Paraxial Gaussian Beam Research Articles

Skew line ray model of nonparaxial Gaussian beam.

Many ray-optics models have been proposed to describe the propagation of paraxial Gaussian beam. However, those paraxial ray-optics models are inapplicable to the beams that violate the paraxial condition. In this paper, we present a skew line ray (SLR) based model to represent the propagation properties of nonparaxial Gaussian beam under the oblate spheroidal coordinates. The free-space evolution of complex wavefront of the light beam including amplitude and phase is derived via this model. Our analysis demonstrates that the SLR model is available for both nonparaxial and paraxial conditions, and can be used to precisely describe the propagation of complex wavefront. Furthermore, this model changes the transverse density of rays while propagating. The behavior influences the transverse intensity distribution and makes the optical rays become concentrated towards the center. We believe that this ray-optics model can be further developed to describe other kind of structured beams such as Laguerre-Gauss and Bessel-Gauss beams.

Time-Harmonic Gaussian Beams: Exact Solutions of the Helmhotz Equation in Free Space

An exact solution of the Helmholtz equation u xx + u yy + u zz + k2u = 0 is presented, which describes propagation of monochromatic waves in the free space. The solution has the form of a superposition of plane waves with a specific weight function dependent on a certain free parameter a. If ka→∞, the solution is localized in the Gaussian manner in a vicinity of a certain straight line and asymptotically coincides with the famous approximate solution known as the fundamental mode of a paraxial Gaussian beam. The asymptotics of the aforementioned exact solution does not include a backward wave.

Modeling physical optics phenomena by complex ray tracing

Physical optics modeling requires propagating optical wave fields from a specific radiometric source through complex systems of apertures and reflective or refractive optical components, or even complete instruments or devices, usually to a focal plane or sensor. The model must accurately include the interference and diffraction effects allowed by the polarization and coherence characteristics of both the initial optical wave field and the components and media through which it passes. Like a spherical wave and a plane wave, a Gaussian spherical wave (or Gaussian beam) is also a solution to the paraxial wave equation and does not change its fundamental form during propagation. The propagation of a Gaussian beam is well understood and easily characterized by a few simple parameters. Furthermore, a paraxial Gaussian beam can be propagated through optical systems using geometrical ray-trace methods. The decomposition of arbitrary propagating wave fields into a superposition of Gaussian beamlets is, thus, an alternative to the classical methods of propagating optical wave fields. This decomposition into Gaussian beamlets has been exploited to significant advantage in the modeling of a wide range of physical optics phenomena.

Fourth order correction to a Gaussian beam focused with a parabolic index lens

We formulate the fourth order correction to a paraxial Gaussian beam propagated along the axis of symmetry of a parabolic index lens. First we examine the evolution of a complex-source-point spherical wave (equivalent paraxially to a Gaussian beam) through the lens in a two-dimensional xz plane. Taking into account the terms of up to fourth order in aperture variables, we find a ray-optical solution to the exit beam that is represented in terms of aberration function. We also analyze the effect of the lens aberration exerted on the degradation in the quality of a Gaussian beam. The fourth order-corrected wave function derived here may be used to evaluate the quality of a Gaussian beam focused with a parabolic index lens. Further it may be applied to the case of an orthogonal system in which the index variations are different in the xz and yz planes.

Modeling highly focused laser beam in optical tweezers with the vector Gaussian beam in the T-matrix method

The vector Gaussian beam with high-order corrections is used to describe accurately the laser beam up to numerical aperture NA=1.20 in the optical tweezers for trapping nanoparticles. The beam is then expanded in the T-matrix method into the vector spherical wave function (VSWF) series using the point matching method with a new selection of the matching points. The errors in the beam description and in the VSWF expansion are much lower than those that occur in the paraxial Gaussian beam model.

