Focal Intensity Distribution Research Articles

Tailoring focal plane component intensities of polarization singular fields in a tight focusing system

The scientific community studies tight focusing of radially and azimuthally-polarized vector beams as it is a versatile solution for many applications. We offer a new method to produce tight focusing that ensures a more uniform intensity profile in multiple dimensions, providing a more versatile and stable solution. We manipulate the polarization of the radially and azimuthally polarized vector beams to find an optimal operating point. We examine in detail optical fields whose polarization states lie on the equator of the relevant Poincaré spheres namely, the fundamental Poincaré sphere, the hybrid order Poincaré sphere (HyOPS), and the higher order Poincaré sphere. We find via simulation that the fields falling on these equators have focal plane intensity distributions characterized by a single rotation parameter α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha$$\\end{document} determining the individual state of polarization. The strengths of the component field distributions vary with α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha$$\\end{document} and can be tuned to achieve equal strengths of longitudinal (z) and transverse (x and y) components at the focal plane. Without control of this parameter (e.g., using α=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha =0$$\\end{document} in radially and α=π\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha =\\pi$$\\end{document} in azimuthally-polarized vector beams) intensity in x and y components are at 20% of the z component. In our solution with α=π/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha =\\pi /2$$\\end{document}, all components are at 80% of the maximum possible intensity of z. In examining the impact of α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha$$\\end{document} on a tightly focused beam, we also found that a helicity inversion of HyOPS beams causes a rotation of 180 degree in the axial intensity distribution.

Tackling the focal shift effect for metalenses

We present a theoretical analysis aimed at comprehending and mitigating the focal shift phenomenon in planar dielectric metalenses. To conduct this analysis, we introduce metalens designs consisting of silicon and germanium nanoblocks on a calcium fluoride substrate, operating in the mid-IR frequency range. The lensing performance of these metalenses is investigated using the finite-difference time-domain method, and they operate at wavelengths of 3 and 4 µm with a polarization conversion efficiency close to unity. Our findings indicate a strong correlation between the focal shift phenomena on dielectric metalenses and the numerical aperture (NA), revealing that increasing the Fresnel number is not always an effective approach to minimizing the focal shift. In contrast to previous studies, we define a critical NA, independent of the lens size, where the focal shift reaches a minimum, resulting in a symmetric focal intensity distribution and ultimately yielding a better-performing metalens. We demonstrate that for NAs greater than the determined critical value, a positive focal shift is observed on planar metalenses, diverging from the conventional negative shift predicted by existing models. Additionally, we show that by selecting a metalens within a specific NA range and with smaller diameters, high focusing efficiencies can be achieved. The focusing efficiency of the studied metalenses is measured as high as 70%, marking one of the best values reported for the IR range to date. These results serve as a guide for improving the agreement between experimental and designed metalens features, enhancing their practical applications.

Soft X-ray wavefront sensing at an ellipsoidal mirror shell

A reliable `in situ' method for wavefront sensing in the soft X-ray domain is reported, developed for the characterization of rotationally symmetric optical elements, like an ellipsoidal mirror shell. In a laboratory setup, the mirror sample is irradiated by an electron-excited (4.4 keV), micrometre-sized (∼2 µm) fluorescence source (carbon K α, 277 eV). Substantially, the three-dimensional intensity distribution I(r) is recorded by a CCD camera (2048 × 512 pixels of 13.5 µm) at two positions along the optical axis, symmetrically displaced by ±21–25% from the focus. The transport-of-intensity equation is interpreted in a geometrical sense from plane to plane and implemented as a ray tracing code, to retrieve the phase Φ(r) from the radial intensity gradient on a sub-pixel scale. For reasons of statistical reliability, five intra-/extra-focal CCD image pairs are evaluated and averaged to an annular two-dimensional map of the wavefront error {\cal W}. In units of the test wavelength (C K α), an r.m.s. value \sigma_{\cal{W}} = ±10.9λ0 and a peak-to-valley amplitude of ±31.3λ0 are obtained. By means of the wavefront, the focus is first reconstructed with a result for its diameter of 38.4 µm, close to the direct experimental observation of 39.4 µm (FWHM). Secondly, figure and slope errors of the ellipsoid are characterized with an average of ±1.14 µm and ±8.8 arcsec (r.m.s.), respectively, the latter in reasonable agreement with the measured focal intensity distribution. The findings enable, amongst others, the precise alignment of axisymmetric X-ray mirrors or the design of a wavefront corrector for high-resolution X-ray science.

