Principle and application of optical information technology (十二) Coherent and incoherent transfer function(相干与非相干传递函数)

Diffraction-limited coherent optical imaging system(衍射受限相干光学成像系统) As shown above , the input surface illumination light of diffraction-limited coherent imaging is coherent optical,

Coherent transfer function （相干传递函数） it means that it has a single wavelength, single polarization （偏振）direction and the phrase different is constantly between any two points during the imaging processes of optical field. 相干光条件：单一波长，单一偏振方向，光场中在成像过程中任意两点之间的光程差（相对位相）恒定。 Equation

频域中描述系统的成像特性的频谱函数称为衍射受限系统的相干传递函数 It indicates that under the coherent illumination of diffraction-limited system is linear space invariant in complex amplitude. The transform property of linear space invariant is more convenient in frequency domain（频率域）. In frequency domain , describe the frequency spectra function of imaging property is called the coherent transfer function of diffraction-limited system , denoted by CTF. 频域中描述系统的成像特性的频谱函数称为衍射受限系统的相干传递函数 .

Intrinsic function and the signal frequency spectra of system （系统的本征函数和信号频谱） Describe the eyeball and the relationship between convolution integral （卷及积分）of coherent imaging system. This convolution integral regarded the object points as element ,the image points was produced by coherent superposition（叠加） of diffraction pattern in this point. 从频域来分析成像过程，系统的本征函数是复指数函数。

Inspect the transfer property in various frequency components , definite the following frequency spectra of the input and output respectively:（输入频谱和输出频谱）

The calculation of coherent transfer function(CTF) transfer function (CTF) is the Fourier transformation of the point-spread function点扩散函数. Point-spread function is itself a pupil function of Fourier transformation（点扩散函数本身是光瞳函数的傅里叶变换）, according to the integral theorem of Fourier transformation.

相干传递函数等于光瞳函数，仅在空域坐标和频域坐标之间存在着一定的坐标缩放关系， And in the previous section is given on the pupil of the coordinate transformation of specific physical meaning , mean spatial frequency In generally, pupil function are all central symmetry （中心对称）, so we can definite the coherent transfer function in same reflect coordinate , get rid of the minus sign , instead the primary coherent transfer function as

Diffraction-limited system is a low-pass filter(低通滤波器) In generally , pupil function always take two results are 0 and 1,so do coherent transfer function is , only the 0 and 1. If the value of pupil coordinate is decided by frequency. In the pupil, the exponential basis of this frequency is the same as that in the distribution of the image, there is no amplitude attenuation and phase change, that is, the transfer function of the value of this frequency is 1.

If decided by frequency , the value of pupil coordinate is out of pupil , system is absolutely not to pass the exponential element of frequency, it means that transfer function turn on the 0 to this frequency. 衍射受限相干光学成像系统是一个低通滤波器，在空间频域中存在一个有限的通频带 It allows the maximum frequency to pass through a cutoff frequency of system,(允许通过的最高频率称为系统的截止频率)utilize the representation of

The calculation of circular pupil coherent transfer function For the circular pupil with a diameter of D , it aperture function is Coherent transfer function and cutoff frequency are

For instance , exit pupil diameter（出瞳直径）is the distance between exit pupil and image plane is the illumination light wavelength is , thus,

The calculation of square pupil coherent transfer function The length of square is of exit pupil（出瞳） , pupil function is Coherent transfer function is

Obviously , different direction have different cutoff frequency , in the direction of ,the cutoff frequency of system is ,the maximum cutoff frequency of the system is in the direction of 45 degree angle with the shaft. 不同方位上的截止频率不相同,系统的最大截止频率在与 轴成45°角方向上。

Examples of the calculation of coherent transfer function The focal length of the lens is the same as that of the two coherent imaging systems. Single aperture lens system in diameter is , dual lens system in order to obtain the same cutoff frequency, the aperture diameter is should be equal to much (relative to write relation)?

