Vectorial Gaussian Beams Research Articles

Spin angular momentum of Gaussian beams with several polarization singularities

We study a paraxial vector Gaussian beam with several polarization singularities located on a circle. Such a beam is superposition of a cylindrically polarized Laguerre-Gaussian beam and a linearly polarized Gaussian beam. It is found that although polarization in the initial plane is linear, alternating regions with the different-sign spin angular momentum density are generated upon free-space propagation, showing that a spin Hall effect arises. For an arbitrary transverse plane, it is shown that the spin angular momentum magnitude is maximal on a certain-radius circle. We obtain an approximate expression for the distance to the transverse plane where the spin angular momentum density is maximal. Besides, we derive an optimal radius of the singularity-containing circle in the initial plane for which the maximal spin angular momentum density can be achieved upon propagation. It is revealed that in this case, the energies of the Laguerre-Gaussian beam and the Gaussian beam are equal to each other. We also obtain an expression for the orbital angular momentum density and find it to be defined by the spin angular momentum density, multiplied by –m/2, with m being the upper index of the Laguerre-Gaussian beam, equal to the number of the polarization singularities. An analogy with plane waves reveals that the spin Hall effect arises due to different divergence rates of the linearly polarized Gaussian beam and the cylindrically polarized Laguerre-Gaussian beam.

Optical vector fields with kaleidoscopic quasicrystal structures by multiple beam interference.

An easily accessible approach is proposed to create structured beams with various quasicrystal structures and polarization distributions based on multi-beam interference. By controlling the azimuthally-dependent polarization for Q evenly and circularly distributed beams to be interfered, the intensity and polarization structures for the generated quasicrystal field with Q-fold rotational symmetry are flexibly adjusted. Using the diffraction theory for interfering Q vector Gaussian beams, an analytical wave function is derived to reconstruct the polarization-resolved intensities and the distributions of Stokes parameters measured in the experiment. With good agreement between the numerical and experimental results, the derived wave function is further employed to characterize the propagation-variant states of polarization, providing fundamentally important information for the vector quasicrystal beams.

Spin Hall Effect in Paraxial Vectorial Light Beams with an Infinite Number of Polarization Singularities.

Elements of micromachines can be driven by light, including structured light with phase and/or polarization singularities. We investigate here a paraxial vector Gaussian beam with an infinite number of polarization singularities residing evenly on a straight line. The intensity distribution is derived analytically and the polarization singularities are shown to exist only in the initial plane and in the far field. The azimuthal angle of the polarization singularities is shown to increase in the far field by π/2. We obtain the longitudinal component of the spin angular momentum (SAM) density and show that it is independent of the azimuthal angle of the polarization singularities. Upon propagation in free space, an infinite number of C-points is generated, where polarization is circular. We show that the SAM density distribution has a shape of four spots, two with left and two with right elliptic polarization. The distance to the transverse plane with the maximal SAM density decreases with decreasing distance between the polarization singularities in the initial plane. Generating such alternating areas with positive and negative SAM density, despite linear polarization in the initial plane, manifests the optical spin Hall effect. Application areas of the obtained results include designing micromachines with optically driven elements.

Spin Hall Effect in the Paraxial Light Beams with Multiple Polarization Singularities.

Elements of micromachines can be driven by light, including structured light with phase and/or polarization singularities. We investigate a paraxial vectorial Gaussian beam with multiple polarization singularities residing on a circle. Such a beam is a superposition of a cylindrically polarized Laguerre-Gaussian beam with a linearly polarized Gaussian beam. We demonstrate that, despite linear polarization in the initial plane, on propagation in space, alternating areas are generated with a spin angular momentum (SAM) density of opposite sign, that manifest about the spin Hall effect. We derive that in each transverse plane, maximal SAM magnitude is on a certain-radius circle. We obtain an approximate expression for the distance to the transverse plane with the maximal SAM density. Besides, we define the singularities circle radius, for which the achievable SAM density is maximal. It turns out that in this case the energies of the Laguerre-Gaussian and of the Gaussian beams are equal. We obtain an expression for the orbital angular momentum density and find that it is equal to the SAM density, multiplied by -m/2 with m being the order of the Laguerre-Gaussian beam, equal to the number of the polarization singularities. We consider an analogy with plane waves and find that the spin Hall affect arises due to the different divergence between the linearly polarized Gaussian beam and cylindrically polarized Laguerre-Gaussian beam. Application areas of the obtained results are designing micromachines with optically driven elements.

