Nonparaxial Corrections Research Articles

Studying near-field propagation of generalized parabolic beams by virtual source method

Generalized parabolic beams have various optical morphologies. They can be used in different research fields, such as component design, aero-optics, and microwave wireless power transmission. Studying the near-field transmission characteristics of these beams is important for improving utilization efficiency. We develop a more accurate theoretical framework to precisely understand the propagation behaviors of complex light fields in the near-field range, especially to break through the limitations of conventional near-axis approximation. This framework fully reveals the propagation mechanism of parabolic beams and their energy transmission modes. Here, based on the principle of independent propagation and the virtual source method, a group of virtual sources are introduced to analyze generalized parabolic beams. These beams can be expanded into the superposition of infinite continuous integer Bessel beams. Then, by combining the Weber integral formula and the Fourier Bessel transform, we rigorously derive an integral expression for generalized parabolic beams during near-field propagation. This expression breaks through the limitation of the traditional paraxial approximation and contains all the key propagation parameters of the family of beams. Based on this integral expression, the intensity distribution and phase characteristics of the generalized parabolic beam along the optical axis are further calculated and analyzed to reveal its energy transfer mode and phase characteristics. By comparing the paraxial approximate solution with the nonparaxial corrected solution for generalized parabolic beams, the far-field propagation of generalized parabolic beams is found to be the same when the propagation distance is sufficiently long. Such simulation results indirectly confirm the correctness of the obtained theoretical solution. The simple paraxial approximation theory can be used conveniently to calculate the far-field propagation of generalized parabolic beams. However, large errors exist when paraxial theory is used to calculate the near-field distribution of generalized parabolic beams. Although calculating nonparaxial propagation is especially complex, the nonparaxial correction solution is necessary when generalized parabolic beams are used in near-field research. Such research results not only deepen the understanding of the propagation mechanism of generalized parabolic beams but also lay a theoretical foundation for studying the precise propagation behaviors of other complex light fields in near-field optics.

Virtual Source of a Swallowtail Beam

AbstractBy applying a virtual source method, the exact and the paraxial solutions of the Swallowtail beams (SBs) that are governed by the inhomogeneous Helmholtz equation are introduced. The paraxial SBs generated experimentally possess both self‐accelerating and self‐focusing features. The parameters controlling the field distributions are discussed in theory. From the spectral representation of the Swallowtail wave, in the appropriate limit, the first four terms of the nonparaxial correction series for the (1+2)‐dimensional Swallowtail waves that simplify to the SBs are obtained.

Nonparaxial corrections for short cavities and fibers

This paper describes optical propagation beyond the paraxial limit in rotational symmetric short Fabry-P\'erot cavities and long fibers with parabolic-index profiles. Frequency shifts due to paraxial and nonparaxial effects, the so-called spectral fine structure and vector corrections, are calculated with perturbation theory and expressed in a single dimensionless expansion parameter. The results obtained for short cavities and fibers have similar functional forms but are surprisingly different. These similarities and differences are pointed out and discussed, including their consequences for mode multiplexing in fibers.

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.

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.

Linearly polarized laser beam with generalized boundary condition and non-paraxial corrections.

Linearly polarized Gaussian beams, under the slowly varying envelope approximation, tightly focused by a perfect parabola modeled with the integral formalism of Ignatovsky are found to be well approximated with a generalized Lax series expansion beyond the paraxial approximation. This allows obtaining simple analytic formulas of the electromagnetic field in both the direct and momentum spaces. It significantly reduces computing time, especially when dealing with the problem of simulating direct laser acceleration. The series expansion formulation depends on integration constants that are linked to boundary conditions. They are found to depend significantly on the region of space over which the integral formulation is fit. Consequently, the net acceleration of electrons initially at rest is extremely sensitive to the chosen set of initial parameters due to the extreme focusing investigated here. This suggests avoiding too tight focusing schemes in order to obtain reliable predictions when the process of interest is sensitive mainly to the field and not the intensity.

Study of the nonparaxial propagation of asymmetric Bessel–Gauss beams by using virtual source method

Abstract The virtual source method, which is also known as the complex source point method was used to study the nonparaxial propagation of asymmetric Bessel–Gauss (aBG) beams. According to optical independence transmission and superposition principles of beam propagation, an aBG beam was expanded into superposition of the infinite Bessel–Gauss beams. Superposition of infinite virtual source points of the expanded aBG beams was used to correspond to electric rings generating an aBG beam. Then Fourier–Bessel transform theory was used to derive the nonparaxial integral expression of aBG beams. Finally, the analytic expression of axial optical field amplitude of an aBG beams was derived when they are kept to third-order nonparaxial correction term.

