From 902dc0c2f6cdf20b5a2611b95cb5dbcb0ccb9863 Mon Sep 17 00:00:00 2001
From: JackHCC <2508074836@qq.com>
Date: Thu, 16 Dec 2021 17:03:35 +0800
Subject: [PATCH] Add Docs and Implement Interference Hologram
1. Add docs to introduce all kinds of holograms
2. Implement Interference Hologram
---
Doc/Fourier_Hologram/README.md | 172 ++++++++++++++++++++++
Doc/Interference_Hologram/README.md | 188 ++++++++++++++++++++++++
Doc/Kinoform/README.md | 175 +++++++++++++++++++++-
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Images/doc_kinoform_p1.png | Bin 0 -> 13545 bytes
Images/doc_kinoform_p2.png | Bin 0 -> 40574 bytes
Matlab/offaxis_interference_hologram.m | 114 ++++++++++++++
Matlab/result/ki_lena_GCH.bmp | Bin 0 -> 1167478 bytes
Matlab/result/ki_lena_recover.bmp | Bin 0 -> 1167478 bytes
Matlab/result/oaih_pku_CGH.bmp | Bin 0 -> 1049654 bytes
Python/fourier_hologram.py | 8 +
Python/kinoforms.py | 12 +-
Python/offaxis_interference_hologram.py | 140 ++++++++++++++++++
Python/result/fh_test_CGH.bmp | Bin 263222 -> 263222 bytes
Python/result/fh_test_large_CGH.bmp | Bin 17334 -> 0 bytes
Python/result/fh_test_recover.bmp | Bin 66358 -> 66358 bytes
Python/result/ki_lena_CGH.bmp | Bin 1167478 -> 1167478 bytes
Python/result/ki_lena_recover.bmp | Bin 1167478 -> 1167478 bytes
Python/result/oaih_pku_CGH.bmp | Bin 0 -> 1049654 bytes
Python/result/oaih_pku_recover.bmp | Bin 0 -> 1049654 bytes
README.md | 50 +++++--
Res/imageO/pku.jpg | Bin 0 -> 93003 bytes
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create mode 100644 Doc/Interference_Hologram/README.md
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create mode 100644 Images/doc_fourier_p2.png
create mode 100644 Images/doc_kinoform_p1.png
create mode 100644 Images/doc_kinoform_p2.png
create mode 100644 Matlab/offaxis_interference_hologram.m
create mode 100644 Matlab/result/ki_lena_GCH.bmp
create mode 100644 Matlab/result/ki_lena_recover.bmp
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delete mode 100644 Python/result/fh_test_large_CGH.bmp
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diff --git a/Doc/Fourier_Hologram/README.md b/Doc/Fourier_Hologram/README.md
new file mode 100644
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--- /dev/null
+++ b/Doc/Fourier_Hologram/README.md
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+# Before Read
+
+注：此文档分中英文撰写，根据需要跳转到相应位置阅读
+
+Note: This document is written in Both Chinese and English, skip to the corresponding location to read if necessary
+
+# 迂回位相型全息图
+
+## 什么是迂回位相型全息图？
+
+1960年，布朗恩( Brown)和罗曼应用迂回位相技术制成二元计算全息图，现在习惯称之为迂回位相型全息图。这种全息图有三个特点：
+
+1. 全息图的透过率是二元的，即只取0或1。
+2. 应用迁回位相编码物光波的位相，全息图可以同时记录复值函数的振幅和位相。
+3. 全息图记录时没有用到参考光波或加偏分量。
+
+这种迁回位相编码制作的二元计算全息图是计算全息的开创篇，正是它的特点使得它在后来的很多编码技术中得到了广泛的使用。
+
+## 原理
+
+二元付里叶变换计算全息图的制作过程一般分为四个步骤：
+
+### 物面和全息面的抽样
+
+在把物面信息输入计算机作离散付里叶变换之前，要先对物面和全息面分别按抽样定理进行抽样，求出各样点的幅值和位相信息。
+
+物面函数：
+$$
+f(x, y)=a(x, y) \exp [j \phi(x, y)]
+$$
+傅里叶变换：
+$$
+F(\mu, v)=A(\mu, v) \cdot \exp [j \Phi(\mu, \nu)]
+$$
+即物函数$f(x,y)$在空间中大小是有限的。同时从物理意义上来说，它在空间域频率也是近似有限的。可以这样来理解，即物函数在空间是缓慢变化的，故其空间频率域中的高频部分可近似为零。
+
+具体的抽样过程如图所示：
+
+
+
+### 计算离散傅里叶变换（DFT）
+
+由于制作的是付里叶变换全息图，故要算出全息图面上谱的复振幅分布$F(\mu, v)$，必须进行离散的付里叶变换。离散的付里叶变换可写成下式：
+$$
+F(u, v)=1 / M N \sum_{x=0}^{M-1}\left[\sum_{y=0}^{N-1} f(x, y) \exp (-j 2 \pi v / N)\right] \exp (-j 2 \pi c x / M)
+$$
+
+### 编码
+
+即将算出的全息图面上样点处的幅值和位相，仍在计算机内进行编码处理，然后把编码后的信息输给绘图仪绘图，或由微密度计显示产生计算全息原图。由编码理论，可知编码的目的是要把复数值转变为实的非负函数。在迂回位相二元全息图中，罗曼等人是用下述精巧的方法分別编码振幅和位相的。在全息图的每个抽样单元中，放置一个通光孔径，改变此通光孔径的面积来调制波面的幅值，改变通光孔径中心距抽样单元中心的位置来编码波面的位相。通光孔径的形状可以是多种多样的，这主要依据绘图仪的作图能力和数学描述的方便来取定。在罗曼等人的工作中，选择了矩形孔径。矩形孔的可变参数可以有好几个，但由于只编码幅值和位相，故孔径的变参数仅取两个。罗曼编码方式分为Ⅰ型、Ⅱ型、Ⅲ型。
+
+以罗曼Ⅲ型为例，处理方法如图：
+
+
+
+### 绘图显示
+
