Hollow elliptical Gaussian beam and its propagation through aligned and misaligned paraxial optical systems

A new mathematical model called hollow elliptical Gaussian beam (HEGB) is proposed to describe a dark-hollow laser beam with noncircular symmetry in terms of a tensor method. The HEGB can be expressed as a superposition of a series of elliptical Hermite-Gaussian modes. By using the generalized diffraction integral formulas for light passing through paraxial optical systems, analytical propagation formulas for HEGBs passing through paraxial aligned and misaligned optical systems are obtained through vector integration. As examples of applications, evolution properties of the intensity distribution of HEGBs in free-space propagation were studied. Propagation properties of HEGBs through a misaligned thin lens were also studied. The HEGB provides a convenient way to describe elliptical dark-hollow laser beams and can be used conveniently to study the motion of atoms in a dark-hollow laser beam.     

Spectral shifts and spectral switches of a pulsed Gaussian beam from a rectangular aperture in the far field

The far-field anomalous spectral behaviours of a space–time-dependent Gaussian pulsed beam passing through a rectangular aperture are studied. By expanding a hard aperture function into a finite sum of complex Gaussian functions and starting from the Fresnel diffraction integral, the approximate analytical expression for the spectral intensity of a space–time-dependent Gaussian pulsed beam passing through a rectangular aperture is derived. Meanwhile, the corresponding closed-forms for the slit and the unapertured cases are also given as special cases of the general results. The red and blue shifts and the spectral intensity distribution are studied and illustrated with numerical calculations. Specifically, it is shown that the spectral switch takes place when the truncation parameter is equal to particular values or the observation position is at the critical diffraction angle. The possibility of tunable spectral switching in the far field with an apertured pulsed beam by varying the size of the rectangular aperture is highlighted.     

Propagation of Laguerre-Gaussian and elegant Laguerre-Gaussian beams in apertured fractional Hankel transform systems

On the basis of the fact that a hard-edged-aperture function can be expanded into a finite sum of complex Gaussian functions, approximate analytical expressions for the output field distribution of a Laguerre-Gaussian beam and an elegant Laguerre-Gaussian beam passing through apertured fractional Hankel transform systems are derived. Some numerical simulation comparisons are done, by using the approximate analytical formulas and diffraction integral formulas, and it is shown that our method can significantly improve the numerical calculation efficiency.     

Fractional Fourier transform for beams generated by Gaussian mirror resonator

Based on the definition of the fractional Fourier transform (FRFT) in the cylindrical coordinate system and the fact that a hard-edged-aperture function can be expanded into a finite sum of complex Gaussian functions, the propagation properties of beams generated by Gaussian mirror resonator passing through FRFT optical systems have been studied in detail, and some typical numerical examples are given to compare the results obtained by the approximate analytical method with those by the numerical integral method, and it is shown that our method can significantly improve the numerical calculation efficiency. Further, the results indicate that the normalized intensity distributions in the FRFT plane are closely related to the fractional order, truncation parameter and initial beam parameters. The variation period of normalized intensity distributions with p is 2 whatever the value of the truncation parameter δ is.     

Near-field anomalous spectral behavior in diffraction of a Gaussian pulsed beam from an annular aperture

Based on the Fresnel diffraction integral and by introducing a hard-aperture function into a finite sum of complex Gaussian functions, the approximate analytical expression for the near-field spectral intensity distribution of a space-time-dependent Gaussian pulsed beam passing through an annular aperture is derived, which permits us to study the on- and off-axis spectral anomalies that are near phase singularities of the diffracted Gaussian pulsed beam in the near-field. The expressions for a circular black screen and a circular aperture are given as special cases of the general results. The relative spectral shift of a space-time-dependent Gaussian pulsed beam versus the different values of the truncation parameters and the position parameters of observation points are also studied and illustrated with numerical calculations. It is shown that the spectral switch appears near phase singularities in the near-field, and the near-field spectral behavior depends on the truncation parameters, the pulse duration tau, and the position parameter. The results of this work have potential applications in free-space information encoding and transmission.     

An intuitive model for on-axis pulse evolution of ultrashort pulsed Gaussian beams diffracted from a circular aperture

The diffraction of ultrashort pulsed Gaussian beams from a circular aperture is studied by means of Fresnel diffraction integral and Fourier transform method. A uniform analytical expression is derived for temporal pulse form of ultrashort pulsed Gaussian beams in two cases, i.e. with constant beam waist and with constant diffraction length. It is shown that the on-axis pulse can be formulated as a superposition of an unapertured pulse and an aperture-induced pulse. The superposition of these two pulses leads to an enhanced pulse intensity for small truncation parameters at certain distances in the near field. Our results may find applications in high-intensity laser waveform control.     

Gaussian beam transfer through hard-aperture optics

We consider Gaussian beam diffraction by hard circular and rectangular-slit apertures. Both numerical results and accurate elementary analytic approximations are derived for the fraction of transmitted power (or energy) contained within the main central lobe of the far-field (or focal-plane) irradiation distribution as a function of the truncation ratio.     

