Comparison of two approximate methods for hard-edged diffracted flat-topped light beams

The integral resulted in an infinite series of Bessel functions and expanding a hard aperture into a complex-Gaussians shape are proposed as two methods for studying the propagation properties of the hard-edged diffraction flat-topped light beam. Using the two methods, the corresponding analytical propagation equations of flat-topped light beams through a circular apertured ABCD optical system are obtained. Some numerical calculations and comparative analyses by using the two methods and the diffraction integral formulae are made. It is shown that the first method of an infinite series of Bessel functions is superior to the second of expanding a hard aperture function into a complex-Gaussians shape at the aspect of calculation accuracy, but the second method is superior to the first method at the aspect of the improvement in the calculation efficiency.     

Elliptical cosh-Gaussian beams

Using tensor method, a new kind of light beams named elliptical cosh-Gaussian beam (EChGB) is introduced in this paper. An analytical propagation expression for the EChGB passing through axially nonsymmetrical ABCD optical systems is derived by using vector integration. The derived formula is easily reduced to the propagation formula of a fundamental Gaussian beam and that of a cosh-Gaussian beam passing through optical systems. Some numerical simulations are illustrated for the propagation properties of EChGBs through the nonsymmetrical optical transforming systems, and further extensions are pointed out.     

Decentered controllable elliptical dark-hollow beams

A new kind of laser beam, named a decentered controllable elliptical dark-hollow beam (DCEDHB), is defined by a tensor method. The propagation formula for a DCEDHB through an axially nonsymmetrical optical system is derived by vector integration. The derived propagation formula could be reduced to the formula for a controllable elliptical dark-hollow beam (CEDHB) and a controllable dark-hollow beam (CDHB) under certain conditions. As an application example of the derived formula, the propagation characteristics of a DCEDHB in free space are calculated and discussed. As another example we study the propagation properties of coherent and incoherent beam combinations with radial array for DCEDHB.     

Off-axial elliptical Hermite-cosh-Gaussian beams

A new kind of light beams named the off-axial elliptical Hermite-cosh-Gaussian beams (EHChGBs) is introduced in this paper by use of tensor method. An analytical propagation expression for an off-axial EHChGB passing through an axially nonsymmetrical optical ABCD system is derived by use of vector integration. The derived formula can be easily reduced to the propagation formulas of off-axial elliptical cosh-Gaussian beams and elliptical Hermite-cosh-Gaussian beams. Some numerical simulations are illustrated for the propagation properties of off-axial EHChGBs passing through a free space and a focusing optical system, and further extensions are suggested.     

Generalized beams in ABCDGH systems

For a generalized beam at the source plane passing through co-located aperture and a propagation path consisting of an off-axis x-y asymmetric ABCDGH system, the receiver plane irradiance expression is derived using the Collins integral. A collection of source beam profiles that are obtainable from the generalized beam formulation are illustrated. Plots are given for viewing the progress of selected generalized beam types along the propagation axis, containing a single thin lens, co-centric and misaligned in the x-direction. The received power falling onto a finite aperture surface is calculated for various misalignment situations.     

Effect of the spiral phase element on the radial-polarization (0, 1) LG beam

Radially-polarized beams can be strongly amplified without significant birefringent-induced aberrations. However, radially-polarized beam is a high-order beam, and therefore has to be transformed into a fundamental Gaussian beam for reduction the beam-propagation factor M². In effort to transform the radially-polarized beam to a nearly-Gaussian beam, we consider effect of a spiral phase element (SPE) on the Laguerre-Gaussian (LG) (0, 1)^∗ beam with radial polarization, and compare this with the case when the input beam is a LG (0, 1)^∗ beam with spiral phase and uniform or random polarization. The LG (0, 1)^∗ beam with radial polarization, despite its identity in intensity profile to the beam with spiral phase, has distinctly different properties when interacting with the SPE. With the SPE and spatial filter, we transformed the radially-polarized (0, 1)^∗ mode with M² = 2.8 to a nearly-Gaussian beam with M² = 1.7. Measured transformation efficiency was 50%, and the beam brightness P/(M²)² was practically unchanged. The SPE affects polarization state of the radially-polarized beam, leading to appearance of spin angular momentum in the beam center at the far-field.     

