Paraxial Gaussian Beam Research Articles

Talbot-like pattern evolution in complex structured light from a unitary transformation.

Astigmatic unitary transformations allow for the adiabatic connections of all feasible states of paraxial Gaussian beams on the same modal sphere, i.e., Hermite-Laguerre-Gaussian (HLG) modes. Here, we present a comprehensive investigation into the unitary modal evolution of complex structured Gaussian beams, comprised of HLG modes from disparate modal spheres, via astigmatic transformation. The non-synchronized higher-order geometric phases in cyclic transformations originate a Talbot-effect-like modal evolution in the superposition state of these HLG modes, resulting in pattern variations and revivals in transformations with specific geodesic loops. Using Ince-Gaussian modes as an illustrative example, we systematically analyze and experimentally corroborate the beamforming mechanism behind the pattern evolution. Our results outline a generic modal conversion theory of structured Gaussian beams via astigmatic unitary transformation, offering a new approach for shaping spatial modal structure. These findings may inspire a wide variety of applications based on structured light.

Photonic hook propagation from eccentric microcylinder

The Photonic hook (PH) is an intricately curved photonic nanojet (PNJ) or a highly intense electromagnetic beam featuring a subwavelength waist, whose principal hallmark lies in its capacity to bend light at the nanoscale. According to existing literature, the origin of PH can be attributed to symmetry breaking, whereas symmetrical microstructures predominantly contribute to PNJ formation. This study presents the novel revelation of PH emergence from an isolated eccentric core–shell dielectric microcylinder, achieved through the illumination of a paraxial Gaussian beam (PGB). The eccentrically structured core–shell microscale geometry introduces an additional degree of freedom, influencing PH formation and directly shaping its characteristic parameters. Much like PNJ, the propagation of PH depends on different parameters such as core and shell refractive indices of the micro-structures, microstructure geometry, incident light type, and direction of propagation. A fascinating outcome from our numerical simulations is the switchable occurrence of PNJ and PH from an eccentric core–shell microcylinder by a simple adjustment of eccentricity, either parallel or perpendicular to the PGB’s propagation direction. This computational investigation emphasizes the impact of eccentricity and the incident wave’s beam waist, maintaining a consistent refractive index contrast between the core and shell. The outcomes are interpreted in terms of key parameters governing PH generation characteristics, encompassing FWHM, maximum electric field enhancement, and focal plane. Notably, we have observed the coexistence of whispering gallery modes (WGM) and PH within this system and these modes exhibit high sensitivity to the excitation wavelength. The potential applications of PH are believed to be far-reaching, including areas like optical trapping, sensing, and functioning as a versatile focusing element. This study contributes to the fundamental understanding of PH and illuminates its potential as a robust tool across diverse optical applications.

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.

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.

On acoustic scattering of beams

When solving scattering problems using T-matrix methods, it is assumed that the incident wavefield can be expanded about a chosen origin as an infinite series of regular spherical wavefunctions. We develop such expansions for a variety of so-called quasi-gaussian beams found in the literature. If the incident wavefield does not solve the governing Helmholtz equation, T-matrix methods are not available, in general. This is the situation if one wants to scatter a paraxial gaussian beam of some kind. We also discuss such beams and their approximations.

Complex source point theory of paraxial and nonparaxial elliptical Gaussian beams.

Using the operator transformation technology, we extend the circular Gaussian beam based virtual (complex) sources method to investigate the paraxial and nonparaxial propagation properties of the elliptical Gaussian beams (EGBs) with planar or cylindrical wavefronts travelling in free space. The paraxial approximation analysis reveals the self-reappearance and self-focusing propagation features for the EGBs with cylindrical wavefront under proper parameter conditions. We further introduce the nonparaxial theory to derive the analytical expressions for the field distribution of an EGB in free space, and confirm that these intriguing propagation features can still be observed with added nonparaxial correction. Comparing with the paraxial approximation results, it is worth noting that there is a clear deviation of the on-axial intensity and phase distributions near the self-focusing position on the basis of nonparaxial correction solution. Our results reveal that the anisotropic diffraction of light propagating through homogeneous medium or free space is possible. The approach in this work can easily be generalized to other beam models with elliptical geometry, which allows us to correctly predict some important information about their near field propagation characteristics for various applications.

Coherence Poincaré sphere of partially polarized optical beams

A Poincar\'e sphere representation of electromagnetic spectral two-point spatial coherence was recently introduced for fully polarized light beams. In this work, we employ the singular-value decomposition of the cross-spectral density matrix and establish a rigorous interpretation for the formalism in the context of partial polarization. At a single point, the construction reduces to the traditional polarization Poincar\'e sphere, which therefore can be regarded as a limiting case of the coherence sphere. The formalism is illustrated with paraxial blackbody radiation and Gaussian Schell-model beams. We expect our results will find use in understanding the coherence-polarization state of optical beams in situations dealing with random light.