Target-plane intensity approximation for apertured Gaussian beams applied to heterodyne backscatter lidar systems

With an application in lidar systems in mind, we investigate the effects of transmit-aperture truncation of Gaussian beams by employing the extended Huygens-Fresnel principle. We derive an approximation to the top-hat aperture-transmission function by defining an abstract Gaussian aperture-transmission function. The two fitting parameters of the latter are found when the beam radius and the on-axis intensity for both aperture cases are equated in the observation plane. Bounds for the applicability of the approximation are established, and its accuracy and usefulness is demonstrated through application to the calculation of the return signal of a heterodyne lidar system.     

Experimental observation of truncated fractional Fourier transform for a partially coherent Gaussian Schell-model beam

The truncated fractional Fourier transform (FRT) is applied to a partially coherent Gaussian Schell-model (GSM) beam. The analytical propagation formula for a partially coherent GSM beam propagating through a truncated FRT optical system is derived by using a tensor method. Furthermore, we report the experimental observation of the truncated FRT for a partially coherent GSM beam. The experimental results are consistent with the theoretical results. Our results show that initial source coherence, fractional order, and aperture width (i.e., truncation parameter) have strong influences on the intensity and coherence properties of the partially coherent beam in the FRT plane. When the aperture width is large, both the intensity and the spectral degree of coherence in the FRT plane are of Gaussian distribution. As the aperture width decreases, the diffraction pattern gradually appears in the FRT plane, and the spectral degree of coherence becomes of non-Gaussian distribution. As the coherence of the initial GSM beam decreases, the diffraction pattern for the case of small aperture widths gradually disappears.     

Approximate method for the generalized M2 factor of rotationally symmetric hard-edged diffracted flattened Gaussian beams

On the basis of the truncated second-order moments method in the cylindrical coordinate systems and the expansion of the hard-edged aperture function into a finite sum of complex Gaussian functions, an approximate method used to calculate the generalized beam propagation factor (M2 factor) is proposed. The approximate analytical expressions of the generalized M2 factor for rotationally symmetric hard-edged diffracted flattened Gaussian beams defined by Gori [Opt. Commun. 107, 335 (1994)] and Li [Opt. Lett. 27, 1007 (2002)] are derived, respectively; we show that it depends on the beam order N and the beam truncation parameter delta. Some typical numerical examples are given to illustrate its applications that we compare by using the obtained analytical method and the numerical integration method.     

Propagation of four-petal Gaussian beams in apertured fractional Fourier transforming systems

Based on the Collins diffraction integral formula and the fact that a hard-edged-aperture function can be expanded into a finite sum of complex Gaussian functions, the propagation of a four-petal Gaussian beam passing through an apertured fractional Fourier transform (FRFT) optical system has been studied in detail. Some typical numerical examples are given to illustrate the properties of the four-petal Gaussian beam in the FRFT plane. The results indicate that the intensity distributions of the beam in the FRFT plane are closely related to not only the fractional order but also the beam parameters and the aperture parameters.     

Propagation characteristics of a diffracted M2 beam

The propagation characteristics of a beam diffracted by a circular aperture are investigated. The beam-quality factor M2 defined by an 86.5% power-content radius is given theoretically and experimentally as a function of the truncation ratio. It is found that the theoretical limit of M2 is 2.37 times as great as that of an incident beam as the truncation ratio approaches 0. For a weakly diffracted beam a simple formula giving M2 is derived. Although M2 does not increase much with diffraction, the influence of diffraction should be taken into account in beam brightness.     

Acousto-optic interaction of a Gaussian laser beam with an ultrasonic wave of cylindrical symmetry

We present experimental studies of the interaction between a narrow Gaussian laser beam and a standing cylindrical ultrasonic wave. As a theoretical approach, a Fourier-optics-based successive diffraction model is used. Depending on the ratio of the Gaussian laser beam diameter to the first nodal diameter of the cylindrical ultrasound, light refraction or diffraction is observed. We experimentally investigate the time-averaged light intensity as well as the modulation of light in the far field of light refraction-diffraction by a cylindrical ultrasound. It is revealed that significant focusing appears if the phase front of the incident light is curved. The focusing effects of the acousto-optic system depend on the width of the laser beam and curvature of the phase front. Finally, possible applications are discussed.     

Phase Retrieval From Fresnel Zone Intensity Measurements by use of Gaussian Filtering

The reconstruction of a complex-valued object from intensity measurements in the Fresnel zone region is considered. The reconstruction is based on the phase-retrieval method from Fresnel zone intensity data obtained by modulation of the object with a known Gaussian function and with its shifted functions along the horizontal and the vertical directions in rectangular coordinates. Two types of reconstruction system are presented. In the two systems a Gaussian amplitude filter and a Gaussian illuminating beam are used as the known Gaussian functions. Computer-simulated examples in two dimensions illustrate the performance of the reconstruction in these systems.     