Fast, Precise, High Contrast Laser Writing for Photonic Chips with Phase Aberrations

AbstractIntegrated photonic chips have significant potential in telecommunications, classic computing, quantum systems, and topological photonics. Direct laser writing offers unique capability for creating three‐dimensional photonic devices in an optical glass chip with quick prototyping. However, it is a challenge for existing laser writing schemes to create index‐modified structures in glass that precisely match the laser focal shape while also achieving high refractive index contrasts and high scanning speeds. Here, we introduce a refractive index modification scheme that combines the advantages of non‐thermal and thermal regime fabrication methods. We also propose a waveguide formation model that is verified through a thorough study on the effects of phase aberrations. The presented new photonic chip fabrication scheme uses a novel focal intensity distribution, where pulse energy is relocated to the bottom of a laser focus by manipulating primary and higher order spherical aberrations. The technique can produce index modifications with high scanning speed (can be 20 mm/s or higher), high index contrast (ranging from 0.009 to 0.021), and high precision to fabricate with arbitrary cross‐sections. This method has potential to expand the capabilities of photonic chips in applications that require small‐scale, high precision, or high contrast refractive index control.

Influence of Primary Coma on the Tightly Focusing Characteristics of Circular Basis Hybrid Order Poincaré Sphere Beams

Analogous to the Poincaré sphere, a hybrid order Poincaré sphere is used to represent the ellipse field singularities (C-points). We study the tight focusing properties of generic bright and dark hybrid order Poincaré sphere beams in the presence of primary coma. The role of the polarization singularity index and handedness of the polarization of the hybrid order Poincaré sphere beams on the focused structure has been discussed. Results have been presented for the total intensity, component intensities, and component phase distributions for left- and right-handed bright and dark star and lemon types singularities. The presence of primary coma distorted the focal plane intensity distributions for both positive and negative index generic C-points. Coma is known to disturb the circular symmetry of the focal plane intensity distribution. Similarly in tight focusing polarization is known to disturb the symmetry. Therefore, a beam with structured and inhomogeneous polarization distribution tightly focused under the influence of coma is a fit case to study. It is found that the presence of primary coma aberration in the focusing system produces a positional shift of the high-intensity peaks and a reduction of the intensity on one side of the center. As the strength of the primary coma increases, the focal plane intensity distributions shift more and more toward the right from the initial position. Unlike the scalar vortex case, in the case of hybrid order Poincaré sphere beams, the focal plane intensity distribution undergoes rotation, as the helicity of the hybrid order Poincaré sphere beams is inverted, in addition to shifting. All the component phase distributions are found to be embedded with phase vortices of charge ±1.

Effect of primary astigmatism on the tight focusing of ellipse field singularities

The tight focusing of ellipse field singularities, namely bright and dark C-points, is studied using the Debye–Wolf diffraction integral in the presence of primary astigmatism. The effect of polarization distribution, the sign of polarization singularity index, amplitude, and handedness of the ellipse field singularities are investigated in the intensity distribution of the focal plane of a high numerical aperture lens system in the presence of primary astigmatism. Ellipse field singularities that are tightly focused can create an additional longitudinal component at the focal plane. At the focal plane, both the longitudinal and transverse components of the ellipse fields are found to be hosted with phase vortices of charge ±1. In the presence of primary astigmatism, the helicity inversion of input C-point does not rotate the focal plane intensity distributions by 180°.

Target Design in SEM-Based Nano-CT and Its Influence on X-ray Imaging.

Nano-computed tomography (nano-CT) based on scanning electron microscopy (SEM) is utilized for multimodal material characterization in one instrument. Since SEM-based CT uses geometrical magnification, X-ray targets can be adapted without any further changes to the system. This allows for designing targets with varying geometry and chemical composition to influence the X-ray focal spot, intensity and energy distribution with the aim to enhance the image quality. In this paper, three different target geometries with a varying volume are presented: bulk, foil and needle target. Based on the analyzed electron beam properties and X-ray beam path, the influence of the different target designs on X-ray imaging is investigated. With the obtained information, three targets for different applications are recommended. A platinum (Pt) bulk target tilted by 25° as an optimal combination of high photon flux and spatial resolution is used for fast CT scans and the investigation of high-absorbing or large sample volumes. To image low-absorbing materials, e.g., polymers or organic materials, a target material with a characteristic line energy right above the detector energy threshold is recommended. In the case of the observed system, we used a 30° tilted chromium (Cr) target, leading to a higher image contrast. To reach a maximum spatial resolution of about 100 nm, we recommend a tungsten (W) needle target with a tip diameter of about 100 nm.

Super-resolution Reflection Microscopy via Absorbance Modulation.

