Analysis of dark spot formation in absorbing liquid media

Abstract

A theoretical analysis based on coupled field-matter equations is given to describe the recently observed phenomenon of a central dark spot formation of a Gaussian beam transmitted through an absorbing defocusing liquid medium. We find that such a pattern formation, which is accompanied by normal defocusing rings in the far field, originates from interplay between the wave-front curvature of the Gaussian beam and strong spatial self-phase medulation arising from thermally induced refractive index change in the medium. Results of numerical analysis for a thin medium are shown to be in a good quantitative agreement with our experimental findings. Further, the dark spot formation is also predicted by using a focused Gaussian beam and self-focusing medium.     

Subwavelength-resolvable focused non-gaussian beam shaped with a binary diffractive optical element.

The diffraction-limited spot size limits the optical disk storage capacity and microscopic resolution. We describe a technique to shape a focused Gaussian beam into a superresolving beam by using a diffractive optical element fabricated by laser-assisted chemical etching. The focused shaped beam has a smaller width and a longer depth of focus than a similarly focused Gaussian beam. Using the diffraction-limited shaped beam along with threshold writing, we achieved a written pit size of less than 0.33 mum at a 695-nm laser wavelength, compared with a 0.7-mum focused Gaussian spot size (full width at e(-2) of the peak) with the same focusing lens. The energy conversion efficiency for the beam shaping was ~81%.     

Variational analysis of z scan of thick medium with an elliptic Gaussian beam

By variational approach, we analyze the characteristics of beam propagation through a cubic optical nonlinear medium using a laser beam that has a transverse elliptic Gaussian profile. The analytic solution to the normalized transmittance at the center of the far field as a function of medium position and the beam characteristics is obtained and compared with the numerical simulation, which is realized by a combination of algorithms. We also analyze the peak-valley transmittance difference as a function of medium length, ellipticity, and a stigmatism. The relationship between peak-valley normalized transmittance difference of the z-scan trace and aperture size or the slit width are obtained. Meanwhile, the comparison of z-scan characteristics with an elliptic Gaussian beam with those using a circular symmetric Gaussian beam is made.     

Determination of waist parameters of a Gaussian beam

A simple method is presented for determining the waist position and the waist size of a Gaussian beam from measured spot sizes. The method does not require any least-squares process, and substitution of measured spot sizes directly into the formulas gives the waist parameters. Only few data are required to determine the parameters accurately as long as measured spot sizes contain small errors.     

A Semi-analytic Approach for Coherent Laser Radar System Efficiency

Abstract

A nearest Gaussian approximation (NGA) is proposed to approximate any shape for a single mode laser beam by a Gaussian shape. The application considered is a determination of the system efficiency in heterodyne coherent laser radar (HCLR). For an actual beam its NGA is defined by three parameters: the waist spot size and location, and an amplitude coefficient. These parameters are computed by a maximization of the norm of the scalar product written for the actual and Gaussian beams. In the case of the truncated Gaussian beam, particularly relevant to HCLR, the waist location can be analytically calculated, and only two parameters remain unknown: the waist spot size and amplitude coefficient. Using numerical applications, it is shown the NGA is in good agreement with Fresnel integral solution. The NGA combines a good accuracy and capability of analytical solutions. It can treat a variation in system efficiency owing to a misalignment angle between the transmitter and local oscillator.     

Acoustooptic 2-D profile shaping of a Gaussian laser beam

Acoustooptic 2-D profile shaping of a Gaussian laser beam has been achieved by two plane ultrasonic waves progressing in orthogonal directions. The spot size W of the Gaussian laser beam must be considerable less than the wavelength lambda of the ultrasonic wave at the acoustooptic interaction region. The ultrasonic cell is dealt with as a Raman-Nath 2-D phase grating but serves as a 2-D beam deflector in time for the interaction scheme of interest. The wave front of the Gaussian laser beam must be almost plane in the interaction region. The profile shaping condition is 0.15 < or = (W/lambda) < or = 0.30 only when the Raman-Nath parameter dependent on the ultrasonic power has values between v = 1.0 and 2.0.     