X-ray vacuum diffraction at finite spatiotemporal offset

We study the nonlinear QED signature of x-ray vacuum diffraction in the head-on collision of optical high-intensity and x-ray free-electron laser pulses at finite spatiotemporal offsets between the laser foci. The high-intensity laser driven scattering of signal photons outside the forward cone of the x-ray probe constitutes a prospective experimental signature of quantum vacuum nonlinearity. Resorting to a simplified phenomenological ad hoc model, it was recently argued that the angular distribution of the signal in the far-field is sensitive to the wavefront curvature of the probe beam in the interaction region with the high-intensity pump. In this work, we model both the pump and probe fields as pulsed paraxial Gaussian beams and reanalyze this effect from first principles. We focus on vacuum diffraction both as an individual signature of quantum vacuum nonlinearity and as a potential means to improve the signal-to-background separation in vacuum birefringence experiments.

Photonic spin Hall effect by anisotropy-induced polarization gradient in momentum space.

We demonstrate theoretically and experimentally a novel photonic spin Hall effect (PSHE), to the best of our knowledge, at an interface between air and uniaxial crystal, whose optical axis is within the interface plane. Owing to the anisotropy of the crystal, partial cross polarization conversion occurs. For a horizontally polarized paraxial Gaussian beam incidence, a linear polarization gradient forms along the in-plane wavevector in the reflected beam, allowing us to achieve spin separation in real space. The spin separation of the reflected beam can be tuned by rotating the optical axis of the crystal. A maximum spin-dependent displacement up to 0.45 times the incident beam waist is obtained at Brewster incidence. This novel anisotropy-induced PSHE deepens the understanding of spin-orbit interaction and provides a new way for control of spin photons.

Influence of optical forces induced by paraxial vortex Gaussian beams on the formation of a microrelief on carbazole-containing azopolymer films.

The paper discusses photoinduced microrelief formation in a film of an azopolymer. A theoretical study of the effect of laser beam polarization on the balance of optical forces acting under the direct action of paraxial Gaussian beams on the irradiated substance was made. We show that taking into account the gradient and scattering components of the force does not allow us to correctly describe the shape of the microasperities obtained on a carbazole-containing azopolymer. An approximation function is presented that describes the dependence of the microasperities' shapes on the non-gradient component of the optical force of laser radiation in the absence and presence of a vortex phase. A comparative analysis of the approximation results and experimentally obtained microreliefs was carried out.

Gouy and spatial-curvature-induced phase shifts of light in two-dimensional curved space

Gouy phase is the axial phase anomaly of converging light waves discovered over one century ago, and is so far widely studied in various systems. In this work, we have theoretically calculated Gouy phase of light beams in both paraxial and nonparaxial regime on two-dimensional curved surface by generalizing angular spectrum method. We find that curvature of surface will also introduce an extra phase shift, which is named as spatial curvature-induced (SCI) phase. The behaviors of both phase shifts are illustrated on two typical surfaces of revolution, circular truncated cone and spherical surface. Gouy phase evolves slower on surface with greater spatial curvature on circular truncated cone, which is however opposite on spherical surface, while SCI phase evolves faster with curvature on both surfaces. On circular truncated cone, both phase shifts approach to a limit value along propagation, which does not happen on spherical surface due to the existence of singularity on the pole. An interpretation is presented to explain this peculiar phenomenon. Finally we also provide the analytical expression of paraxial Gaussian beam on general SORs. By comparing the result with the exact method we find the analytical expression is valid under the approximation that beam waist and scale of surface are beyond order of wavelength. We expect this work will enhance the comprehension about the behavior of electromagnetic wave in curved space, and further contribute to the study of general relativity phenomena in laboratory.

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.

Investigation of paraxial and nonparaxial self-focusing of Gaussian beam in chalcogenide glass medium using NLSE

Variational method is used for the investigation of paraxial and nonparaxial self-focusing of Gaussian beam in bulk chalcogenide glass (Kerr media). Stationary self-focusing for paraxial Gaussian beam with distinct singularity and focusing-defocusing cycles for nonparaxial propagation of Gaussian beam is observed. Emphasis is laid on the study of variation in beam width and intensity of the beam with propagation distance.

X-ray photon scattering at a focused high-intensity laser pulse

We study x-ray photon scattering in the head-on collision of an XFEL pulse and a focused high-intensity laser pulse, described as paraxial Laguerre-Gaussian beam of arbitrary mode composition. For adequately chosen relative orientations of the polarization vectors of the colliding laser fields, this gives rise to a vacuum birefringence effect manifesting itself in polarization flipped signal photons. Throughout this article the XFEL is assumed to be mildly focused to a waist larger than that of the high-intensity laser beam. As previously demonstrated for the special case of a fundamental paraxial Gaussian beam, this scenario is generically accompanied by a scattering phenomenon of x-ray energy signal photons outside the forward cone of the XFEL beam, potentially assisting the detection of the effect in experiment. Here, we study the fate of the x-ray scattering signal under exemplary deformations of the transverse focus profile of the high-intensity pump.