Propagation of a decentered elliptical Gaussian beam through apertured aligned and misaligned paraxial optical systems

By expanding the hard aperture function into a finite sum of complex Gaussian functions, approximate analytical formulas for a decentered Gaussian beam (DEGB) passing through apertured aligned and misaligned paraxial apertured paraxial optical systems are derived in terms of a tensor method. The results obtained by using the approximate analytical expression are in good agreement with those obtained by using the numerical integral calculation. Furthermore, approximate analytical formulas for a decentered elliptical Hermite-Gaussian beam (DEHGB) through apertured paraxial optical systems are derived. As an application example, approximate analytical formulas for a decentered elliptical flattened Gaussian beam through apertured paraxial optical systems are derived. Our results provide a convenient way for studying the propagation and transformation of a DEGB and a DEHGB through apertured paraxial optical systems.     

Dynamics of the linearly polarized fundamental Gaussian light wave

A continuous planar array of dipoles that are oriented in a particular direction and have an amplitude distribution that is Gaussian in the paraxial limit is introduced as a source for the fundamental Gaussian light wave. The radiation intensity of the Gaussian light wave is determined and its characteristics are analyzed. The universal Gaussian beam factor is deduced and identified as the radiation intensity of the scalar Gaussian wave. The total radiated power, the mean center of the localized wave, and the beam widths of the intensity distribution are obtained. The ratio of the power in the Gaussian wave to that in the corresponding paraxial Gaussian beam is used as a measure of the quality of the paraxial beam approximation. A limiting factor for the power ratio is introduced as an indicator for the acceptability of the paraxial beam approximation. The cross section and the beam widths of the localized light wave are investigated in the large and small kw0 limits, where k is the wavenumber and w0 is the beam waist at the input plane. The beam width of the full Gaussian wave is found to be less than that of the corresponding paraxial Gaussian beam both for the scalar Gaussian wave and for the Gaussian light wave.     

Fresnel and Far-field Diffraction of a Truncated Elliptical Gaussian Beam Due to an Elliptical Aperture

Abstract

Using the Hopkins algorithm, expressions are derived for the intensity patterns, in both the Fresnel and far-field regions, associated with the diffraction of a plane-wave elliptical Gaussian beam truncated by an elliptical aperture. This is accomplished by evaluating the diffraction integral subject to the Fresnel approximation. Numerical results are presented that indicate how the truncation parameter affects the side-lobe level in the Fresnel and far-field regions.     

Decentered Gaussian beams, ray bundles, and Bessel-Gauss beams

The concept of ray parameters for decentered Gaussian beams is developed on a formal basis. When the beam propagates through first-order optical systems, these parameters are transformed as the ray parameters of geometrical optics. It is shown how this feature helps one to understand the behavior of more sophisticated beams that can be considered as bundles of decentered Gaussian beams. In particular, the case of Bessel-Gauss beams and their recently introduced generalizations is analyzed, and simple transformation formulas are obtained.     

Collimating cylindrical diffractive lenses: rigorous electromagnetic analysis and scalar approximation

Practical collimating diffractive cylindrical lenses of 2, 4, 8, and 16 discrete levels are analyzed with a sequential application of the two-region formulation of the rigorous electromagnetic boundary-element method (BEM). A Gaussian beam of TE or TM polarization is incident upon the finite-thickness lens. F/4, F/2, and F/1.4 lenses are analyzed and near-field electric-field patterns are presented. The near-field wave-front quality is quantified by its mean-square deviation from a planar wave front. This deviation is found to be less than 0.05 free-space wavelengths. The far-field intensity patterns are determined and compared with the ones predicted by the approximate Fraunhofer scalar diffraction analysis. The diffraction efficiencies determined with the rigorous BEM are found to be generally lower than those obtained with the scalar approximation. For comparison, the performance characteristics of the corresponding continuous Fresnel (continuous profile within a zone but discontinuous at zone boundaries) and continuous refractive lenses are determined by the use of both the BEM and the scalar approximation. The diffraction efficiency of the continuous Fresnel lens is found to be similar to that of the 16-level diffractive lens but less than that of the continuous refractive lens. It is shown that the validity of the scalar approximation deteriorates as the lens f-number decreases.     

Focusing Properties of the Gaussian Beam Generated by Optical Resonators with Gaussian Reflectivity Mirrors

Abstract

This paper presents a parametric study concerning the focusing properties of the light beam generated by optical resonators with Gaussian reflectivity mirrors. The diffraction integral has been evaluated and analytical expressions are derived for intensity patterns near the focus in systems of different Fresnel numbers. Universal curves have been plotted against dimensionless parameters to show the variations in the intensity patterns due to the change in focusing geometry and the incident beam intensity profile. Modification to the well known diffraction formula are considered in order to extend the conventional Gaussian formalism to the case of Gaussian mirror resonators.