The Relationship Between Rayleigh-Sommerfeld And Fractional Fourier Transform

Light is a part of the electromagnetic wave.We applied the theory of scalar diffraction to study the light diffraction problems,namely ignoring each component of the electromagnetic field coupling each other.Diffraction problem for a lot of optical instruments,the theory of scalar diffraction has high calculation precision.In most cases,we study the diffraction aperture paraxial light beam Angle is small,the results using paraxial scalar diffraction theory is very accurate.But as the microcavity laser and the development of the photonic crystal technology,with large diffraction Angle of non-paraxial beam research is becoming more and more important.Fractional Fourier transform in the field of applied optics for the first time in 1993,the formation of fractional Fourier optics.The relations of fractional Fourier transform and Fresnel diffraction in paraxial scalar diffraction theory are studied,and it is concluded that the fractional Fourier transform and Fresnel diffraction are equivalence.Based on the relationship between the paraxial scalar diffraction theory and the fractional Fourier transform,extract the analysis and research the relationship of non-paraxial scalar diffraction theory and the fractional Fourier transform.In the paper,on the basis of optical fractional Fourier transform and Rayleigh-sommerf--eld diffraction,the analysis,physical mechanism,the applicable scope,the difference between the conditions of use,and many other factors of two types of Rayleigh-sommerfeld diffraction was studied.Under the condition of paraxial approximation and the non-paraxial approximation,deducing the Rayleigh-sommerfeld diffraction integral formula,and stating its physical meaning,demonstrating the relationship of the Rayleigh-sommerfeld diffraction and conventional Fourier transform,fractional Fourier transform.Obtained the following conclusion:Fresnel approximation paraxial Rayleigh-sommerfeld diffraction and fractional Fourier transform is equivalent;Fraunhofer approximation paraxial Rayleigh-sommerfeld diffraction and Fourier transform is equivalent;Non-paraxial Rayleigh-sommerfeld diffraction and fractional Fourier transform is equivalent,the complex amplitude of non-paraxial Rayleigh-sommerfeld diffraction can be viewed as fractional Fourier transform with the modulation of z/?r2exp(ikr-i?/2)exp[i?cos2?/?r(x2+y2],In particular sphere,the light intensity distribution of non-paraxial Rayleigh-sommerfeld diffraction is consistent with the light intensity distribution of fractional Fourier transform.The conclusion makes the fractional Fourier transform has perfect effect on the modernization of teaching content.In addition,the instance of plane wave rectangular hole diffraction and computer simulation,proved the conclusion is reliable and feasible in the paper.