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.     

Design of a mirror aberration corrector and a beam separator for LEEM

A SPLEEM (spin polarized low energy electron microscope) has been designed with a numerical simulation of electrostatic and magnetic field distributions and electron ray trajectories. Highly (more than 90%) spin polarized electron source has been used. A Wien type spin manipulator and a magnetic lens type spin rotator are used to align spin direction. A magnetic field free objective lens is designed to observe magnetic domain structure of magnetic materials. High or low magnification mode can be selected by using a combined electrostatic and magnetic objective lens for a high spatial resolution and a wide imaging area observation. An electrostatic mirror aberration corrector is installed after the image forming objective lens. A double deflection 45° beam separator is used to bend the direction of electrons from the source to the objective lens and from the objective lens to the mirror aberration corrector.     

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.     

Systematic engineering of paraxial coherent fields with spherical optical systems

The concept of the Iwasawa decomposition [1] of ABCD ray-matrices is used for a systematic calculation of field distributions appearing in spherical optical systems. Examples of optimization for applications are calculated.     

Generation of dark hollow beams by using a fractional radial Hilbert transform system

The radial Hilbert transform has been extend to the fractional field, which could be called the fractional radial Hilbert transform (FRHT). Using edge-enhancement characteristics of this transform, we convert a Gaussian light beam into a variety of dark hollow beams (DHBs). Based on the fact that a hard-edged aperture can be expanded approximately as a finite sum of complex Gaussian functions, the analytical expression of a Gaussian beam passing through a FRHT system has been derived. As a numerical example, the properties of the DHBs with different fractional orders are illustrated graphically. The calculation results obtained by use of the analytical method and the integral method are also compared.     

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.     

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.     

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.     

Acceptance and transmission of a quadrupole mass filter with a prefilter

We have proposed an analytical model of a quadrupole mass filter with an additional rf quadrupole (prefilter) at its entry, which is based on the model of a transient field in the region between the edges of the mass filter electrodes and the prefilter electrodes. The characteristics of the complex mass filter have been investigated by the methods of phase space dynamics. In the presence of the entrance prefilter, instead of the entrance diaphragm, the y acceptance increases by 1.5 times relative to the x acceptance. Upon an increase in the ion residence time in the fringing field, the y acceptance contour exhibits the tendency to rotation on the phase plane, while the x acceptance contour is broadened in the x coordinate. The transmission function increases and oscillates upon an increase in the ion residence time in the fringing field. The result of numerical experiment using the SIMION 8 software differs from the result of analytical calculation by 14% for the x acceptance and by 35% for the y acceptance.     

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.     

Vectorial structures of non-paraxial linearly polarized Gaussian beam and their beam propagation factors

Based on the vectorial structure of non-paraxial electromagnetic beam and non-paraxial vectorial moment theory, the relationship of the beam waists, the divergence angles and the beam propagation factors among non-paraxial linearly polarized Gaussian beam, its TE and TM terms have been presented, respectively. The analytical beam propagation factors are given and further discussed at the highly non-paraxial case. The maximum divergence angles in the x-direction of non-paraxial linearly polarized Gaussian beam, its TE and TM terms are all 54.7°, and those in the y-direction are limited to be 63.4°, 67.7° and 39.2°, respectively. As TE and TM terms are orthogonal and can be detached at the far field, the potential applications of the isolated TE and TM terms are deserved further investigation.     

Off-Axis Astigmatic Gaussian Beam Combination Beyond the Paraxial Approximation

Taking the off-axis astigmatic Gaussian beam combination as an example, the beam-combination concept is extended to the nonparaxial regime. The closed-form propagation expressions for coherent and incoherent combinations of nonparaxial off-axis astigmatic Gaussian beams with rectangular geometry are derived and illustrated with numerical examples. It is shown that the intensity distributions of the resulting beam depend on the combination scheme and beam parameters in general, and in the paraxial approximation （i.e., for the small f-parameter） our results reduce to the paraxial ones.     

Analytical formulas for a circular or non-circular flat-topped beam propagating through an apertured paraxial optical system

Propagation of a flat-topped beam of circular or non-circular (rectangular or elliptical) symmetry through an apertured optical system is investigated. By expanding the hard aperture function as a finite sum of complex Gaussian functions, some approximate analytical propagation formulas are derived for a flat-topped beam of circular or non-circular (rectangular or elliptical) symmetry propagating through an apertured paraxial general astigmatic (GA) optical system or an apertured paraxial misaligned stigmatic (ST) optical system. The derived formulas are very fast to compute. The results obtained by using the approximate analytical expressions are in a good agreement with those obtained by direct numerical integration. The present analytical formulas provide a convenient and effective way for studying the propagation and transformation of a circular or non-circular flat-topped beam through an apertured general optical system.     

Off-axis-Gaussian beam intensity in random medium

The receiver intensity profile of an off-axis-Gaussian beam travelling in random medium is formulated. By examining the related exponential terms of this intensity expression, the rules governing the receiver plane displacements are deduced. Off-axis-Gaussian beam is characterized by introducing into a Gaussian beam, complex displacement parameters that exhibit transverse source coordinate dependent attenuation and phase shifts. Our results are applied to turbulent horizontal links. Intensity plots describing the dependence on the source and propagation parameters both on the source and the receiver planes are provided. Even though the normalized intensities of the off-axis-Gaussian beam having the same source sizes but differing displacements in x- and y-directions may look the same on the source plane, they will differentiate after propagation. The views from the progress of an off-axis-Gaussian beam along the turbulent link length show that a source displaced beam will act according to rules set by related exponential terms. An asymmetrical (ellipsoidal) off-axis-Gaussian beam will initially be converted into symmetric (circular) shape at the intermediate link lengths, then it will start to expand in the other direction, thus reverting to an ellipsoid shape whose major axis is now along the transverse coordinate opposite to that of the source plane.     

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.     

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.     

Far-Field Distributions of Double-Heterostructure Diode Lasers： an Improved Non-Equiphase Model

A non-equiphase Gaussian model is proposed to simulate the far-field distributions ofdouble-heterostructure diode lasers, which is physically reasonable because the phase along the junction of diode lasers could not be equal, A comparison of the numerically calculated intensity profiles in using the equal phase and non-equiphase models with the experimentally measured intensity profiles given by Nemoto shows that in the x direction perpendicular to the junction plane, the non-equiphase Gaussian model is as good as the equal phase Gaussian model. Specifically, in the y direction parallel to the junction plane and the 45° direction with respect to the x axis in the xoy plane, the numerical results by using the non-equiphase model are in good agreement with the experimental data, as the propagation distance is larger than a certain value.     

Optical necklaces generated by the diffraction on a stack of dielectric wedges

We demonstrate that the regular ring-shaped arrays of Gaussian beams, or optical necklaces, can be generated using diffraction on a stack of dielectric wedges. A condition for self-similarity and structural stability of the beams has been derived and shows good comparison with experimental data.     

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.