Principle and application of optical information technology (11) Coherent illumination diffraction limited system

Aperture effect of lens(孔径效应) 输入面紧贴透镜的情况比较简单，可直接利用透镜孔径作为变换积分域进行计算。 For the object in the rear of the lens, the area of the object surface is illuminated by a projection of the aperture of the lens on the object surface. The diffraction effect of the aperture of the lens can be used as an equivalent alternative to the diffraction effect of aperture projection on the surface of the lens. 对于物在透镜后方，透镜孔径的衍射效应可以用在物面上孔径投影的衍射效应做等效替代。 The product function is increased by a factor:

In front of the lens, approximate geometric optics, is considered the distance between the object surface and the lens with respect to the diameter is not large. （几何光学近似，也就是考虑物面与透镜之间的距离相对于透径直径而言不是很大的情况） When the light from the lens to the spread between can be seen as a straight line, and ignore aperture diffraction of lens. Such conditions, in the vast majority of practical problems can be met. Then there

Classroom practice The optical path shown below the spatial frequency spectrum analysis of a one-dimensional objects, the lowest spatial frequency which contains for 20 weeks /mm, the highest spatial frequency for 200 weeks /mm, lighting wavelength of 0.6 microns, if the spectrum of the surface between the lowest spatial frequency with the highest spatial frequency is 50 mm, the focal length of the lens we should take much?

Answer

General model of imaging system

Any imaging system can be divided into three parts: 1、 from the plane to the entrance pupil plane is the first part; （从物平面到入瞳平面为第一部分） 2、from the entrance pupil plane to the pupil plane into second parts; （从入瞳平面到出瞳平面为第二部分） 3、from the exit pupil plane to the image plane into third parts. （从出瞳平面到像平面为第三部分）

光波在一、三两部分空间的传播可按菲涅耳衍射处理。第二部分的透镜系统，在等晕条件下可当做一个“黑箱”来处理。 Black box two edge are the entrance pupil and exit pupil, as long as the black box to determine the two edges of the end of the nature, the nature of the whole system can be determined, not necessarily to its internal structure. 黑箱的两个边端分别是入瞳和出瞳，只要能够确定这黑箱的两个边端的性质，整个系统的性质便可确定，不必深究其内部结构。

The optical system of the "black box" the edge of nature In order to determine the impulse response of the system, need to know the transformation of spherical wave and the black box on the point of light, when the pupil plane input emission spherical wave, the wave field characteristics of pupil plane transmission. 当入瞳平面上输入发射球面波时，出瞳平面透射的波场特性。 The actual light group can be divided into two categories ：the diffraction limited system and the aberration of the system. （衍射受限系统和有像差的系统）

当像差很小或者系统的孔径和视场都不大，实际光学系统就可近似看做衍射受限系统。 At this time, the nature of the edge is relatively simple, the surface of a point source emitted by a point source of divergent spherical wave projected onto the pupil, the light group is transformed into the pupil on the convergence of the spherical wave. 有像差系统的边端条件是，点光源发出的发散球面波投射到入瞳上，出瞳处的透射波场明显偏离理想球面波，偏离程度由波像差决定。

Abel that is due to the limited diffraction effect of the pupil size, the Rayleigh diffraction effect from the exit pupil of limited size. Because a pupil is just another pupil of geometry, the two views are equivalent. The diffraction effect can be attributed to the restriction of the pupil and the pupil to the imaging light wave. 衍射效应可以归结为入瞳或出瞳对于成像光波的限制 This book uses Rayleigh's statement.

Imaging analysis of coherent illumination diffraction limited system 任何平面物场分布都可以看做是无数小面元的组合，而每个小面元都可看做一个加权的函数。 For a lens or an imaging system, if we can understand clearly the object plane took a light vibration facet through imaging system, as in the light vibration distribution caused by the plane, through the linear superposition principle can be formed by any surface light field distribution through the system after the like optical field distribution, and then obtain the image plane intensity distribution. 这就是相干照明下的成像过程。这里关键是求出任意小面元的光振动所对应的像场分布。

The corresponding facet of the light vibration of arbitrary point spread function like field distribution When the light vibration of the surface element δ is a unit impulse（单位脉冲）, the field distribution function is called the point spread function or the impulse response. 像场分布函数叫做点扩散函数或脉冲响应。 The optical field distribution of the point in the plane of the object plane is usually represented by the , and the light field produced by the point in the image plane is produced by the imaging system.

