Nonparaxial Regime Research Articles

Effect of evanescent wave on vector vortex beams on higher-order Poincaré sphere

Abstract Based on the vector angular spectrum representation, the contributions of the propagating and evanescent waves to the vector vortex Laguerre–Gaussian beams on the higher-order Poincaré sphere are investigated in the nonparaxial regime. The evolution properties of the propagating and evanescent waves are closely related to the topological charge, the radial order, the ellipticity angle, and the waist width. The influences of the evanescent waves on the intensity, the spin and orbital angular momenta of the vector vortex beams are discussed in detail.

Effects of aberrations on 3D optical topologies

Optical knots and links are nontrivial three-dimensional topologies consisting of trajectories of phase or polarisation singularities. They are theoretically predicted and experimentally observed in paraxial and nonparaxial regimes and in random and speckle fields. The topological nature of optical knots suggests that environmental disturbances should not alter their topology, hence becoming a resilient vector of information. However, the robustness of optical knots under typical disturbances encountered in optical experiments has not been investigated. Here, we provide the experimental analysis of the effects of optical phase aberrations on optical knots and links. We demonstrate that Hopf links, trefoil and cinquefoil knots are robust to misalignment and phase aberrations. The observed knots are obliterated for high aberration strengths and defining apertures close to the characteristic optical beam size. Our observations indicate these photonic topological structures as viable alternatives for both classical and quantum information processing noisy channels, where optical modes are not applicable.

Simple algorithm for the design of accelerating Bessel-like beams with adjustable features along their propagation.

We present an extremely simple method for designing self-accelerating non-diffracting beams having arbitrary trajectories while their intensity, width and orbital angular momentum are modulated in a prescribed way along their propagation. Different beams constructed with this method are demonstrated experimentally in the paraxial regime and numerically in the non-paraxial regime.

Remarkable robustness of self-imaging distance for a misaligned paraxial Gaussian input and variation in the nonparaxial regime

The variation of focusing distance in a parabolic graded-index slab with the width of a one-dimensional Gaussian input fed at its waist, both axially and misaligned, into the waveguide is studied in paraxial and beyond-paraxial regimes. We obtain analytical expressions, scalable in terms of material parameters, for input coupling coefficients for such a Gaussian input. The focusing distance shows remarkable stability for an axially fed input for beam width exceeding the fundamental mode width of the waveguide. There is a smooth variation for the other regime of beam width. In the paraxial domain, we identify a unique beam width of ∼0.76 times the fundamental mode width for which the self-imaging distance is nearly independent of misalignment. The stability, a well-known sharp shift of the focusing point for an axially fed beam of width around that of the fundamental mode, and remarkable stability of self-imaging distance with misalignment at the unique beam width should be useful for efficiency enhancement of device interconnects, sensing, and lensing applications.

Geometric spin Hall effect of a laser beam beyond the paraxial approximation

An intriguing spin-orbital angular momentum interaction, called the geometric spin Hall effect of light (GSHEL), has aroused much interest and manifests itself as a transverse shift. The GSHEL of the classical optical field is independent of the beam size. In this paper the GSHELs of the vortex vector continuous wave (VVCW) and vortex ultrashort pulsed beam (VUPB) are systematically analyzed beyond the paraxial approximation. The analytical expression of the transverse shift of the VVCW is obtained. Evidently, the magnitude of the transverse shift depends on the ratio of the beam waist to the wavelength. The transverse shift can be effectively enlarged by narrowing the beam waist even to the order of more than $10\phantom{\rule{4pt}{0ex}}\ensuremath{\mu}\text{m}$. This intriguing phenomenon is caused by the stronger spin-orbital interaction induced energy flow in the nonparaxial regime. Furthermore, the GSHEL of the VUPB is also explored. The pulse duration time strongly affects the distribution of the phase. The singularities play a crucial role in altering the orientation of the electric field. As a result, the VUPBs with different pulse duration time have different magnitudes of the GSHEL. These results may find significant applications in the investigation of the particle manipulation, microscopic imaging, and ultrashort optics.

One more time on the helicity decomposition of spin and orbital optical currents

The helicity representation of the linear momentum density of a light wave is well understood for monochromatic optical fields in both paraxial and non-paraxial regimes of propagation. In this note we generalize such representation to nonmonochromatic optical fields. We find that, differently from the monochromatic case, the linear momentum density, aka the Poynting vector divided by c 2, does not separate into the sum of right-handed and left-handed terms, even when the so-called electric–magnetic democracy in enforced by averaging the electric and magnetic contributions. However, for quasimonochromatic light, such a separation is approximately restored after time-averaging. This paper is dedicated to Sir Michael Berry on the occasion of his 80th birthday.