In recent years, fluorescence microscopy has been revolutionized. Reversible switching of fluorophores has enabled circumventing the limits imposed by diffraction. Thus, resolution down to the molecular scale became possible. However, to the best of our knowledge, the application of the principles underlying super-resolution fluorescence microscopy to reflection microscopy has not been experimentally demonstrated. Here, we present the first evidence that this is indeed possible. A layer of photochromic molecules referred to as the absorbance modulation layer (AML) is applied to a sample under investigation. The AML-coated sample is then sequentially illuminated with a one-dimensional (1D) focal intensity distribution (similar to the transverse laser mode TEM01) at wavelength λ1 = 325 nm to create a subwavelength aperture within the AML, followed by illumination with a Gaussian focal spot at λ2 = 633 nm for high-resolution imaging. Using this method, called absorbance modulation imaging (AMI) in reflection, we demonstrate a 2.4-fold resolution enhancement over the diffraction limit for a numerical aperture (NA) of 0.65 and wavelength (λ) of 633 nm.

Flexible tailoring of multifocal spot arrays of partially coherent radially polarized beam with periodical spatial coherence

In this work, a new kind of partially coherent vector beam, namely, partially coherent radially polarized multi-Gaussian Schell-model array (PCRP MGSMA) beam, is introduced and the focusing properties of such beam passing through a thin lens are characterized. One of the most remarkable features of such beam is its capability of producing tunable multifocal spot arrays in the focal plane, and each array lobe can be shaped into dark hollow, Gaussian-like and flat-topped profile by modulating the initial spatial coherence width More interesting, the modulation effect of multiple off-axis vortices can also be acted on each array lobe, thus the creation of triangle-like, square-like and pentagon-like hollow structure arrays can be realized for the high coherence case, but this remarkable characteristic will be vanished in the case of a very low coherence. In addition to the focal spot arrays tailoring, the focal intensity distribution of such beam can be flexibly shaped into diversified flat-topped profiles, such as linear, square and rectangular, by elaborately choosing the matching parameters of the source. These interesting results will be useful for optical multi-particle manipulation, high resolution imaging and material surface processing.

Focused polarization ellipse field singularities: interaction of spin-orbital angular momentum and the formation of optical Möbius strips

We study here the intensity distribution and formation of optical polarization Möbius strips by tightly focusing of C-point singularity beams. These beams are characterized by a central circular polarization point (C-point) surrounded by a spatially varying elliptic polarization. Under tight focusing conditions, the different polarization components of the beam interfere and exhibit clear difference between left-handed and right handed input beams. The transverse polarization distribution at the focal plane is similar to the input distribution for left-handed lemon beam, but exhibits 180° rotation for right handed lemon beam. Moreover, the longitudinal polarization component exhibits spiral phase distribution, owing to spin-orbit angular momentum conversion at the focal plane, with opposite winding directions for the left-handed and right-handed input beams. We show that the shape of the resulting Möbius strip is determined by the helicity of the C-point and by the polarization singularity index, which is the contour integral of polarization ellipse angle around the singularity. It is found that inverting the helicity leads to 180° rotation in the focal plane intensity distribution, accompanied by handedness inversion for the polarization ellipses. The number of separatrices in the input polarization distribution is equivalent to the number of twist points of the Möbius strip in the focal plane, as well as to the number of intensity zeros in the z-component of the focused field. These phenomena are observed for beams with a bright C-point, but also for dark C-point, in which the electric field is zero at the center of the beam.

Tight focusing of orthogonal C-point polarization states

Recently it is shown that spatially varying polarization distribution embedded with polarization singularity can be used to form an orthogonal basis pair. In this paper we show tight focusing of orthogonal basis pairs of bright and dark C-points. We observe that unlike orthogonal basis pairs of homogeneous polarization distribution, the orthogonal basis pairs of spatially varying polarization distributions embedded with C-points, show same intensity distribution at the focal plane upon tight focusing. We show that the focal plane intensity distribution is depend upon the intensity distribution of the beam under tight focusing. The longitudinal and transverse field components are hosted with phase vortices of charge ±1. Though the position of phase singularities remains same in a given orthogonal C-point pair, but the sign of phase vortices undergo reversal. The current study may reveal various other fundamental concepts related to polarization singularities.

