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.     

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.     

Propagation of decentered elliptical Gaussian beams in apertured and nonsymmetrical optical systems

A hard-edged elliptical aperture is described approximately by a tensor form, which can be expanded as a finite sum of complex Gaussian functions. An analytical propagation expression for a decentered elliptical Gaussian beam (DEGB) through an axially nonsymmetrical optical system with an elliptical aperture is derived by using vector integration. The approximate analytical results are compared with numerically integral ones, and it is shown that this method can significantly improve the efficiency of numerical calculation. Some numerical simulations are illustrated for the propagation properties of DEGBs through apertured and nonsymmetrical optical transforming systems.     

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.     

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 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 of a random electromagnetic beam through a misaligned optical system in turbulent atmosphere

On the basis of the generalized diffraction integral formula for misaligned optical systems in the spatial domain, an analytical propagation expression for the elements of the cross-spectral density matrix of a random electromagnetic beam passing through a misaligned optical system in turbulent atmosphere is derived. Some analyses are illustrated by numerical examples relating to changes in the state of polarization of an electromagnetic Gaussian Schell-model beam propagating through such an optical system. It is shown that the misalignment has a significant influence on the intensity profile and the state of polarization of the beam, but the influence becomes smaller for the beam propagating in strong turbulent atmosphere. The method in this paper can be applied for sources that are either isotropic or anisotropic. It is shown that the isotropic sources and the anisotropic sources have different polarization properties on beam propagation.     

Propagation of high-order Bessel–Gaussian beams through a hard-aperture misaligned optical system

In this paper, we consider the propagation of high-order Bessel–Gaussian beams (HBGBs) passing through a hard-aperture misaligned optical system. By expanding the hard-aperture function into a finite sum of complex Gaussian functions, a general propagating formula of HBGBs is derived in terms of the generalized diffraction integrals. Based on the derived formula, the diffraction properties of HBGBs propagating through a simple misaligned lens system are numerically illustrated. This method provides a convenient tool for studying the propagation and transformation properties of a high-order Bessel–Gaussian beam through an apertured misaligned optical system.     

Paraxial propagation of a partially coherent Hermite-Gaussian beam through aligned and misaligned ABCD optical systems

Paraxial propagation of a partially coherent Hermite-Gaussian beam through aligned and misaligned ABCD optical systems is investigated based on the generalized Collins formula for treating the propagation of a partially coherent beam through such optical systems. Analytical formulas for the cross-spectral density of a partially coherent Hermite-Gaussian beam propagating through such optical systems are derived. As an application example, we derive the propagation formulas for a partially coherent flattened Gaussian beam by expressing it as a superposition of a series of partially coherent Hermite-Gaussian beams by using polynomial expansion. The focusing properties of a partially coherent Hermite-Gaussian beam focused by a thin lens are studied as a numerical example.     

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.     

Changes in the polarization and coherence of a random electromagnetic beam propagating through a misaligned optical system

On the basis of the generalized diffraction integral formula for misaligned optical systems in the spatial domain, an analytical propagation expression for the elements of the cross-spectral density matrix of a random electromagnetic beam passing through a misaligned optical system is derived. Some analyses are illustrated by numerical examples relating to changes in the spectral degree of polarization and in the spectral degree of coherence of an electromagnetic Gaussian-Schell-model beam propagating through such an optical system. We find that the degree of polarization in the neighboring areas of the focal plane is oscillating, and the effect of misalignment on coherence is not so evident as that on polarization.     

Generalized huygens-fresnel diffraction integral for misaligned asymmetric first-order optical systems and decentered anisotropic Gaussian Schell-model beams

The generalized Huygens-Fresnel diffraction integral for misaligned asymmetric first-order optical systems is derived by using the canonical operator method, which enables us to study propagation properties of anisotropic Gaussian Schell-model (AGSM) beams through misaligned asymmetric first-order optical systems. It is shown that under the action of misaligned asymmetric first-order optical systems AGSM beams do not preserve the closed property. Therefore generalized partially coherent anisotropic Gaussian Schell-model beams called decentered anisotropic Gaussian Schell-model (DAGSM) beams are introduced, and AGSM beams can be regarded as a special case of DAGSM beams.     

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.     

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.     

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.     

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.     

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.