Spatially controlled nano-structuring of silicon with femtosecond vortex pulses

Engineering material properties is key for development of smart materials and next generation nanodevices. This requires nanoscale spatial precision and control to fabricate structures/defects. Lithographic techniques are widely used for nanostructuring in which a geometric pattern on a mask is transferred to a resist by photons or charged particles and subsequently engraved on the substrate. However, direct mask-less fabrication has only been possible with electron and ion beams. That is because light has an inherent disadvantage; the diffraction limit makes it difficult to interact with matter on dimensions smaller than the wavelength of light. Here we demonstrate spatially controlled formation of nanocones on a silicon surface with a positional precision of 50 nm using femtosecond laser ablation comprising a superposition of optical vector vortex and Gaussian beams. Such control and precision opens new opportunities for nano-printing of materials using techniques such as laser-induced forward transfer and in general broadens the scope of laser processing of materials.

Extreme ultraviolet polarization vortex beam based on high harmonic generation

Polarization is a property of vector beam that is widely used in many areas of science and technology. And vector beam is also called polarization vortex beam. Radially polarized beam and azimuthally polarized beam are the paradigm of vector beam. Extreme ultraviolet (EUV) vector beam could be applied in many fields such as diffractive imaging, Extreme Ultraviolet Lithography (EUVL), or ultrafast control of magnetic properties. In our experiments, a home-made EUV spectrometer was used to generate a tunable ultrafast EUV coherent light source based on high-order harmonic generation (HHG) by intense femtosecond laser. The apparatus features by using the plane grating in conical diffraction. The radially polarized vector beam and Gaussian beam with 800 nm, 35 fs laser pulses were applied to interact with Argon atoms, respectively. The high harmonic spectrums with a polarization singularity and a Gaussian distribution were observed. The experimental results demonstrate that the EUV vector beam could be transfered from near-infrared driven laser during the highly nonlinear interaction. The short-wavelength radiation with a polarization singularity can reach a photon flux of 10<sup>8</sup> per second. And the harmonic orders produced by Gaussian beam are significantly higher than that of vector field. The mechanism of macroscopic phase matching was discussed. It indicates that the phase matching for vector harmonic yields is similar with that driven by a Gaussian beam. In this case, EUV vector beam through HHG has been obtained, which provides one important method for attosecond vector pulses and opens new possibilities for exploring and manipulating the time-dependent evolution of quantum states in atom and molecule.

Mathematics of vectorial Gaussian beams

Since the development of laser light sources in the early 1960s, laser beams are everywhere. Laser beams are central in many industrial applications and are essential in ample scientific research fields. Prime scientific examples are optical trapping of ultracold atoms, optical levitation of particles, and laser-based detection of gravitational waves. Mathematically, laser beams are well described by Gaussian beam expressions. Rather well covered in the literature to date are basic expressions for scalar Gaussian beams. In the past, however, higher accuracy mathematics of scalar Gaussian beams and certainly high-accuracy mathematics of vectorial Gaussian beams were far less studied. The objective of the present review then is to summarize and advance the mathematics of vectorial Gaussian beams. When a weakly diverging Gaussian beam, approximated as a linearly polarized two-component plane wave, say (Ex,By), is tightly focused by a high-numerical-aperture lens, the wave is “depolarized.” Namely, the prelens (practically) missing electric field Ey,Ez components suddenly appear. This is similar for the prelens missing Bx,Bz components. In fact, for any divergence angle (θd<1), the ratio of maximum electric field amplitudes of a Gaussian beam Ex:Ez:Ey is roughly 1:θd2:θd4. It follows that if a research case involves a tightly focused laser beam, then the case analysis calls for the mathematics of vectorial Gaussian beams. Gaussian-beam-like distributions of the six electric–magnetic vector field components that nearly exactly satisfy Maxwell’s equations are presented. We show that the near-field distributions with and without evanescent waves are markedly different from each other. The here-presented nearly exact six electric–magnetic Gaussian-beam-like field components are symmetric, meaning that the cross-sectional amplitude distribution of Ex(x,y) at any distance (z) is similar to the By(x,y) distribution, Ey(x,y) is similar to Bx(x,y), and a 90° rotated Ez(x,y) is similar to Bz(x,y). Components’ symmetry was achieved by executing the steps of an outlined symmetrization procedure. Regardless of how tightly a Gaussian beam is focused, its divergence angle is limited. We show that the full-cone angle to full width at half-maximum intensity of the dominant vector field component does not exceed 60°. The highest accuracy field distributions to date of the less familiar higher-order Hermite–Gaussian vector components are also presented. Hermite–Gaussian E-B vectors only approximately satisfy Maxwell’s equations. We have defined a Maxwell’s-residual power measure to quantify the approximation quality of different vector sets, and each set approximately (or exactly) satisfies Maxwell’s equations. Several vectorial “applications,” i.e., research fields that involve vector laser beams, are briefly discussed. The mathematics of vectorial Gaussian beams is particularly applicable to the analysis of the physical systems associated with such applications. Two user-friendly “Mathematica” programs, one for computing six high-accuracy vector components of a Hermite–Gaussian beam, and the other for computing the six practically Maxwell’s-equations-satisfying components of a focused laser beam, supplement this review.

