傅立叶光学基本原理

实验二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(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 madeusing 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1～2mm 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）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 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 :[]dxdye y x E v v y x E v v E y v x v i y x y x y x ⎰⎰+∞∞-+∞∞-+-=ℑ=)(2,).()(),(),(~πWhere vxandvy: spatial frequencies . 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 planeZ = 0 with the following transmission distribution),(y x τ:),(),(),(y x E y x y x E e ∙=τwhere),(y x E e : electric field distribution of the incident wave.The 图 2further expansion can be described by the assumption that a spherical wave emanates from each point ( x , y , 0 ) behind the diffracting structure ( Huygens , principle ) . This leads to Kirchhoff ’s diffraction integral :dxdyr n r e y x E i z y x E ikr ),cos(),(1),,(⎰⎰∞+∞-∞+∞-=''λ (2)W ithλ＝ spherical wave length ; n= normal vector of the plane ( x , y ) ;k = wave number λπ/2Equation ( 2 ) corresponds to an accumulation of spherical waves , where the factor λi 1 is a phase and amplitudefactor and),cos(r n, 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 the optical axis ( 2 axis ) , i.e .zy x 〈〈, andzy x 〈〈'',. In this case , the directional factor can be neglectedand 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 in2222222)()(1)()(z y y z x x z z y y x x r -'+-'+∙=+-'+-'=are expanded into a series and one obtains :()()zy y z x x z r 2222-'+-'+=This results in the Fresnel approximation of the diffraction integral()()()()()dxdy e y x E i e z y x E y y x x z ikikz 222,,,-'+-'∞+∞-∞+∞-⎰⎰=''λ (3)For long distances from the diffracting plane with concurrent finite expansion of the diffracting structure , one obtains the FRAUNHOFER APPROXIMAT1ON :()()()dxdyey x E z y x C z y x E z z y x i ⎪⎪⎭⎫ ⎝⎛'+'-∞+∞-∞+∞-⎰⎰∙''=''λλπ2,,,,,(4)with()()22,,y x z i ikz ez i e z y x C '+'∙=''λπλwith the spatial frequencies as new coordinates : 图 3zx x λυ'=;zy y λυ'=Consequently the field distribution in the plane of observation ( x ' , y ' , z ) is shown by the following :()()()[]()()y x y x y x E y x E z z z C z y x E νννννλνλ,~,,,,,,=ℑ=''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 plane after multiplication with a quadratic phase factorexp()()()22y x z i +π. The spatial frequencies are proportional to the corresponding diffraotion angles ( see Fig .3 ) , where :λλλν∂≈∂='=tan z x x ; λβλλν≈∂='=tan z y yThrough the making of a photographic recording or through observation of the diffraction image with one eye , the intensity formation disappears due to the phase information of the light in the plane （x ’,y ’z ）. 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 :()()[]()222,,1,yx y x y x E z I ννλννℑ=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 strength distribution ) lies in the front focal plane ( see Fig . 4 ) . In this process , the coordinates νand u correspond to the anglesβandαwith the following correlations :Bx f u z x λλαλν=='=; Byf uz y λλβλν=='=(8) This means that the lens projects the image of the remote radiation field in the rear focal plane :()()dxdyey x E f u A u E y f x f u B B B⎪⎪⎭⎫ ⎝⎛+-∞+∞-∞+∞-⎰⎰∙=λυλπνν2,,,),(~(9)The phase factor A becomes independent of u and v ，if the entry field distribution is positioned exactly in the front focal p lane . Thus , the complex amplitude spectrum results :()ν,~u E ～()[]()ν,,u y x E ℑAgain the power spectrum is recorded or observed :()()2,~,ννu E u I =～()[]2,y x E ℑ (10)It , too , is independent of the phase factor A and thus becomes independent of the position of the diffraction structure in the front focal plane . Additionally , equation ( 8 ) shows thatthe larger the图 4 focal length of the lens is , the more extensive the diffraction image in the （u,v ）plane is .傅立叶谱的实例 a) 平面波A plane wave which propagates itself in the direction of the optical axis ( z axis ) ( Fig . 5 ) 15 distinguished in the object plane –( x , y ) plane- by a constant amplitude . Thus , the following results for the Fourier transform :()0,E y x E = (11)and()[]()()()y x yx i E dxdy eE y x E y x νδνδννπ∙==ℑ+-+∞∞-+∞∞-⎰⎰020,This is a point on the focal plane at （vx,vy ）=(0,0) which shifts at slanted incidence by an angle αto the optical axison the rear focal plane(see Fig . 