Logarithmic Axicon Research Articles

Long focusing range and self-healing Bessel vortex beam generator.

Here a continuous axial-spiral phase microplate (CAsPP), based on combining a logarithmic axicon and a spiral phase plate, was proposed for generating high-quality higher-order Bessel vortex beams. The novel optical component implemented via femtosecond laser direct writing possesses compact geometry and unique optical properties. The CAsPP with a diameter of 80 µm possesses a controllable long focus ranging from 50 to 600 µm and exhibits a good self-healing ability after free transmission of about 45 µm. Unique optical properties were demonstrated in both experiments and simulations, which were well matched to each other. This Letter provides new opportunities for applications in integrated optics, optical trapping, laser machining, and information reconstruction.

Propagation of fully and partially coherent flat-topped multi-Gaussian beams through axicons.

Theoretically, we have investigated the propagation of fully and partially coherent flat-topped multi-Gaussian beams through classical and logarithmic axicons. The stationary phase method applied to the Fresnel diffraction integral for intensity distribution has led to applicable relationships. The influences of axicon base angle, beam width, beam flatness, and coherence length with intensity distribution were studied. A number of results have been analyzed and discussed revealing some important conclusions. During this investigation, a specific interest has been given toward producing uniform intensity and long focal depth.

Kummer laser beams with a transverse complex shift

Here, we study asymmetric Kummer beams, whose complex amplitude is proportional to the Kummer function (a degenerate hypergeometric function). These beams are exact solutions of the paraxial equation (Schrödinger-type equation) and are obtained from the conventional symmetric hypergeometric beams by complex shift of the transverse coordinates. On propagation, the asymmetric Kummer beams change their intensity weakly and rotate around the optical axis. These beams are an example of vortex laser beams with a fractional orbital angular momentum (OAM), which depends on four parameters: the vortex topological charge, the shift magnitude, the logarithmic axicon parameter, and the degree of the radial factor. Changing these parameters, it is possible to control the beam OAM, continuously increasing or decreasing it.

Asymmetric hypergeometric laser beams

Here we study asymmetric Kummer beams (aK-beams) with their scalar complex amplitude being proportional to the Kummer function (a degenerate hypergeometric function). These beams are an exact solution of the paraxial propagation equation (Schrödinger-type equation) and obtained from the conventional symmetric hypergeometric beams by a complex shift of the transverse coordinates. On propagation, the aK-beams change their intensity weakly and rotate around the optical axis. These beams are an example of vortex laser beams with a fractional orbital angular momentum (OAM), which depends on four parameters: the vortex topological charge, the shift magnitude, the logarithmic axicon parameter and the degree of the radial factor. Changing these parameters, it is possible to control the beam OAM, either continuously increasing or decreasing it.

Vortex axicons for hypergeometric beams formation

A method for obtaining multi-ring distributions by using combinations of diffractive optical elements (DOE) for the formation of hypergeometric (HG) vortices is described. The calculated vortex axicons form a superposition of two HG vortices due to a combination of two logarithmic axicons in the phase function. Experimental results on the optical capture and transport of polystyrene microparticles with a diameter of 5 μm using calculated DOEs are presented.

Development of spot size and lateral intensity distribution generated by exponential, logarithmic, and linear axicons.

Axicons are known to produce a nearly Bessel beam transverse intensity distribution. It has been previously recognized that linear axicons and logarithmic axicons with a predefined distant depth of field (DOF) and no blocking present a development region characterized by an enlarged central spot size. In this paper, we aim to obtain a better insight on the formation of lateral light distribution in this development region. We also examine the spot size and transverse intensity distribution of the recently developed exponential axicon. We present experiments and detailed nonparaxial numerical simulations for a plane wave passing through these optical elements and show that the lateral intensity distribution they generate differs from that of a Bessel beam for a significant part of their DOF. This anomaly/irregularity in the formation of nondiffracting Bessel beams has to be taken into account in applications of these axicons, such as imaging or optical trapping.

Characterization of a refractive linear axicon with distant depth of field and no central blocking

We report the manufacturing and characterization of a refractive linear axicon producing a linearly increasing axial intensity Bessel-type beam over a predetermined range starting away from the axicon and without central blocking when illuminated by a plane wave. This is in contrast to a classical axicon that generates a diffraction-free beam starting from the axicon tip and extending to a range limited by the input beam aperture. The measured characteristics of the beam produced by the linear axicon, including its intensity distribution and spot size, are in good agreement with the theoretical predictions. Together with logarithmic axicon and exicon, this is another element of the axicon family that can generate a prescribed intensity distribution over a chosen range/depth of field.

Imaging properties of three refractive axicons

The imaging properties of three types of refractive axicons are examined by using them in an imaging system. A linear axicon, a logarithmic axicon, and a Fresnel axicon are characterized by determining their point spread functions (PSFs) experimentally and by numerical simulation. The PSFs, which vary along the depth of field for the cases considered in the present investigation, are used in digital filters to denoise the images. A comparison of the imaging performance of these three optical elements is presented.

Logarithmic axicon characterized by scanning optical probe system

A scanning optical probe system is proposed to measure a logarithmic axicon (LA) with subwavelength resolution. Multiple plane intensity profiles measured by a fiber probe are interpreted by solving an optimization problem to get the phase retardation function (PRF) of the LA. Experimental results show that this approach can accurately obtain the PRF with which the optical path difference of the generated quasi-nondiffracting beam in the propagation is calculated.