Virtual source for Lommel–Gauss beams

We introduce a group of virtual sources for generating Lommel–Gauss beams based on beam superposition to analyze nonparaxial propagation. We typically derive the paraxial approximation and integral representations of the nth-order Lommel–Gauss beams. The first three orders of the nonparaxial corrections for the on-axis field of the Lommel–Gauss beams are analytically obtained. The on-axis intensity distribution of the corresponding nonparaxial corrections is also provided.

Relativistic vortex electrons: Paraxial versus nonparaxial regimes

A plane-wave approximation in particle physics implies that a width of a massive wave packet $\sigma_{\perp}$ is much larger than its Compton wavelength $\lambda_c = \hbar/mc$. For Gaussian beams or for packets with the non-singular phases (say, the Airy beams), corrections to this approximation are attenuated as $\lambda_c^2/\sigma_{\perp}^2 \ll 1$ 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 enhanced and $|\ell|$ can already be as large as $10^3$. We describe the relativistic wave packets, both for vortex bosons and fermions, which transform correctly under the Lorentz boosts, are localized in a 3D space, and represent a non-paraxial generalization of the massive Laguerre-Gaussian beams. We compare such states with their paraxial counterpart paying specific attention to the relativistic effects and to the differences from the twisted photons. In particular, a Gouy phase is found to be Lorentz invariant and it generally depends on time rather than on a distance $z$. By calculating the electron packet's mean invariant mass, magnetic moment, etc., we demonstrate that the non-paraxial corrections can already reach the relative values of $10^{-3}$. These states and the non-paraxial effects can be relevant for the proper description of the spin-orbit phenomena in relativistic vortex beams, of scattering of the focused packets by atomic targets, of collision processes in particle and nuclear physics, and so forth.

Short pulse laser beam beyond paraxial approximation

Nonparaxial perturbative equations are derived from the scalar wave equation by taking into account spatiotemporal couplings. General solutions are obtained in Fourier space and further transformed back in direct space. They depend on parameters that can be used to match various boundary conditions and the perturbative expansion of any nonparaxial exact solutions. This parametrization is used to study the sensitivity of direct electron acceleration off an ultrashort tightly focused laser pulse to nonparaxial corrections of radially polarized electromagnetic fields.

Virtual source for a Mathieu–Gauss beam

We introduce a group of virtual sources for generating 2nth-order even Mathieu–Gauss beams based on the beam superposition. Integral and differential representations are derived for a 2nth-order even Mathieu–Gauss wave and the solution yields a corresponding 2nth-order even paraxial Mathieu–Gauss beam in an appropriate limit. The first three orders of nonparaxial corrections for the on-axis field of the 2nth-order even paraxial Mathieu–Gauss beam are obtained using the integral representation.

S-shaped non-paraxial corrections to general astigmatic beams.

Non-paraxial perturbation wave equations are solved for general astigmatic Gaussian beams for the first time, to the best of our knowledge, in the angular spectrum representation by taking into account generic boundary conditions. Expressions for second-order corrections are derived and exemplified with an optical cavity made of two cylindrical mirrors. Non-paraxial corrections can lead, depending on the choice of boundary conditions, to a transverse S-shaped beam mode, which has been qualitatively been observed in a highly divergent non-planar four-mirror cavity.

Odd version Mathieu-Gaussian beam based on Green function

Like the theoretical pattern of non-diffracting Bessel beams, ideal non-diffracting Mathieu beams also carry infinite energy, but cannot be generated as a physically realizable entity. Mathieu-Gaussian beams can be experimentally generated by modulating ideal Mathieu beams with a Gaussian function, and thus they are a kind of pseudo-non-diffracting beams with finite energy and finite transverse extent. The research of Mathieu-Gaussian beam propagating characteristics in free space is of great significance. In order to analytically study the propagation of Mathieu-Gaussian beams, the Mathieu function is expanded into the superposition of a series of Bessel functions in polar coordinates based on the superposition principle of light waves. It means that the Mathieu-Gaussian beam can be converted into accumulation of the infinite terms of the Bessel beams with different orders. According to the properties of the Bessel function, the free-space propagation properties of Mathieu-Gaussian beams can be studied in the circular cylindrical coordinates. Thus, a group of virtual optical sources are introduced to generate the odd Mathieu-Gaussian beams of the first kind, i.e., (2n+2)th-order, which is a family of Mathieu-Gaussian beams. Using the virtual source technique and the Green function, we derive the rigorous integral formula for the odd Mathieu-Gaussian beams of the first kind. Taking for example the first three orders with non-paraxial corrections, the analytical solution of the on-axis field of odd Mathieu-Gaussian beams of the first kind is further obtained from the integral formula. The axial intensity distribution of the odd Mathieu-Gaussian beams of the first kind is numerically calculated by the integral formula. The simulation results show that the calculation results obtained with the paraxial theory and the rigorous integral expressions of non-paraxial Mathieu-Gaussian beams are obviously different when the propagation distance of the odd Mathieu-Gaussian beams of the first kind is small. The calculation results of the two methods are coming closer and closer with the increasing propagation distance. The results indicate that the correct results can be obtained with the paraxial theory when we study the propagation of Mathieu-Gaussian beams in the far-field, but the non-paraxial theory must be used to obtain correct results when we study the propagation of Mathieu-Gaussian beams in the near-field. Owing to the complexity of the non-paraxial theory, it is difficult to obtain the exact analytic solutions of Mathieu-Gaussian beams in the near-field with the classical diffraction theory. Based on the superposition principle of light waves, by introducing the virtual source technique and the Green function, the complex Mathieu-Gaussian function can be expanded into the superposition of a series of simple Bessel functions, and the axial intensity distributions of Mathieu-Gaussian beams in the far-field and the near-field can be studied well. It will also provide a feasible method to study other complex beams propagating in free space.