Passage of a vortex electron over an inclined grating

We study Smith-Purcell radiation from a conducting grating generated by an inclined passage of a shaped electron wave packet with an electric quadrupole moment in the nonparaxial regime. Spreading of an asymmetric wave packet induces quadrupole corrections to the radiation field. Although the nonparaxial corrections stay small, they are dynamically enhanced during the interaction of the electron with the grating whose length exceeds the Rayleigh length of the packet. To simplify the possible experimental setup where such effects could be measured, we study the dependence of these effects on the inclination angle, i.e., the angle between the mean velocity of the packet and the surface of the grating. There is a minimal angle such that the multipole expansion always stays valid at the grating surface. In such a regime, the quadrupole contribution to the Smith-Purcell radiation can become the leading one, which represents a quantum effect impossible for classical pointlike electrons. Thus, the impact of the wave-packet shape (vortex structure or nonspherical shape) can be observed experimentally by comparing the radiation for different orientations of the grating in the single-electron regime.

Stability of nonparaxial gap-soliton bullets in waveguide gratings

We investigate the formation and propagation of gap-soliton bullets in nonlinear periodic waveguides at frequencies close to the gap for Bragg reflection beyond the paraxial approximation. Using a multiple-scales analysis, we derive a two-dimensional (2D) nonlinear Schrödinger equation with higher-order correction terms that consider the nonparaxial regimes in the slowly-varying envelope approximation. In addition, a fully numerical simulation of the newly derived model equation demonstrates that the mutual balancing between Kerr, dimensionality, higher-order dispersions and nonparaxiality allows shape-preserving propagation of gap-soliton bullets in a grating waveguide.

Gain-assisted optical tweezing of plasmonic and large refractive index microspheres

We have theoretically investigated optical tweezing of gain-functionalized microspheres using a highly focused single beam in the nonparaxial regime. We employ the Mie–Debye theory of optical tweezers to calculate the optical force acting on homogeneous and core–shell Mie microspheres with gain. We demonstrate that the optical gain plays a crucial role in optical manipulation, especially to optimize the restoring force and thus allowing for trapping of large refractive index and plasmonic particles. Indeed, we demonstrate that one can trap such particles, which is usually not possible in the case of passive media, by functionalizing them with an optical gain material. We show that by varying the value of the gain, which can be realized by changing the pump power, one can not only achieve trapping but also manipulate the equilibrium position of the tweezed particle. Altogether, our findings open new avenues for gain-assisted optomechanics, where gain functionalized systems could facilitate optical trapping and manipulation of plasmonic nanoparticles in particular, with potential applications in self-assembling of nanoparticle suspensions and on a chip.

Polychromatic electric field knots

The polarization of a monochromatic optical beam lies in a plane, and in general, is described by an ellipse, known as the polarization ellipse. The polarization ellipse in the tight focusing (non-paraxial) regime forms non-trivial three-dimensional topologies, such as M\"obius and ribbon strips, as well as knots. The latter is formed when the dynamics of specific polarization states, e.g., circular polarization states, are studied upon propagation. However, there is an alternative method to generate optical knots: the electric field's tip can be made to evolve along a knot trajectory in time locally. We propose an intuitive technique to generate and engineer the path traced by the electric field vector of polychromatic beams to form different knots. In particular, we show examples of how tightly focused beams with at least three frequency components and different spatial modes can cause the tip of the electric field vector to follow, locally, a knotted trajectory. Furthermore, we characterize the generated knots and explore different knot densities upon free-space propagation in the focal volume. Our study may provide insight for designing current densities when structured polychromatic electromagnetic fields interact with materials.

Backward energy flow in simple four-wave electromagnetic fields

Electromagnetic energy backflow is a phenomenon that occurs in regions where the direction of the Poynting vector is opposite to that of the propagation of the wave field. It is particularly remarkable in the nonparaxial regime and has been exhibited in the focal region of sharply-focused beams, for vector Bessel beams, and vector-valued spatiotemporally localized waves. A detailed study is undertaken of this phenomenon and the conditions for its appearance are examined in detail in the case of a superposition of four plane waves in free space, the simplest electromagnetic arrangement for the observation of negative energy flow, as well as its comprehensive and transparent physical interpretation. It is shown that the state of polarization of the constituent components of the electromagnetic plane wave quartet determines whether or not energy backflow takes place and what values the energy flow velocity assumes. Depending on the polarization angles, the latter can assume any value from c (the speed of light in vacuum) to −c in certain spatiotemporal regions of the field.

Microsphere kinematics from the polarization of tightly focused nonseparable light.

Recently, it was shown that vector beams can be utilized for fast kinematic sensing via measurements of their global polarization state [Optica2, 864 (2015)10.1364/OPTICA.2.000864]. The method relies on correlations between the spatial and polarization degrees of freedom of the illuminating field which result from its nonseparable mode structure. Here, we extend the method to the nonparaxial regime. We study experimentally and theoretically the far-field polarization state generated by the scattering of a dielectric microsphere in a tightly focused vector beam as a function of the particle position. Using polarization measurements only, we demonstrate position sensing of a Mie particle in three dimensions. Our work extends the concept of back focal plane interferometry and highlights the potential of polarization analysis in optical tweezers employing structured light.