Generation of thin and hollow beams by the axicon with a large open angle

We propose a method to produce diffraction-free thin and hollow beams. The method is based on Laguerre-Gaussian (LG) beams incident on a large open-angle axicon. We use the vector diffraction integrals and stationary phase method to deduce a simple and analytical formula of the propagating field of the linearly polarized LG beams through an axicon. The numerical results show that the hollow beams of whose diameter is in the order of the wavelength can be obtained by using the axicon with the refractive index n = 2 and the open angle α = 25°. These diffraction-free thin and hollow beams may be very useful to accurately trap and manipulate atoms. However, when the open angle is over large, the conversion efficiency from the LG beam to the diffraction-free hollow beam will decrease obviously.     

Fractional Fourier transform for off-axis elliptical Gaussian beams

The fractional Fourier transform (FRT) is applied to off-axis elliptical Gaussian beam (EGB). An analytical formula is derived for the FRT of off-axis EGB in terms of the tensor method. The corresponding tensor ABCD law for performing the FRT of off-axis EGB is also obtained. By using the derived formulae, numerical examples are given. The derived formulae provide a convenient way for analyzing and calculating the FRT of off-axis EGB.     

Fractional Fourier transforms of elliptical Hermite-cosh-Gaussian beams

A new kind of laser beams named the elliptical Hermite-sinusoidal-Gaussian beams (EHSGBs) is introduced and defined by use of tensor method, and the elliptical Hermite-cosh-Gaussian beam (EHChGB) can be regarded as special case of EHSGBs. The fractional Fourier transform (FRFT) is applied to treat the propagation of EHChGBs. An analytical expression for an EHChGB passing through an FRFT system is derived by using vector integration. Some numerical simulations are illustrated for the propagation properties of EHChGBs through FRFT systems.     

Generalized M factor of hard-edged diffracted controllable dark-hollow beams

On the basis of the truncated second-order moments method on the cylindrical coordinate systems, an analytical expression of the generalized beam propagation factor ( factor) of hard-edged diffracted controllable dark-hollow beams is derived and illustrated numerically. It is shown that the factor of truncated controllable dark-hollow beams is dependent on the beam truncation parameter δ and the beam parameters N and ε. The result can be reduced to that for the non-truncated case as the truncation parameter approaches to be infinite. The power fraction is also discussed analytically and numerically.     

Propagation characteristics of the rectangular flattened Gaussian beams through circular apertured and misaligned optical systems

Based on the fact that a hard aperture function can be expanded into a finite sum of complex Gaussian functions, the approximate analytical expression for the output field distribution of a rectangular flattened Gaussian beam passing through a circular apertured and misaligned paraxial ABCD system is derived. The result brings more convenient for studying its propagation than the usual way by using diffraction integral directly. Some numerical simulations are also given for illustrating the propagation properties of a rectangular flattened Gaussian beam through a circular apertured and misaligned optical system.     

The symmetry properties of stable polynomial Gaussian beam profiles

We consider the symmetry properties of polynomial Gaussian beam profiles (intensity distributions) that remain stable during propagation, apart from being scaled and possibly rotated. These beams are expressed as special linear combinations of the Laguerre-Gaussian modes. Two kinds of symmetries are present: discreet rotational symmetries and mirror symmetries. The symmetry properties are shown to depend on the particular subset of Laguerre-Gaussian modes that is used to construct the stable beam. We demonstrate the symmetry properties of a few examples of stable beams through numerical simulations.     

Propagation properties of anomalous hollow beams in a turbulent atmosphere

Propagation of circular and elliptical anomalous hollow beams in a turbulent atmosphere is investigated in detail. Based on the extended Huygens–Fresnel integral, analytical formulae for the average irradiance of circular and elliptical anomalous hollow beams propagating in a turbulent atmosphere are derived. The irradiance and spreading properties of circular and elliptical anomalous hollow beams in a turbulent atmosphere and in free space are studied numerically. It is found that circular and elliptical anomalous hollow beams at short propagation distance in turbulent atmosphere have similar propagation properties to those of free space, while at long propagation distance, circular and elliptical anomalous hollow beams eventually become circular Gaussian beams in a turbulent atmosphere, which is much different from their propagation properties in free space. The conversion from an anomalous hollow beam to a circular Gaussian beam becomes quicker and the beam spot spreads more rapidly for a larger structure constant, a shorter wavelength and a smaller waist size of the initial beam.     