Analysis of characteristics of paraxial vector Gaussian beams affecting the formation of microstructures in an azopolymer

The paper considers paraxial Gaussian laser beams with different polarization states and their characteristics, such as the Poynting vector, gradient force, and force density, which are important for evaluating the effects of radiation on matter. An analysis of the considered characteristics of laser radiation from the point of view of the formation of photoinduced microrelief in an azopolymer film is carried out.

Electron acceleration by a tightly focused cylindrical vector Gaussian beam

We have studied the electron acceleration by a tightly focused cylindrical vector Gaussian beam. Different from the Lax series field, cylindrical vector Gaussian beams are vector-beam solutions of Maxwell’s equations and its focusing property can be numerically analyzed by the Richards–Wolf vectorial diffraction theory. Field differences exist between the cylindrical vector Gaussian beam and the Lax series field. The cylindrical vector Gaussian beam increases the asymmetry of the electromagnetic fields, which is more beneficial to the electron acceleration. When the beam waist falls down to the order of the wavelength, the high laser intensity zone is more proper to define the reflection, capture and transmission conditions of the electrons. The injection energy and the injected angle of the electron and the initial phase of the laser beam play important roles for the electron to enter and be trapped by the high laser intensity zone.

Nonparaxial Propagation of Vectorial Elliptical Gaussian Beams

Based on the vectorial Rayleigh-Sommerfeld diffraction integral formulae, analytical expressions for a vectorial elliptical Gaussian beam’s nonparaxial propagating in free space are derived and used to investigate target beam’s propagation properties. As a special case of nonparaxial propagation, the target beam’s paraxial propagation has also been examined. The relationship of vectorial elliptical Gaussian beam’s intensity distribution and nonparaxial effect with elliptic coefficientαand waist width related parameterfωhas been analyzed. Results show that no matter what value of elliptic coefficientαis, when parameterfωis large, nonparaxial conclusions of elliptical Gaussian beam should be adopted; while parameterfωis small, the paraxial approximation of elliptical Gaussian beam is effective. In addition, the peak intensity value of elliptical Gaussian beam decreases with increasing the propagation distance whether parameterfωis large or small, and the larger the elliptic coefficientαis, the faster the peak intensity value decreases. These characteristics of vectorial elliptical Gaussian beam might find applications in modern optics.

Modeling highly focused laser beam in optical tweezers with the vector Gaussian beam in the T-matrix method

The vector Gaussian beam with high-order corrections is used to describe accurately the laser beam up to numerical aperture NA=1.20 in the optical tweezers for trapping nanoparticles. The beam is then expanded in the T-matrix method into the vector spherical wave function (VSWF) series using the point matching method with a new selection of the matching points. The errors in the beam description and in the VSWF expansion are much lower than those that occur in the paraxial Gaussian beam model.

Goos–Hänchen shift for higher-order Hermite–Gaussian beams

We study the reflection of a Hermite-Gaussian beam at an interface between two dielectric media. We show that unlike Laguerre-Gaussian beams, Hermite-Gaussian beams undergo no significant distortion upon reflection. We report Goos-H\"anchen shift for all the spots of a higher order Hermite Gaussian beam near the critical angle. The shift is shown to be insignificant away from the critical angle. The calculations are carried out neglecting the longitudinal component along the direction of propagation for a spatially finite, s-polarized, full 3-d vector beam. We briefly discuss the difficulties associated with the paraxial approximation pertaining to a vector Gaussian beam.

Decoding divergent series in nonparaxial optics

A theoretical analysis aimed at investigating the divergent character of perturbative series involved in the study of free-space nonparaxial propagation of vectorial optical beams is proposed. Our analysis predicts a factorial divergence for such series and provides a theoretical framework within which the results of recently published numerical experiments concerning nonparaxial propagation of vectorial Gaussian beams find a meaningful interpretation in terms of the decoding operated on such series by the Weniger transformation.

Propagation characteristics of nonparaxial Gaussian beams

Based on the rigorous definition of light intensity of nonparaxial vectorial beams in the transverse plane, beam width, far-field divergence angles and M2 factors are defined by the power in the bucket. Taking nonparaxial vectorial Gaussian beam as an example, the detailed theoretical analyses and numerical calculations about the propagation characteristics are performed, and the results are compared with those from the nonparaxial scalar theory. Showing that when the half width of light source ω0 is less than Λ, there are obvious differences between the propagation characteristics described by the nonparaxial scalar and the vectorial theory, and their propagation characteristics tend to be consistent with ω0 increasing, change to that obtained in the paraxial scalar approximation gradually.