Optical Manipulation in Liquid Crystals: Effects of Tight Focusing on Nonlinear Optical Reorientation

We report a detailed analysis of the problem of nonlinear optical reorientation induced in a nematic liquid crystal by a tightly focused Gaussian beam under experimental conditions that prevent the effect of conventional trapping originated by optical gradient forces. Novel features arise under these conditions: no threshold effect; reduced reorientation if compared the one induced by a paraxial Gaussian beam, orientational singularity at the focal waist.

On the longitudinal component of paraxial fields**This paper is dedicated to the memory of our colleague and friend Professor Pedro M Mejías who recently died unexpectedly.

The analysis of paraxial Gaussian beams features in most undergraduate courses in laser physics, advanced optics and photonics. These beams provide a simple model of the field generated in the resonant cavities of lasers, thus constituting a basic element for understanding laser theory. Usually, uniformly polarized beams are considered in the analytical calculations, with the electric field vibrating at normal planes to the propagation direction. However, such paraxial fields do not verify the Maxwell equations. In this paper we discuss how to overcome this apparent contradiction and evaluate the longitudinal component that any paraxial Gaussian beam should exhibit. Despite the fact that the assumption of a purely transverse paraxial field is useful and accurate, the inclusion of the above issue in the program helps students to clarify the importance of the electromagnetic nature of light, thus providing a more complete understanding of the paraxial approach.

The behavior of the instantaneous Poynting vector of symmetrical laser beams

In this work we investigate the behavior of the instantaneous Poynting vector of symmetrical paraxial laser beams, namely the modification of the instantaneous Poynting vector and the radiation pattern during propagation in free space for a variety of such beams. As an example, we have investigated in detail the behavior of the instantaneous Poynting vector and the radiation pattern of the paraxial Gaussian and Bessel beams.

Fourth order correction to a paraxial Gaussian beam in a rotationally symmetric system of non-spherical surfaces

AbstractWe formulate the fourth order correction to a paraxial Gaussian beam propagating through a non-spherical surface system that is rotationally symmetric. First we represent the Gaussian beam by a complex-source-point spherical wave (CSPSW). Next we examine the evolution of the CSPSW through the non-spherical surface system by applying a Fresnel–Kirchhoff diffraction integral. We find a ray-optical solution to the diffraction integral in terms of both the conic constant and the aspheric coefficient of fourth order. Then we numerically evaluate the quality factor of the fourth order-corrected beam propagating through either an aspheric mirror or a thin lens made up of two non-spherical surfaces. The fourth order formulas derived here can be useful in determining the quality factor of the Gaussian beam degraded by a rotationally symmetric system of non-spherical surfaces.

Analysis of acoustic quasi-Gaussian beams and scattering by spheres at the focus of this type of beam.

A paraxial Gaussian beam is an approximation sometimes used to describe the focusing of ultrasonic waves. In this research an exact solution of the linear Helmholtz equation is examined that is similar to a Gaussian beam when weakly focused. This superposition has advantages for relatively tightly focused beams not describable with a paraxial approximation. The quasi-Gaussian beam considered is an appropriate superposition of zero-order Bessel beams [P. L. Marston (unpublished)]. This way of representing a focused beam has the additional attribute that an analytical result is known for the scattering by an isotropic sphere placed on the axis of a Bessel beam [P. L. Marston, J. Acoust. Soc. Am. 121, 753–758 (2007)]. In the focused case the scattering by a sphere centered on the focal point of the quasi-Gaussian beam follows by superposition. In comparison with the usual case of plane-wave illumination, mode amplitudes in the scattering are modified by a function depending on a ratio of incomplete gamma functions and the size of the focus of the beam. The analysis is useful for predicting the level of increase in scattering when a sphere is placed in a strongly focused beam. [Work supported by ONR.]

Scattering of acoustic quasi‐Gaussian beams and Bessel beams by fluid‐loaded spheres: Exact solutions and physical interpretation.

