Elliptical Gaussian Beam Research Articles

Elliptical Gaussian beam propagation in nonlinear fibres with focusing and defocusing refractive profiles

The paper shows systematic analysis of elliptical Gaussian beam (GB) propagating in self-focusing/self-defocusing media of Kerr type with focusing/defocusing linear refractive profiles. Instead of solving commonly accepted Nonlinear Schrödinger Equation (NLS) we propose equations of geometrical optics: complex eikonal equation and complex transport one. Eikonal equation let us derive immediately the ordinary differential equations for principle widths omitting this way complicated variational process and virial theory used in nonlinear optics. From the transport equation we obtain first order ordinary differential equation for complex amplitude evolution and conservation principle for energy flux in the elliptical GB cross-section. For combined effects of diffraction, nonlinear refraction and focusing/defocusing graded-index factor we observe three types of the solutions for the elliptical radius of principle beam widths: self-focusing or self-defocusing together with diffraction widening, stationary and oscillatory ones under and without collapse effect. We prove here that principle curvatures of the wave front can introduce into our description the additional ellipticity of the beam, which can effectively limit collapse effect. Moreover, we discuss the evolution of elliptical Gaussian beam in nonlinear self-defocusing fibre with focusing refractive profile presenting conditions for existence of elliptical spatial solitons. We discuss here also the influence of initial wave front curvatures on elliptical Gaussian beam evolution in nonlinear inhomogeneous fibres of Kerr-type.

Statistical properties of partially coherent radially and azimuthally polarized rotating elliptical Gaussian beams in oceanic turbulence with anisotropy.

A new class of partially coherent radially and azimuthally polarized rotating elliptical Gaussian (PCRPREG and PCAPREG) beams is introduced. The analytical expressions of the PCRPREG and PCAPREG beams propagating through anisotropy oceanic turbulence are derived based on the extended Huygens-Fresnel principle and the spatial power spectrum of oceanic turbulence. The effects of beam waist size w0, coherence width σ0, propagation distance z and oceanic turbulence parameters on the evolution statistics properties of PCRPREG and PCAPREG beams are studied in detail by numerical simulation. Our results indicate that with the increase of the propagation distance in the far field region, the normalized initial profile with a doughnut-like distribution of PCRPREG and PCAPREG beams gradually converts into a flat-topped one, and finally evolves into a Gaussian-like beam profile. We also find that the salinity-induced turbulence fluctuation makes a greater contribution to the decrease of beam quality compared with the temperature-induced turbulence fluctuation. Furthermore, the full width at half maximum becomes wider for the larger propagation distance z and wavelength λ or the smaller dissipation rate ε. Our work will pave the way for the development of underwater optical communication and underwater laser radar in oceanic environment.

Vortex astigmatic Fourier-invariant Gaussian beams.

We find a two-parameter family of astigmatic elliptical Gaussian (AEG) optical vortices, which are free space modes up to scale and rotation. We calculate total normalized orbital angular momentum of AEG vortices, which can be an integer, fractional and zero, and which is equal to the algebraic sum of two terms reflecting the contribution of the vortex and astigmatic components of the light field. In any transverse plane, such a beam has an isolated n-fold degenerate intensity null on the optical axis (an optical vortex) embedded into an elliptical Gaussian beam. In addition to the quadratic elliptical phase, a beam has the phase of a cylindrical lens rotated by an angle of 45 degrees with respect to the principal axes of the ellipse of the Gaussian beam intensity distribution. The degenerated central intensity null in these beams does not split it into n spatially separated intensity nulls, as is usually assumed for elliptical astigmatic beams.

Multidirectional digital scanned light-sheet microscopy enables uniform fluorescence excitation and contrast-enhanced imaging

Light-sheet fluorescence microscopy (LSFM) has emerged as a powerful method for rapid and optically efficient 3D microscopy. Initial LSFM designs utilized a static sheet of light, termed selective plane illumination microscopy (SPIM), which exhibited shadowing artifacts and deteriorated contrast due to light scattering. These issues have been addressed, in part, by multidirectional selective plane illumination microscopy (mSPIM), in which rotation of the light sheet is used to mitigate shadowing artifacts, and digital scanned light-sheet microscopy (DSLM), in which confocal line detection is used to reject scattered light. Here we present a simple and passive multidirectional digital scanned light-sheet microscopy (mDSLM) architecture that combines the benefits of mSPIM and DSLM. By utilizing an elliptical Gaussian beam with increased angular diversity in the imaging plane, mDSLM provides mitigation of shadowing artifacts and contrast-enhanced imaging of fluorescently labeled samples.