Coherent transfer function calculation examples of Solutions These two systems are systems with a lateral magnification（横向放大率） of 1, so they do not have to be distinguished from the cutoff frequency and the cutoff frequency of the image . the cutoff frequency of single lens system（单透镜系统） is 凡是物面上各面元发出的低于空间频率的平面波均能无阻挡地通过此成像系统

For the double lens imaging system, the aperture pupil in the spectrum plane, so the entrance pupil and exit pupil at the object side and the image side infinity. For the amplification of 1 systems, can also through the entrance pupil and exit pupil must through the highest spatial frequency aperture, the cutoff frequency of the system can be calculated through an aperture size. 对于双透镜成像系统，系统的截止频率可通过光阑的尺寸来计算 要保证4f系统物面上每一面元发出的低于某一空间频率的平面波均都毫无阻挡地通过此成像系统，则要求光阑直径应不小于透镜直径与物面直径之差 Thus , the cutoff frequency is

Because the ,so 尽管表面上看第二个系统的光栏孔径可以比第一个系统的透镜孔径要小，但是由于要求光阑直径应不小于透镜直径与物面直径之差第二个系统的透镜孔径并不小,因此是一个常用的光学信息处理系统. The two lens focal lengths of the second systems are not necessarily equal. In optical information processing, the magnification（放大率） ratio is sometimes required.

Characteristics of incoherent illumination The amplitude and phase of each point on the surface of the incoherent illumination are independent and statistically independent of each other. 非相干照明时物面上各点的振幅和相位随时间变化的方式是彼此独立、统计无关的 Although on the surface of each point can be obtained through the system after the complex amplitude distribution of a corresponding, but because the illumination is incoherent, should be the first of these complex amplitude distribution（复振幅分布） were obtained corresponding to the intensity distribution（强度分布）,

and then the intensity distribution of superposition (incoherent superposition) whereas the image plane intensity distribution. The incoherent superposition of the propagation （传播）time is linear to the intensity, so the incoherent imaging system is a linear system of intensity. 在传播时光的非相干叠加对于强度是线性的，因此非相干成像系统是强度的线性系统。 The imaging of the optical system in the corona zone（等晕区） is space invariant, so the incoherent imaging system is a linear space invariant system of intensity.

Convolution integral of incoherent linear space invariant imaging system Incoherent linear space invariant imaging system, the relationship between object and image convolution integral In the formula, the intensity distribution of the geometrical optics ideal image（几何光学理想像） is , is the intensity distribution, is constant, because it does not affect the distribution form, so there is no need to give a specific expression.

h1 is intensity impulse response (or incoherent impulse response, intensity point-spread function). h1强度脉冲响应（或称非相干脉冲响应、强度点扩散函数），是是复振幅点扩散函数模的平方 , so 线性空不变成像系统的像强度分布是理想像的强度分布与强度点扩散函数的卷积，系统的成像特性由强度点扩散函数表示，而强度点扩散函数又复振幅点扩散函数由决定。

The strength of linear space invariant system frequency domain image relation The convolution integral constants on both sides of the Fourier transform and the omission of no great importance, On type, , And are all intensity distribution,

其傅里叶变换必有一个常数分量即零频分量，而且它的幅值大于任何非零分量的幅值。 决定像的清晰与否的，主要不是包括零频分量在内的总光强有多大，而在于携带有信息那部分光强相对于零频分量的比值有多大 so the more significant is that, , and compared to the ratio of their zero frequency component.

The transfer function of the object image and normalized spectral intensity By using the zero frequency component, they are normalized and the normalized frequency spectrum is obtained

Normalized frequency spectrum satisfaction formula 其中H为非相干成像系统的光学传递函数（OTF），它描述非相干成像系统在频域的效应。 The three are generally complex functions, can be used with its mode（模） and radial angle（径向角）, so the optical transfer function can be expressed as

Among these In generally , is modulation transfer function(MTF)调制传递函数, is phase transfer function(PTF)相位传递函数. 前者描写了系统对各频率分量对比度的传递特性，后者描述了系统对各频率分量施加的相移

The intrinsic function of the system: cosine function The intensity distribution of the object, the intensity distribution of the image and the point-spread function of the intensity is a non negative real function, and the cosine function is the intrinsic function（本征函数） of the system. That is, the cosine component of the intensity is still in the same frequency, and the change of the contrast and phase is determined by the mode and the amplitude of the system transfer function. OTF唯一的影响是改变这些基元的对比度和相对相位。