Study on Refractive-Diffractive Logarithm Axicons in mm/submm Range

Large depth of field and uniform axial intensity distributions are important in many practical applications. Lots of optical elements have been proposed for this end, such as full-aperture logarithmic axicons (LAs), annular-aperture LAs and apodized annular-aperture LAs. In millimeter/submillimeter (mm/submm) range, the refractive-diffractive behaviors (including transverse and longitudinal intensity distributions) behind these axicons illuminated by a plane wave are calculated by rigorous electromagnetic analysis method in our paper. Many important features of these axicons are concluded by comparing their refractive-diffractive field distributions

Study on Refractive-Diffractive Logarithm Axicons in mm/submm Range

Large depth of field and uniform axial intensity distributions are important in many practical applications. Lots of optical elements have been proposed for this end, such as full-aperture logarithmic axicons (LAs), annular-aperture LAs and apodized annular-aperture LAs. In millimeter/submillimeter (mm/submm) range, the refractive-diffractive behaviors (including transverse and longitudinal intensity distributions) behind these axicons illuminated by a plane wave are calculated by rigorous electromagnetic analysis method in our paper. Many important features of these axicons are concluded by comparing their refractive-diffractive field distributions

Electromagnetic Analysis of Apodized Annular-Aperture Logarithmic Axicons in Wave Imaging Systems

Electromagnetic Analysis of Apodized Annular-Aperture Logarithmic Axicons in Wave Imaging Systems

Electromagnetic Analysis of Apodized Annular-Aperture Logarithmic Axicons in Wave Imaging Systems

An apodized annular-aperture logarithmic axicon (3ALA) can be applied potentially to improve imaging resolution in quasi-optical and/or millimeter wave imaging systems, due to its property of long depth of field (DOF). Electromagnetic analyses of forward and backward 3ALAs are conducted respectively at mm- and submm-wavebands. Their lateral and axial intensity distributions are evaluated by rigorous electromagnetic analysis method. And the comparative conclusions are drawn at last in our present work

Diffraction of a Gaussian beam by a logarithmic axicon

We derive an explicit analytical relationship to describe the axial light intensity when a Gaussian beam is diffracted by the logarithmic axicon (LA). An evaluation formula for the effective radius of the diffraction pattern that we deduce shows the said radius to be in inverse proportion to the LA "force" parameter. The finite-difference time-domain-based simulation has shown that using the LA makes it possible to go beyond the diffraction limit: in the LA vicinity, the FWHM of the light beam can be as small as one fifth of the illumination wavelength.

Characterization of a refractive logarithmic axicon

We show that it is feasible to design and manufacture a refractive logarithmic axicon that generates a quasi-diffraction-free/Bessel beam with nearly constant beam size and intensity over a predetermined range. The novel optical element was characterized with both coherent and incoherent light, and good correspondence with the predicted behavior of the intensity distribution and spot size was found. The energy flow was also found to be nearly constant over most of the designed range. Logarithmic axicons may find applications in situations where large depth of field and uniform axial intensity/energy distributions are important.

Family of hypergeometric laser beams

We discuss a three-parameter family of paraxial coherent light fields that originate from a complex amplitude composed of the following four cofactors: the Gaussian beam, a logarithmic axicon, a spiral phase plate (angular harmonic), and an amplitude power function with a possible singularity at the origin of coordinates. For such types of beams, the near-field complex amplitude is proportional to the degenerate hypergeometric function, prompting the beams' name--hypergeometric (HyG) beams. When the Gaussian beam is replaced by a plane wave, the above beams change to generalized HyG modes that preserve their structure up to scale upon propagation. The intensity profile of the HyG beams is similar to that of the Bessel modes, forming a set of alternating bright and dark rings. However, the thickness of the rings of the HyG beams decreases with increasing ring number.

Spheric long-focal-depth lens

Spheric lens designed by nonlinear phase fitting method is proposed to duplicate the performance of logarithmic axicon, which can obtain a beam with long-focal-depth, small focal spot and side lobe. Moreover, the spheric lens is easier and less expensive for manufacture than logarithmic or aspherical elements. Based on this method, a simple lens is designed for the plasma diagnostic system in inertial confinement fusion. It is verified through Fresnel diffraction integral and an initial experiment that the performance of the lens can satisfy the requirements of the plasma measurements.

Imaging with extended focal depth by means of lenses with radial and angular modulation

The paper presents imaging properties of modified lenses with the radial and the angular modulation. We analyze three following optical elements with moderate numerical apertures: the forward logarithmic axicon and the axilens representing the radial modulation as well as the light sword optical element being a counterpart of the axilens with the angular modulation. The abilities of the elements for imaging with extended depth of focus are discussed in detail with the help of structures of output images and modulation transfer functions corresponding to them. According to the obtained results only the angular modulation of the lens makes possible to maintain the acceptable resolution, contrast and brightness of the output images for a wide range of defocusing. Therefore optical elements with angular modulations and moderate numerical apertures seem to be especially suitable for imaging with extended focal depth.

Optical functionalities of dielectric material deposits obtained from a Lambertian evaporation source

The thickness profile of deposits obtained from Lambertian evaporation sources is highlighted concerning its transmission optical functionality, in the case of dielectric materials. Fresnel diffraction is used to characterize the lateral resolution and intensity on the optical axis of an input gaussian laser beam. Functionality similar to logarithmic axicons, with uniform lateral resolution and also uniform on-axis intensity, is theoretically derived. It is also shown for this particular optical structure that the intensity slope along the optical axis can be changed from positive to negative values by only changing the input beam width.

Lens axicons illuminated by Gaussian beams for generation of uniform-axial intensity Bessel fields

A lens axicon (a converging lens with spherical aberration) illuminated by a Gaussian beam is studied. It is demonstrated that, when a backward-type lens axicon is illuminated with an appropriate beam radius of the incident Gaussian beam, one can achieve a focal segment with uniform axial intensity, which is similar to that generated by the logarithmic axicon. It is shown that this approach is very simple and efficient. A forward-type lens axicon illuminated by a Gaussian beam is also investigated.