We investigate in detail the focusing of a circularly polarized Laguerre-Gaussian laser beam ($\hbar \ell$ orbital angular momentum per photon; $σ=1/-1$ for left/right-handed polarization) by a high numerical aperture objective. The diffraction-limited focused beam has unexpected properties, resulting from a strong interplay between the angular spatial structure and the local polarization in the non-paraxial regime. In the region near the beam axis, and provided that $|\ell|\ge 2$ and $\ell$ and $σ$ have opposite signs, the energy locally counter-propagates and the projection of the electric field onto the focal plane counter-rotates with respect to the circular polarization of the incident beam. We explicitly show that the total angular momentum flux per unit power is conserved after focusing, as expected by rotational symmetry, but the spin and orbital separate contributions change.
It is shown that the momentum density of free electromagnetic field splits into two parts. One has no contribution to the net momentum due to the transversality condition. The other yields all the momentum. The angular momentum that is associated with the former part is spin, and the angular momentum that is associated with the latter part is orbital angular momentum. Expressions for the spin and orbital angular momentum are given in terms of the electric vector in reciprocal space. The spin and orbital angular momentum defined this way are used to investigate the angular momentum of nonparaxi…
We introduce and experimentally demonstrate a method for measuring at the same time the mean and the variance of the photonic orbital angular momentum (OAM) distribution in any paraxial optical field, without passing through the acquisition of its entire angular momentum spectrum. This method hence enables one to reduce the infinitely many output ports required in principle to perform a full OAM spectrum analysis to just two. The mean OAM, in turn, provides direct access to the average mechanical torque that the optical field in any light beam is expected to exert on matter, for example in the…
In this paper we will analyze the propagation of the angular spectrum of the electromagnetic field through a finite length uniaxial crystal. We solve the boundary conditions of the fields at an interface of an isotropic medium and an uniaxial anisotropic medium with the optical axis in an arbitrary direction and soon after we choose the optical axis in a plane formed by the direction of propagation (z-axis) and the x-axis . We show how the couplings occur between the components of the field and in what situations we can give a vector or scalar treatment for the propagation
We analytically and numerically investigate the propagation of ultrashort tightly focused laser pulses in vacuum, with particular emphasis on Hermite-Gaussian and Laguerre-Gaussian modes. We revisit the Lax series approach for forward-propagating linearly-polarized laser pulses, in order to obtain Maxwell-consistent and symmetry-preserving analytical solutions for the propagation of all field components beyond the paraxial approximation in four-dimensional geometry (space and time). We demonstrate that our solution conserves the energy, which is set by the paraxial-level term of the series. Th…
We derive focused laser pulse solutions to the electromagnetic wave equation in vacuum. After reproducing beam and pulse expressions for the well-known paraxial Gaussian and axicon cases, we apply the method to analyse a laser beam with Lorentzian transverse momentum distribution. Whilst a paraxial approach has some success close to the focal axis and within a Rayleigh range of the focal spot, we find that it incorrectly predicts the transverse fall-off typical of a Lorentzian. Our vector-potential approach is particularly relevant to calculation of quantum electrodynamical processes in weak l…
We investigate the properties of a single-view fluorescence microscope in a 4$f$ geometry when imaging fluorescent dipoles without using the monopole or scalar approximations. We show that this imaging system has a spatio-angular band limit, and we exploit the band limit to perform efficient simulations. Notably, we show that information about the out-of-plane orientation of ensembles of in-focus fluorophores is recorded by paraxial fluorescence microscopes. Additionally, we show that the monopole approximation may cause biased estimates of fluorophore concentrations, but these biases are smal…
Fundamental and applied concepts concerning the ability of light beams to carry a certain mechanical angular momentum with respect to the propagation axis are reviewed and discussed. Following issues are included: Historical reference; Angular momentum of a paraxial beam and its constituents; Spin angular momentum and paradoxes associated with it; Orbital angular momentum; Circularly-spiral beams: examples and methods of generation; Orbital angular momentum and the intensity moments; Symmetry breakdown and decomposition of the orbital angular momentum; Mechanical models of the vortex light bea…