Vectorial Nonparaxial Four-Petal Gaussian Beams and Their Propagation in Free Space

The vectorial nonparaxial four-petal Gaussian beam （FPGB） is introduced. The closed-form propagation expressions for the free-space propagation of FPGBs are derived and their more general applicable advantages are illustrated analytically and numerically. Some special interesting cases, in particular the paraxial one, are discussed. It is found that the parameter f = 1/kwo with the k being the wave number and wo being the waist width plays a crucial role in determining the nonparaxiallity of FPGBs. For small values of the f parameter the paraxial approximation is allowable. In the nonparaxial regime the beam order n additionally affects the vectorial and nonparaxial behaviour of FPGBs.     

Global parameters for characterizing the radial and azimuthal polarization content of totally polarized beams

Several global parameters are proposed to characterize the radial and azimuthal polarization content of non-uniformly totally polarized beams. Such figures of merit can be written and measured in terms of two Stokes parameters, and also from the data at the output of either a radial or an azimuthal dichroic polarizer, integrated throughout the beam profile. The measurability of the proposed parameters has also been experimentally checked.     

Propagation characteristics of laser beams with amplitude modulations and phase fluctuations in apertured fractional Fourier transforming system

The apertured fractional Fourier transforming system is introduced and applied to treat the propagation of Gaussian beams with amplitude modulations and phase fluctuations. Based on the treatment that a rectangular function can be expanded into an approximate sum of complex Gaussian functions with finite numbers, the analytical expressions for the mutual intensity distribution of laser beams with amplitude modulations and phase fluctuations passing through the apertured fractional Fourier transforming system are obtained. Some numerical simulations are illustrated for their propagation properties.     

Propagation characteristics of stochastic electromagnetic array beams

A stochastic electromagnetic array beam that is generated by an electromagnetic Gaussian Schell-model source is introduced by using tensor method. An analytical expression for the cross-spectral density matrix of the stochastic electromagnetic array beam propagating in a paraxial ABCD optical system is derived after performing vector integration. Some numerical calculations are illustrated for the propagation characteristics of such an array beam in free space and fractional Fourier transform (FRFT) system as application examples.     

Effects of the frequency chirp on the fields of a chirped Gaussian pulse passing through a hard-edged aperture

Based on the Huygens-Fresnel diffraction integral and Fourier transform, propagation expression of a chirped Gaussian pulse passing through a hard-edged aperture is derived. Intensity distributions of the pulse with different frequency chirp in the near-field and far-field are analyzed in detail by numerical calculations. In the near-field, amplitudes of the intensity peaks generated by the modulation of the hard-edged aperture decrease with increasing the frequency chirp, which results in the improving of the beam uniformity. A physical explanation for the smoothing effect brought by increasing the frequency chirp is given. The smoothing effect is achieved not only in the pulse with Gaussian transverse profile but also in the pulse with Hermite-Gaussian transverse profile when the frequency chirp increases.     

Optical vortices generated by multi-level achromatic spiral phase plates for broadband beams

We analyze the characterizations of the optical vortices generated by a multi-level achromatic spiral phase plate. The effective topological charge of the multi-level fractional spiral phase plate is expanded into Fourier series and the analytical formula of the relative intensity of each component is obtained. It is shown that the fractional part of the topological charge sharply reduces the purity of vortices. Using the 36 levels achromatic spiral phase plate, we can obtain a vortex beam with purity larger than 95% across a bandwidth exceeding 140 nm in the visible region of the spectrum.     

Conversion of high-order Laguerre-Gaussian beams into Bessel beams of increased, reduced or zeroth order by use of a helical axicon

We propose a new method for transformation of a Laguerre-Gaussian beam of azimuthal index l and radial index n = 0 (LG_l,0) into a vortex, diverging or nondiverging Bessel beam, which can have increased or decreased phase singularity order, or into a zeroth order Bessel beam, by means of a helical axicon. The Bessel beam divergence or nondivergence depends upon the waist position of the input Laguerre-Gaussian beam, regarding the plane where the helical axicon is situated.The expressions for the amplitude and the intensity distribution of the diffracted wave field, in the process of Fresnel diffraction, are deduced using the stationary phase method. The theoretical analysis for the vortex radius and the maximum propagation distance of the Bessel beams obtained is presented.