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1、 傅立叶光学基本原理2f和4f系统实验目的观测和了解2f系统中一个透镜对物平面的光场的傅立叶变换作用，计算光栅的栅格常数。观测和了解4f系统中两个透镜对物平面的光场的傅立叶变换作用及光学滤波，测量小孔直径。实验元件HeNe激光，平面镜，透镜，f=+100mm ,白屏，光栅1，光栅，衍射物1，衍射物2，物镜（objective），20x，支架，尺子，实验步骤下文括号中的数字表示的坐标仅适用于开始阶段的粗调。如图1摆放器件。初期的调整，不需要E20x扩束系统（1,6）和透镜L0(1,3)。使用M1(1,8)和M2(1,1)调整光路时，要让光线沿平台的x=1和y=1的直线走。放置E20x和透镜L0(
2、F=+150mm)在光路中，调整器件的位置以保证从透镜发出的光是平行光线，即随距离增大，光点不会发散。用尺子在 透镜L0后0.5m范围内不同距离处测量光点的直径。 检验其平行度，应保证不同距离处的圆形光斑的直径基本保持不变。摆放另外的光学元件。其中P1为物平面，屏幕SC放在透镜L1(F=+100mm)的后焦距处，即构成2f系统。 图1 2f系统a) 实验的第一步观察平面波（光斑），此时物平面没有放置衍射物体。依据理论， 在透镜L1后的傅立叶面SC应该出现的一个光点。也称焦点。b) 将可调狭峰在物平面P1上，调整高度和截面的方位，使光点通过狭峰。在屏幕上可以看到狭峰的傅立叶变换，即典型的单峰衍射
3、图样（与理论比较）。c) 将光栅1（diffraction grating）放在P1，透镜L1后的傅立叶面SC上即为衍射图（the slit separation can be made using the separation of the diffraction maxima in the Fourier planes SC behind the lens L1）。计算该光栅的光栅常数。 将2f系统扩展为4f系统将提供的支架P2、透镜L2（f=+100mm）和白屏SC分别放置在距透镜L1一倍、二倍和三倍的焦距处，此时即构成4f系统。（如右图）不带滤波器时的衍射图象1， 将带有箭头的衍射物2
4、放在P1，调整其位置，使得光照在图形的箭头处，记录下在屏上观测到的反置图形，并予理论解释；调整图象的位置，将其旋转90，重复上述步骤。2， 将箭头的图片换成国王的图片（衍射物3），让光束照亮脸部的轮廓，此时在屏上的图象是什么样的？3， 将光栅2安装在P1，观测在P2、SC的位置处的图象，（在SC时，可将屏绕轴旋转（接近平行与光的传播方向，能否在屏上观测到光栅图象）？滤波后的图象1， 将光栅2安装在P1,在P2放置带小孔的圆盘（直径12mm的小孔），让中间的衍射最大通过。观测小孔的直径渐小时，对SC上光栅图象的影响。当小孔直径小到某一值时，光栅像应基本消失。2， 保持光路不变，将国王像（衍射物3
5、）和光栅2（4lines/mm）的图片一起装在P1，在屏上能够观测到的合成图象和去掉小孔圆盘的图象相比有什么区别？3， 将激光直接照射到该小孔上，由其在墙上的衍射斑，计算出小孔的直径，该尺寸与光栅2的4lines/mm的物理条件的关系如何？应用傅立叶变换的知识解释上述现象。 实验原理 The Fourier transform plays a major role in the natural sciences . In the majority of cases , one deals with Fourier transforms in a time range ; they supply
6、 us with the spectral composition of a time signal . This concept can be extend - ed in two aspects : 1 . In our case a spatial signal and not a temporal signal is transformed .2 . A two- dimensional transform is performed . From this , the following is obtained :Where vxandvy: spatial frequencies .
7、标量衍射理论（scalar diffraction theory）In Fig . 2 we observe a plane wave which is diffracted in one plane . For this wave in the xy plane directly behind the plane Z = 0 with the following transmission distribution : where : electric field distribution of the incident wave.The 图 2 further expansion can b
8、e described by the assumption that a spherical wave emanates from each point ( x , y , 0 ) behind the diffracting structure ( Huygens , principle ) . This leads to Kirchhoffs diffraction integral : (2)W ith spherical wave length ; = normal vector of the plane ( x , y ) ;k = wave number Equation ( 2
9、) corresponds to an accumulation of spherical waves , where the factor is a phase and amplitude factor and , a directional factor which results from the Maxwell field equations .The Fresnel approximation ( observations in a remote radiation field ) considers only rays which occupy a small angle to t
10、he optical axis ( 2 axis ) , i.e . and . In this case , the directional factor can be neglected and the 1/r dependence becomes : l/r =1/z . In the exponential function , this cannot be performed as easily since even small changes in r result in large phase changes . To achieve this , the roots inare
11、 expanded into a series and one obtains : This results in the Fresnel approximation of the diffraction integral (3)For long distances from the diffracting plane with concurrent finite expansion of the diffracting structure , one obtains the FRAUNHOFER APPROXIMAT1ON :(4)with with the spatial frequenc
12、ies as new coordinates : 图 3; Consequently the field distribution in the plane of observation ( x , y , z ) is shown by the following : The electric field distribution in the plane （x,y） for z = const is thus established by a Fourier transform of the field strength disiribution in the diffracting pl
13、ane after multiplication with a quadratic phase factor exp. The spatial frequencies are proportional to the corresponding diffraotion angles ( see Fig . 3 ) , where : ; Through the making of a photographic recording or through observation of the diffraction image with one eye , the intensity formati
14、on disappears due to the phase information of the light in the plane （x,yz）. As a consequence , only the intensity distribution ( this corresponds to the power spectrum ) can be observed . As a consequence of the phase factor C , ( Equation6 ) drops out of the operation . Therefore , the following results :一个透镜的傅立叶变换A biconvex lens exactly performs a two-dimensional Fourier transform from the front to the rear focal plane if the diffracting structure (entry field s