One of the reasons for the tremendous success of a plane-wave approximation in particle physics is that the non-paraxial corrections to such observables as energy, magnetic moment, scattering cross section, and so on are attenuated as $λ_c^2/σ_{\perp}^2 \ll 1$ where $σ_{\perp}$ is a beam width and $λ_c = \hbar/mc$ is a Compton wavelength. This amounts to less than $10^{-14}$ for modern electron accelerators and less than $10^{-6}$ for electron microscopes. Here we show that these corrections are $|\ell|$ times enhanced for vortex particles with high orbital angular momenta $|\ell|\hbar$, which…
We study paraxial beam propagation parallel to the screw axis of a dislocated amorphous medium that is optically weakly inhomogeneous and isotropic. The effect of the screw dislocation on the beam's orbital angular momentum is shown to change the optical vortex strength, rendering vortex annihilation or generation possible. Furthermore, the dislocation is shown to induce a weak \textit{biaxial} anisotropy in the medium due to the elasto-optic effect, which changes the beam's spin angular momentum as well as causing precession of the polarization. We derive the equations of motion of the beam a…
In light beams with circular or elliptic polarization, the transverse energy flow consists of the "spin" and "orbital" parts. Both of them can induce the orbital motion of microparticles suspended within the field of a light beam, and this should be taken into account in experiments on the spin-to-orbital angular momentum conversion. The character of the spin, orbital and total transverse energy flows in circular Laguerre-Gaussian beams is studied analytically; graphical representations of the flows in the beam cross section (flow maps) are calculated and analyzed. The spin circulatory flow ca…
We construct solutions of the paraxial and Helmholtz equations which are polynomials in their spatial variables. These are derived explicitly using the angular spectrum method and generating functions. Paraxial polynomials have the form of homogeneous Hermite and Laguerre polynomials in Cartesian and cylindrical coordinates respectively, analogous to heat polynomials for the diffusion equation. Nonparaxial polynomials are found by substituting monomials in the propagation variable $z$ with reverse Bessel polynomials. These explicit analytic forms give insight into the mathematical structure of…
In the study of laser-driven electron acceleration, it has become customary to work within the framework of paraxial wave optics. Using an exact solution to the Helmholtz equation as well as its paraxial counterpart, we perform numerical simulations of electron acceleration with a high-power TM01 beam. For beam waist sizes at which the paraxial approximation was previously recognized valid, we highlight significant differences in the angular divergence and energy distribution of the electron bunches produced by the exact and the paraxial solutions. Our results demonstrate that extra care has t…
Longitudinal electromagnetic fields generally become comparable with the usually dominant transverse components in strongly-focussed, non-paraxial beams. For optical vortex modes it is highlighted here how their angular momentum properties produce longitudinal fields that in general must be accounted for, even within the paraxial regime. First-order longitudinal components of quantized Laguerre-Gaussian modes are derived and numerically studied with respect to the paraxial parameter, highlighting light-matter and spin-orbit interactions that stem from longitudinal fields of weakly-focussed, pa…
Spin angular momentum, an elementary dynamical property of classical electromagnetic fields, plays an important role in spin-orbit and light-matter interactions, especially in near-field optics. The research on optical spins has led to the discovery of phenomena such as optical spin-momentum locking and photonic topological quasiparticles, as well as applications in high-precision detection and nanometrology. Here, we investigate spin-momentum relations in paraxial optical systems and show that the optical spin angular momentum contains transverse and longitudinal spin components simultaneousl…
It is well-known that electric spin angular momentum and electric orbital angular momentum are conserved under paraxial propagation of travelling waves in free-space. Here we study the electric and magnetic angular momentum in counter-propagating waves and show both theoretically and experimentally that neither component alone is conserved except in special cases. We attribute this non-conservation to spin-spin and orbit-orbit coupling between the electric and magnetic fields. This work generalises previous findings based on travelling waves, explains the apparent spin-orbit coupling in counte…
The prevailing notion is that the orbital angular momentum (OAM) of an optical vortex can only be transferred to the internal degrees of freedom (i.e. electronic motion) of materials through electric quadrupole and higher-order multipole interactions. Here it is highlighted how this is an artefact of the paraxial approximation and that optical OAM can be transferred to electronic motion via dipole transitions under the correct experimental conditions where non-paraxial, longitudinal fields must be accounted for.
