Propagation of off-axial cosine-Gaussian beams passing through an apertured and misaligned optical system

A closed-form expression for the one-dimensional off-axial cosine-Gaussian beams passing through an apertured and misaligned paraxially ABCD optical system is derived. As special cases, the corresponding closed-forms for the off-axial or non off-axial cosine-Gaussian beams passing through apertured or unapertured and misaligned or aligned optical systems are also given. The obtained results could be straightforward to the two-dimensional case or sine-Gaussian beams.     

Off-axial elliptical Hermite-cosh-Gaussian beams

A new kind of light beams named the off-axial elliptical Hermite-cosh-Gaussian beams (EHChGBs) is introduced in this paper by use of tensor method. An analytical propagation expression for an off-axial EHChGB passing through an axially nonsymmetrical optical ABCD system is derived by use of vector integration. The derived formula can be easily reduced to the propagation formulas of off-axial elliptical cosh-Gaussian beams and elliptical Hermite-cosh-Gaussian beams. Some numerical simulations are illustrated for the propagation properties of off-axial EHChGBs passing through a free space and a focusing optical system, and further extensions are suggested.     

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.     

Focal shift in focused off-axial Hermite-cosh-Gaussian beams

Based on the encircled-power criterion, the focal shift of off-axial Hermite-cosh-Gaussian beams passing through an apertured lens is studied and illustrated with detailed numerical examples. The dependence of focal shift on the beam and optical system parameters is discussed.     

Approximate analytical expressions of Laguerre-Gaussian beams passing through a paraxial optical system with an annular aperture

On the basis of the expansion of the hard aperture function into a finite sum of complex Gaussian functions, the approximate analytical expression of Laguerre-Gaussian beams passing through an annular apertured paraxial ABCD optical system is derived. Meanwhile, the corresponding closed-forms for the unapertured or circular apertured or circular black screen cases have also been given. Numerical examples are given to illustrate the propagation characteristics of Laguerre-Gaussian beams.     

Changes in the state of polarization of apertured stochastic electromagnetic modified Bessel–Gauss beams in free-space propagation

By using the generalized Huygens–Fresnel diffraction integral, the analytical expressions for the cross-spectral density matrix, spectral degree of polarization, orientation angle and degree of ellipticity of polarization ellipse of apertured stochastic electromagnetic modified Bessel–Gauss beams (MBGBs) through a paraxial optical ABCD system are derived, and used to study the changes in the state of polarization of apertured stochastic electromagnetic MBGBs propagating in free space. The invariance of the on-axis state of polarization of unapertured stochastic electromagnetic MBGBs propagating through paraxial optical ABCD systems is illustrated analytically and numerically. For apertured stochastic electromagnetic MBGBs, the on-axis spectral degree of polarization, orientation angle and degree of ellipticity of polarization ellipse increase with increasing propagation distance, and approach asymptotic values when the propagation distance is large enough. There is a uniform distribution region of the state of polarization around the center of the beams whose range decreases with increasing truncation parameter. In addition, the state of polarization of apertured stochastic electromagnetic MBGBs upon propagation can be modulated by controlling the truncation parameter.     

Approximate analytical expression for the kurtosis parameter of off-axial Hermite-cosine-Gaussian beams propagating through apertured and misaligned ABCD optical systems

The approximate analytical expression for the kurtosis parameter of off-axial Hermite-cosine-Gaussian beams (HCosGBs) propagating through apertured and misaligned ABCD optical systems is derived based on the approximate propagation equation of off-axial HCosGBs and an example is given to illustrate for its application. The method used in this paper could be extended to studying the kurtosis parameter of the Hermite-sinusoidal-Gaussian beams passing through apertured and misaligned ABCD optical systems.     

Propagation of the two-dimensional elegant Hermite-cosh-Gaussian beams in strongly nonlocal nonlinear media

In this paper, two-dimensional elegant Hermite-cosh-Gaussian beams are introduced and an analytical expression for the beam propagation through an optical ABCD system is derived by using Collins formula. By considering the transfer matrix of the strongly nonlocal nonlinear media, the propagation properties of a two-dimensional elegant Hermite-cosh-Gaussian beam through these media are investigated.     