+当计算机已经完成幅值和位相编码后，下一步就是制作计算全息原图并把它缩小到合适的尺寸。绘图仪、阴极射线管、或计算机控制的微密度计都可用来制作原图。光学缩版后就变成实际可用的计算全息片了。如果采用特殊的输出（如电子束）直接把全息图记录在胶片上，就可省去制作原图和缩版的步骤，这样全息图的制作再现精度会更高些。
+
+
+
+## 如何制作迂回位相型全息图？
+
+在这个实验中，我们使用这个图像进行测试:
+
+
+
+这里提供了两种进行实验的方法，Matlab或Python，您可以选择熟悉的方法来运行代码。
+
+- 运行[`fourier_hologram.m`](../../Matlab/fourier_hologram.m) 或 [`fourier_hologram.py`](../../Python/fourier_hologram.py)，它们分别在Python和Matlab文件夹中。
+
+- 然后你会在下面得到Circuitous Phase Type Hologram/Fourier Hologram。
+
+
+
++ 最后，运行程序后，可以得到复现的图像，并将其与原始图像进行比较。
+
+
+
+
+
+
+
+# Circuitous Phase Type Hologram
+
+## What is Circuitous Phase Type Hologram?
+
+In 1960, Brown and Roman used devolution phase technology to make binary computed holograms, now commonly known as devolution phase holograms. This hologram has three characteristics:
+
+1. The transmittance of holograms is binary, that is, 0 or 1.
+
+2. The hologram can record both the amplitude and the phase of the complex function by reverting to the phase of the object light wave.
+
+3. No reference light wave or offset component is used in hologram recording.
+
+This kind of binary CGH is the first part of CGH. It is its characteristic that makes it widely used in many coding technologies.
+
+## Principle
+
+The production process of binary Fourier transform CGH is generally divided into four steps:
+
+### Sampling of object and holographic surfaces
+
+Before input the information of object plane into computer for discrete Fourier transform, the object plane and holographic plane should be sampled respectively according to sampling theorem, and the amplitude and phase information of each point can be obtained.
+
+Object surface function:
+
+$$
+f(x, y)=a(x, y) \exp [j \phi(x, y)]
+$$
+
+Fourier transform:
+
+$$
+F(\mu, v)=A(\mu, v) \cdot \exp [j \Phi(\mu, \nu)]
+$$
+
+That is, the matter function $f(x,y)$is finite in space. At the same time, in a physical sense, its frequency in the space domain is approximately finite. It can be understood that the matter function changes slowly in space, so the high frequency part of its spatial frequency domain can be approximately zero.
+
+The specific sampling process is shown in the figure:
+
+
+
+### Compute the Discrete Fourier Transform (DFT)
+
+A discrete Fourier transform is necessary to calculate the complex amplitude distribution $F(\mu, v)$of the hologram. The discrete Fourier transform can be written as follows:
+
+$$
+F(u, v)=1 / M N \sum_{x=0}^{M-1}\left[\sum_{y=0}^{N-1} f(x, y) \exp (-j 2 \pi v / N)\right] \exp (-j 2 \pi c x / M)
+$$
+
+### Codes
+
+The amplitude and phase of the sample point on the hologram plane to be calculated are still coded in the computer, and then the coded information is transferred to the plotter for drawing, or the original cHOLogram is generated by the micro-densitometer display. From coding theory, we know that the purpose of coding is to transform complex values into real non-negative functions. In a roundabout phase binary hologram Roman et al. coded the amplitude and phase separately in the following ingenious way. In each sampling element of the hologram, an aperture is placed, the amplitude of the wave surface is modulated by changing the area of the aperture, and the phase of the wave surface is encoded by changing the position between the center of the aperture and the center of the sampling element. The shape of aperture can be various, which is mainly based on the plotting ability of plotter and the convenience of mathematical description. In the work of Roman et al., rectangular apertures were chosen. There can be several variable parameters of rectangular hole, but only two variable parameters of aperture are selected because only amplitude and phase are encoded. Romanesque coding is divided into type ⅰ, type ⅱ and type ⅲ.