5 ) withλανsin =x ·图 5（b ）有限宽度的无限长狭缝If the diffracting structure is an infinite slit which is transilluminated by a plane wave , this slit is mathematicallydescribed by a reotangular function " rect " perpendicular to the slit direction and having the same width a :(){210,a x for otherwiseEa x rect y x E 〈=⎪⎭⎫⎝⎛=In the rear focal plane the following spectrum then results :()[]()()()()()x y xx y a a yx i a c a E a E dxdy eE y x E yx ννδπννπνδννπsin sin ,002220∙=∙==ℑ⎰⎰∞+∞-+-+- (12)W ith the definition of the Slit function "sinc " :()()x x x c ∙∙=ππsin sinfor infinitely long extension of the slit , one obtains no extension in the slit direction in the spectrum . This changes for a finite length of the slit . The zero points of the " sinc " function are located at …-2/a,-1/a,1/a,2/a …(see Fig . 6 ) .图 6(c) 栅格A grid is a composite diffracting structure . It consists of a periodic sequence ( to be represented by a so-called comb function " comb " ) of individual identical slit functions " SinC " .The grid consists of M slits having a width a and a slit separation d ( > a ) in the x direction . As a result , the field strength distribution in the front focal plane can be represented as follows :()⎪⎭⎫⎝⎛*⎥⎦⎤⎢⎣⎡∙-=⎪⎭⎫ ⎝⎛∙-=∑∑==a x rect d m x E a d m a x rect E y x E M m Mm 1010(,δ where the Fourier transform of a convolution product （E1*E2）is given by :()()[]()()[]()[]()y x y x y x y x E y x E v y x E E ννννν,),(,,,,2121ℑ∙ℑ=*ℑUsing the calculation rules for Fourier transforms , the following spectrum results in the rear focal plane of the lens . :[]()()()()()()()x x M id x y Mm dv mi xx y d d M e a c a E e a E E x x νπνπννδπννπνδνπ∙∙∙∙∙∙=∙∙=ℑ+-=⋅-∑sin sin sin sin 10120 (13)As a consequence of the intensity formation , the phase factor cancels out :()()()()()x x x y y x d d M a c a E v I νπνπννδν∙∙∙=222220sin sin sin , (14)In Fig . 7 , a grid with its corresponding spectrum ( and the corresponding intensity distributions ) is presented . One sees from the spectrum that the envelope curve is formed by the spectrum of the individual slit which has a width a . The finer structure is produced by the periodicity , which is determined by the grid constant Md .图 7Coherent optical filtrationBy intervening in the Fourier spectrum,optical filtration can be performed; this can result in image improvement, etc . The appropriate operation for making the original image visible is again the inverse Fourier transform . Howeve it cannot be used here due to diffraction. The Fourier transform is again used; this leads to the 4f set- up ( see figure 3) Using the lst lens(L1), the spectrum with the appropriate spatial frequencies is generated in the Fourier plane from the original diffraction structure.In this plane,the spectrum can be altered by fading out specific spatial frequency fraction. A modifiec spectrum is created;it is again Fourier transformed by the 2nd lens(L2).If the spectrum is not altered,one obtains the original image in the inverse direction in the image plane (right focal plane of the 2nd lens,see also partial experiment(a) with the arrow diaphragm).This follows from the calculation of the twofold Fourier transformation:The simplest applications for optical fitration are the high and low-pass filtration .Low-pass raster eliminationIn the experiment,the photographic slide was provided with a grid by superimposing grid lines on it in one direction .The scanning theory states that a non-raster image(in this case :Emperor Maximilian) can be exactly reconstructed if the image is band-limited in its spectrum,i.e if it only contains spatial frequencies in the Fourier plane up to an upper limitingfrequencies.The raster image can be described mathematically as follows:This describes the grid lines and the non-raster image.The slit separation of the grid is b.The Fourier spectrum B(vx,vy) of the entire image becomes the following with the convolution law:Where G(vx,vy) is the Fourier transformation of the non-raster image. In addition we made use of the fact that the Fourier transformation of an comb function is also a comb function. This means that the Fourier spectrum is again a grid which is formed by the reiteration of the spectrum of the non-raster image. Each grid point with its immediate surrounding contains the total information of the non-raster image g(x,y).It is important, that the distance of the grid points in the Fourier plane is far enough, that the spectrum of the non-rastered picture (the slide) don’t overlap. However, only in this cases is it possible to filter out a single image point with a pinhole diaphragm. Fig,4(d) shows a situation, where the individual spectrums overlap too much, so that a filtering cannot be successful. This spatial frequency filtration can be considered as multiplication of the spectrum by an aperture function A(vx,vy)(pinhole diaphragm in the Fourier spectrum).In this case, an appropriate measuring dimension would be a diameter of ~1/b. Therefore, we obtain。