首先研究在相干照明下，一个消像差的正薄透镜对透明物成实像的情况。 Sketch is the lens point spread function

Discussion method The distance before object is placed on the input plane of the lens, and the imaging condition is observed on the conjugate surface after the lens distance is . Assume that the distribution of complex amplitude close to the object (紧靠物体后的复振幅分布) is , point emit the unite pulse .

沿光波传播方向，逐面计算三个特定平面上的场分布: Field distribution on the plane close to the lens Field distribution on a plane close to the lens Field distribution on the observation plane h In this way, the input and output relations of a point source can be finally derived. 最终可导出一个点源的输入输出关系。

The complex amplitude distribution of light field near the lens Using the Fresnel formula, close to the plane in front of the lens on the field distribution（紧靠透镜前的平面上的场分布）

Omit the apostrophe, omitting the constant phase factor on formula This wave through the aperture function ,the focal length of lens is , complex amplitude

Complex amplitude distribution caused by unit pulse is point spread function 由透镜后表面到观察面，光场的传播满足菲涅耳衍射，于是物平面上的单位脉冲在观察面上引起的复振幅分布即点扩散函数 Constant phase factor expression substitution and omit including -1 to

The use of image plane conjugate relation satisfies the Gauss formula, get 单位脉冲引起的复振幅分布即点扩散函数。

Simplification of point spread function Phase factor before integral sign ,The intensity distribution of the final detection is not affected. You can omit. 积分前的相位因子不影响最终探测的强度分布，略去。 Phase factor related to the object plane coordinate , For each point on the surface of the image surface light field contributions, to participate in the integration, can not be neglected. 与物面坐标有关的相位因子在求物面上各点对像面光场的贡献时，要参与积分，不可随意略去。

当透镜的孔径比较大时，物面上每一物点产生的脉冲响应是一个很小的像斑，能够对于像面上点光场产生有意义贡献的，必定是物面上以几何成像所对应的以物点为中心的微小区域。 In this region can be approximated to think that the coordinates of the same value. Its size is the same as the conjugate coordinate of the point, and can be used as the following approximation is the transverse magnification of the imaging lens （横向放大率）

The final form of the point spread function By approximating the two phase factors are no longer dependent on the material surface coordinates So it will not affect the intensity distribution on the image plane , All can be omitted. In this way, the form of the point spread function becomes

The lateral magnification is is the object point corresponding to the image point coordinates.

The physical meaning of point spread function(1) Under the condition of near axis imaging, the point spread function is only related to the difference of image plane coordinate. This shows that the lens imaging system is spatial invariant. 近轴条件下，点扩散函数只与像面坐标差有关，这说明透镜成像系统是空不变的。 透镜的脉冲响应就等于透镜孔径的夫琅和费衍射图样，其中心位于理想像点处。

透镜孔径的衍射作用明显与否，是由孔径线度相对于波长和像距的比例决定的,为此对孔径平面上的坐标做如下变换： By the point spread function equation

The physical meaning of point spread function(2) When the aperture size is much larger than the , In coordinates , The value of is 1 in the infinite region. So At this point, the object point is imaged as a point, namely, geometrical optics ideal image（几何光学理想像）, so the geometrical optics is the limit of the wave optics, the condition of the "infinite" aperture. 几何光学是波动光学的极限情况，“无限大”孔径的情况。

Point spread function of diffraction limited system The point spread function of the lens is a special case of the general optical system, and we extend it to the diffraction limited optical system. 所谓衍射受限是指不考虑系统的几何像差，仅仅考虑系统的衍射限制。 如果忽略衍射效应，点物成理想的点像， Diffraction limited systems in general can be composed of several coaxial spherical lens, the lens can be both positive or negative lens, lens, and the lens is not necessarily thin.

系统对光束大小的限制是由系统的孔径光阑决定的，即在考察衍射受限系统时，实际上主要是考察孔径光阑的衍射作用。 The aperture in the object space and the image is called the entrance pupil, referred to as the pupil; the aperture in the image space of the image is called the exit pupil, referred to as the exit pupil . The whole optical system, the entrance pupil and exit pupil keep the conjugate relation.（入瞳和出瞳保持物像共轭关系。） 由入瞳控制的物方光束通过系统后成为被出瞳控制的像方光束，下面建立模型：

General model of imaging system

Any imaging system can be divided into three parts: 1、 from the plane to the entrance pupil plane is the first part; （从物平面到入瞳平面为第一部分） 2、from the entrance pupil plane to the pupil plane into second parts; （从入瞳平面到出瞳平面为第二部分） 3、from the exit pupil plane to the image plane into third parts. （从出瞳平面到像平面为第三部分）