Exact solutions are available for the scattering of sound by spheres placed on the axis of certain types of acoustic beams. In the case of an acoustic Bessel beam the scattering for various elastic spheres is greatly modified relative to the standard case of plane‐wave illumination [P. L. Marston, J. Acoust. Soc. Am. 122, 247–252 (2007)]. Symmetry and geometric considerations explain why selected sphere modes can be suppressed in a way that depends on the Bessel beam parameters. The quasi‐Gaussian case is treated by an appropriate superposition of zero‐order Bessel beams displaying the focal properties of a Gaussian beam [P. L. Marston, submitted]. This superposition, unlike the case of a paraxial Gaussian beam, is an exact solution of the Helmholtz equation. The sphere is centered on the focal point of the beam and the scattering follows by superposition from the Bessel beam case. While not prone to suppress specific modes of the sphere, the quasi‐Gaussian case can also modify the scattering for a wide range of beam parameters and frequencies. Mode amplitudes are modified by a function depending on a ratio of incomplete gamma functions and the size of the focus of the beam. [Work supported by ONR.]

Quasi-Gaussian Bessel-beam superposition: Application to the scattering of focused waves by spheres

A superposition of zero-order Bessel beams is examined that closely resembles an idealized paraxial Gaussian beam, provided the superposition is not tightly focused. Plots compare wavefield properties in the focal region and in the far field for different values of kw(0), the product of the wavenumber k, and the focal-spot-radius w(0). The superposition (which is an exact solution of the Helmholtz equation) has the important property that the scattering by an isotropic sphere can be calculated without any approximations for the commonly considered case of linear waves propagating in an inviscid fluid. The nth partial wave amplitude is similar to the case of plane-wave illumination except for a weighting factor that depends on incomplete gamma functions. An approximation for the weighting factor is also discussed based on a generalization of the Van de Hulst localization principle for a sphere of radius a at the focus of a Gaussian beam. Examples display differences between the directionality of the scattering with the plane wave case even though for the cases displayed, ka does not exceed 2 and w(0)∕a is not less than 2. Properties of tightly focused wavefields and the partial wave weighting factors are discussed.

Evolution of a complex-source-point spherical wave through a symmetric optical system with spherical aberration

Abstract We discuss the evolution of a complex-source-point spherical wave (CSPSW) along the axis of a symmetric optical system with spherical aberration. The CSPSW is equivalent to the sum of all the higher order corrections to a paraxial Gaussian beam. First we formulate the diffracted CSPSW through spherical boundaries, where the terms of up to fourth order in aperture variables are taken into account. We then find a ray-optical solution for the diffracted beam that can be represented in terms of the coefficient of spherical aberration. Next we formulate the beam spot size, the far-field divergence angle, and the beam quality factor which are characteristic of the aberrated beam. For the case of a thick lens, we examine the effect of lens aberration on the beam parameters. The derived formulas can be used to determine the parameters of the Gaussian beam degraded by an arbitrary symmetric optical system with spherical aberration.

The Gouy phase anomaly for harmonic and time-domain paraxial Gaussian beams

SUMMARY The Gouy phase anomaly resulting from the focusing of wave solutions is illustrated using 2-D paraxial Gaussian beams. For harmonic Gaussian beams, this gives rise to a continuous variation of the Gouy phase as a function of propagation distance. This is in contrast to the discontinuous phase anomaly at caustics for ray solutions. However, as the beam-width of a Gaussian beam at a focus gets smaller, the Gouy phase anomaly becomes more concentrated near the focus and approaches that of the ray solution. The Gouy phase for a harmonic Gaussian beam is first illustrated in a homogeneous medium, and then in a quadratic velocity waveguide where the beam can pass through multiple focus points. However for multiple focus points, care must be taken to ensure that the phase remains continuous. Finally, an example is shown of the Gouy phase for a time-domain signal using a Gabor wavelet. This is validated using the finite difference method, and illustrates the progressive phase advance of a time-domain signal modifying the pulse shape with distance. Intuitively, as a wave solution gets ‘squeezed’ at a focus, it ‘squirts’ forward by slightly increasing its apparent speed in the propagation direction and modifying the pulse shape. However, this is a phase advance and not a group or energy advance and does not violate causality. Nonetheless, this could potentially influence the interpretation of travel-times using correlation techniques when using sources that generate beamed signals, for example from transducer sources in the laboratory.

