Propagation of vectorial apertured cosh-Gaussian beams beyond the paraxial approximation

By using the vectorial Rayleigh-Sommerfeld diffraction integrals, the propagation equation of vectorial nonparaxial cosh-Gaussian (ChG) beams in the presence of an aperture is derived and expressed in a closed form. The on-axis, far-field and paraxial cases are studied as special cases. Analytical results are illustrated with numerical examples.     

Phase-flipped Hermite-Gaussian beams and their propagation beyond the paraxial approximation

An extension of phase-flipped Gaussian (PFG) beams to the multi-mode nonparaxial case is made. The recurrence propagation expressions for phase-flipped Hermite-Gaussian (PFHG) beams beyond the paraxial approximation are derived and used to study nonparaxial propagation properties of PFHG beams in free space and through a knife edge and an aperture, and to compare nonparaxial results with paraxial ones. The propagation of paraxial PFHG beams and PFG beams and nonparaxial PFG beams is treated as special cases of nonparaxial PFHG beams. Numerical examples are given to illustrate the theoretical results.     

Anomalous spectral behavior of vectorial apertured ultrashort pulsed beams beyond the paraxial approximation

Based on the vectorial Rayleigh diffraction integrals, the analytical expression for the spectral intensity of a vectorial nonparaxial ultrashort pulsed Gaussian beam diffracted at a circular hard-aperture is derived. The effect of f-parameter (f = 1/k₀w₀) on the spectral anomalies near phase singularities of the vectorial nonparaxial ultrashort pulsed beams is studied. It is shown that the spectral switch near the phase singularity of diffracted vectorial nonparaxial ultrashort pulsed beam still exists beyond paraxial regime, but disappears when the f-parameter is larger than a certain value.     

Propagation of a vectorial Laguerre–Gaussian beam beyond the paraxial approximation

Based on the vectorial Rayleigh–Sommerfeld integrals, the analytical propagation expression of a vectorial Laguerre–Gaussian beam beyond paraxial approximation is presented. The far field expression and the scalar paraxial result are obtained as special cases of the general formulae. According to the analytical representation, the light intensity distribution of the vectorial Laguerre–Gaussian beam is depicted in the reference plane. The light intensity distribution of a vectorial Laguerre–Gaussian beam with cos m is also compared with that of a vectorial Laguerre–Gaussian beam with sin m.     

Vectorial Hermite--Laguerre--Gaussian beams beyond the paraxial approximation

Starting from the vectorial Rayleigh--Sommerfeld integrals, the free-space propagation expressions for vectorial Hermite--Laguerre--Gaussian (HLG) beams beyond the paraxial approximation are derived. The far-field expressions and the scalar paraxial results are given as special cases of our general expressions. The intensity distributions of vectorial nonparaxial HLG beams are studied and illustrated with numerical examples.     

Propagation properties of vectorial elliptical Gaussian beams beyond the paraxial approximation

Starting from the vectorial Rayleigh diffraction integral formula and without using the far-field approximation, a solution of the wave equation beyond the paraxial approximation is found, which represents vectorial non-paraxial elliptical Gaussian beams in free space. The far-field expressions for non-paraxial Gaussian beams and elliptical Gaussian beams can be regarded as special cases treated in this paper. Some basic propagation properties of vectorial non-paraxial elliptical Gaussian beams, including the irradiance distribution, phase term, beam widths and divergence angles are studied. Numerical results are given and illustrated.     

Partially coherent vectorial cosh-Gaussian beams beyond the paraxial approximation

The concept of partially coherent vectorial nonparaxial cosh-Gaussian (ChG) beams is introduced, and their analytical propagation expressions for the cross-spectral density matrix in free space are derived by using the generalized vectorial Rayleigh diffraction integrals. Some interesting cases, in particular, the vectorial nonparaxial Gaussian-Schell-model (GSM) beams are discussed and treated as special cases of our general expressions. It is shown that the f and f_σ parameters play a crucial role in determining the vectorial property and nonparaxiality of partially coherent ChG beams, but the decentered parameter additionally affects their behavior.     

Specular and vortical Airy beams

A new kind of truncated Airy beams is investigated and discussed. These beams are a superposition of shifted and truncated Airy functions and its specular counterparts, where zeroes or extremal points of the Airy function are chosen as a truncation point. The specular Airy beams are smooth at the truncation point and produce a diffraction pattern similar to Hermite-Gaussian modes. Under propagation in Fresnel zone, specular Airy beams demonstrate a symmetrical acceleration in opposite sides and the beam divergence is proportional to the traveled distance squared. The astigmatic mode converter transforms a two-dimensional specular Airy beam into a quasi-annular field with a nonzero orbital angular momentum. Vortical Airy beams based on truncated Airy functions are also discussed. These beams are similar to Laguerre-Gaussian modes, while their annular structure is changed during propagation.     

Relation between Airy beams and triple-cusp beams

Airy beams and triple-cusp beams are two kinds of accelerating beams. The propagation characteristics and internal topological structures of accelerating Airy beams are well understood because of the developed mathematical theory about Airy function. However, limited information is available about the optical characteristics of accelerating triple-cusp beams. In this work, the relationship between Airy beams and triple-cusp beams is examined theoretically and experimentally. Results reveal some important optical characteristics of triple-cusp beams based on the optical characteristics of Airy beams. These findings are expected to provide a foundation for future applications of triple-cusp beams.     