+
+Taking Roman ⅲ type as an example, the treatment method is shown as follows:
+
+
+
+### Drawing and Display
+
+Once the computer has encoded the amplitude and phase, the next step is to make the CGH original and reduce it to the right size. Plotters, cathode-ray tubes, or computer-controlled microdensitometers can be used to make original drawings. The optical version is reduced to a practical CGH. If a special output (such as an electron beam) is used to record the hologram directly on the film, the process of making the original image and reducing the size of the hologram can be eliminated and the reproduction of the hologram will be more accurate.
+
+## How to make Circuitous Phase Type Hologram?
+
+Circuitous Phase Type Hologram/Fourier Hologram is Binary hologram.
+
+For this experiment we used this image for testing
+
+
+
+Two ways to do experiments are provided here, Matlab or Python, and you can choose the familiar way to run the code.
+
+- Run the `fourier_hologram.m` or `fourier_hologram.py`, They are in the Python and Matlab folders respectively.
+- Then you will get Circuitous Phase Type Hologram/Fourier Hologram below.
+
+
+
++ Finally, after running the program, you can get the reproduced graph and compare it with the original image.
+
+
+
+
+
+
+
+
+
+
+
+
+
diff --git a/Doc/Interference_Hologram/README.md b/Doc/Interference_Hologram/README.md
new file mode 100644
index 0000000..b98920d
--- /dev/null
+++ b/Doc/Interference_Hologram/README.md
@@ -0,0 +1,188 @@
+# Before Read
+
+注：此文档分中英文撰写，根据需要跳转到相应位置阅读
+
+Note: This document is written in Both Chinese and English, skip to the corresponding location to read if necessary
+
+# 计算全息干涉图
+
+## 什么是计算全息干涉图？
+
+干涉是光学中的基本现象，相干的双光束或多光束叠加就形成干涉条纹。以往的干涉都是应用光学方法来实现，但光学的干涉图却完全可用计算来仿真( simulation)，即计算机产生的干涉图。全息摄像主要经历光学全息，数字全息，和计算全息这样几个过程，其基本原理都是通过干涉纪录及衍射重建。当两束光波进行干涉时，记录物体的振幅和相位信息，通过衍射计算，重构物体的振幅和相位信息。
+
+用计算机模拟产生光学干涉图，只预先知道干涉图样的数学模式即可。但是一般的光学干涉图，其透过率是连续变化的函数，而用计算机输出的数字讯号经绘图仪或CRT显示来制作的计算全息，干涉图，常常希望只得到二元透过率(取0或1值)，即二元计算全息干涉图。为了达到这一目的，就必须把连续透过率函数进行非线性变换，转化成二元形式，而且此二元形式的干涉图仍应保存物波的全部信息，在再现时能得到所需的物光波。
+
+## 原理
+
+普通的离轴型光学全息图，可以把它看成是物光和参考光“双光束”相叠加而形成的干涉图，其透过率为
+$$
+\begin{aligned}
+h(x, y) &=|R \exp (j 2 \pi x / T)+A(x, y) \exp (j \phi(x, y))|^{2} \\
+&=R^{2}+|A(x, y)|^{2}+2 R A(x, y) \cos (2 \pi x / T-\phi(x, y))
+\end{aligned}
+$$
+其中，$R \exp (j 2 \pi x / T)$为参考光复振幅，$A(x, y) \exp (j \phi(x, y))$为物光复振幅。
+
+当物光仅仅只有相位变化时，上式可简化为：
+$$
+h(x, y)=\frac{1}{2}(1+\cos \{2 \pi x / T-\phi(x, y)\})
+$$
+根据上式可以得出振幅最大值和最小值位置，得到干涉条纹。但是透过率是连续的，干涉条纹的对比是不高的。如果只是为了要提高条纹的对比，以致在极端的情况下能产生二元值化(0,1)的条纹，一种方法是可以用多光束相干涉，当参加干涉的光束数越多时，条纹就越锐，逐渐趋向于二元形式，这时的透过率函数可写成多光束迭加求和的形式。另一常规的方法是用高反差的胶片来记录此干涉条纹，这种高反差胶片的硬限幅特性能把连续色调的干涉条纹变成二元干涉图。后一种方法启示我们，若要计算机来制作二元计算全息干涉图，可以借用电讯系统中的非线硬限幅器的模型来进行处理。
+
+我们举三个例子来说明二元计算全息图的制作技术及其再现情况。这三个例子都是记录的位相型波面，它们在光学干涉量度中有很大的应用价值。其中第一个例子是球面波，它通常在光学中是由透镜或波带板状得到的；第二个例子是锥形波面，也可由光学的旋转三棱镜获得；而第三个例子是螺旋形波面，其位相在方位方向上是线性变化的，这种波面用通常的光学元件却是得不到的。故最后一个例子说明了，由于计算机仿真干涉图的灵活性大，我们完全可以得到用光学制作技术难以得的特殊位相型变化波前。这使得计算全息干涉图很适用于光学的量度术中去检特殊的光学波面。
+
+**球面波**
+
+球面波的位相变化为：
+$$
+\phi(x, y)=\frac{k}{2 f}\left(x^{2}+y^{2}\right)
+$$
+其中，$k=\frac{2 \pi}{\lambda}$，$f$是球面波曲率半径。
+
+计算得到条纹数：
+$$
+N_{p}=\left.\phi(x, y)\right|_{\max }=\frac{\pi \cdot \frac{D^{2}}{4}}{\lambda \cdot f \cdot 2 \pi}=\frac{D^{2}}{8 \lambda f}=\frac{N}{16}
+$$
+**锥形波面**
+
+锥形面波的位相变化为：
+$$
+\phi(x, y)=2 \pi r / r_{0}
+$$
+其中，$r=\left(x^{2}+y^{2}\right)^{1 / 2}$。
+
+计算得到条纹数：
+$$
+N_{\mathrm{p}}=\frac{D}{2 \cdot r_{0}}=\frac{N}{8}
+$$
+**螺旋形波面**
+
+相对变化：
+$$
+\phi(x, y)=2 \pi \theta / \theta_0
+$$
+其中，$\theta=\tan ^{-1} y / x$
+
+
+
+## 如何制作计算全息干涉图？
+