We present a method for the realization of radially and azimuthally polarized nonparaxial Bessel beams in a rigorous but simple manner. This result is achieved by using the concept of Hertz vector potential to generate exact vector solutions of Maxwell's equations from scalar Bessel beams. The scalar part of the Hertz potential is built by analogy with the paraxial case as a linear combination of Bessel beams carrying a unit of orbital angular momentum. In this way we are able to obtain spatial and polarization patterns analogous to the ones exhibited by the standard cylindrically polarized pa…
This work is the second part of an investigation aiming at the study of optical wave equations from a field-theoretic point of view. Here, we study classical and quantum aspects of scalar fields satisfying the paraxial wave equation. First, we determine conservation laws for energy, linear and angular momentum of paraxial fields in a classical context. Then, we proceed with the quantization of the field. Finally, we compare our result with the traditional ones.
This paper is devoted to study the propagation of light beams carrying orbital angular momentum in optically anisotropic media. We first review some properties of homogeneous anisotropic media, and describe how the paraxial formalism is modified in order to proceed with a new approach dealing with a general setting of paraxial propagation along uniaxial inhomogeneous media. This approach is suitable for describing the space-variant-optical-axis phase plates.
We analyze the paraxial beam transformation upon reflection and refraction at a plane boundary. In contrast to the usual approach dealing with the beam angular spectrum, we apply the continuity conditions to explicit spatial representations of the electric and magnetic fields on both sides of the boundary. It is shown that the polarization-dependent distortions of the beam trajectory (in particular, the "longitudinal" Goos-Hänchen shift and the "lateral" Imbert-Fedorov shift of the beam center of gravity) are directly connected to the incident beam longitudinal component and appear due to its …
We present a set of paraxial light beams with cylindrical symmetry, smooth and localized transversal profile carrying finite power, that develop intensity singularities when they are focused in a linear medium, such as vacuum. They include beams with orbital angular momentum and with radial polarization, in which case they develop punctual phase and polarization singularities surrounded by infinitely bright rings, along with singular longitudinal fields. In practice, these effects are manifested in focal intensities and spot sizes, vortex bright ring intensities and radii, and strengths of the…
For paraxial and non-paraxial light, numerous measures of electromagnetic attribute are expressible in terms of photon annihilation and creation. Accordingly, energy, angular momentum and chirality measures acquire a consistent interpretation. From the photonic nature of light, it emerges that an infinite hierarchy of spin-type measures depend on a difference in number operators for modes of opposing helicity, pure circular polarization giving maximal values. Measures of orbital angular momentum are determined by a sum of corresponding number operators. By analysis of electric and magnetic fea…
The analytical vectorial structure of non-paraxial four-petal Gaussian beams(FPGBs) in the far field has been studied based on vector angular spectrum method and stationary phase method. In terms of analytical electromagnetic representations of the TE and TM terms, the energy flux distributions of the TE term, the TM term, and the whole beam are derived in the far field, respectively. According to our investigation, the FPGBs can evolve into a number of small petals in the far field. The number of the petals is determined by the order of input beam. The physical pictures of the FPGBs are well …
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 para…
A new set of vector solutions to Maxwell's equations based on solutions to the wave equation in spheroidal coordinates allows laser beams to be described beyond the paraxial approximation. Using these solutions allows us to calculate the complete first-order corrections in the short-wavelength limit to eigenmodes and eigenfrequencies in a Fabry-Perot resonator with perfectly conducting mirrors. Experimentally relevant effects are predicted. Modes which are degenerate according to the paraxial approximation are split according to their total angular momentum. This includes a splitting due to co…
We investigate how the orbital angular momentum (OAM) of a paraxial light beam is affected upon reflection at a planar interface. Theoretically, the unavoidable angular spread of the (paraxial) beam leads to OAM sidebands which are found to be already significant for modest beam spread (0.05). In analogy to the polarization Fresnel coefficients we develop a theory based upon spatial Fresnel coefficients; this allows straightforward prediction of the strength of the sidebands. We confirm this by experiment.