Modified Fresnel zone plates that produce sharp Gaussian focal spots

A modified Fresnel zone plate that can produce an approximate Gaussian focal spot is proposed for the focusing and imaging of soft x rays and extreme ultraviolet radiation. The selection conditions for the positions and the widths of the concentric open rings are analytically presented. The focal spot size can be much smaller than the width of the narrowest open ring, and the sidelobes and the higher orders can be effectively suppressed. Through numerical experiments, we confirm that a Gaussian focal spot with a beam width of 7.7 nm can be produced by a modified Fresnel zone plate with a minimum structure size of 30 nm.     

Effective beam parameters and the turbulent beam waist for convergent Gaussian beams

Expressions are developed for the location and the size of the beam waist for a convergent Gaussian beam in statistically homogeneous and isotropic atmospheric turbulence. Subsidiary expressions are presented that lead to the maximum distance from the transmitter at which the beam waist can be located under given optical turbulence conditions and the optimal initial radius of curvature required for placing the beam waist at a desired location. The free-space beam radius W of a Gaussian beam satisfies the relationship ?W/?z = - W/R, where z represents the path length and R is the phase-front radius of curvature at z. By enforcing this relation on the effective beam spot size in turbulence W(e), we can define an effective radius of curvature R(e). In addition to specifying the beam waist, R(e) leads to a pair of effective beam parameters θ(e) and Λ(e) that provide a natural extension to the complex amplitude plane. Within this context, general propagation characteristics may be described, including the coherence properties of a Gaussian beam in both weak and strong optical turbulence.     

环形孔径高斯光束的远场特性

讨论环形孔径高斯光束的远场光斑能量分布，并给出拟合表达式。首先推导光束没有抖动时远场光斑的光能分布表达式，然后研究各种不同中心遮拦和不同光束截断比，在有以及没有激光束抖动的情况下，对高斯光束的远场环围能量分布变化的影响。     

Reducing the uncertainty in laser beam size measurement with a scanning edge method

A modification in the analysis of a conventional laser beam spot size measurement method has been developed. The new analysis significantly decreases the uncertainty in the estimation of the beam-spot size. A conventional beam scanning approach was used in the measurement, but instead of differentiating the data and fitting the result to a Gaussian function, the data were fit to an analytical approximation to the complementary error function. As a result, fitted parameters were obtained that were consistent with the standard differentiation approach, but with considerably smaller uncertainty.     

Spot size, depth-of-focus, and diffraction ring intensity formulas for truncated Gaussian beams

Simple polynomial formulas to calculate the FWHM and full width at 1/e2 intensity diffraction spot size and the depth of focus at a Strehl ratio of 0.8 and 0.5 as a function of a Gaussian beam truncation ratio and a system f-number are presented. Formulas are obtained by use of the numerical integration of a Huygens-Fresnel diffraction integral and can be used to calculate the number of resolvable spots, the modulation transfer function, and the defocus tolerance of optical systems that employ laser beams. I also derived analytical formulas for the diffraction ring intensity as a function of the Gaussian beam truncation ratio and the system f-number. Such formulas can be used to estimate the diffraction-limited contrast of display and imaging systems.     

Aberration limits for annular Gaussian beams for optical storage

Annularly apodized beams have been suggested for use in optical storage because of their potential to go beyond the conventional spot size and depth-of-focus limits. One concern for such applications is the effects of small aberrations on beams in which the energy is concentrated in a small annular ring. We present calculations and experimental results that show that annular apodization of a Gaussian beam reduces the sensitivity to defocus as well as balanced spherical and coma aberrations. The sensitivity to astigmatism is increased by a small amount.     

Tunable beam focusing based on optical bistability with a composite Kretschmann configuration involving a Kerr medium

A beam focuser with a composite Kretschmann configuration involving a Kerr medium is investigated theoretically. The structure employs a silver film with four slits filled with the Kerr medium. Optical bistability and beam focusing are demonstrated, and changes to the incident intensity result in optical bistability characteristics in the reflection, focused intensity, spot size, and depth of focus. The proposed structure has the potential to be applied for optical switching and nano-illumination.     

Transmission of a two-dimensional Gaussian beam into a uniaxial crystal.