+在这个实验中，我们使用这个图像进行测试：
+
+
+
+这里提供了两种进行实验的方法，Matlab或Python，您可以选择熟悉的方法来运行代码。
+
+- 运行[`offaxis_interference_hologram.m`](../../Matlab/offaxis_interference_hologram.m) 或 [`offaxis_interference_hologram.py`](../../Python/offaxis_interference_hologram.py)，它们分别在Python和Matlab文件夹中。
+
+- 然后你会在下面得到Off Axis Interference Hologram。
+
+
+
+- 最后，运行程序后，可以得到复现的图像，并将其与原始图像进行比较。
+
+
+
+
+
+# Computational Interference Holograms
+
+## What is Computational Interference Holograms？
+
+Interference is a basic phenomenon in optics. Interference fringes are formed when two or more coherent beams are superimposed. In the past, interference is realized by optical method, but the optical interferogram can be completely simulated by calculation, that is, the interferogram generated by computer. Holography mainly goes through optical holography, digital holography and computational holography. The basic principle of holography is interference recording and diffraction reconstruction. When two light waves interfere, the amplitude and phase information of the object are recorded, and the amplitude and phase information of the object are reconstructed by diffraction calculation.
+
+To produce optical interferograms by computer simulation, only the mathematical pattern of interference pattern is known in advance. However, the transmittance of general optical interferogram is a function of continuous change, while the computed hologram (CGH), which is made by using the digital signal output by computer and displayed by plotter or CRT, usually hopes to get only binary transmittance (0 or 1), namely binary CGH interferogram. In order to achieve this goal, it is necessary to transform the continuous transmittance function into a binary form by nonlinear transformation, and the binary interferogram should still preserve all the information of object wave, so as to obtain the desired object wave when reproducing.
+
+## Principle
+
+The common off-axis optical hologram can be regarded as an interferogram formed by the superposition of object light and reference light "double beam", and its transmittance is
+
+$$
+\begin{aligned}
+
+h(x, y) &=|R \exp (j 2 \pi x / T)+A(x, y) \exp (j \phi(x, y))|^{2} \\
+
+&=R^{2}+|A(x, y)|^{2}+2 R A(x, y) \cos (2 \pi x / T-\phi(x, y))
+
+\end{aligned}
+$$
+
+Where,$R \exp (j 2 \pi x / T)$is the reference light recovery amplitude and i$A(x, y) \exp (j \phi(x, y))$s the object light recovery amplitude.
+
+When the object light only changes in phase, the above equation can be simplified as:
+
+$$
+h(x, y)=\frac{1}{2}(1+\cos \{2 \pi x / T-\phi(x, y)\})
+$$
+
+According to the above formula, the position of the maximum and minimum amplitude can be obtained, and the interference fringe can be obtained. But the transmittance is continuous and the contrast of interference fringes is not high. If only in order to improve the contrast of stripe, so that in extreme cases can generate binary value (0, 1) stripes, a kind of method is you can use multiple beam interference, when attend the more the number of beam interference stripe is more sharp, gradually tend to be binary form, the transmittance function can be written in the form of multiple beam superposition summation. Another conventional method is to record the interference fringes on high-contrast film. The hard limiting property of this high-contrast film can transform the continuous tone interference fringes into binary interferograms. The latter method suggests that the model of non-linear hard limiter in telecommunication system can be used to make binary CGH interferogram by computer.
