Misaligned Optical Systems Research Articles

Propagation of a Laguerre–Gaussian beam through a slightly misaligned paraxial optical system

Based on the generalized diffraction integral formula for treating the propagation of a laser beam through a slightly misaligned optical system in a cylindrical coordinate system, an analytical formula for a Laguerre–Gaussian beam passing through such an optical system is derived. Furthermore, an approximate analytical formula is derived for a Laguerre–Gaussian beam passing through an apertured slightly misaligned optical system by expanding the hard aperture function as a finite sum of complex Gaussian functions. Some analytical formulas are also given for a flattened Gaussian beam by expanding its field as a superposition of a finite series of Laguerre–Gaussian beams.

The elliptical elegant Laguerre–Gaussian beam and its propagation through aligned and misaligned paraxial optical systems

A new kind of laser beam called the elliptical elegant Laguerre–Gaussian beam (EELGB) is defined by using tensor method. By using the generalized diffraction integral formulas for light beam passing through paraxial optical system, the analytical propagation formulas for EELGB passing through paraxial aligned and misaligned optical systems are obtained through vector integration. As examples of applications, the propagation properties of EELGBs in free space propagation and through a misaligned thin lens are studied.

Propagation of hollow Gaussian beam through a misaligned first-order optical system and its propagation properties

Propagation properties of hollow Gaussian beam through a misaligned first-order ABCD system is studied using the generalized Huygens-Fresnel diffraction integral, augmented matrix. It is shown that, as a hollow Gaussian beam passes through the misaligned first-order ABCD system, the beam shape is not preserved, the out-put beams have differences when passing different misaligned optical systems. We can adjust the size of dark region through adjusting the misaligned transverse vector E.

Propagation of a hollow Gaussian beam through a paraxial misaligned optical system

Based on the generalized diffraction integral formula for treating the propagation of a laser beam through a paraxial misaligned optical system in the cylindrical coordinate system, we obtain an analytical formula for a hollow Gaussian beam passing through a paraxial misaligned optical system. Furthermore, we also obtain the approximate analytical formula for a hollow Gaussian beam passing through a paraxial circularly apertured misaligned optical system by expanding the hard aperture function into a finite sum of complex Gaussian functions. As a numerical example, the propagation properties a hollow Gaussian beam through a misaligned thin lens are studied numerically.

Propagation of the Hermite–Gaussian beams through misaligned optical system with a circular aperture

By means of expanding a hard-edged aperture into a finite sum of complex Gaussian functions, the approximate analytical formula of one kind of higher-order Gaussian beams called the Hermite–Gaussian beams (HGBs) passing through circular apertured and misaligned optical system is obtained in this paper. The result provides more convenience for studying its propagation than the usual way by using diffraction integral directly. Some numerical simulations are also given for illustrating the propagation properties of the HGBs through the circular apertured optical systems.

Propagation characteristics of the rectangular flattened Gaussian beams through circular apertured and misaligned optical systems

Based on the fact that a hard aperture function can be expanded into a finite sum of complex Gaussian functions, the approximate analytical expression for the output field distribution of a rectangular flattened Gaussian beam passing through a circular apertured and misaligned paraxial ABCD system is derived. The result brings more convenient for studying its propagation than the usual way by using diffraction integral directly. Some numerical simulations are also given for illustrating the propagation properties of a rectangular flattened Gaussian beam through a circular apertured and misaligned optical system.

Fractional Hankel transform and the diffraction of misaligned optical systems

Under the condition of rotational symmetry fractional Fourier transforms become fractional Hankel transforms. The relationships between the fractional Hankel transform and the diffraction of a misaligned optical system are established in this paper. By settling and discussing the diffraction integral formulae of a misaligned ABCD optical system, the fractional Hankel transform of an input function can be obtained by selecting a suitable reference plane under given conditions.

A partially coherent elliptical flattened Gaussian beam and its propagation

A new type of model beam named a partially coherent elliptical flattened Gaussian beam (EFGB) is proposed. This beam can be expressed as a finite series of partially coherent anisotropic Gaussian Schell-model beams with different parameters. The respective analytical propagation formulae for a partially coherent EFGB through both aligned and misaligned optical systems are derived. The propagation properties of a partially coherent EFGB through free space and a misaligned thin lens are illustrated numerically. Our method provides a convenient way to describe a partially coherent beam with elliptical flat-topped profile and treat its propagation and transformation.

Four-beamlets laser array and its propagation

A new type of laser array named four-beamlets laser array is proposed. Based on the diffraction integrals, analytical propagation formulae for the four-beamlets laser array passing through paraxial aligned and misaligned optical systems are derived, respectively. The four-beamlets laser array can also be expressed as a superposition of a series of Hermite Gaussian modes by using polynomial expansion. As a numerical example, the propagation properties of a four-beamlets laser array in free space are illustrated graphically. The fractional Fourier transform of the four-beamlets laser array is also studied. Our model provides a convenient way to describe laser array with four-beamlets.

Partially coherent flat-topped multi-Gaussian–Schell-model beam and its propagation

A new type of laser beam named partially coherent flat-topped multi-Gaussian–Schell-model beam whose profile is flat-topped is proposed. Analytical propagation formulae for partially coherent flat-topped multi-Gaussian–Schell-model beam passing through paraxial aligned and misaligned optical systems are derived respectively. As a numerical example, the propagation properties of partially coherent flat-topped multi-Gaussian–Schell-model beam in free space are illustrated graphically. Fractional Fourier transform for partially coherent flat-topped multi-Gaussian–Schell-model beam is also studied.