We study the transmission of a two-dimensional (2-D) TM Gaussian beam through a plane interface between an isotropic medium (e.g., air) and a uniaxially anisotropic crystal. The optic axis of the crystal is taken to be in the plane of incidence but is arbitrarily oriented relative to the interface normal. We show that, in the paraxial approximation, a nontruncated transmitted 2-D TM Gaussian beam inside a uniaxial crystal can be expressed in a form similar to that of a scalar Gaussian beam that propagates in a homogeneous medium. We also show that the transmitted beam corresponding to an incident 2-D TM Gaussian beam with its main propagation direction along the interface normal is tilted inside the crystal by the same angle as is the transmitted axial ray that corresponds to a normally incident ray.     

Beam-divergence control of excimers with plane-parallel Gaussian cavities

Plane-parallel cavities with Gaussian-reflectivity-profile mirrors as full reflectors were applied to a XeCl laser, and the near- and the far-field characteristics of the laser radiation were analyzed. It is shown that excimer lasers fitted with plane-parallel Gaussian cavities deliver laser radiation with a beam-quality factor M(2) that is more than 50% smaller than that of laser beams delivered by conventional plane-parallel cavities. The effect of the Gaussian mirror spot size on M(2) was also investigated, and it is shown that the narrowing of the Gaussian mirror spot size reduces the beam-quality-factor value.     

Precise recursive formula for calculating spot size in optical waveguides and accurate evaluation of splice loss

The spot size of a single-mode waveguide was defined from the viewpoint of a least-squares fit of the field profile to a Gaussian profile. The field profile was expanded in terms of Hermite-Gaussian functions, and a new precise recursive formula for calculating the spot size was derived. It was shown that our formula is equivalent to the best fitting of the offset coupling loss to that of a Gaussian profile and keeps its form against the Fourier transform that corresponds to the diffraction in the same manner as the Gaussian profile. The accuracy of conventional formulas and our new spot-size formula was compared with the exact value defined from the viewpoint of a least-squares fit to a Gaussian profile, and it was shown that our recursive formula is the most accurate of the approximate formulas. Next we proposed a new formula for calculating the splice loss between two waveguides and showed that our formula is more accurate than the conventional one, which involves only the spot size.     

Optimized coupling of a Gaussian beam into an optical waveguide with a grating coupler: comparison of experimental and theoretical results

The coupling efficiency of grating couplers is derived for a Gaussian incident beam. Its optimum value depends on the beam waist and on the position of a light spot with respect to the coupler edge for given grating parameters. The characteristic coupling length has been experimentally determined for the grating coupler studied. Relative measurements of the coupling efficiency as a function of incident beam characteristics are in good agreement with the numerical results.     

Simple and rigorous analytical expression of the propagating field behind an axicon illuminated by an azimuthally polarized beam

A simple and rigorous analytical expression of the propagating field behind an axicon illuminated by an azimuthally polarized beam has been deduced by use of the vector interference theory. This analytical expression can easily be used to calculate accurately the propagation field distribution of azimuthally polarized beams throughout the whole space behind an axicon with any size base angle, not just restricted inside the geometric focal region as does the Fresnel diffraction integral. The numerical results show that the pattern of the beam produced by the azimuthally polarized Gaussian beam that passes through an axicon is a multiring, almost-equal-intensity, and propagation-invariant interference beam in the geometric focal region. The number of bright rings increases with the propagation distance, reaching its maximum at half of the geometric focal length and then decreasing. The intensity of bright rings gradually decreases with the propagation distance in the geometric focal region. However, in the far-field (noninterference) region, only one single-ring pattern is produced and the dark spot size expands rapidly with propagation distance.     

Extended focus diffractive optical element for Gaussian laser beams

The depth of focus of the Gaussian beam is extended by introducing a wavefront phase correction with properly designed diffractive optical elements. Results of the computer simulations show that, compared with other methods, the presented method demonstrates a reduced focal spot size and low sidelobes in a focal domain, within a considerable range of defocusing distances. Experimental results for the visible range diffractive optical element with a focus of 40 mm and a depth of focus that extends to 1 mm agree with the theory.