光波在一、三两部分空间的传播可按菲涅耳衍射处理。第二部分的透镜系统，在等晕条件下可当做一个“黑箱”来处理。 Black box two edge are the entrance pupil and exit pupil, as long as the black box to determine the two edges of the end of the nature, the nature of the whole system can be determined, not necessarily to its internal structure. 黑箱的两个边端分别是入瞳和出瞳，只要能够确定这黑箱的两个边端的性质，整个系统的性质便可确定，不必深究其内部结构。

The optical system of the "black box" the edge of nature In order to determine the impulse response of the system, need to know the transformation of spherical wave and the black box on the point of light, when the pupil plane input emission spherical wave, the wave field characteristics of pupil plane transmission. 当入瞳平面上输入发射球面波时，出瞳平面透射的波场特性。 The actual light group can be divided into two categories ：the diffraction limited system and the aberration of the system. （衍射受限系统和有像差的系统）

当像差很小或者系统的孔径和视场都不大，实际光学系统就可近似看做衍射受限系统。 At this time, the nature of the edge is relatively simple, the surface of a point source emitted by a point source of divergent spherical wave projected onto the pupil, the light group is transformed into the pupil on the convergence of the spherical wave. 有像差系统的边端条件是，点光源发出的发散球面波投射到入瞳上，出瞳处的透射波场明显偏离理想球面波，偏离程度由波像差决定。

Abel that is due to the limited diffraction effect of the pupil size, the Rayleigh diffraction effect from the exit pupil of limited size. Because a pupil is just another pupil of geometry, the two views are equivalent. The diffraction effect can be attributed to the restriction of the pupil and the pupil to the imaging light wave. 衍射效应可以归结为入瞳或出瞳对于成像光波的限制 This book uses Rayleigh's statement.

The point spread function of exit pupil Spherical wave emitted by the point of the object, in which the image is obtained is a spherical wave, which is restricted by the exit pupil. 由于出射光瞳的限制作用，在像平面上将产生以理想像点为中心的出瞳孔径的夫琅和费衍射花样。 On the surface of the unit pulse through the diffraction limited system after conjugation and object surface as the complex amplitude distribution on the surface, the point spread function

In the formula, and is independent and is a complex constant; is pupil function (often called the pupil function), at the pupil in the value of 1, at the pupil outside its value is 0 ; is the pupil plane to image plane distance, image distance is not common sense. 还要说明，在推导公式时，同样略去了关于和的二次相位因子 The center of the pupil in the Fraunhofer diffraction pattern of geometric optics as ideal point .

General expression of point spread function of diffraction limited system The same on the object plane coordinates and the pupil plane coordinates coordinate transformation, So

If the pupil is large enough for the , in the coordinates, the pupil function is 1, and the point spread function becomes When the diffraction of the pupil can be neglected, the point of the pulse can be obtained by the diffraction limited system on the image plane, which is still a point pulse.(忽略光瞳的衍射时，点的脉冲通过衍射受限系统后在像面上得到的仍然是点脉冲)

Imaging law of diffraction limited system under coherent illumination A deterministic distribution can be easily decomposed into a linear combination of numerous functions, and each function can be determined by the point spread function type. However, in the image plane, the results of the synthesis of the numerous impulse responses are related to the illumination of the material surface. Illumination is coherent, then the response of numerous pulses on the image plane is coherent superposition. 在像平面上将这些无数个脉冲响应合成的结果和物面照明情况有关。物面上照明是相干的，则这无数个脉冲在像平面上的响应便是相干叠加。

Like the complex amplitude distribution像的复振幅分布 can be expressed as the superposition integral formula for the complex amplitude distribution of the and the pulse response function （脉冲响应函数）superposition integral.

Image distribution of ideal imaging The convolution integral can be rewritten as 这个坐标的转换意义是使物面上的坐标和像面上坐标归一化 Impulse response with ideal imaging

We can get the ideal convolution integral image distribution The distribution form of the ideal image is the same as the distribution of the object, but it is amplified by M times. 理想像的分布形式与物的分布形式是一样的，只是放大了M倍

Convolution integral of diffraction limited imaging system In order to represent the imaging process in the form of a standard convolution, the point spread function is first defined. Then the convolution integral becomes

Therefore, after the diffraction limited system, the image distribution is the convolution of the ideal image and the point spread function, which indicates that the diffraction limited imaging system can still be regarded as a linear space invariant system in a more general situation. 因此，物通过衍射受限系统后的像分布是的理想像和点扩散函数的卷积，这就表明，对于更普遍的情形，衍射受限成像系统仍可看成线性空不变系统。

The relation between point spread function and pupil function For the point spread function of the coordinate transformation 经过坐标变换的点扩散函数

This shows that the point spread function is the Fourier transform of pupil function, so the importance of pupil function for the imaging of diffraction limited system is seen. 点扩散函数是光瞳函数的傅里叶变换，由此可见光瞳函数对于衍射受限系统成像的重要性。 Since it is null, the impulse response of the origin can be used to represent the characteristics of the imaging system.

