristics in nonpiezoelectric cladded fibers of cubic crystal symmetry. The fiber axis coincides with the crystalline Z axis. This approximate method utilizes coupled mode equations and the exact solutions for the corresponding cladded isotropic fiber are used as the approximation base. We concentrate our analysis on cubic fibers with weak anisotropy where the first order approximation obtained from the coupled mode equations can provide accurate results. Dispersion curves of a few lower order flexural, torsional and radial-axial modes in such a fiber with infinite cladding are presented for the first time. Higher order approximations are also discussed. We show the accuracy of the perturbation formula, reveal that the torsional and radial-axial modes are not pure in cubic fibers, and inspect the modal birefringence phenomenon     

Trapping forces in a multiple-beam fiber-optic trap

Transverse and axial trapping forces are calculated in the ray optics regime for a multiple-beam fiber-optic light-force trap for dielectric microspheres located both on and off axis relative to the beam axis. Trap efficiencies are evaluated as functions of the effective index of refraction of the microspheres, normalized sphere radius, and normalized beam waist separation distance. Effects of the linear polarization of the electric field and of beam focusing through microlenses are considered. In the case of a counterpropagating two-beam fiber-optic trap, using microlenses at the distal ends of the fiber to focus the beams may somewhat increase the trapping volume and the axial stability if the fiber spacing is sufficiently large but will greatly reduce the stiffness of the transverse force. Trapping forces produced in a counterpropagating two-beam fiber-optic trap are compared with those generated in multiple-beam fiber-optic gradient-force traps. Multiple-beam fiber-optic traps use strong gradient forces to trap a particle; therefore they stabilize the particles much more firmly than do counterpropagating two-beam traps.     

Direct writing of microlenses in polycarbonate with excimer laser ablation

A method for fabricating microlenses in polycarbonate material is reported. Using a direct-write technique based on scanning excimer laser ablation with a circular beam, we can etch an arbitrary shape in the polymer material. The beam is obtained by imaging a circular aperture onto the polymer surface, and scanning is realized by the translation stage carrying the sample, which makes successive contours with well-chosen diameters and scan velocities. Afterward, to smooth the ablated surface and release it from debris, a large beam aperture covering the full lens area is used to ablate the lens deeper into the substrate. The fabrication process and the characterization method are described, including calculation of the contour set for a desired lens shape. The optical performance is evaluated by Mach-Zehnder interferometry, showing that aberrations below lambda/10 are possible for slow lenses.     

Evaluation of microlens properties in the presence of high spherical aberration

Microlenses can be generated with various fabrication technologies. Some of these technologies cause large spherical aberrations in the resulting microlenses. We describe an algorithm based on Rayleigh's quarter-wave criterion, which allows the evaluation of lens parameters for those microlenses. Specifically, we investigate numerical aperture, focal length, and space-bandwidth product with respect to applications in optical microsystems. We apply our algorithm to different types of microlenses, three gradient-index lenses, and one surface-relief lens. The experimental results demonstrate that our algorithm provides a helpful characterization method for microlenses with large aberrations.     

Propagation of vectorial Lorentz beam beyond the paraxial approximation

Based on the vectorial Rayleigh–Sommerfeld integrals, the analytical propagation expression of vectorial Lorentz beam beyond paraxial approximation is presented. The far field expression and the scalar paraxial result are obtained as special cases of the general formulae. According to the analytical representation, the light intensity distribution of vectorial Lorentz beam is depicted at the different reference planes. This research provides an approach to further investigate the propagation of Lorentz beam beyond the paraxial approximation.     

Gaussian content as a laser beam quality parameter

We propose the Gaussian content (GC) as an optional quality parameter for the characterization of laser beams. It is defined as the overlap integral of a given field with an optimally defined Gaussian. The definition is especially suited for applications where coherence properties are targeted. Mathematical definitions and basic calculation procedures are given along with results for basic beam profiles. The coherent combination of an array of laser beams and the optimal coupling between a diode laser and a single-mode fiber are elaborated as application examples. The measurement of the GC and its conservation upon propagation are experimentally confirmed.     