Intensity distribution of a cosine input The output light intensity distribution is obtained through the incoherent optical system 余弦条纹的变化取决于系统的光学传递函数在该频率处的取值

The contrast of cosine intensity distribution The definition of contrast or modulation （对比度或调制度）is The modulation of object(or ideal image)and image is

Merge above two formula is The amplitude of transfer function obviously is the phase difference between cosine image and cosine object: 传递函数辐角显然是余弦像和余弦物的相位差 MTF给出了对比度的降低，PTF给出了相应的相移。

The relationship between OTF and CTF Optical transfer function and coherent transfer function are described the transfer function of incoherent and coherent illumination respectively in the same system, both of them are determined by the physical properties of the system, and they are related to each other. By using the autocorrelation theorem and the theorem of

光学传递函数等于相干传递函数的自相关归一化函数。这一结论对有像差的系统和没有像差的系统都完全成立

Diffraction limited system OTF The diffraction-limited system of coherent illumination has The corresponding non coherent illumination diffraction limited OTF is Order the , the substitution of the integral variable does not affect the result.

Geometric interpretation of optical transfer function Under normal circumstances the pupil function is only 1 and 0 of the two value , in the denominator is the total area of the pupil S0, The overlap area (重叠面积) between the translational pupil and the pupil of the molecular representation Center is on ,For diffraction-limited system OTF is only calculated normalized overlap area（归一化重叠面积） 重叠面积取决于两个错开的光瞳的相对位置，也就是和频率有关。

Geometric interpretation of optical transfer function

Some properties of diffraction-limited system OTF 衍射受限的非相干成像系统只改变各频率余弦分量的对比，而不改变它们的相位。即只需考虑MTF而不必考虑PTF。 When , the two pupil overlap completely, and the normalized overlap area is 1, which is the result of OTF normalization, which does not mean that the average light intensity of object and image is the same.由于吸收、反射、散射及光阑挡光等原因，像面平均(背景)光强总要弱于物面光强. But in contrast, the image side zero frequency contrast is the unit value, no attenuation（衰减）,so

,this conclusion is very easy from the two pupil are staggered overlapping area less than completely overlapping area. have a cutoff frequency , when the is big enough, when the two pupil is completely separated, the overlapping area is zero . it means the , that is, the optical transfer function is zero, the frequency component does not appear on the surface of the cutoff frequency in image.

Example 1: the calculation of the diffraction-limited OTF of a square pupil As for pupil diffraction-limited side length of the square optical incoherence imaging system transfer function.

Square pupil function is Pupil total area is When the in the direction of respectively displacement ,the overlapping area can be obtained.

Optical transfer function is is the same system that uses the cutoff frequency of coherent illumination, and the frequency of incoherent system is two times as much as that of the axis（轴）.

Example two: the calculation of the diffraction-limited OTF of the circular pupil

The optical transfer function of the diffraction-limited system with the diameter of the pupil is obtained. Since the pupil is round, the OTF should be round and symmetrical. As long as the axis can be calculated. See the top of the chart, in the direction of moving along the axis, the overlapping area is divided into two equal area AB. According to the geometric formula, the overlapping area is Among the

The cutoff frequency is two times of the cutoff frequency of the corresponding coherent transfer function. In the polar coordinates, the expression of the circular pupil OTF is Among

Classroom practice In the coherent imaging system as shown in the figure below, the complex amplitude of the object is In order to make it look like a face, ask if the circular light bar is used, the diameter should be greater than the number? if using the rectangular light bar, the length should be greater than the number?

Answer The object is a two dimensional cosine grating （二维余弦光栅）at the spatial frequency of X and Y in two different directions, to be like the need to make it the full spectrum of light from the bar through the circular aperture, for its radius should be at least is larger by the spatial frequency spectrum, the rectangular aperture of the long side should at least make spectrum the spatial frequency larger by the short side should at least make smaller by the spatial frequency spectrum. The specific calculation is omitted（省略）.

Exercises The third chapter exercises 3.2, 3.3, 3.4.