Tight Focusing Properties of Ring Pearcey Beams with a Cross Phase

We theoretically investigated the properties of tightly focused ring Pearcey beams with a cross phase (CPRPB). The expressions of the distributions of both electric field and magnetic field in the focal region of an objective were first derived from the vectorial Debye theory, and then numerical calculations were carried out to obtain the focused intensity distribution and the Poynting vector of CPRPB near the focus. Numerical calculations indicate that as CPRPB is focused on an objective of high numerical aperture (NA), two nonuniform self-focusing spots occur at both sides of the geometrical focus of the objective symmetrically, and the angle between their directions is 90 degrees. The stronger is the strength of cross-phase modulation, the flatter are the ellipses of the self-focusing spots, and the smaller is the intensity at the geometrical focus of the objective. Numerical calculations also demonstrate that the optical gradient force produced by tightly focused CPRPB in the focal region can be manipulated in magnitude and in direction by tuning the strength of cross-phase modulation. Due to these properties of tightly focused CPRPB, they might find applications in the manipulation of micro- and nanoparticles and so on.

Measurement of shock roughness due to phase plate speckle imprinting relevant for x-ray diffraction experiments on 3rd and 4th generation light sources

Laser-shock compression experiments at 3rd and 4th generation light sources generally employ phase plates, which are inserted into the beamline to achieve a repeatable intensity distribution at the focal plane. Here, the laser intensity profile is characterized by a high-contrast, high-frequency laser speckle. Without sufficient smoothing, these laser non-uniformities can translate to a significant pressure distribution within the sample layer and can affect data interpretation in x-ray diffraction experiments. Here, we use a combination of one- and two-dimensional velocity interferometry to directly measure the extent to which spatial frequencies within the laser focal spot intensity pattern are smoothed out during propagation within the laser plasma and a polyimide ablator. We find that the use of thicker polyimide layers results in spatially smoother shock fronts, with the greatest degree of smoothing associated with the highest spatial frequencies. Focal spots with the smallest initial speckle separation produce the most rapid smoothing. Laser systems that employ smoothing by spectral dispersion techniques to rapidly modulate the focal plane intensity distribution are shown to be the most effective ones in producing a spatially smooth shock front. We show that a simple transport model combined with the known polyimide Hugoniot adequately describes the extent of shock smoothness as a function of polyimide thickness. Our results provide a description of spatial structure smoothing across a shock front, which can be used to design targets on x-ray free electron laser facilities.

Designing chromatic optical retarder stacks for segmented next-generation easySTED phase plates.

Fluorescence nanoscopy methods based on the RESOLFT principle, such as beam-scanning STED nanoscopy, require the co-alignment of optical beams for molecular state (on/off) switching and fluorescence excitation. The complexity and stability of the beam alignment can be drastically simplified and improved by using a single-mode fibre as the sole light source for all required laser beams. This in turn then requires a chromatic optical element for shaping the off-switching beam into a focal-plane donut while simultaneously leaving the focal intensity distributions at other wavelengths shaped as regular focal spots. Here we describe novel designs of such so-called 'easySTED phase plates' and provide a rationale how to find the desired spectral signature for combinations of multiple wavelengths.

Spin Hall effect of fractional order radially polarized beam in its tight focusing

Different from the traditional radially polarized beams, the fractional order radially polarized beam possess the symmetry broken structure of the polarization state distribution. The symmetry broken of the optical beams’ intrinsic structure can lead to the spin–orbit coupling of light. In this paper, we investigate the tight focusing of the fractional order radially polarized beam and show some interesting phenomena induced by the symmetry broken structure of the polarization state distribution. Firstly, the intensity and the spin angular momentum density are split in the focal plane, and the focal field intensity distribution can be controlled (split or focused) by modulating the initial phase of the polarization state. Furthermore, just like the spin angular momentum, the orbital angular momentum density also is split into two opposite parts (positive and negative). Though, the evolutions of the spin and the orbital angular momentum are different: as the change of the initial phase of the polarization state, the spin angular momentum distribution just is rotated slightly, yet the orbital angular momentum distribution is reversed. All these phenomena can be considered as the manifestations of the spin Hall effect of the fractional order radially polarized beam. Our results provide further potential applications of the vector beams in the focal field shaping.