Microlens-array-based exit-pupil expander for full-color displays

Two-dimensional arrays of microlenses can be used in wearable display applications as numerical aperture expanders or exit-pupil expanders (EPEs) to increase the size of the display exit pupil. A novel EPE approach that uses two microlens arrays (MLAs) is presented. The approach is based on cascading two identical microlens arrays spaced precisely at one focal-length distance with submicrometer registration tolerances relative to each other. The ideal MLA for this application requires a 100% fillfactor, sharp seams between microlenses, and a perfect spherical profile. We demonstrate a dual-MLA-based EPE that produces excellent exit-pupil uniformity and better than 90% diffraction efficiency for all three wavelengths in a color-display system. Two-MLA registration is performed with submicrometer precision by use of far-field alignment techniques. Fourier optics theory is used to derive the analytical formulas, and physical optics beam propagation is used for numerical computations. Three MLA fabrication technologies, including gray-scale lithography, photoresist reflow, and isotropic etching, are evaluated and compared for an EPE application.     

Actual focal length of a symmetric biconvex microlens and its application in determining the transmitted beam waist position

The actual focal length of a three-dimensional continuous profile symmetric biconvex microlens with normal monochromatic plane wave illumination is theoretically determined using a full-field separation of variables method in the oblate spheroidal coordinate system. The investigations are performed for microlenses of 5, 10, and 20 wavelength diameters by calculating the electromagnetic field distributions inside of and adjacent to the microlenses. The importance and potential application of the microlens actual focal length in the design of microlens optical systems are demonstrated by showing that for normal monochromatic TEM00 mode Gaussian beam illumination, the transmitted beam waist position through a single microlens, calculated using Self's beam waist position transformation formula [Appl. Opt.22, 658 (1983)] with the microlens actual focal length, closely matches the exact value given by the separation of variables method.     

Propagation of light trapped within a set of lowest-order modes of graded-index multimode fiber undergoing bending

We describe experimental results and a theoretical analysis for propagation in graded-index multimode fiber when diode laser light is launched into the lowest-order propagation modes and the fiber undergoes severe bending perturbations. Experimentally, near-field modal interference images and transmission loss measurements were obtained for different loop diameters. The data indicate that, when the fundamental mode is excited, the light remains in lowest-order modes even for small bend diameters. This is consistent with analysis which predicts that, in a parabolic-index multimode fiber subject to constant diameter bending, the light tends to oscillate between lowest-order modes and remains trapped therein rather than diffusing to high-order modes. Implications of these results for an intrusion-resistant communication system with graded-index multimode fiber are discussed.     

Gaussian-Schell-model beams propagating through rough gratings

In this work we analyze the near-field intensity distribution produced by a rough grating illuminated with a Gaussian-Schell-model beam. This kind of grating is formed by rough and smooth slits. Statistical techniques are used to describe the grating, and the Fresnel approach is used to perform the propagation of light. Two kinds of coherence affect the light propagation. One of them comes from the light beam, since it is not totally coherent. The other one comes from the rough topography of the grating surface. We have found that the Talbot effect is not present just after the grating, but it gradually increases. In addition, the contrast of the self-images decreases from a certain distance due to the coherence properties of the illumination beam. Then, the self-imaging process is only present between two specific distances from the grating. To corroborate the analytical results, we have performed numerical simulations for the mean intensity distribution based on the Sommerfeld-Rayleigh approach, showing their validity.     