+
+Three examples are given to illustrate the fabrication and reproduction of binary CGH. These three examples are all recorded phase type wavefronts which have great application value in optical interferometry. The first example is spherical waves, which are usually obtained in optics by lenses or waveband plates; The second example is the conical wavefront, also obtained by optical rotating prisms; A third example is a helical wave surface, whose phase varies linearly in orientation, which is not available with ordinary optical elements. Therefore, the last example illustrates that, due to the flexibility of computer simulation interferogram, we can get special phase change wavefront which is difficult to obtain by optical fabrication technology. This makes CGH very suitable for optical measurement to detect special optical wavefronts.
+
+**Spherical wave**
+
+The potential transformation of spherical waves is transformed into:
+
+$$
+\phi(x, y)=\frac{k}{2 f}\left(x^{2}+y^{2}\right)
+$$
+
+Where, $k=\frac{2 \ PI}{\lambda}$, $f$is the radius of spherical wave curvature.
+
+Calculate the number of fringe:
+
+$$
+N_{p}=\left.\phi(x, y)\right|_{\max }=\frac{\pi \cdot \frac{D^{2}}{4}}{\lambda \cdot f \cdot 2 \pi}=\frac{D^{2}}{8 \lambda f}=\frac{N}{16}
+$$
+
+**Conical wave surface**
+
+The potential phase transformation of the conical surface wave transforms into:
+
+$$
+\phi(x, y)=2 \pi r / r_{0}
+$$
+
+Where，$r= left(x^{2}+y^{2})^{1/2}$
+
+
+
+Calculate the number of fringe:
+
+$$
+N_{\mathrm{p}}=\frac{D}{2 \cdot r_{0}}=\frac{N}{8}
+$$
+
+**Spiral wave surface **
+
+
+
+Relative change:
+
+$$
+\phi(x, y)=2 \pi \theta / \theta_0
+$$
+
+Where，$\theta=\tan ^{-1} y/x$
+
+## How to make Computational Interference Holograms？
+
+For this experiment we used this image for testing
+
+
+
+Two ways to do experiments are provided here, Matlab or Python, and you can choose the familiar way to run the code.
+
+- Run the `offaxis_interference_hologram.m` or `offaxis_interference_hologram.py`, They are in the Python and Matlab folders respectively.
+- Then you will get Off Axis Interference Hologram below.
+
+
+
+- Finally, after running the program, you can get the reproduced graph and compare it with the original image.
+
+
+
+
+
+
+
diff --git a/Doc/Kinoform/README.md b/Doc/Kinoform/README.md
index 9dd872c..cd8e3b7 100644
--- a/Doc/Kinoform/README.md
+++ b/Doc/Kinoform/README.md
@@ -1,10 +1,181 @@
-# Kinoform
+# Before Read
+
+注：此文档分中英文撰写，根据需要跳转到相应位置阅读
+
+Note: This document is written in Both Chinese and English, skip to the corresponding location to read if necessary
+
+# 相息图
+
+## 什么是相息图？
+
+相息图是假设在整个记录平面内光波振幅为常数的条件下，仅记录波前位相信息的元件。因此，相息图再现时也只给出物波的位相信息，而丢失了物波的振幅信息（振幅为常数）。随着相息图技术的发展，后来的相息图不仅能保存物波的位相信息，而且同时能记录和再现物波的振幅信息，这种扩展了的相息图就完全保存了物体的全部信息，因而是一种新型的全息图。
+
+光学全息和计算全息图都将光波信息转化为全息图的透过率变化或干涉条纹图形而记录在胶片上，但相息图却只将光波的位相信息以浮雕形式记录在胶片上。光学全息和计算全息的再现复数波场都是建立在光的衍射理论的基础上，而相息图却似菲涅尔透镜那样，只是通过改变光学厚度去变化照射光波的位相分布，从而再现出原始的物光波。因此相息图可以看成是一块由计算机制作的复杂透镜，其表面形状，对同轴再现的相息图来说，很像光学菲涅尔透镜，而对离轴像的再现来讲，相息图则像一块精密制作的闪耀光栅。
+
+相息图的最大优点也许是其衍射效率特高。一张相息图完全是一透明片，不会衰减入射光强。照明相息图只得到单一的衍射级，再现单一的图像，没有共轭像或杂散图像叠加其上。因此它既不像光学全息中的由于实、虚共轭像的存在而减弱再现像的光强，也不像二元计算全息中有多余的衍射级存在而分散入射光强。在理想情况下，相息图可把全部入射光用来重建这个单个像，故衍射效率可达到100%。
+
+## 原理
+
+相息图是在个基本假定下制作的，即认为待记录的物体是理想的漫射体，从物体上发出的漫射光均匀地照射在整个相息图平面上，因此相息图平面内的光振幅为常数，而且物体的全部信息都均匀地扩展到整个相息图平面。这样，要构造这个漫射性质的物体，仅仅只要记录散射光波的位相信息就可以了。相息图平面内的光波复振幅分布为：
+$$
+U(x, y)=A \cdot \exp (j \phi(x, y))
+$$