Summary Relates to the strict theory mainly has two aspects, one is the approximate integral variable within a must, two is due to the different object coordinates must coordinate transformation. 一是由于积分内的变量必须的近似，二是由于物像坐标不同必须的坐标变换 The research method used is the result of the diffraction of the convergent light which is given in the last section, which is the Fourier transform of the aperture. 会聚光照明的菲涅耳衍射的结果就是孔径的傅里叶变换

Results are obtained, the image distribution is the convolution of the ideal image and the point spread function after the diffraction limited system. This shows that, for a more general situation, the diffraction limited imaging system can still be regarded as a linear space invariant system. 物通过衍射受限系统后的像分布是的理想像和点扩散函数的卷积。这就表明，对于更普遍的情形，衍射受限成像系统仍可看成线性空不变系统。

Examples A cosine type amplitude grating, the complex amplitude transmittance （复振幅透过率） On the surface of the imaging system, the plane wave propagation direction is in the plane, and the angle between the plane wave and the Z axis is in the plane. Lens focal length is , aperture is . (1) the frequency spectrum of the transmission light field of the object（物体透射光场的频谱）; (2) how much is the maximum angle of the stripe in the image plane? Image surface strength distribution.

Answer (1) monochromatic inclined illumination light can be expressed as The transmission light field of the object can be expressed as The spectrum of transmitted light field is

Answer continued (2) the image surface intensity distribution can be applied to the convolution formula 几何像 The point spread function is

The independent variable of the pupil function is actually the spatial frequency, or the pupil function is the pupil function with the spatial frequency as the independent variable. The normalized point spread function is the Fourier transform to the spatial frequency of the pupil function of the independent variable. 光瞳函数的自变量实际上就是空间频率，这里的光瞳函数已经是以空间频率为自变量的光瞳函数。归一化的点扩散函数是以空间频率为自变量的光瞳函数的傅氏变换。

Answer continued(2) The convolution theorem can be applied to calculate the intensity distribution of the image surface. The spectrum of Fourier transform geometric image available previously computed object is expressed as

The frequency spectrum of the point spread function is

Answer continued(3) Fourier transform geometry is like three functions, the inverse transform is plane wave. The spectrum of the point spread function is the pupil function（点扩散函数的频谱就是光瞳函数）. In order to make the image appear stripe, at least two functions of the spectrum can be generated from the pupil function, and the transform becomes a plane wave to the surface of the object . 要求与出瞳函数乘积不为零就要求

The position of the other function is further than the first one, which requires only two functions. When the near axis optical approximation is used, the sine of the angle is equal to the value of its arc, and the left and right sides of the above two inequalities are required. 采用近轴光学近似时，角度的正弦等于其弧度值

Answer continued(4) The maximum value reached previously given instant , like the Fourier transform geometry in three of only two, the two functions and the pupil function of the product or function, but also because the pupil function in the pupil range as a function of two , before the coefficient is constant. 反变换可以得到像面上的光场分布 The intensity distribution is the square of the light field distribution.

Answer continued(5) (2) in the spectrum of the transmitted light field, at least two of the spectrum of the transmitted light field can be passed through the pupil of the lens, and the image of the object surface is imaged. In the imaging optical path, in the back focal plane of the light bar and the diameter of the lens frame formed by light bar equivalent (光栏等效). 后焦面作为观察平面时，光场复振幅除一个相位因子外，就是物体透射光场的傅里叶变换即其频谱

so in the back focal plane of optical field for the three use function in the spot the three spot in the spatial frequency（空间频率）. the coordinates of the rear focal plane corresponding to the spatial frequency are 考虑近轴光学近似，角度的正弦等于其弧度值

Answer continued(6) (2) in the spectrum of the transmitted light field, at least two of the spectrum of the transmitted light field can be passed through the pupil of the lens, and the image of the object surface is imaged. Clearly required 使像平面出现条纹的要求： Two spot can be passed in the rear focal plane diameter of the equivalent light bar.

因为这时只可能有一个光点通过等效光栏射到物面上成像，形成等效的平行光照明的现象。

Exercises The third chapter exercises 3.1 One of the third questions can be set to a domain value, to determine the approximate conditions.