Propagation of relative phase shift of apertured Gaussian beams through axisymmetric optical systems

When a Gaussian beam is apertured, it undergoes a phase shift as well as a focal shift. The relative phase shift of an apertured Gaussian beam through an axisymmetric optical system written in ABCD matrix is analyzed by applying Collins' diffraction integral formula. And, more important, the condition of dispeling the relative phase shift is obtained, which is related with the Fresnel number. At last, some extensions are given.     

Propagation of a general-type beam through apertured aligned and misaligned ABCD optical systems

By expanding the hard-aperture function into a finite sum of complex Gaussian functions, analytical formulae for the electric field of a general-type beam propagating through apertured aligned and misaligned ABCD optical systems are derived using the generalized Collins formulae, which provide a convenient way of studying the propagation of a variety of laser beams, such as Gaussian, cos-Gaussian, cosh-Gaussian, sine-Gaussian, sinh-Gaussian, flat-topped, Hermite-cosh-Gaussian, Hermite-sine-Gaussian, higher-order annular Gaussian, Hermite-sinh-Gaussian and Hermite-cos-Gaussian beams, through such optical systems. As numerical examples, the propagation properties of a cos-Gaussian beam through an apertured aligned or misaligned thin lens are studied.     

Analytical formulas for a circular or non-circular flat-topped beam propagating through an apertured paraxial optical system

Propagation of a flat-topped beam of circular or non-circular (rectangular or elliptical) symmetry through an apertured optical system is investigated. By expanding the hard aperture function as a finite sum of complex Gaussian functions, some approximate analytical propagation formulas are derived for a flat-topped beam of circular or non-circular (rectangular or elliptical) symmetry propagating through an apertured paraxial general astigmatic (GA) optical system or an apertured paraxial misaligned stigmatic (ST) optical system. The derived formulas are very fast to compute. The results obtained by using the approximate analytical expressions are in a good agreement with those obtained by direct numerical integration. The present analytical formulas provide a convenient and effective way for studying the propagation and transformation of a circular or non-circular flat-topped beam through an apertured general optical system.     

Rectangular hard-edged aperture diffracted Laguerre–Gaussian beams

Based on the relations between Laguerre–Gaussian (LG) and Hermite–Gaussian (HG) modes and by introduced the complex Gaussian expansion method for two dimensional rectangular aperture, the approximate analytical propagation expressions of the rotational symmetrical LG beams along with their even and odd modes through a paraxial ABCD optical system with rectangular hard-edged aperture are derived. As special cases of the results, the corresponding closed-forms of the circular aperture diffracted LG beams and non-truncated LG beams are also given. Numerical examples are given to prove the validity of this approximate analytical method and illustrate the propagation properties of the rectangular hard-edged aperture diffracted LG beams.     

Matrix formulation of axis-symmetric laser beams through a paraxial ABCD system containing hard-edged apertures: A comparative study

A matrix formulation is presented, which enables us to study the propagation of axis-symmetric beams through a paraxial optical ABCD system containing hard-edged aperture. Numerical calculation results of super-Gaussian beams passing through a multi-aperture-lens system are given to illustrate the advantage of the method. A comparison of the matrix formulation, complex Gaussian expansion and direct numerical integration of the Collins formula is made, where the propagation of apertured Laguerre-Gaussian beams is chosen as an illustrative example. It is shown that the matrix formulation provides satisfactory results in both Fraunhofer and Fresnel regions, and reduces the computational time greatly in comparison with the direct integration. However, this method is suited only to axis-symmetric optical beams and systems. By using the complex Gaussian expansion discrepancies exist in the near zone closer to the aperture, but usually its computational efficiency is higher than the matrix formulation.     

A detailed study of Mathieu-Gauss beams propagation through an apertured ABCD optical system

Using Collins formula and the expansion of Mathieu beams in terms of Bessel beams we derive the exact propagation equations of Mathieu-Gauss beams through an apertured paraxial ABCD optical system. A comparison between the exact propagation equations and the approximated ones, which are derived by expanding the circ function into a finite sum of Gaussian functions, is presented. The propagation characteristics of zeroth-order Mathieu-Gauss beams in (y-z) and (x-z) planes are analyzed with detailed numerical calculations. It is found that the profile of the propagated Mathieu-Gauss beam is similar to that of Bessel-Gauss beam. Furthermore, the focalization of the Mathieu-Gauss beams through a thin lens is illustrated and analyzed with numerical results.     