+然而在通常的情况下，相息图平面内的光振幅A不可能总是常数，因此常数振幅的假设是有问題的。但我们可以改变原始物体的位相函数，努力做到此物体的散射光到达相息图平面时实质上是均匀分布的，即光振幅可为常数。在物理上，相当于通过漫射体（如毛玻璃）来照明物体，这个漫射体扩展了照明光波，并把入射光波的位相随机地散布在物体的各个部分，结果是相息图上的每一部分都接收到了物体的全部信息。
+
+现在我们从数学处理上来寻找物体为漫射体的合适描述。常用的方法是把物体表示成二维（平面）或三维（立体）点孔阵列的集合。毎个小孔波场的复振幅为$T_k$,并向空间发射球面次波。
+$$
+T_{\mathrm{k}}=t_{\mathrm{k}} \exp \left(j \alpha_{\mathrm{k}}\right)
+$$
+其中，$t_k$表示第$k$个点孔的透射率，$t_k$=0表示点孔阻拦光波，$t_k$=1表示点孔完全透光，$t_k$在(0,1)之间取值，表示不同的透光情况。$\alpha_k$是各点的位相变化函数。把物体表示成点孔集合这一模式，优点在于以后作数学运算时比较方便，物理意义地很清楚，它自动地保证了光信息的空间漫射（毎个点源向整个空间发射球面波）.
+
+给出了物体的数学表达式，下面就可说明相息图的实际计算和制作过程，为了简化，我们以一维为例，但它可以很方便地推广到二维或三维情况。下图表示物面到相息图的菲涅尔衍射。
+
+
+
+物面上中心在$a_k$,线度为$\delta$的孔径，发射光波传播到距离为$E$处的的相息图平面$x$.根据克希霍夫衍射理论，相息图平面上的光波复振幅为
+$$
+U\left(x, a_{\mathrm{k}}\right)=C \int_{a_{\mathrm{k}}-\delta / 2}^{a_{\mathrm{k}}+\delta / 2} \exp \left[j \frac{2 \pi}{\lambda}\left\{(x-a)^{2}+z^{2}\right\}^{\frac{1}{2}}\right] d a
+$$
+如果$\delta$的孔径无限大，同时光波是一种电磁波，它满足麦克斯韦方程组，将光波表示成球面波的形式，根据推导可以获得标量场下光波的菲涅尔衍射积分的表达式如下：
+$$
+U(x, y)=\frac{\exp (j k d)}{j \lambda d} \iint_{-\infty}^{\infty} U_{0}\left(x_{0}, y_{0}\right) \exp \left(\frac{j k}{2 d}\left[\left(x-x_{0}\right)^{2}+\left(y-y_{0}\right)^{2}\right]\right) d x d y
+$$
+将上式改写为离散的傅里叶变换的形式为：
+$$
+\begin{array}{c}
+U(p \Delta x, q \Delta y)=\frac{\exp (j k d)}{j \lambda d} \exp \left[\frac{j k}{2 d}\left(p \Delta x^{2}\right)+q \Delta y^{2}\right] \\
+\times I\left[U_{0}\left(m \Delta x_{0}, n \Delta y_{0}\right) \times \exp \left(\frac{j k}{2 d}\left(m \Delta x_{0}\right)^{2},\left(n \Delta y_{0}\right)^{2}\right)\right]_{\frac{p \Delta x}{\lambda d}, \frac{p \Delta y}{\lambda d}}
+\end{array}
+$$
+相息图平面上的复振幅分布（离散形式）为
+$$
+U(x, y)=U_{\mathrm{R}}(x, y)+j U_{\mathrm{I}}(x, y)
+$$
+其相角为：
+$$
+\phi(x, y)=\tan ^{-1}\left(U_{\mathrm{I}}(x, y) / U_{\mathrm{R}}(x, y)\right)
+$$
+由于上述的相息图不能记录光场的振幅信息，这总会给再现像的逼真度产生影响。但是相息图又有比全息图衍射效率高，同轴再现时无共轭像干扰等的优点，故自然就想到，要在保持相息图优点的条件下，进一步对相息图改进，消除其不足之处。下面几种相息图的扩展技术，它们都能同时记录物波的位相和振幅全部信息。
+
+**无参考光同轴复式全息图 ROACH**
+
+这种方法是想法寻找特殊的全息照相记录介质，它不仅能用漂白技术来形成浮雕像，达到调制光波位相的目的，而且也能引入光波的振幅调制。这种照相记录介质就是用多层乳胶的柯达Ⅱ彩色胶片( Kodacolor Ⅱ),其中不同的乳胶层只对不同的色光曝光。当用单色光照明处理以后的胶片时，一层乳胶将吸收光能，以调制入射波的振幅，而其余两层仍是透明的，只能产生入射光的位相信息。其工作原理图下图：
-Todo
+
+
+**纯位相型全息图**
+
+另外一种相息图扩展技术是对相息图记录的位相再调制。制作的办法是在相息图上不直接记录物波经付里叶变换后的位相，而是记录由物波的付里叶变换振幅重新调制的位相。通常是在原先的位相项中加进额外的位相项，再现时用一项位相变化去调物波振幅。这祥制作的相息图也叫做纯位相型全息图。尽管全息图面上的振幅仍为常数，但这种纯位相型全息图同时包含了物波的全部信息。当然，由于在再现波中引进了振幅变化，再现时光波必然产生偏离，这将减少再现像的衍射效率。
+
+## 如何制作相息图？
+
+在这个实验中，我们使用这个图像进行测试:
+
+
+
+这里提供了两种进行实验的方法，Matlab或Python，您可以选择熟悉的方法来运行代码。
+
+- 运行[`kinoforms.m`](../../Matlab/kinoforms.m) 或 [`kinoforms.py`](../../Python/kinoforms.py)，它们分别在Python和Matlab文件夹中。
+
+- 然后你会在下面得到Kinoform。
+
+
+
+- 最后，运行程序后，可以得到复现的图像，并将其与原始图像进行比较。
+
+
+
+
+
+# Kinoform
## What is Kinoform?
+A kinoform is an element that records only the phase information of the wave front under the assumption that the amplitude of the light wave in the whole recording plane is constant. Therefore, only the phase information of the object wave is given, but the amplitude information of the object wave is lost (the amplitude is constant). With the development of kinoform technology, the later kinoform can not only save the phase information of object wave, but also record and reproduce the amplitude information of object wave. This extended kinoform completely preserves all the information of object, so it is a new type of hologram.
+
