Propagation of elliptical Gaussian beams modulated by an elliptical annular aperture

An analytical propagation expression for a generalized astigmatic elliptical Gaussian beam (EGB) modulated by an elliptical annular aperture and passing through an axially nonsymmetrical optical system is obtained by the use of vector integration. The derived analytical results provide more convenience for studying the propagation and transformation of EGBs than the usual method of using the diffraction integral directly, and the efficiency of the numerical calculation is significantly improved. Some numerical simulations are illustrated for the propagation properties of EGBs passing through a free space with an elliptical annular aperture, an elliptical screen, or an elliptical aperture. Further extensions are also pointed out.     

Propagation of elliptical Gaussian beams in apertured and misaligned optical systems

On the basis of the fact that a hard-edged elliptical aperture can be expanded approximately as a finite sum of complex Gaussian functions in tensor form, an analytical propagation expression for an elliptical Gaussian beam (EGB) through a misaligned optical system with an elliptical aperture is derived by use of vector integration. The approximate analytical results provide more convenience for studying the propagation and transformation of EGBs than the usual way by using a diffraction integral directly, and the efficiency of numerical calculation is improved. Some numerical simulations are illustrated for the propagation properties of EGBs through apertured optical transforming systems with misaligned thin lenses.     

Decentered elliptical Gaussian beam

A new kind of laser beam, called a decentered elliptical Gaussian beam (DEGB), is defined by a tensor method. The propagation formula for a DEGB passing through an axially nonsymmetrical paraxial optical system is derived through vector integration. The derived formula can be reduced to the formula for a fundamental elliptical Gaussian beam and a decentered Gaussian beam under certain conditions. As an example application of the derived formula, the propagation characteristics of a DEGB in free space are calculated and discussed. As another example we study the properties of a generalized laser beam array constructed by use of a DEGB as the fundamental mode.     

Decentered elliptical Hermite-Gaussian beam

A new kind of laser beam called the decentered elliptical Hermite-Gaussian beam (DEHGB) is defined by use of a tensor method. The propagation formula of the DEHGB passing through a nonsymmetrical paraxial optical system is derived through vector integration. The derived formula can be easily reduced to the propagation formula of an aligned elliptical Hermite-Gaussian beam and that of a decentered elliptical Gaussian beam under certain conditions. By use of this formula, the propagation characteristics of the DEHGB through free space are presented graphically. As application examples, we construct a generalized laser array using the DEHGB as the fundamental mode. We also obtain the decentered elliptical flattened Gaussian beam by expressing it as superposition of a series of DEHGBs by using polynomial expansion.     

Fractional Fourier transform of truncated elliptical Gaussian beams

Based on the fact that a hard-edged elliptical aperture can be expanded approximately as a finite sum of complex Gaussian functions in tensor form, an analytical expression for an elliptical Gaussian beam (EGB) truncated by an elliptical aperture and passing through a fractional Fourier transform system is derived by use of vector integration. The approximate analytical results provide more convenience for studying the propagation and transformation of truncated EGBs than the usual way by using the integral formula directly, and the efficiency of numerical calculation is significantly improved.     

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.     

Elliptical Hermite-cosine-Gaussian beam and its propagation characteristics

A new kind of light beam named the elliptical Hermite-cosine-Gaussian beam (EHCosGB) is introduced in this paper by using a tensor method. An analytical propagation expression for an EHCosGB passing through an axially nonsymmetrical optical system is derived by using vector integration. By virtue of numerical simulations, the propagation properties of EHCosGBs through a free space and a focusing system are illustrated, respectively. The results indicate that an EHCosGB on the input plane can be regarded as the elliptical Hermite-Gaussian beam (EHGB) modulated by the cosine curves with certain oscillating frequency. Some of the neighboring lobes are combined and TEM-type Hermite-cosh-Gaussian beams are formed during the propagation.     

Decentred elliptical flat-topped light beams and their coherent and incoherent combinations

A new kind of laser beam called the decentred elliptical flat-topped light beam (DEFTB) is introduced in this paper by using a tensor method. Based on the generalized Collins integral, the propagation formula for a DEFTB passing through an axially nonsymmetrical paraxial optical system is derived through vector integration. The derived formula can be reduced to the formulae for the generalized decentred elliptical Gaussian beam and elliptical flat-topped light beam under certain conditions. As a typical application example of the derived formula, the propagation characteristics of a DEFTB in a thin lens system are calculated and discussed. Furthermore, the generalized laser beam arrays with rectangular symmetry and radial symmetry by using DEFTBs as fundamental modes are constructed and the propagation properties of both a coherent and an incoherent combination of DEFTBs in the thin lens system with numerical examples are investigated.     

Propagation of hollow Gaussian beams through apertured paraxial optical systems

On the basis of the generalized Collins formula and the expansion of the hard-aperture function into a finite sum of complex Gaussian functions, an approximate analytical formula for a hollow Gaussian beam propagating through an apertured paraxial stigmatic (ST) ABCD optical system is derived. Some numerical examples are given. Furthermore, by using a tensor method, we derive approximate analytical formulas for a hollow elliptical Gaussian beam propagating through an apertured paraxial general astigmatic ABCD optical system and an apertured paraxial misaligned ST ABCD optical system. Our results provide a convenient way for studying the propagation and transformation of a hollow Gaussian beam and a hollow elliptical Gaussian beam through an apertured general optical system.     

Propagation characteristics of Gaussian beams through a paraxial ABCD optical system with an annular aperture

By introducing a hard aperture function into a finite sum of complex Gaussian functions, an approximate analytical expression for a Gaussian beam passing through a paraxial ABCD optical system with an annular aperture has been derived. The results could be reduced to the case of circular black screen or circular aperture. Some numerical simulations are also performed and illustrated for the propagation characteristics of a Gaussian beam through a paraxial ABCD optical system with an annular aperture, a circular black screen or a circular aperture.     