Spatial distribution of the longitudinal field component of a circularly polarized optical beam depends on the polarization handedness, which causes the lateral shift of the beam "center of gravity" when its polarization toggles. We present the generalized theory of this effect, which demonstrates its relation with the angular irradiance moments of the beam. The theory is applicable to arbitrary paraxial beams and shows that the lateral shift is the same for all cross sections of the beam.
Light beams can carry a discrete, in principle unbounded amount of angular momentum. Examples of such beams, the Laguerre-Gauss modes, are frequently expressed as solutions of the paraxial wave equation. There, they are eigenstates of the orbital angular momentum (OAM) operator. The paraxial solutions predict that beams with large OAM could be used to resolve arbitrarily small distances - a dubious situation. Here we show how to solve that situation by calculating the properties of beams free from the paraxial approximation. We find the surprising result that indeed one can resolve smaller dis…
A plane-wave approximation in particle physics implies that a width of a massive wave packet $σ_{\perp}$ is much larger than its Compton wavelength $λ_c = \hbar/mc$. For Gaussian beams or for packets with the non-singular phases (say, the Airy beams), corrections to this approximation are attenuated as $λ_c^2/σ_{\perp}^2 \ll 1$ and usually negligible. Here we show that this situation drastically changes for particles with the phase vortices associated with an orbital angular momentum $\ell\hbar$. For highly twisted beams with $|\ell| \gg 1$, the non-paraxial corrections get $|\ell|$ times enha…
We show that an analog of the physics at the Planck scale can be found in the propagation of tightly focused laser beams. Various equations that occur in generalized quantum mechanics are formally identical to those describing the nonlinear nonlocal propagation of nonparaxial laser beams. The analysis includes a generalized uncertainty principle and shows that the nonlinear focusing of a light beam with dimensions comparable to the wavelength corresponds to the spontaneous excitation of the so-called maximally localized states. The approach, driven by the ideas of the quantum gravity physics, …
A novel optical device is designed and fabricated in order to overcome the limits of the traditional sorter based on log-pol optical transformation for the demultiplexing of optical beams carrying orbital angular momentum (OAM). The proposed configuration simplifies the alignment procedure and significantly improves the compactness and miniaturization level of the optical architecture. Since the device requires to operate beyond the paraxial approximation, a rigorous formulation of transformation optics in the non-paraxial regime has been developed and applied. The sample has been fabricated a…
There is significant interest in tailoring wave packets that transversely accelerate during propagation. The first realisation was in the optical domain, the Airy beam. Valid in the paraxial approximation, such beams were shown to transversely accelerate while following a parabolic caustic, even if the centroid itself travelled along a rectilinear path. Later, non-paraxial optical fields in the form of Weber and Mathieu beams were demonstrated, as well as non-optical wave packets with electrons. Such fields have found a plethora of applications from particle manipulation, spatial-temporal beam…
In this paper, we present a systematic study of beam shifts and angular momenta of paraxial vortex beams at air-left-handed material (LHM) interfaces. It is shown that, compared to their counterparts at air-right-handed material (RHM) interfaces, the spatial Goos-Hänchen (GH) and Imbert-Fedorov (IF) shifts remains the same, while the angular GH and IF shifts are reversed, at air-LHM interfaces. The spatial and angular shifts of paraxial vortex beams have their respective origins in transverse angular momenta and transverse linear momenta. The spatial GH and IF shifts remain unreversed as a res…
Orbital Angular Momentum (OAM) waves were first recognized as those specific vortex solutions of the paraxial Helmholtz equation for which the orbital contribution to the total angular momentum of the beam yields an integer multiple of $\hbar$ along the propagation direction. However, this class of solutions can be generalized to include more sophisticated vector vortex waves with coupled polarization and spatial complexity, that are eigenfunctions of the third component of the angular momentum operator. In this work, a rigorous framework is proposed for the analysis of all the possible famili…