+Both optical holography and CGH record the light wave information on film by converting the transmittance change or interference fringe pattern of hologram, but the phase pattern only records the light wave phase information in relief form on film. The reconstruction of complex wave field of optical holography and computational holography is based on the diffraction theory of light, while the kinoform is just like Fresnel lens, which only changes the phase distribution of the irradiating light wave by changing the optical thickness, so as to reproduce the original object light wave. Thus, a kinoform can be thought of as a complex computer-generated lens whose surface shape looks much like an optical Fresnel lens for a coaxial representation of the kinoform, and like a precisely fabricated sparkle grating for an off-axis representation.
+
+Perhaps the greatest advantage of the kinoform is its high diffraction efficiency. A kinoform is completely transparent and does not attenuate the incident light intensity. Illumination kinoform only gets a single diffraction order, reproduces a single image, no conjugate image or stray image superimposed on it. Therefore, it is not like optical holography which weakens the light intensity of the reconstructed image due to the existence of real and virtual conjugated images, nor like binary CGH which disperses the incident light intensity due to the existence of redundant diffraction levels. Under ideal conditions, the kinoform can be used to reconstruct the single image with all incident light, so the diffraction efficiency can reach 100%.
+
## Principle
+The kinoform is made under the basic assumption that the object to be recorded is an ideal diffuse body, the diffuse light from the object is uniformly illuminated on the whole kinoform plane, so the amplitude of the light in the kinoform plane is constant, and all the information of the object is uniformly extended to the whole kinoform plane. Thus, to construct the diffuse object, it is only necessary to record the phase information of the scattered light wave. The complex amplitude distribution of light wave in the kinoform plane is:
+$$
+U(x, y)=A \cdot \exp (j \phi(x, y))
+$$
+In general, however, the optical amplitude A in the kinoform plane cannot always be constant, so the assumption of constant amplitude is problematic. However, we can change the phase function of the original object and try to achieve that the scattered light of the object is essentially evenly distributed when it reaches the plane of the kinoform, that is, the amplitude of the light can be constant. In physics, it is equivalent to illuminating an object with a diffuse object (such as ground glass), which expands the illumination wave and randomly disperses the phase of the incident light wave over various parts of the object, so that each part of the kinoform receives all the information of the object.
+
+Now let's look at the mathematical process to find a suitable description of the object as diffuse. A common method is to represent an object as a collection of two-dimensional (flat) or three-dimensional (solid) point-hole arrays. Each keyhole wave field has a complex amplitude of $T_k$and emits spherical secondary waves into space.
+$$
+T_{\mathrm{k}}=t_{\mathrm{k}} \exp \left(j \alpha_{\mathrm{k}}\right)
+$$
+$t_k$=0; $t_k$=1; $t_k$=0; $t_k$=1; $\alpha_k$is the phase change function of each point. The advantage of this model is that the object is represented as a set of point holes, which is convenient for mathematical calculation and has a clear physical meaning. It automatically ensures the spatial diffusion of optical information (every point source emits spherical waves to the whole space).