Vectorial Gaussian beam expansion for high-frequency wave propagation

Wide-waisted Gaussian beam (GB) expansion and point matching methods are discussed in this study as promising techniques for the expansion of electromagnetic waves into GBs. These methods combined with the GB tracing method provide an appropriate scheme for wave propagation analysis in large-scale problems. Details of the wide-waisted GB expansion and point matching methods are provided, and their accuracy and validity regions are discussed. A combination of the aforementioned expansion and tracing methods is utilised to analyse a simple outdoor high-frequency wave propagation problem. The results are compared to the results of the well-known ray tracing method, showing good agreement.

RIGOROUS 3D VECTORIAL GAUSSIAN BEAM MODELING OF DEMULTIPLEXING PERFORMANCE OF VIRTUALLY-IMAGED-PHASED-ARRAYS

We extend our previously-derived generalized closed-form representation for spectral dispersing performance of the Virtually- Imaged-Phased-Array (VIPA) based on a 3D vectorial Gaussian beam formulation to demultiplexing application. To analyze VIPA in the demultiplexer scheme, a spherical lens is added after the VIPA, so that the device plane is superimposed on the focal plane of the lens. The calculated output proflle at previous step is reformulated > in a matrix form in this step. Finally, the derived closed-form is simulated, and the numerical outcomes are compared with the previous results. The 3D output radiation of VIPA demultiplexer pattern is also depicted and found to be very intuitive and promising for some applications especially WDM demultiplexer and optical Code Division Multiple Access (CDMA).

The rigorous electromagnetic theory of the diffraction of vector beams by a circular aperture

The rigorous electromagnetic theory of the diffraction of vector beams by an aperture is proposed and numerically evaluated by the diffraction of vector Gaussian beams by a circular aperture. The results are compared with those, by using the vector Rayleigh–Sommerfeld diffraction integrals and angular-spectrum expression theory, showing a good consistency. The numerical calculation shows that the result calculated by the rigorous theory is much more precise than those calculated by the integral method in diffraction near field, and a highly consistency reaches by the three methods in diffraction far field. A further extension of the theory is discussed.

Effect of vectorial nature on spectral anomalies of ultrashort pulsed beams from a hard-edged aperture

Based on the vectorial Rayleigh–Sommerfeld diffraction integrals, the analytical expression for the spectral intensity of a vectorial nonparaxial ultrashort pulsed Gaussian beam diffracted at a rectangular hard-aperture is derived. The results of the far-field and paraxial cases can be regarded as special cases of the general expression. Effect of vectorial nature on spectral anomalies of ultrashort pulsed beams passing through a hard-edged aperture is analyzed in detail. It is shown that the spectral switch near the phase singularity, which is predicated under the condition of the scalar paraxial theory, disappears with the increasing effect of the vectorial and nonparaxial nature.

A Full-Wave 2.5D Volume Integral Equation Solver for 3D Millimeter-Wave Scattering by Large Inhomogeneous 2D Objects

A two-and-a-half dimensional full-wave forward solver to compute the three-dimensional (3D) electromagnetic field scattered by an infinitely long inhomogeneous (lossy) dielectric cylinder with arbitrary cross-sectional shape under a given 3D time-harmonic illumination is presented. The relevant set of linear equations, obtained after performing a spatial Fourier transform of the fields along the axial direction and applying a method of moments discretization to the two-dimensional contrast source volume integral equation, is solved iteratively with a stabilized biconjugate gradient fast Fourier transform method. In this way, objects with cross-sectional dimensions of several to many wavelengths can be handled in a very fast way. Furthermore, a vectorial 3D Gaussian beam illumination, usually employed in active millimeter-wave (mm-wave) imaging systems, is implemented using a complex source Gaussian beam formulation. The validity of the method is proved by comparison with analytic results and with the results from a full-wave 3D solver. Finally, the method is applied to a millimeter-wave imaging example, showing the scattering from an object hidden under clothing on a human body.

Rigorous vectorial Gaussian beam modeling of spectral dispersing performance of virtually imaged phased arrays

We have developed a generalized closed-form representation of the virtually imaged phased array (VIPA) output profile based on a 3D vectorial Gaussian beam formulation. Tracing the input vectorial elliptic Gaussian beam (GB), the output profile of the VIPA is represented as an array of virtual images. Furthermore, employing GB propagation and summing up all the virtual image profiles, the VIPA output profile has been calculated on a vertical plane at an arbitrary distance from the device. Finally, the derived closed-form representation is applied to the different sources (i.e., circular and elliptic GBs) and the results are compared with previous work.