We have proposed a unified framework towards the dynamics of optical and electron vortex beams from the perspective of geometric phase and the associated Hall effects. The unification is attributed to the notion that the spin degrees of freedom of a relativistic particle, either massive or massless, is associated with a vortex line. %The formulation in cylindrical coordinates leads to a local vortex structure related to orbital angular momentum(OAM). Based on cylindrical corodinate formulation, which leads to a local vortex structure related to orbital angular momentum (OAM), it can be shown t…
The angular spectrum of a vectorial laser beam is expressed in terms of an intrinsic coordinate system instead of the usual Cartesian laboratory coordinates. This switch leads to simple, elegant and new expressions such as for the angular spectrum of the Hertz vector corresponding to the electromagnetic fields. As an application of this approach, we consider axially symmetric vector beams, showing non-diffracting properties of these beams, without invoking the paraxial approximation. Further, we indicate the relevance of the method for analyzing nonparaxial resonators, such as microresonators
Our goal is to study optical signatures of quantum vacuum nonlinearities in strong macroscopic electromagnetic fields provided by high-intensity laser beams. The vacuum emission scheme is perfectly suited for this task as it naturally distinguishes between incident laser beams, described as classical electromagnetic fields driving the effect, and emitted signal photons encoding the signature of quantum vacuum nonlinearity. Using the Heisenberg-Euler effective action, our approach allows for a reliable study of photonic signatures of QED vacuum nonlinearity in the parameter regimes accessible b…
Spatially accelerating beams that are solutions to the Maxwell equations may propagate along incomplete circular trajectories, after which diffraction broadening takes over and the beams spread out. Taking these truncated Bessel wave fields to the paraxial limit, some authors sustained that it is recovered the known Airy beams (AiBs). Based on the angular spectrum representation of optical fields, we demonstrated that the paraxial approximation rigorously leads to off-axis focused beams instead of finite-energy AiBs. The latter will arise under the umbrella of a nonparaxial approach following …
We derive expressions for the energy, linear momentum, and angular momentum content of circularly polarized Laguerre-Gaussian wave-packets propagating in free space. The vectorial nature of the electromagnetic field is taken into account, and the various consequences of paraxial approximation, which is typically invoked in theoretical treatments of the Laguerre-Gaussian beams, are examined.
The orbital angular momentum Hall effect and spin Hall effect of electron vortex beams (EVB) have been studied for the EVBs interacting with laser field. In the scenario of paraxial beam, the cumulative effect of the orbit-orbit interaction of EVBs and laser fields drives the orbital Hall effect, which in turn produces a shift of the center of the beam from that of the field-free case towards the polarization axis of photons. Besides, for non-paraxial beams one can also perceive a similar shift of the center of the beam owing to spin Hall effect involving spin-orbit interaction. Our analysis s…
We demonstrate that the orbital angular momentum flux density of a paraxial light beam can change on propagation in free space. These changes are entirely due to the spatial coherence state of the source, and the effect is analogous to correlation-induced changes in the intensity, spectrum and polarization of a beam. The use of the source coherence state to control the width, shape, and transverse shifts of the OAM flux density is demonstrated with numerical examples.
We explore experimentally if the light's orbital angular momentum (OAM) interacts with chiral nematic polymer films. Specifically, we measure the circular dichroism of such a material using light beams with different OAM. We investigate the case of strongly focussed, non-paraxial light beams, where the spatial and polarization degrees of freedom are coupled. Within the experimental accuracy, we cannot find any influence of the OAM on the circular dichroism of the cholesteric polymer.