Oblique-incidence, arbitrary-profile wave injection for electromagnetic simulations.

In an electromagnetic code, a wave can be injected in the simulation domain by prescribing an oscillating field profile at the domain boundary. The process is straightforward when the field profile has a known analytical expression (typically, paraxial Gaussian beams). However, if the field profile is known at some other plane, but not at the boundary (typically, nonparaxial beams), some preprocessing is needed to calculate the field profile after propagation back to the boundary. We present a parallel numerical technique for this propagation between an arbitrary tilted plane and a given boundary of the simulation domain, implemented in the Maxwell-Vlasov particle-in-cell code Smilei.

Photonic tractor beams: a review

Usually, an unfocused light beam, such as a paraxial Gaussian beam, can exert a force on an object along the direction of light propagation, which is known as light pressure. Recently, however, it was found that an unfocused light beam can also exert an optical pulling force (OPF) on an object toward the source direction; the beam is accordingly named an optical tractor beam. In recent years, this intriguing force has attracted much attention and a huge amount of progress has been made both in theory and experiment. We briefly review recent progress achieved on this topic. We classify the mechanisms to achieve an OPF into four different kinds according to the dominant factors. The first one is tailoring the incident beam. The second one is engineering the object’s optical parameters. The third one is designing the structured material background, in which the light–matter interaction occurs, and the fourth one is utilizing the indirect photophoretic force, which is related to the thermal effect of light absorption. For all the methods, we analyze the basic principles and review the recent achievements. Finally, we also give a brief conclusion and an outlook on the future development of this field.

Changes of phase structure of a paraxial beam due to spin-orbit coupling

We analytically derive an expression for the optical field describing the spin Hall effect of a paraxial Gaussian beam in a lenslike inhomogeneous medium using the matrix optics method. Interestingly, we found changes in the phase structure and rotation of the polarization of the Gaussian beam induced by spin-orbit coupling, in addition to a polarization-dependent transverse deflection of the entire beam. This rotation of the polarization of the beam is not the result of the Berry phase. In addition, we show that the intrinsic optical angular momentum density of the beam is changed due to the spin-orbit coupling, and we also know that polarization-dependent rotation of the beam occurs when a circularly polarized nonaxisymmetric beam propagates in a lenslike inhomogeneous medium. Our results provide further potential applications of the spin Hall effect of light in optical signal processing.

A Gaussian beam method for ultrasonic non-destructive evaluation modeling

The propagation of high-frequency ultrasonic body waves can be efficiently estimated with a semi-analytic Dynamic Ray Tracing approach using paraxial approximation. Although this asymptotic field estimation avoids the computational cost of numerical methods, it may encounter several limitations in reproducing identified highly interferential features. Nevertheless, some can be managed by allowing paraxial quantities to be complex-valued. This gives rise to localized solutions, known as paraxial Gaussian beams. Whereas their propagation and transmission/reflection laws are well-defined, the fact remains that the adopted complexification introduces additional initial conditions. While their choice is usually performed according to strategies specifically tailored to limited applications, a Gabor frame method has been implemented to indiscriminately initialize a reasonable number of paraxial Gaussian beams. Since this method can be applied for an usefully wide range of ultrasonic transducers, the typical case of the time-harmonic piston radiator is investigated. Compared to the commonly used Multi-Gaussian Beam model [1], a better agreement is obtained throughout the radiated field between the results of numerical integration (or analytical on-axis solution) and the resulting Gaussian beam superposition. Sparsity of the proposed solution is also discussed.

Gaussian beams with multiple polarization singularities

In recent years, alongside light beams with phase singularities (so-called optical vortices), polarization vortices (including beams with radial and azimuthal polarization) have also been studied. In this work, we analyze paraxial Gaussian beams that have an array of arbitrarily located polarization singularities. A closed analytical expression to describe their complex amplitude is derived. A particular case of singularities localized at the vertices of a regular polygonal is discussed. If the beam has one or two singularities, these points will have radial polarization. If, however, there are four singularities, two of them will be characterized by azimuthal polarization. As the beam propagates, the polarization singularities found in the initial plane are shown to reoccur only in a discrete set of planes, unlike phase singularities, which occur at any transverse plane. In the case of two singularities, radial polarization in the initial plane is found to convert into azimuthal polarization in the far-field. The research findings may find uses in optical data transmission.