Cylindrically polarized Bessel–Gauss beams

We present a study of radially and azimuthally polarized Bessel–Gauss (BG) beams in both the paraxial and nonparaxial regime. We discuss the validity of the paraxial approximation and the form of the nonparaxial corrections for BG beams. We show that independently on the ratio between the Bessel aperture cone angle and the Gaussian beam divergence , the nonparaxial corrections are alway very small and therefore negligible. The explicit expressions for the nonparaxial vector electric field components are also reported.

Virtual source of a Pearcey beam.

A virtual source that yields a family of a Pearcey wave is demonstrated. A closed-form expression is derived for the Pearcey wave that simplifies to the paraxial Pearcey beam (PB) in the appropriate limit. From the perturbative series representation of a complex-source-point spherical wave, an infinite series nonparaxial correction expression for a PB is obtained. The infinite series expression of a PB can give accuracy up to any order of the diffraction angle. By applying the integral representation of the Pearcey wave, the first three terms in the nonparaxial correction series to the paraxial PB are provided.

Direct electron acceleration with tightly focused TM_0,1 beams: boundary conditions and non-paraxial corrections

Non-paraxial corrections for a scalar optical field that follows the Helmotz equation are extracted for the first time, to the best of our knowledge, in the angular spectrum representation by taking into account generic boundary conditions. Those integration constants are compared with closed-form solutions and approximate series expansions usually obtained by other authors. This method is particularized to the direct electron acceleration with a tightly focused TM(0,1) laser beam to demonstrate that these constants have a strong effect on the final average energy and quality of the electron beam.

Virtual source for an Airy beam

We identify a virtual source for generating an Airy wave. A spectral integral expression is derived to describe the Airy wave, which, in the paraxial limit, yields the freely accelerating, nondiffracting, and finite energy Airy beam. From the spectral representation of the Airy wave, the first two orders of nonparaxial corrections to the paraxial Airy beam are determined. Also, a connection between the obtained Airy wave and the well-known complex source point spherical wave is given.

Relativistic Electron Vortex Beams: Angular Momentum and Spin-Orbit Interaction

Motivated by the recent discovery of electron vortex beams carrying orbital angular momentum (AM), we construct exact Bessel-beam solutions of the Dirac equation. They describe relativistic and nonparaxial corrections to the scalar electron beams. We describe the spin and orbital AM of the electron with Berry-phase corrections and predict the intrinsic spin-orbit coupling in free space. This can be observed as a spin-dependent probability distribution of the focused electron vortex beams. Moreover, the magnetic moment is calculated, which shows different g factors for spin and orbital AM and also contains the Berry-phase correction.

On the nonparaxial corrections of Bessel-Gauss beams

The nonparaxial corrections for Bessel-Gauss beams were derived recently using two different approaches [Borghi et al., J. Opt. Soc. Am. A 18, 1618 (2001) and Vaveliuk et al., J. Opt. Soc. Am. A 24, 3297 (2007)]. However, the two obtained results do not agree, so it is necessary to determine which method is correct. In the most recent of those papers, Vaveliuk et al. claimed that their method is correct while the method described by Borghi et al. is incorrect. In the present work, just by solving the rigorous propagation problem, we show that exactly the converse is true.

Exact nonparaxial propagation of a hollow Gaussian beam

A group of virtual sources that generate a hollow Gaussian wave are determined on the basis of the superposition of beams. A closed-form expression is derived for the hollow Gaussian wave that in the appropriate limit yields the paraxial hollow Gaussian beam (HGB). From the perturbative series representation of a complex-source-point spherical wave, an infinite series nonparaxial correction expression for a HGB is derived. The infinite series expression of a HGB can provide accuracy up to any order of diffraction angle. The radiation intensity of the hollow Gaussian wave is ascertained, and the radiation intensity pattern is characterized. The total time-averaged power is evaluated. The characteristics of the quality of the paraxial beam approximation to the full hollow Gaussian wave are discussed.