Effects of the turbulent atmosphere and the oceanic turbulence on the propagation of a rotating elliptical Gaussian beam

Based on the extended Huygens–Fresnel integral, the average intensity, the gradient force and the effective beam width in the x- and y-directions of a rotating elliptical Gaussian (REG) beam in the turbulent atmosphere and the oceanic turbulence are explored analytically and numerically. It is shown that the intensity patterns of the REG beam in the atmospheric and the oceanic turbulences spin anticlockwise within a certain propagation range and always keep unimodal distributions during the propagation. Meanwhile, the numerical experiments of the REG beam are carried out to verify the theoretical analytical results. By comparative analysis, the effective beam widths can be adjusted by varying the turbulent characteristic parameters and the initial beam waist of the REG beam. The intensity modulation of the REG beam through the turbulent atmosphere and the oceanic turbulence can be realized.

Calculation of light scattering of an elliptical Gaussian beam by a spherical particle.

The use of an elliptical Gaussian beam (EGB) for applications relying on light scattering depends much on the ability to evaluate the beam shape coefficients (BSC) effectively and accurately. Based on the angular spectrum decomposition (ASD) of the radial components of the beam field, we present the formulations of the BSCs for the EGB. Numerical calculations of the BSCs, the beam reconstruction, and light scattering are performed. The fidelity of the reconstructed field to the given one is discussed so as to examine the BSC calculation. A comparison of the ASD method with the quadrature and localized approximation methods leads to the conclusion that the ASD method is much faster than the quadrature method, and it is very powerful for acquiring reliable and accurate results of BSCs, even for far off-axis locations and high ellipticities of the EGB.

Beam shape coefficients calculation for an elliptical Gaussian beam with 1-dimensional quadrature and localized approximation methods

The use of a shaped beam for applications relying on light scattering depends much on the ability to evaluate the beam shape coefficients (BSC) effectively. Numerical techniques for evaluating the BSCs of a shaped beam, such as the quadrature, the localized approximation (LA), the integral localized approximation (ILA) methods, have been developed within the framework of generalized Lorenz-Mie theory (GLMT). The quadrature methods usually employ the 2-/3-dimensional integrations. In this work, the expressions of the BSCs for an elliptical Gaussian beam (EGB) are simplified into the 1-dimensional integral so as to speed up the numerical computation. Numerical results of BSCs are used to reconstruct the beam field and the fidelity of the reconstructed field to the given beam field is estimated. It is demonstrated that the proposed method is much faster than the 2-dimensional integrations and it can acquire more accurate results than the LA method. Limitations of the quadrature method and also the LA method in the numerical calculation are analyzed in detail.

Analysis of lateral binding force exerted on a bi-sphere induced by an elliptic Gaussian beam.

Based on the generalized Lorentz-Mie theory (GLMT) and the localized approximation of the beam shape coefficients, we derived the expansions of incident elliptic Gaussian (EG) beams in terms of spherical vector wave functions (SVWFs). Utilizing multiple scattering (MS) equations and electromagnetic momentum (EM) theory, the lateral binding force (BF) exerted on a bi-sphere induced by an EG beam is calculated. Numerical effects of various parameters such as beam waist widths, beam polarization states, incident wavelengths, particle sizes, and material losses are analyzed and compared with the results of a circular Gaussian (CG) beam in detail. The observed dependence of the separation of optically bound particles on the incidence of an EG beam is in agreement with earlier theoretical predictions. Accurate investigation of BF induced by an EG beam could provide an effective test for further research on BF between more complex particles, which plays an important role in using optical manipulation on particle self-assembly.

Astigmatic laser beams with a large orbital angular momentum.

We show that an elliptic Gaussian beam, focused by a cylindrical lens, can be represented as a linear combination of a countable number of only even angular harmonics with both positive and negative topological charge. For the orbital angular momentum (OAM) of the astigmatic Gaussian beam, an exact expression is obtained in a form of a converging series of the Legendre functions of the second kind. It is shown that at some conditions only the terms with the positive or negative topological charge are remained in this series. Using a hybrid numeric-experimental approach, we obtained the normalized OAM of the astigmatic beam, equal to 109, which is just 6% different from the exact OAM of 116, calculated by the equation. To generate such laser beams, there is no need in special optical elements such as spiral phase plates. The OAM of such beams can be adjusted by varying the waist radius of the Gaussian beam and the focal length of the cylindrical lens. The OAM of such beams can reach large values.

A variety of Fourier-invariant Gaussian beams

A two-parameter family of astigmatic elliptical Gaussian (AEG) optical vortices is found, which are free space modes up to scale and rotation. We calculate the total normalized orbital angular momentum of AEG vortices, which can be integer, fractional and zero, and which is equal to the algebraic sum of two terms corresponding to the contribution of the vortex and astigmatic components of the light field. In any transverse plane, such a beam has an isolated n-fold degenerate intensity null on the optical axis (an optical vortex) embedded into an elliptical Gaussian beam. In addition to the quadratic elliptical phase, the beam has a phase of a cylindrical lens rotated by an angle of 45 degrees relative to the principal axes of the Gaussian beam intensity ellipse.