Traceable x-ray focal spot reconstruction by circular edge analysis: from sub-microfocus to mesofocus

The size and shape of the focal spot, i.e. the area where x-rays are generated in an x-ray tube, is a key figure of merit of a radiographic imaging system, because it governs the achievable resolution and, if the shape is irregular, can introduce anisotropic blur in the radiographs. Size and shape are mainly determined by the diameter of the accelerated electron-beam and electron interactions with the target material. Since focal spot parameters change over time, due to target wear and drift in the electron optics, traceable methods to monitor the focal spot size and shape are required. Here, we present a method to map the 2D intensity distributions of focal spots, ranging from sub-microfocus (0.1 µm) to mesofocus (>50 μm). The method evaluates radial profiles of a circular aperture and applies filtered back projection to reconstruct the focal spot intensity distribution. Special emphasis is placed on traceability and influence factors, such as image noise and aperture geometry. The method was validated over a focal spot range from 0.3 µm to 200 µm. Experimental results were compared to complementary methods, using line pair gauges (JIMA) for microfocus (1 µm–10 µm), e-beam characterisation for sub-microfocus (0.3 µm–1.0 µm), and pinhole camera imaging for mesofocus spots (50 µm–200 µm). Good agreement between the focal spot reconstruction and the complementary methods was achieved. Limitations due to edge penetration, caused by the finite thickness of the circular aperture gauge, were experimentally verified. From these findings, we derived recommendations for an improved gauge geometry, regarding its thickness and the aperture diameters. The method perfectly complements the well-established pinhole imaging, which is limited to focal spots larger than ∼50 µm.

Characteristics of the beam smoothing using the combination of induced spatial incoherence and continuous phase plate

Beam smoothing effect of induced spatial incoherence combined with continuous phase plate is reported. General theoretical models that can be used to describe the spatiotemporal characteristics of the focal region in beam smoothing is proposed. The focal region evolution using induced spatial incoherence is studied and compared with that of using smoothing by spectral dispersion. The result shows that, when using induced spatial incoherence combined with continuous phase plate for beam smoothing, the focal region spatiotemporal evolution is chaotic and random, and the size of speckle on spacetime is determined by the coherence time of the laser. The comparison shows that the smoothing speed of induced spatial incoherence is higher. Focal spot time-resolved spectrum and focal spot intensity distribution using induced spatial incoherence are measured experimentally. The results show that the beam smoothing method has an excellent ability to obtain the desired low coherence in focal region, and can completely suppress the middle-high frequency nonuniformity of the focal spot.

Squared Focal Intensity Distributions for Applications in Laser Material Processing.

Tailored intensity profiles within the focal spot of the laser beam offer great potential for a well-defined control of the interaction process between laser radiation and material, and thus for improving the processing results. The present paper discusses a novel refractive beam-shaping element that provides different squared intensity distributions converted from the Gaussian output beam of the utilized femtosecond (fs) laser. Using the examples of surface structuring of stainless-steel on the micro- and nano-scale, the suitability of the beam-shaping element for fs-laser material processing with a conventional f-Theta lens is demonstrated. In this context, it was shown that the experimental structuring results are in good agreement with beam profile measurements and numerical simulations of the beam-shaping unit. In addition, the experimental results reveal the improvement of laser processing in terms of a significantly reduced processing time during surface nano-structuring and the possibility to control the ablation geometry during the fabrication of micro-channels.

Increasing the brightness of harmonic XUV radiation with spatially-tailored driver beams

Bright high harmonic sources can be produced by loosely focussing high peak power laser pulses to exploit the quadratic scaling of flux with driver spot size at the expense of a larger experimental footprint. Here, we present a method for increasing the brightness of a harmonic source (while maintaining a compact experimental geometry) by spatially shaping the transverse focal intensity distribution of a driving laser from a Gaussian to supergaussian. Using a phase-only spatial light modulator we increase the size and order of the supergaussian focal profiles, thereby increasing the number of harmonic emitters more efficiently than possible with Gaussian beams. This provides the benefits of a loose focussing geometry, yielding a five-fold increase in harmonic brightness, whilst maintaining a constant experimental footprint. This technique can readily be applied to existing high harmonic systems, opening new opportunities for applications requiring bright, compact sources of coherent short wavelength radiation.

Shaping the focal intensity distribution using a partially coherent radially polarized beam with multiple off-axis vortices.

A new kind of partially coherent vector vortex beam, namely, the partially coherent radially polarized (PCRP) beam with multiple off-axis vortices, is introduced, and the average intensity distributions of such vortex beam focused by a thin lens are investigated theoretically. It is novelty that the off-axis vortices will induce the focal intensity redistribution and reconstruction, while this remarkable characteristic will be vanished in the case of a very low coherence. In view of this distinctive feature, a new method has been put forward to shape or modulate the focal intensity distribution by elaborately tailoring the multiple off-axis vortices as well as the coherence length. More importantly, some peculiar focal fields with novel structures, such as bar-shaped, triangle-shaped, square-shaped, and pentagon-shaped hollow profiles or flat-top foci, are obtained. Our results indicate that modulating the multiple off-axis vortices provides an additional degree of freedom for focus shaping.