Focal Shift of Paraxial Gaussian Beams in a Left-Handed Material Slab Lens

Focal shift is inevitable in conventional lens systems due to the Fresnel number and angular aperture. In this Letter, we demonstrate that there is no focal shift when a paraxial Gaussian beam passes through a left-handed material slab lens without absorption or gain. However, the effect is exhibited in the presence of absorption or gain, and becomes larger as the absorption or gain increases. When the absorption is equal to the gain, the phenomenon of the focal shift caused by the gain is more obvious. In addition, the field distribution is not affected by the absorption or gain and always remains Gaussian both in internal and external focus planes.

A Cell Lens Model for Transverse Modes in Optofluidic Intracavity Spectroscopy

Optofluidic intracavity spectroscopy of individual cells from canine hemangiosarcoma (HSA) and canine lymphoma cancer cell lines exhibit relatively uniformly spaced multiple transverse modes repeated in each free spectral range of a microfluidic Fabry-Pérot cavity, while similar spectra of healthy canine monocytes and lymphocytes have minimal or no transverse-mode peaks. Modeling of the cells as thin lenses allows paraxial Gaussian beam resonator analysis that produces spectral features that quantitatively match the frequencies of transverse modes and qualitatively agree with the trends in maximum transmission of the modes when aperture losses are included. Cell focal lengths of 600 and 300 μm, respectively, are extracted for canine HSA and lymphoma cells from the ratio of transverse to longitudinal frequency spacing, while the spectra of canine monocytes indicate that they have a focal length ≥ 580 μm. Under an assumption of constant refractive index, the focal lengths imply a radius of curvature for the top surface of the cells that is much greater than their lateral radius.

Transformation relations for second-order derivatives of travel time in anisotropic media

In the computation of paraxial travel times and Gaussian beams, the basic role is played by the second-order derivatives of the travel-time field at the reference ray. These derivatives can be determined by dynamic ray tracing (DRT) along the ray. Two basic DRT systems have been broadly used in applications: the DRT system in Cartesian coordinates and the DRT system in ray-centred coordinates. In this paper, the transformation relations between the second-order derivatives of the travel-time field in Cartesian and ray-centred coordinates are derived. These transformation relations can be used both in isotropic and anisotropic media, including computations of complex-valued travel times necessary for the evaluation of Gaussian beams.

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.

An automated submicron beam profiler for characterization of high numerical aperture optics

An automated, interferometrically referenced scanning knife-edge beam profiler with submicron resolution is demonstrated by directly measuring the focusing properties of three aspheric lenses with numerical aperture (NA) between 0.53 and 0.68, with spatial resolution of 0.02 microm. The results obtained for two of the three lenses tested were in agreement with paraxial Gaussian beam theory. It was also found that the highest NA aspheric lens, which was designed for 830 nm, was not diffraction limited at 633 nm. This process was automated using motorized translation stages and provides a direct method for testing the design specifications of high numerical aperture optics.

A study of the Thomson scattering of radiation by a relativistic electron of a tightly-focused, co-propagating femtosecond laser beam

The paraxial approximation to a laser field is no longer valid when the beam waist of the laser is of the order of its wavelength. In such a case, a laser beam propagating even in free space is not purely transverse but contains a significant longitudinal field. The present study reveals that the characteristics of the Thomson scattering by a relativistic electron (80 MeV) of a tightly-focused, co-propagating laser field is sensitive to the electron's initial transverse position and phase with respect to the laser field: the electron radiates more strongly when it is initially located off the laser axis by about the beam waist than when on the laser axis. An enhancement of about 2000 times in the radiation power is noticed for the focused (beam waist w0 of 5 μm) laser intensity of 5×1018 W cm- 2 compared to a paraxial Gaussian beam case.