Numerical Simulation Of The Propagation Of Laser Beams Through An Optical System With Hard-edged Apertures

In Optic, the edge or frame of optical element is defined as an aperture. In the production and propagation of lasers there are more or less limiting apertures. In practice apertures with different shapes are applied to suppress higher-order oscillating modes, and to realize spatial filtering and beam shaping etc. The soft-edged aperture that can be expressed by mathematical function, for example the numerical simulation of Gaussian aperture is easy. But for hard-edged aperture, and analytical propagation equation can' t be obtained when beams through such types of optical system. We often calculate from straightforward integral of the Collins formula. But in complicated optical systems, integral will become very difficult and even can' t be finished in individual computer. Therefore, the numerical simulation of the transformation of Laser beams through hard-edged apertured optical systems is of practical importance, and a variety of fast algorithms which are mode expansion method, matrix representation, expansion of complex Gaussian functions, analytic formula and recurrence algorithm have been developed. The matrix representation and expansion of complex Gaussian functions are widely applied, so importance is attached to them.The present thesis is devoted to Studying laser beams passing through optical systems with hard-edged apertures. The main results achieved in the thesis can be summarized as follows:1. The propagation of Gaussian beams through a paraxial ABCD optical system with an annular aperture is studied. The analytical propagation equation is derived by means of the expansion of the circ function into a finite sum of complex Gaussian functions. The propagation equations of Gaussian beams through circular aperture and circular screen can be regarded as special case in our theoretical model. Numerical calculation examples are given for the focusing of Gaussian beams by an aperture lens system.2. By expanding the aperture function into a finite sum of complex Gaussian functions and using the method of matrix decomposition, the closed-form propagation equations of Laguerre~{jaussian(L-G) beams through a multi-apertured imaging system of #=0 and multiple spatial filters are derived. The propagation of Gaussian beams through such types of optical system can be regarded as a special case in our theoretical model. Numerical calculation examples by using our formula are given, which are found a consistent with those by straightforward integral of the Collins formula, and computing time is greatly reduced.3. The focusing property of Hermite-Gaussian(H-G) beams whose on-axis intensity is zero passing through an optical system with an aperture is studied in detail. The approximate analytical propagation equation of beam width def inited by Green-Hall method is derived by means of expansion of the hard-edged aperture function into a finite sum of complex Gaussian functions. The influence of the optical system and beam parameters on the focusing property of beams is also studied with numerical examples.4. By using the method of matrix representation, the propagationequations of Bessel-Gauss beams through a multi-apertured complex optical system are derived. Numerical calculation examples of Bessel-Gauss beams propagation through a multi-aperture lens system are given. The numerical results find an agreement with those by straightforward integral of the Collins formula. The advantage of matrix representation method is very accurate even in the Fresnel region or Fraunhofer region.5. Using the methods of matrix representation and complex Gaussian function expansion, respectively, derives the recurrence propagation formulae of a flattened Gaussian beam through multi-apertured optical ABCD systems. Numerical examples are given. It is shown that the matrix formulation provides satisfactory results in both Fraunhofer and Fresnel regions. 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. Both of the two methods reduce the computational time greatly in comparison with the direct integration.