+
+Given the mathematical expression of the object, the actual calculation and production process of the kinoform can be illustrated below. For simplicity, we use one dimension as an example, but it can be easily generalized to two or three dimensions. The figure below shows Fresnel diffraction from the surface to the kinoform.
+
+
+
+The aperture centered at $a_k$and linear at $\delta$is transmitted to the kinoform plane $X $at distance $E$. According to kirchhoff diffraction theory, the complex amplitude of light wave on the kinoform plane is
+$$
+U\left(x, a_{\mathrm{k}}\right)=C \int_{a_{\mathrm{k}}-\delta / 2}^{a_{\mathrm{k}}+\delta / 2} \exp \left[j \frac{2 \pi}{\lambda}\left\{(x-a)^{2}+z^{2}\right\}^{\frac{1}{2}}\right] d a
+$$
+If the aperture of $\delta$is infinite and the light wave is a kind of electromagnetic wave, which satisfies Maxwell's equations and is expressed in the form of spherical wave, the Fresnel diffraction integral of the light wave under scalar field can be obtained as follows:
+
+$$
+U(x, y)=\frac{\exp (j k d)}{j \lambda d} \iint_{-\infty}^{\infty} U_{0}\left(x_{0}, y_{0}\right) \exp \left(\frac{j k}{2 d}\left[\left(x-x_{0}\right)^{2}+\left(y-y_{0}\right)^{2}\right]\right) d x d y
+$$
+
+The formula above can be rewritten as the discrete Fourier transform:
+
+$$
+\begin{array}{c}
+
+U(p \Delta x, q \Delta y)=\frac{\exp (j k d)}{j \lambda d} \exp \left[\frac{j k}{2 d}\left(p \Delta x^{2}\right)+q \Delta y^{2}\right] \ \
+
+\times I\left[U_{0}\left(m \Delta x_{0}, n \Delta y_{0}\right) \times \exp \left(\frac{j k}{2 d}\left(m \Delta x_{0}\right)^{2},\left(n \Delta y_{0}\right)^{2}\right)\right]_{\frac{p \Delta x}{\lambda d}, \frac{p \Delta y}{\lambda d}}
+
+\end{array}
+$$
+
+The complex amplitude distribution (discrete form) on the kinoform plane is
+
+$$
+U(x, y)=U_{\mathrm{R}}(x, y)+j U_{\mathrm{I}}(x, y)
+$$
+
+Its phase Angle is:
+
+$$
+\phi(x, y)=\tan ^{-1}\left(U_{\mathrm{I}}(x, y) / U_{\mathrm{R}}(x, y)\right)
+$$
+
+Since the above kinoform cannot record the amplitude information of the light field, it will always affect the fidelity of the reconstructed image. However, the phase and kinoform has the advantages of higher diffraction efficiency than hologram and no conjugate image interference in coaxial reproduction. Therefore, it naturally occurred to us to further improve the phase and kinoform to eliminate its shortcomings while maintaining the advantages of the phase and kinoform. The following are several extended kinoform techniques that simultaneously record all phase and amplitude information of object waves.
+
+**Coaxial compound hologram without reference light ROACH**
+
+This method is to find a special hologram recording medium, which can not only use bleaching technology to form floating statues to achieve the purpose of modulation of light wave phase, but also can introduce the amplitude modulation of light wave. The photographic recording medium is Kodacolor ii, which uses multiple layers of latex, in which different layers of latex are exposed only to different colors. When the resulting film is illuminated with monochromatic light, one layer of latex absorbs light energy to modulate the amplitude of the incident wave, while the other two layers remain transparent, producing only phase information about the incident light. Its working principle diagram is as follows:
+
+
+
+**Pure phase hologram **
+
+Another phase - and - kinoform extension technique is phase - modulation of the phase - and - kinoform record. The method is to record the phase of the object wave modulated by the Fourier transform amplitude of the object wave on the kinoform instead of directly recording the phase after the Fourier transform. Usually, an additional phase term is added to the original phase term and a phase change is used to modulate the amplitude of the matter wave during reproduction. Such a kinoform is also called a pure phase hologram. Although the amplitude of the hologram is constant, the hologram of pure phase contains all the information of matter wave. Of course, due to the introduction of amplitude changes in the reconstructed wave, the light wave will inevitably deviate during the reconstruction, which will reduce the diffraction efficiency of the reconstructed image.
+
## How to make Kinoform?
+For this experiment we used this image for testing
+
+
+
+Two ways to do experiments are provided here, Matlab or Python, and you can choose the familiar way to run the code.
+
+- Run the [`kinoforms.m`](../../Matlab/kinoforms.m) or [`kinoforms.py`](../../Python/kinoforms.py), They are in the Python and Matlab folders respectively.
+- Then you will get Kinoform below.
+
+
+
+- Finally, after running the program, you can get the reproduced graph and compare it with the original image.
+
+
+
+###
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