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 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 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 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 apertured Bessel beams

The propagation features of several apertured Bessel beams are numerically calculated. The calculations show that the relations of axial intensity versus propagation distance are similar to the radial distribution of the aperture functions, which may be helpful in choosing the proper aperture functions in experiments.     

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 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.     

Propagation of twisted gaussian Schell-model beams through a misaligned first-order optical system

Abstract

The propagation of twisted Gaussian Schell-model (TGSM) beams passing through a misaligned first-order optical system is studied. The explicit expressions for the cross-spectral density function and Wigner distribution function of the output beam are derived. As a result, generalized partially coherent beams called the decentred twisted Gaussian Schell-model (DTGSM) beams are introduced and their properties are discussed.     

Propagation of broadband gaussian Schell-model beams in the apertured fractional Fourier transformation systems

On the basis of the fact that a hard-edged aperture function can be expressed as finite matrices with different weighting coefficients, we obtain the analytical formula for the propagation of the broadband gaussian Schell-model (BGSM) beam through the apertured fractional Fourier transformation (AFrFT) system. It is shown by numerical examples that the intensity distribution in the plane of a small fractional order is obviously influenced by the bandwidth when the BGSM beams propagate through the AFrFT system. Further extensions are also pointed out.     

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.     

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.     

Relative phase shifts of apertured Gaussian beams and transformation of a Gaussian beam through a phase aperture

When a Gaussian beam is apertured, it undergoes a focal shift as well as a phase shift. The focal shift has been investigated extensively although the phase shift has seldom been discussed. We analyze the phase shift of the apertured Gaussian beam. Furthermore we point out that the phase aperture may be used to transform the intensity distribution of the Gaussian beam to obtain a more concentrated beam, to realize uniformity of the Gaussian beam, and to obtain a ring beam.     

Propagation and focusing of Gaussian beams generated by Gaussian mirror resonators

The propagation and focusing properties of a class of Gaussian beams generated by optical resonators with Gaussian reflectivity mirrors are investigated. Attention is concentrated on the following two beams in this class: (a) the annular Gaussian beam (the Gaussian doughnut mode) and (b) the flat-topped Gaussian beam. A class of flat-topped Gaussian beams is introduced. All analysis is limited to a coherent superposition scheme of the lowest-order Gaussian modes (TEM00) that have different parameters.     

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.     

Wigner distribution function of Hermite-cosine-Gaussian beams through an apertured optical system

By introducing the hard-aperture function into a finite sum of complex Gaussian functions, the approximate analytical expressions of the Wigner distribution function for Hermite-cosine-Gaussian beams passing through an apertured paraxial ABCD optical system are obtained. The analytical results are compared with the numerically integrated ones, and the absolute errors are also given. It is shown that the analytical results are proper and that the calculation speed for them is much faster than for the numerical results.     

Propagation of polychromatic Gaussian beams through thin lenses

The transformation by a lens of a polychromatic laser beam composed of on-axis superposed monochromatic TEM00 Gaussian modes in the paraxial approximation is studied. The chromatic aberrations are described by allowing the waist position on the z axis and the Rayleigh range to depend on wavelength. The beam radius, the far-field divergence, the Rayleigh range, the beam product, the beam propagation factor, and the kurtosis parameter are calculated. The relationship between the fourth-order and the second-order moments of Hermite-Gaussian and Laguerre-Gaussian modes is obtained and is used for calculating kurtosis parameter. The results are generalized to polychromatic modes of higher orders. It is shown that the on-axis superposition of monochromatic TEM00 modes with no chromatic aberration is leptokurtic.     

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.     

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.