Kurtosis parameters of super Lorentzben Gauss beams through a paraxial and real ABCD optical system

Based on the propagation equation of higher-order intensity moments, analytical propagation expressions for the kurtosis parameters of a super Lorentz-Gauss (SLG) SLG₀₁ beam through a paraxial and real ABCD optical system are derived. By replacing the parameters in the expressions of the kurtosis parameters of the SLG₀₁ beam, the kurtosis parameters of the SLG₁₀ and SLG₁₁ beams through a paraxial and real ABCD optical system can be easily obtained. The kurtosis parameters of an SLG₀₁ beam through a paraxial and real ABCD optical system depend on two ratios. One is the ratio of the transfer matrix element B to the product of the transfer matrix element A and the diffraction-free range of the super-Lorentzian part. The other is the ratio of the width parameter of the super-Lorentzian part to the waist of the Gaussian part. As a numerical example, the properties of the kurtosis parameters of an SLG₀₁ beam propagating in free space are illustrated. The influences of different parameters on the kurtosis parameters of an SLG₀₁ beam are analysed in detail.     

Comparison of two approximate methods for hard-edged diffracted flat-topped light beams

The integral resulted in an infinite series of Bessel functions and expanding a hard aperture into a complex-Gaussians shape are proposed as two methods for studying the propagation properties of the hard-edged diffraction flat-topped light beam. Using the two methods, the corresponding analytical propagation equations of flat-topped light beams through a circular apertured ABCD optical system are obtained. Some numerical calculations and comparative analyses by using the two methods and the diffraction integral formulae are made. It is shown that the first method of an infinite series of Bessel functions is superior to the second of expanding a hard aperture function into a complex-Gaussians shape at the aspect of calculation accuracy, but the second method is superior to the first method at the aspect of the improvement in the calculation efficiency.     

Propagation properties of the beam generated by Gaussian mirror resonator passing through a paraxial ABCD optical system

By expanding a hard aperture function into a finite sum of complex Gaussian functions, approximate propagation formula is derived in the situation that the beam generated by Gaussian mirror resonator passes through a paraxial ABCD optical system with an annular aperture. The corresponding forms for a circular aperture and a circular black screen are also given. Some numerical simulations are shown to illustrate propagation properties and focusing properties of the beam passing through a paraxial ABCD optical system with the three different kinds of aperture.     

ABCD law for partially coherent Gaussian light,propagating through first-order optical systems

This paper shows under what condition the well-knownABCD law — which can be applied to describe the propagation of one-dimensional Gaussian light through first-order optical systems (orABCD systems) — can be extended to more than one dimension. It is shown that in the two-dimensional (or higher-dimensional) case anABCD law only holds for partially coherent Gaussian light for which the matrix of second-order moments of the Wigner distribution function is proportional to a symplectic matrix. Moreover, it is shown that this is the case if we are dealing with a special kind of Gaussian Schell model light, for which the real parts of the quadratic forms that arise in the exponents of the Gaussians are described by the same real, positive-definite symmetric matrix.     

Propagation of flat-topped beams

The propagation of flat-topped beams passing through paraxial ABCD optical system is investigated based on the propagation formulas of Gaussian beam. The focal shift of focused coherent flat-topped beam is also studied in detail. Analytical expressions of the M² factor and the far-field intensity distribution for flat-topped beams are derived on the basis of second-order moments.     

Hermite hyperbolic/sinusoidal Gaussian beams in ABCD systems

For a Hermite hyperbolic/sinusoidal Gaussian beam with focusing properties, passing through an arbitrarily shaped rectangular aperture on the source plane and an on-axis x–y asymmetric ABCD system, the receiver plane expression is derived using the Collins integral. The specific example of a single thin lens placed on the propagation path is examined at selected source, propagation and optical element parameters. Viewing the progress of the beam in propagation, we find that subjecting the source beam to an aperture will give rise to excessive spreading during propagation. The lens setup will act to concentrate the energy of the beam around its focal point as expected, while in some circumstances it will also execute beam profile changes. By adjusting the aperture opening in the shape of a narrow slit, the beam will become aligned in the opposite direction after propagating after having traveled sufficiently. The results are presented as intensity graphs in the form of contour plots and 3D illustrations.