The deflected angle and reflected displacement of Airy beams

The deflected angle of airy beams is defined to describe their bending degree. Based on this parameter, we analyze the lateral displacement of airy beams and the changes of deflected angles during reflection process. In multi-reflections of airy beams, we find that deflected angles increase linearly with the times of reflection, which is useful to study the propagation properties of airy beam in parallel plate waveguides and other waveguides. Also we can use this parameter to control the incident and reflected routes to realize airy beam bypassing obstacles and reflecting back at predictable location.     

Coherent and incoherent combinations of partially coherent beams beyond the paraxial approximation

The concept of coherent and incoherent combinations of partially coherent beams is described and interpreted physically. Taking the Gaussian Schell-model (GSM) beam as a typical example of a partially coherent beam, the analytical propagation expressions for the coherent and incoherent combinations of GSM beams beyond the paraxial approximation are derived and illustrated numerically. Some special cases of the general results are analyzed.     

Propagation of vectorial Gaussian beams behind a circular aperture

Based on the vectorial Rayleigh diffraction integral and the hard-edge aperture function expanded as the sum of finite-term complex Gaussian functions, an approximate analytical expression for the propagation equation of vectorial Gaussian beams diffracted at a circular aperture is derived and some special cases are discussed. By using the approximate analytical formula and diffraction integral formula, some numerical simulation comparisons are done, and some special cases are discussed. We find that a circular aperture can produce the focusing effect but the beam becomes the shape of ellipse in the Fresnel region. When the Fresnel number is equal to unity, the beam is circular and the focused spot reaches a minimum.     

Propagation of Airy beams from right-handed material to left-handed material

Based on the ABCD matrix formalism,the propagation property of an Airy beam from right-handed material(RHM) to left-handed material(LHM) is investigated.The result shows that when the Airy beam propagates in the LHM,the intensity self-bending due to its propagation in the RHM can be compensated.In particular,if the propagation distance in the RHM is equal to that in the LHM and the refractive index of the LHM is n L =-1,the transverse intensity distribution of the Airy beam can return to its original state.     

Airy光束在Kerr介质中的传输特性

通过对非线性薛定谔方程的研究,得出Airy光束在Kerr介质中的崩塌功率及有效束宽演化的解析表达式。经过数值计算发现,Airy光束在聚焦的Kerr介质中,其主瓣在开始传播时始终是会聚的;当输入功率小于临界崩塌功率时,Airy光束主瓣的中心部分出现局部崩塌。在不同的Kerr介质中, Airy光束的形状和传输轨道均能保持不变,如同在自由空间中传播,但光场大小的分布,随着不同的Kerr介质会发生改变:在Kerr的聚焦介质中,光场向中心聚焦;而在散焦的Kerr介质中,光场会发散。     

The optical Airy transform and its application in generating and controlling the Airy beam

We propose an optical Airy transform in this paper, and obtain the analytical expressions for the Airy transform of fundamental Gaussian beams and finite energy Airy beams. The setup for performing the optical Airy transform is presented. The Airy transform for Gaussian beams and finite energy Airy beams are theoretically calculated and analyzed. Our results show that the Airy beam can be conveniently generated and controlled through the optical Airy transform of the Gaussian beam. The optical Airy transform also can be used to directly modulate the beam parameters of the incident Airy beam, and it can transform the incident Airy beam into the Gaussian beam.     

Nonparaxial Lorentz and Lorentz-Gauss beams

The Lorentz and Lorentz-Gauss beams are extended to the nonparaxial regime. Analytical propagation expressions of nonparaxial Lorentz and Lorentz-Gauss beams in free space are derived, and the propagation of paraxial Lorentz and Lorentz-Gauss beams is treated as a special case of our general results. The propagation properties of Lorentz and Lorentz-Gauss beams are illustrated and compared with numerical examples.     

Physical interpretation of the paraxial estimator

The paraxial estimator (PE) is a parameter quantifying the paraxiality of a light beam. Even if some of its features were previously tackled, key details on its behavior were not fully presented. This paper robustly presents the physical meaning of the PE in a global way, enlarging its interpretation out of the paraxial region what permits to get a first view of the beam propagation dynamics from the value of this parameter. The physical interpretation is given in the spatial domain and in the spectral domain as well. In the first one, the value of PE is related to the competition between the fast oscillations and the remaining oscillations of a propagating field. Looking at spectral domain, the PE deals with the spectral dispersion (or width) of the plane waves forming the field. In this context, a negative value of PE concerns the effective contribution of the evanescent waves what only happens in a strong nonparaxial regime. The PE also accounts the geometric and physical features on the concept of the paraxial approximation in a natural way. An analysis performed for beams propagating through a spherical thin lens reveals that the loss of paraxiality is due to the geometric effect of ray bending by the lens and by another physical effect, concerning the nonideal collimation of the beam.     

Evolution of self-healing characteristic on optical Airy beam

We investigate and analyze the evolution of self-healing characteristic on diffraction-free Airy beams. We also show that different internal structure of Airy beams contribute to self-healing by breaking the integrity of Airy beams. A numerical simulation is performed and demonstrate that each independent structure undertakes different roles. It is believed that the intriguing characteristic of Airy beams can be applied in many fields such as optical tweezers, atom trapping and manipulating.