Analytic treatment of nonparaxial full-Poincaré fields: singularity structure and trapping properties

An analytic extension to the nonparaxial regime of the full-Poincaré (FP) beams is presented. Instead of the stereographic mapping used in the paraxial case, these FP fields are defined in terms of a mapping from the polarization Poincaré sphere onto the sphere of plane-wave directions. It is shown that multipolar fields with complex arguments can be used to implement this mapping and provide closed-form expressions. The three-dimensional polarization singularities of the resulting fields are studied with the help of auxiliary fields presenting vortices at points where the polarization is circular or linear. Finally, the Mie scattering and trapping properties of the FP fields are studied, both of which are greatly simplified by the choice of fields.

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.

Double freeform surfaces design for beam shaping with non-planar wavefront using an integrable ray mapping method.

We propose a method to design double smooth freeform surfaces applied in beam shaping with a ray mapping method in the paper. We couple the calculation of ray mapping and the construction of freeform surfaces to approach the surface normal field integrability condition based on the symplectic flow mapping scheme. In this paper, the incident beam wavefront is not limited to be planar or spherical. Several challenging design examples are presented that include transforming a circular Gaussian beam to an unconventional beam with variously shaped contour, and transforming an elliptic beam to a convergent beam with complex irradiance distribution in non-paraxial regime. The results show the high efficiency and feasibility of the proposed method in designing freeform optics for beam shaping applications.

Non-paraxial design and fabrication of a compact OAM sorter in the telecom infrared

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 as 256-level phase-only diffractive optics with high-resolution electron-beam lithography, and tested for the demultiplexing of OAM beams at the telecom wavelength of 1310 nm. The designed sorter can find promising applications in next-generation optical platforms for mode-division multiplexing based on OAM modes both for free-space and multi-mode fiber transmission.

Morphological properties of 2D symmetric Airy beams extracted from the stationary wave approximation

We explore the morphological properties of symmetric Airy beams in the paraxial and nonparaxial regimes. We consider a 2D electromagnetic realization with a single transverse component of the electric field, and in the nonparaxial regime, the longitudinal component along the optic axis. The general structure of these beams is analyzed with the combination of several approaches: geometrical optics through the use of caustics, the asymptotic wave properties of the light field using the stationary wave approximation and numerical integration. The geometrical optics approach involves locating the critical points that are later used in the stationary phase approximation. In the paraxial regime the highest order of the roots is 3, while in the nonparaxial regime, the order can be of up to 6. The technique yields conditions to identify interesting features on the beam, like the number of waves interfering constructively/destructively at the critical positions. The results are confirmed by the numerical simulations. In this way it is possible to distinguish and classify phase singularities like optical vortices and dislocations. The developed algorithm could be used to study any structured light field.

Nonparaxial Propagation Properties of Specially Correlated Radially Polarized Beams in Free Space

A specially correlated radially polarized (SCRP) beam with unusual physical properties on propagation in the paraxial regime was introduced and generated recently. In this paper, we extend the paraxial propagation of an SCRP beam to the nonparaxial regime. The closed-form 3 × 3 cross-spectral density matrix of a nonparaxial SCRP beam propagating in free space is derived with the aid of the generalized Rayleigh–Sommerfeld diffraction integral. The statistical properties, such as average intensity, degree of polarization, and spectral degree of coherence, are studied comparatively for the nonparaxial SCRP beam and the partially coherent radially polarized (PCRP) beam with a conventional Gaussian–Schell-model correlation function. It is found that the nonparaxial properties of an SCRP beam are strikingly different from those of a PCRP beam. These nonparaxial properties are closely related to the correlation functions and the beam waist width. Our results may find potential applications in beam shaping and optical trapping in nonparaxial systems.

Talbot effect with a grating of period 1 μm

Abstract Under coherent plane wave illumination, the classical Talbot effect yields exact replica of the periodic input pattern in planes periodically distributed in the free propagation direction (paraxial regime). But when the period of the object is comparable with the wavelength, the Talbot phenomenon is considered to enter its non-paraxial regime(non-paraxial Talbot effect). In this contribution an experimental and theoretical report of this phenomenon is presented at non-paraxial regime with a amplitude grating of period 1μm, using a plane wave with wavelength of 616nm. The experimental self-images are registered by a confocal microscope and these self-images are contrasted with those obtained by the theory of diffraction in representation of plane waves. Contrary to what was expected, the remarkable results obtained is that it does not present quasi-self-images phenomenon and a detailed discussion of this behavior is included.

Nonparaxial propagation of vector vortex beams diffracted by a circular aperture.

In terms of the vectorial Rayleigh-Sommerfeld diffraction integral, the analytical expressions of the arbitrary vector vortex Laguerre-Gaussian beams on the higher-order Poincaré sphere diffracted by a circular aperture propagating in the nonparaxial and paraxial regimes are presented. The cylindrical vector beam, circularly polarized vortex beam, and elliptically polarized vortex beam are viewed as the special cases of our general result. The analyses show that the nonparaxial evolution properties of the apertured vector vortex beams are determined by the waist width, the truncation parameter, the topological charge, and the ellipticity angle.