Analyses of vector Gaussian beam propagation and the validity of paraxial and spherical approximations

The analysis of many systems in optical communications and metrology utilizing Gaussian beams, such as free-space propagation from single-mode fibers, point diffraction interferometers, and interference lithography, would benefit from an accurate analytical model of Gaussian beam propagation. We present a full vector analysis of Gaussian beam propagation by using the well-known method of the angular spectrum of plane waves. A Gaussian beam is assumed to traverse a charge-free, homogeneous, isotropic, linear, and nonmagnetic dielectric medium. The angular spectrum representation, in its vector form, is applied to a problem with a Gaussian intensity boundary condition. After some mathematical manipulation, each nonzero propagating electric field component is expressed in terms of a power-series expansion. Previous analytical work derived a power series for the transverse field, where the first term (zero order) in the expansion corresponds to the usual scalar paraxial approximation. We confirm this result and derive a corresponding longitudinal power series. We show that the leading longitudinal term is comparable in magnitude with the first transverse term above the scalar paraxial term, thus indicating that a full vector theory is required when going beyond the scalar paraxial approximation. In spite of the advantages of a compact analytical formalism, enabling rapid and accurate modeling of Gaussian beam systems, this approach has a notable drawback. The higher-order terms diverge at locations that are sufficiently far from the initial boundary, yielding unphysical results. Hence any meaningful use of the expansion approach calls for a careful study of its range of applicability. By considering the transition of a Gaussian wave from the paraxial to the spherical regime, we are able to derive a simple expression for the range within which the series produce numerically satisfying answers.     

A bound for the range of a narrow light beam in the near field

We investigate the possibility of light beams that are both narrow and long range with respect to the wavelength. On the basis of spectral electromagnetic field representations, we have studied the decay of the evanescent waves, and we have obtained some bounds for the width and range of a light beam in the near-field region. The range determines the spatial bound of the near field in the direction of propagation. For a number of representative examples we found that narrow beams have a short range. Our analysis is based on the uncertainty relations between spatial position and spatial frequency.     

Optimization of the arrangement of fiber laser array in beam coherent combining

Based on the expanding of the fundamental mode of a step-index fiber in terms of Laguerre–Gaussian modes, the accurate expression for the beam propagation of a fiber laser in free space is obtained. Thereby, the coherent combining beam of three fiber laser arrays including circular arrangement, square arrangement, and diamond arrangement are numerically analyzed. The study shows that all the beams gradually concentrate centrally on the propagation axis and the highest far-field peak intensity can be obtained by using the circular arrangement. Meanwhile, the far-field intensity of the circular arrangement by using the Laguerre–Gaussian approximation is also compared with that by using the pure Gaussian approximation, which indicates that the pure Gaussian approximation will induce much error in the far-field intensity. Finally, the influence of the radius of a circular fiber laser array on the far-field intensity is studied, of which the result shows that the far-field intensity decreases with increasing radius. Therefore, the fiber laser elements are suggested to be placed as close as possible.     

Surface waves and atomic force microscope probe-particle near-field coupling: discrete dipole approximation with surface interaction

Evanescent waves on a surface form due to the collective motion of charges within the medium. They do not carry any energy away from the surface and decay exponentially as a function of the distance. However, if there is any object within the evanescent field, electromagnetic energy within the medium is tunneled away and either absorbed or scattered. In this case, the absorption is localized, and potentially it can be used for selective diagnosis or nanopatterning applications. On the other hand, scattering of evanescent waves can be employed for characterization of nanoscale structures and particles on the surface. In this paper we present a numerical methodology to study the physics of such absorption and scattering mechanisms. We developed a MATLAB implementation of discrete dipole approximation with surface interaction (DDA-SI) in combination with evanescent wave illumination to investigate the near-field coupling between particles on the surface and a probe. This method can be used to explore the effects of a number of physical, geometrical, and material properties for problems involving nanostructures on or in the proximity of a substrate under arbitrary illumination.     

Failures of local approximation in finite-element methods

It is shown that standard finite-element discretizations of second-order differential equations (i.e. Galerkin and subdomain methods) using conforming linear elements may fail to approximate the original equation locally if the finite-element grid is irregular or if subdomains are chosen improperly. This failure of local approximation can lead to spurious computational results when subdomain methods are used, but these difficulties can be averted by a judicious choice of subdomains. The conditions which the subdomains must satisfy in order for local approximation to hold are derived and used to construct an algorithm for choosing them properly. The relation of these local results to the global convergence properties of the Galerkin method is discussed.