Fractional Fourier transform for partially coherent beam in spatial-frequency domain

By using Fourier transform and the tensor analysis method, the fractional Fourier transform (FRT) in the spatial-frequency domain for partially coherent beams is derived. Based on the FRT in the spatial-frequency domain, an analytical transform formula is derived for a partially coherent twisted anisotropic Gaussian–Schell model (GSM) beam passing through the FRT system. The connections between the FRT formula and the generalized diffraction integral formulae for partially coherent beams through an aligned optical system and a misaligned optical system in the spatial-frequency domain are discussed, separately. By using the derived formula, the intensity distribution of partially coherent twisted anisotropic GSM beams in the FRT plane are studied in detail. The formula derived provide a convenient tool for analysing and calculating the FRTs of the partially coherent beams in spatial-frequency domain.

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 the two-dimensional off-axial Hermite-cosh-Gaussian beams through rectangular apertured and misaligned optical systems

By introducing a hard aperture function into a finite sum of complex Gaussian functions, an approximate analytical expression for the two-dimensional off-axial Hermite-cosh-Gaussian beams passing through a rectangular apertured and misaligned paraxially ABCD optical system has been derived. The results provide more convenience for studying their propagation and transformation than the usual way by using diffraction integral directly. Some numerical simulations are also illustrated for the propagation characteristics of a two-dimensional off-axial Hermite-cosh-Gaussian beam through a rectangular apertured ABCD optical system.

Propagation of flattened Gaussian beams passing through a misaligned optical system with finite aperture

The propagation of flattened Gaussian beams (FGBs) passing though misaligned optical systems with aperture has been studied in detail. The corresponding approximate equation has been derived, where the rectangular aperture function is expanded into a finite sum of complex Gaussian functions. Numerical examples are given to illustrate the application of the propagation equation and the properties of flattened Gaussian beams passing through a misaligned optical system with aperture are discussed.

Transformation and spectrum properties of partially coherent beams in the fractional Fourier transform plane.

An analytical and concise formula is derived for the fractional Fourier transform (FRT) of partially coherent beams that is based on the tensorial propagation formula of the cross-spectral density of partially coherent twisted anisotropic Gaussian-Schell-model (GSM) beams. The corresponding tensor ABCD law performing the FRT is obtained. The connections between the FRT formula and the generalized diffraction integral formulas for partially coherent beams passing through aligned optical systems and misaligned optical systems are discussed. With use of the derived formula, the transformation and spectrum properties of partially coherent GSM beams in the FRT plane are studied in detail. The results show that the fractional order of the FRT has strong effects on the transformation properties and the spectrum properties of partially coherent GSM beams. Our method provides a simple and convenient way to study the FRT of twisted anisotropic GSM beams.

Propagation of elliptical Hermite–Gaussian beam through misaligned optical system

Based on the generalized diffraction integral formula in terms of tensor form for paraxial misaligned optical systems, an analytical propagation formula is derived for the elliptical Hermite–Gaussian beam (EHGB) passing through misaligned optical systems by using vector integration. By using the derived formula, the propagation properties of EHGB through a misaligned thin lens are illustrated. As an application example, we derived the analytical propagation formula for elliptical flattened Gaussian beam through misaligned optical system by expressing it as superposition of a series of EHGB by using polynomial expansion.

Decentered twisted Gaussian Schell-model beams and their propagation through a misaligned first-order optical system

The propagation properties of decentered twisted Gaussian Schell-model (DTGSM) beams passing through a misaligned first-order optical system are studied. The explicit expressions for the cross-spectral density function and Wigner distribution function of the output beam are derived, which retain their form unchanged. It is shown that the DTGSM beams preserve their closed property. The second-order moments matrix and the Wigner distribution function evolve with the usual laws, whereas the first-order moments matrix varies, as if a ray passes through such system. The propagation of DTGSM beams through an aligned first-order optical system is treated as the limiting case that corresponds to the vanishing misalignment parameters.

Propagation of partially coherent twisted anisotropic Gaussian–Schell model beams through misaligned optical systems

The generalized diffraction integral formulae for partially coherent beams through misaligned optical systems are derived in both spatial and spatial-frequency domain. Based on these formulae, the analytical propagation formulae for partially coherent twisted anisotropic Gaussian–Schell model (GSM) beams through misaligned optical systems are derived. The derived formulae provide a powerful and convenient tool for treating the propagation and transformation of partially coherent twisted anisotropic GSM beams through misaligned optical systems in spatial domain and spatial-frequency domain. As an application example, the intensity distribution properties of partially coherent twisted GSM beams through a misaligned thin lens are studied.

Propagation of elliptical Gaussian beam through misaligned optical systems in spatial domain and spatial-frequency domain

The propagation formulae of elliptical Gaussian beam through misaligned optical systems in spatial domain and spatial-frequency domain are investigated analytically by using tensor method. The derived formulae provide a powerful tool for treating the propagation and transformation of elliptical Gaussian beam through misaligned optical systems. As an application example, the propagation properties of elliptical Gaussian beam through a misaligned thin lens are illustrated.

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.