Study On Numerical Calculation Of Fresnel Diffraction And Its Application

The Fresnel diffraction is a paraxial approximation of Kirchhoff diffraction and Rayleigh-Sommerfeld diffraction in the region of the near field.In the fields of digital holography phase retrieval,and design of diffractive optical elements,the calculation of Fresnel,diffraction integral is involved.Generally speaking,there is no analytical solution for the Fresnel diffraction integral,which has to be solved by numerical algorithms.Since the Fresnel diffraction is a space-invariant linear system,Fourier analysis is usually employed to solve the Fresnel diffraction problem.The discrete Fourier transform has fast algorithms that can effectively improve the computing efficiency,so the Fourier transform algorithm is a common approach for numerically calculating the Fresnel diffraction.This paper mainly investigates three fast algorithms to calculate the Fresnel diffraction based on Fourier transform: the single Fourier transform algorithm,convolution method and angular spectrum method.The concrete implementation process of three numerical calculations is clarified,and the issues of sampling interval,computation window size and zero padding are discussed.For the single algorithm,the method of selecting the ideal sampling interval of the quadratic phase factor and the computation window size is given through the concept of local spatial frequency and the sampling theorem.The influence of the object light bandwidth on the sampling interval and computation window is also considered.The specific implementation process of the convolution method is described from the perspective of spatial filtering,and the zero-padding method of the object light matrix and the impulse response function matrix is given to avoid wraparound error caused by the circular convolution.The specific implementation process of the angular spectrum method is described from the perspective of frequency domain filtering.A translation zero padding method is proposed to avoid wraparound error and control aliasing.It is pointed out that the range of the local spatial frequency of the transfer function is actually the support of the corresponding impulse response function in the spatial domain,and then the selection method of the ideal sampling interval of the transfer function and the computation window size is clarified.The theoretical analytical solution of the Fresnel diffraction of the cosine grating illuminated by the Gaussian beam is derived,and the numerical solutions given by the three algorithms are compared with the analytical solutions to verify the effectiveness and the computational accuracy of these algorithms.In this paper,several optical problems are studied by the above three algorithms.The single Fourier transform algorithm was used to simulate the Fresnel diffraction distribution of the cross-aperture illuminated by the collimated laser beam.The convolution method and the angular spectrum method were used to simulate the Talbot self-imaging of a two-dimensional periodic object.These numerical simulation results are very consistent with these experimental results,which verifies the effectiveness of these algorithms.Finally,numerical simulation of the correlation reconstruction of the hologram is implemented by the single Fourier transform algorithm which verifies the feasibility of the correlation reconstruction of the pure phase hologram under incoherent illumination.