Paraxial ABCD Optical System Research Articles

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.

Studies of propagation characteristics of elliptical Gaussian beams through circular apertured optical systems

An approximate analytical propagation formula of the elliptical Gaussian beams through a paraxial ABCD optical system with a circular aperture is obtained on the basis of the expansion of a hard-edged aperture into a finite sum of complex Gaussian functions. And the parameter characteristics of the truncated beams including the beam propagation factor and the kurtosis parameter are also studied in detail by using the second-order-moments method.

Propagation of various dark hollow beams through an apertured paraxial ABCD optical system

Propagation of a dark hollow beam (DHB) of circular, elliptical or rectangular symmetry through an apertured paraxial ABCD optical system is investigated. Approximate analytical formulas for various DHBs propagating through an apertured paraxial optical system are derived by expanding the hard-aperture function into a finite sum of complex Gaussian functions in terms of a tensor method. Some numerical results are given. Our formulas provide a convenient way for studying the propagation of various DHBs through an apertured paraxial optical system.

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

Abstract 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.

Models for an aperture function and corresponding analytical propagation equations of a flattened Gaussian beam through an annular apertured optical system

Beam profiles that consist of a sum of complex-Gaussian functions and a sum of polynomial-Gaussian functions offset by some fixed amount are proposed as two types of models for a hard aperture function. By expanding an aperture function into two types of models, the approximate analytical expression of a flattened Gaussian beam through a paraxial ABCD optical system with an annular aperture is derived. As special cases, the corresponding closed forms for the unaperture or circular aperture or circular black screen cases are also obtained. The results provide a more convenient way of studying propagation and transformation than the usual method using diffraction integral formula directly. Numerical examples are given to illustrate the propagation properties of flattened Gaussian beams.

Four-petal Gaussian beams and their propagation

A new form of laser beams called four-petal Gaussian beams is introduced. Based on the Collins integral, two kinds of analytical propagation expressions for this new kind of beams through a paraxial ABCD optical system are derived. The propagation properties of the four-petal Gaussian beams are studied and illustrated with numerical examples. At the source plane the beam has four-petals; the space among the petals is determined by the beam order. In the far field the beam evolves into a number of mirror symmetric petals and the petals of higher order beams can be equally spaced.

Paraxial propagation of partially coherent flat-topped beam

The propagation of flat-topped beam passing through a paraxial optical ABCD system is extended to partially coherent case and has been studied in detail. Based on Wigner distribution function (WDF), the irradiance distribution of partially coherent flat-topped beam is derived. Furthermore, two propagation parameters, such as M2-factor and Power in Bucket (PIB) are illustrated analytically and numerically.

Propagation and focal shift of J0-correlated Schell-model beams

Based on the propagation law of partially coherent beams, the propagation equations of J 0-correlated Schell-model (JSM) beams through a paraxial optical ABCD system are derived and expressed in closed-form. The free-space propagation and focusing of JSM beams are regarded two special cases of our result. The focusing properties, in particular, the focal shift of JSM beams are studied, and illustrative numerical results are given. It is found that the focal shift of JSM beams, which is determined by an algebraic equation of 3rd order, increases with decreasing Fresnel number N and increasing parameter σ.

Propagation of a phase flipped Gaussian beam through a paraxial optical ABCD system

Using the Huygens–Fresnel diffraction integral, we obtain a closed-form analytic expression for the amplitude distribution of a phase flipped Gaussian (PFG) beam passing through a paraxial optical ABCD system. The resulting expression is related in a simple way to the corresponding expression for a Gaussian beam and contains error functions of complex argument. Both ideal and non-ideal PFGs are considered. Numerical results are presented for the special case of free-space propagation. Intensity patterns of the propagating PFG are explained by means of the power series and asymptotic expansions of the error function.

Propagation properties of beam generated by Gaussian mirror resonator

Based on scalar diffraction theory, propagation properties of beam generated by Gaussian mirror resonator were investigated. Explicit expressions for the field distribution of the beam that propagate through a paraxial optical ABCD system were derived and numerical examples were given to illustrate our analytical results.