Controlling orbital angular momentum of an optical vortex by varying its ellipticity

An exact analytical expression is obtained for the orbital angular momentum (OAM) of a Gaussian optical vortex with a different degree of ellipticity. The OAM turned out to be proportional to the ratio of two Legendre polynomials of adjoining orders. It is shown that if an elliptical optical vortex is embedded into the center of the waist of a circularly symmetrical Gaussian beam, then the normalized OAM of such laser beam is fractional and it does not exceed the topological charge n. If, on the contrary, a circularly symmetrical optical vortex is embedded into the center of the waist of an elliptical Gaussian beam, then the OAM is equal to n. If the optical vortex and the Gaussian beam have the same (or matched) ellipticity degree, then the OAM of the laser beam is greater than n. Continuous varying of the OAM of a laser beam by varying its ellipticity degree can be used in optical trapping for accelerated motion of microscopic particles along an elliptical trajectory as well as in quantum informatics for detecting OAM-entangled photons.

Astigmatic transforms of an optical vortex for measurement of its topological charge.

We obtain analytical expressions for the complex amplitudes of optical vortices deformed by astigmatic transforms, i.e., passed either through a cylindrical lens or through an inclined spherical lens. We also obtain similar analytical expressions describing propagation of an optical vortex generated when a Gaussian beam illuminates an inclined spiral phase plate (SPP) or when an elliptic Gaussian beam illuminates a SPP (not inclined). All these optical vortices with a topological charge (TC) n are described by the n-th order Hermite polynomial with a complex argument. It is shown that the argument is real only on a straight line in the transverse plane of the laser beam. There are n intensity nulls on this line. The treated here astigmatic transforms are used to determine the integer TC of optical vortices. We conduct a comparative experimental study of different astigmatic transforms and we show that the transform with a cylindrical lens is the best for determining the TC. Unlike other similar works, in this study we achieve transformation of n-degenerate intensity null of an optical vortex with the TC n=100 into n isolated first-order intensity nulls.

Orbital angular momentum of an astigmatic Gaussian laser beam

We show that an elliptic Gaussian beam focused by a cylindrical lens can be represented as a linear combination of a countable number of only even angular harmonics with both positive and negative topological charge. For the orbital angular momentum of an astigmatic Gaussian beam, an exact expression is obtained in the form of a converging series of the Legendre functions of the second kind. It is shown that at some conditions only terms with the positive or negative topological charge are retained in this series.

Z-scan analysis of elliptical Gaussian beams in Kerr media through the variational method: Toward integrative analytic solutions of Z-scan

Previous studies on Z-scan have paid little attention to Z-scan transmittance about elliptical Gaussian beams in thick Kerr media. The purpose of this study is twofold. First, it proposes a new analytical approach to the Z-scan on-axis transmittance for elliptical Gaussian beams through the variational method. The analytical approach yields an explicit analytical solution (not numerically analyzed solution) for the Z-scan on-axis transmittance in arbitrary thick media, and then the explicit solution is generalized to circularly or elliptically symmetric Gaussian beams. With the analytical solutions, we explain the mechanism of the elliptical beam waist changes in both positive and negative Kerr media. Second, the proposed solutions are experimentally tested in the photorefractive LiNbO3:Fe crystal with a cw Ti:sapphire elliptical laser beam (λ=800nm), showing congruence between experimental data and the predicted values from the solutions.

Analytical solutions for three-dimensional modeling of temperature rise inside solid material induced by laser irradiation

In this paper, analytical solutions for three-dimensional(3-D) modeling of temperature rise inside solid material induced by laser irradiation is presented. Firstly, the theoretical physical model of temperature rise for a material surface irradiated by laser is established based on classical heat conduction theory. Then, integral transform method is used to solve the heat conduction equation and its analytical solutions are obtained. In the end, the square and elliptical Gaussian laser beam is selected as examples, and temperature distributions of silicon material irradiated by the two lasers are simulated. Results show that, when silicon material is irradiated by laser, the shape of temperature distributions on the surface of material are similar to that of the laser intensity due to direct absorption mechanism of laser energy on the material surface. The temperature distributions in a certain depth below the material surface also contain the shape information of laser intensity on the material surface.

Selectable light-sheet uniformity using tuned axial scanning.