Propagation and transformation of flat-topped multi-Gaussian beams in a general nonsymmetrical apertured double-lens system

On the basis of expanding a hard-edged aperture function as a finite sum of complex Gaussian functions, an approximate analytical expression for the propagation of an input complex amplitude distribution passing through a general nonsymmetrical apertured double-lens system is derived. Then, the propagation result for two-dimensional flat-topped multi-Gaussian beams is given. It is shown that the apertured Lohmann's symmetrical double-lens system for fractional Fourier transform is a special case of the general apertured double-lens system. The numerical calculation, graphical illustration, and some discussions for the transformation of the two-dimensional flat-topped multi-Gaussian beam in apertured Lohmann's symmetrical double-lens systems are also presented.     

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.     

Propagation of anomalous hollow beams through a circular-apertured optical system

In this paper, we consider the propagation of anomalous hollow beams through a circular-apertured paraxial ABCD optical system. Based on the Huygens–Fresnel integral and the expansion of the circular aperture function into a finite sum of Gaussian functions, the approximation analytical expression of an anomalous hollow beam passing through a circular-apertured paraxial ABCD optical system is derived. Based on the approximation analytical expression, the propagation properties in free space are numerically illustrated. This method provides a convenient tool for studying the propagation and transformation properties of anomalous hollow beams passing through a circular-apertured paraxial ABCD optical system.     

Controllable elliptical dark-hollow beams

A new (to our knowledge) kind of light beam called the controllable elliptical dark-hollow beam (CEDHB), is introduced to describe dark-hollow beams with axially rotational asymmetry by using the tensor method. The propagation formulas of CEDHBs through paraxial aligned and misaligned nonsymmetrical optical systems are derived through vector integration. With the derived formulas, the propagation properties of CEDHBs in free-space propagation and through a misaligned thin lens are studied graphically. The CEDHBs provide a convenient model to describe and treat dark-hollow beams with axially rotational asymmetry and can be used conveniently to analyze atoms manipulated with a dark-hollow beam.     

Vectorial nonparaxial propagation equation of elliptical Gaussian beams in the presence of a rectangular aperture

Based on the vectorial Rayleigh-Sommerfeld diffraction integrals, an analytical propagation equation of vectorial, nonparaxial, elliptical Gaussian beams through a rectangular aperture is derived. Unlike in previous work, the aperture effect and nonrotational symmetry of the beam and aperture are considered in our theoretical model. The results of the far-field and paraxial approximation for the apertured case are treated as special cases of our general expression. It is found that two f parameters fx, fy and two truncation parameters deltax, deltay in the x and y directions, respectively, have to be introduced that affect the beam nonparaxial evolution behavior in both the near field and the far field.     

Double Wigner distribution function of a first-order optical system with a hard-edge aperture

The effect of an apertured optical system on Wigner distribution can be expressed as a superposition integral of the input Wigner distribution function and the double Wigner distribution function of the apertured optical system. By introducing a hard aperture function into a finite sum of complex Gaussian functions, the double Wigner distribution functions of a first-order optical system with a hard aperture outside and inside it are derived. As an example of application, the analytical expressions of the Wigner distribution for a Gaussian beam passing through a spatial filtering optical system with an internal hard aperture are obtained. The analytical results are also compared with the numerical integral results, and they show that the analytical results are proper and ascendant.     

Different models for a hard-aperture function and corresponding approximate analytical propagation equations of a Gaussian beam through an apertured optical system

Beam profiles that consist of a sum of complex-Gaussian functions, a sum of polynomial-Gaussian functions and a sum of multi-Gaussian functions offset by some fixed amount are proposed as three types of model for a hard-aperture function. By expanding an aperture function into these models, approximate analytical propagation equations for a Gaussian beam through an apertured ABCD optical system are obtained. Comparison among these models themselves and among propagation characteristics of a Gaussian beam through these models are made. It is shown that the first and third types of model for a hard-aperture function are more suitable than the second type, in terms of calculation efficiency and simulation results, for application to such diffraction problems. Moreover, there are some differences in the applicability of the first and the third models.     

Hollow Gaussian beams with the power-exponent-phase vortex

A kind of hollow Gaussian beams with the power-exponent-phase vortex is introduced. Based on the Collins integral, an analytical expression of a hollow Gaussian beam with the power-exponent-phase vortex passing through a paraxial optical system described by the ABCD matrix approach is derived. The analytical expressions for the beam propagation factors and the orbital angular momentum density of such hollow vortex Gaussian beam passing through a paraxial optical system described by the ABCD matrix approach are also derived, respectively. As a numerical example, the propagation properties of a hollow Gaussian beam with the power-exponent-phase vortex are demonstrated in free space. The evolutions of the normalized intensity, the phase and the orbital angular momentum density distributions are investigated, respectively. The influences of the power order and the topological charge on the beam propagation factors in the x- and y-directions are analysed. The introduced hollow Gaussian beam has potential applications in the atom manipulation and the optical trapping.     

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.     

Direct focusing modifications of elliptical gaussian beam by non-circular apertures

Abstract

For focusing the elliptical Gaussian beam directly, the effects of a non-circular aperture on the focusing properties are studied. The focusing properties for different shapes of apertures, which include a circle, an ellipse and a rectangle, are calculated and compared. Moreover, for different elliptical Gaussian beams, an empirical aperture selection rule that can be used to circularize the focusing spot is proposed. The energy transmission ratios are also considered in this paper.