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 Bessel-modulated Gaussian beams through a paraxial ABCD optical system with an annular aperture

Based on the generalized Collins diffraction integral and the expansion of the hard aperture function into a finite sum of complex Gaussian functions, the approximate analytical expressions of Bessel–Gaussian beams and QBG beams passing through a paraxial ABCD optical system with an annual aperture are derived. As special cases, the corresponding closed-forms for the unaperture or circular aperture or circular black screen cases are also given. The results provide more convenience for studying their propagation and transformation than the usual way by using diffraction integral formula directly. Numerical examples are given to illustrate the propagation properties of Bessel–Gaussian and QBG beams.

Rotating the intensity distribution of Gaussian beams propagating through paraxial ABCD systems with parallelogram apertures

By using the method of expanding the aperture function into a finite sum of complex Gaussian functions, approximate closed-form propagation equations of Gaussian beams through a paraxial ABCD optical system with parallelogram apertures are derived and illustrated with numerical examples. It is found that the intensity distribution pattern rotates round the propagation axis during the propagation.

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.

Paraxial propagation of Mathieu beams through an apertured ABCD optical system

Based on the relationship between separable solutions of the Helmholtz equation, we expanded the field’s amplitude associated with Mathieu beams in terms of Bessel beams. By using the expansion of the circ function into a finite sum of complex Gaussian functions, we derived an approximated analytical expression of the amplitude distribution of Mathieu beams at the output plane of any apertured paraxial ABCD optical system. Some numerical calculations are performed and it is shown that our approach can substitute the usual numerical integration of the Helmholtz equation or the numerical calculation of the diffraction integral.

Approximate analytical representation of Wigner distribution function for Gaussian beams passing through ABCD optical systems with hard aperture

Based on the treatment that a rectangular function can be expanded as an approximate sum of complex Gaussian functions with finite numbers, the analytical expression of the Wigner distribution function for a Gaussian beam passing through a paraxial ABCD optical system with hard-edged aperture is obtained. By numerical simulation, it is shown that the effect of the aperture on the Wigner distribution function is prominent. By comparing the analytical results with the numerical integral results, it is shown that this method of expanding hard-edged aperture into Gaussian functions with finite numbers is proper and ascendant. This method could be extended to study the Wigner distribution functions of other light beams passing through a paraxial ABCD optical system with hard-edged aperture.

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.

Propagation of Hermite–Laguerre–Gaussian beams through a paraxial optical ABCD system with rectangular hard-edged aperture

Based on a recurrence approach the analytical equations of the propagation of Hermite–Laguerre–Gaussian (HLG) beams through a paraxial optical ABCD system with rectangular hard-edged aperture are derived. The results are strict and can be used to study the propagation of HLG beams of any orders. The effect of the aperture on the propagation of HLG beams and the focal shifts are analyzed with numerical examples.

Propagation properties of partially coherent modified Bessel–Gauss beams

Starting from the propagation law of partially coherent light, the analytical propagation equations of partially coherent modified Bessel–Gauss beams (MBGBs) through a paraxial optical ABCD system are derived and illustrated with typical application examples. Furthermore, by using the intensity moments method and integral transformation technique, the important characteristic parameters, including the beam width, far-field divergence angle, M 2 factor and kurtosis parameter of partially coherent MBGBs, are expressed in a closed and simple form. As a result, some basic properties of MBGBs and the dependence of the M 2 factor and kurtosis parameter on the spectral degree of coherence and beam order are illustrated both analytically and numerically.

Algorithms for flattened Gaussian beams passing through apertured and unapertured paraxial ABCD optical systems

The propagation property of flattened Gaussian beams (FGBs) is studied based on the Collins formula, and analytical propagation equations for FGBs with and without an aperture are derived. It is shown that the closed-form expressions for the FGB of order N passing through paraxial ABCD optical systems without an aperture can be written as the combinations of Confluent Hyper-geometric Functions, and that the analytical equations for the FGB of order N passing through apertured paraxial ABCD optical systems may involve Error Functions. The proposed algorithms for the considered beams can save computing time, provide intuitive physical insights, and help to design optical systems. Our results are confirmed and illustrated with numerical examples.