Light-sheet fluorescence microscopy (LSFM) is an optical sectioning technique capable of rapid three-dimensional (3D) imaging of a wide range of specimens with reduced phototoxicity and superior background rejection. However, traditional light-sheet generation approaches based on elliptical or circular Gaussian beams suffer an inherent trade-off between light-sheet thickness and area over which this thickness is preserved. Recently, an increase in light-sheet uniformity was demonstrated using rapid biaxial Gaussian beam scanning along the lateral and beam propagation directions. Here we apply a similar scanning concept to an elliptical beam generated by a cylindrical lens. In this case, only z-scanning of the elliptical beam is required and hence experimental implementation of the setup can be simplified. We introduce a simple dimensionless uniformity statistic to better characterize scanned light-sheets and experimentally demonstrate custom tailored uniformities up to a factor of 5 higher than those of unscanned elliptical beams. This technique offers a straightforward way to generate and characterize a custom illumination profile that provides enhanced utilization of the detector dynamic range and field of view, opening the door to faster and more efficient 2D and 3D imaging.

Compact formulation of the beam shape coefficients for elliptical Gaussian beam based on localized approximation.

It has been proved that localized approximation (LA) is the most efficient way to evaluate the beam shape coefficients (BSCs) in generalized Lorenz-Mie theory. The BSCs are usually expressed in the form of multiple summations of an infinite series of terms, which is cumbersome to calculate, and the infinite series is frequently slowly convergent. In this paper, we present a compact expression of the BSCs for an elliptical Gaussian beam based on the LA that is more convenient and efficient for numerical computations. A comparison with the integral LA is made, showing the reliability, stability, and efficiency of the presented formulation.

Propagation characteristics of partially coherent anomalous elliptical hollow Gaussian beam propagating through atmospheric turbulence along a slant path

Based on the extended Huygens–Fresnel principal and the Wigner distribution function, the root mean square (rms) angular width and propagation factor (M2-factor) of partially coherent anomalous elliptical hollow Gaussian (PCAEHG) beam propagating through atmospheric turbulence along a slant path are studied in detail. Analytical formulae of the rms angular width and M2-factor of PCAEHG beam are derived. Our results show that the rms angular width increases with increasing of wavelength and zenith angle and with decreasing of transverse coherence length, beam waist sizes and inner scale. The M2-factor increases with increasing of zenith angle and with decreasing of wavelength, transverse coherence length, beam waist sizes and inner scale. The saturation propagation distances (SPDs) increase as zenith angle increases. The numerical calculations also indicate that the SPDs of rms angular width and M2-factor for uplink slant paths with zenith angle of π/12 are about 0.2 and 20 km, respectively.

Quantitative Fluorescence Sensing Through Highly Autofluorescent, Scattering, and Absorbing Media Using Mobile Microscopy.

Compact and cost-effective systems for in vivo fluorescence and near-infrared imaging in combination with activatable reporters embedded inside the skin to sample interstitial fluid or blood can enable a variety of biomedical applications. However, the strong autofluorescence of human skin creates an obstacle for fluorescence-based sensing. Here we introduce a method for quantitative fluorescence sensing through highly autofluorescent, scattering, and absorbing media. For this, we created a compact and cost-effective fluorescence microscope weighing <40 g and used it to measure various concentrations of a fluorescent dye embedded inside a tissue phantom, which was designed to mimic the optical characteristics of human skin. We used an elliptical Gaussian beam excitation to digitally separate tissue autofluorescence from target fluorescence, although they severely overlap in both space and optical spectrum. Using ∼10-fold less excitation intensity than the safety limit for skin radiation exposure, we successfully quantified the density of the embedded fluorophores by imaging the skin phantom surface and achieved a detection limit of ∼5 × 10(5) and ∼2.5 × 10(7) fluorophores within ∼0.01 μL sample volume that is positioned 0.5 and 2 mm below the phantom surface, corresponding to a concentration of 105.9 pg/mL and 5.3 ng/mL, respectively. We also confirmed that this approach can track the spatial misalignments of the mobile microscope with respect to the embedded target fluorescent volume. This wearable microscopy platform might be useful for designing implantable biochemical sensors with the capability of spatial multiplexing to continuously monitor a panel of biomarkers and chronic conditions even at patients' home.

Chopper z-scan technique for elliptic Gaussian beams.

This paper reports an improvement to the chopper z-scan technique for elliptic Gaussian beams. This improvement results in a higher sensitivity by measuring the ratio of eclipsing time to rotating period (duty cycle) of a chopper that eclipses the beam along the main axis. It is shown that the z-scan curve of the major axis is compressed along the z-axis. This compression factor is equal to the ratio between the minor and major axes. It was found that the normalized peak-valley difference with respect to the linear value does not depend on the axis along which eclipsing occurs.