Theory and application of modern microscopy Oct23
 Basic concepts 显微成像的基本问题
 Basic Principles for Microscopic Imaging 显微成像的基本原理
 Resolution Limit Imposed by Wave Nature of Light for conventional microscopy
 Basic Principles for Modern Microscopic Imaging 现代显微成像的原理
Basic concepts 显微成像的基本问题
\[E_{s}=\frac{E_{i}}{\sqrt {2}}r_{s}e^{i\left( 2kz_{s}+\varphi \right) } \]
Imaging 成像
Imaging is an important property of any optical system. To define what is an image of an abject fraud by an optical system, first we need to know what is an object in optics. In optics, object can be defined as a spatial distribution of the light intensity in the incoherent illumination or light field in the coherent case. So the image can be defined as a spatial distribution of the light that in most similar to the one on the object. How to predict the image by an object system in terms of the geometrical optics or wave optics or physical optics?
Using light rays trace.
Microscopic imaging 显微成像
Microscopic imaging is an imaging technique to see the microstructure of an object in cellular resolution.
Resolution 分辨率
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The resolution of an optical microscope is defined as the shortest distance between two points on a specimen that can still be distinguished by the observer or camera system as separate entities. An example of this important concept is presented in the figure below (Figure 1), where point sources of light from a specimen appear as Airy diffraction patterns at the microscope intermediate image plane.
Magnification 放大率
Magnification, in optics, the size of an image relative to the size of the object creating it. Linear (sometimes called lateral or transverse) magnification refers to the ratio of image length to object length measured in planes that are perpendicular to the optical axis. A negative value of linear magnification denotes an inverted image. Longitudinal magnification denotes the factor by which an image increases in size, as measured along the optical axis. Angular magnification is equal to the ratio of the tangents of the angles subtended by an object and its image when measured from a given point in the instrument, as with magnifiers and binoculars.
Field of View 视场
 Field of view FOV: the diameter of the object/image

演示 Field of View Diameter
\[Field Size = \frac{Field Number(fn)}{Objective Magnification(M_0)}\] 
The field of view FOV of an optical system is often expressed as the maximum angular size of the object as seen from the entrance pupil. The maximum image height is also used. For finite conjugate systems, the maximum object height is useful.
Concepts of object, image and resolution 物体、像、分辨率的概念
 Optical definition of object and image 物体和像的光学定义
 In optics, object can be defined as a spatial distribution of the light intensity in the incoherent illumination or light field in the coherent case.
 So the image can be defined as a spatial distribution of the light that in most similar to the one on the object.

Resolution limit 传统的极限分辨率概念
 The lateral resolution of an optical system can be defined in terms of its ability to resolve images of two adjacent, selfluminous points.
 When two Airy disks are too close, they form a continuous intensity distribution and cannot be distinguished. The Rayleigh resolution limit is defined as occurring when the peak of the Airy pattern arising from one point coincides with the first minimum of the Airy pattern arising from a second point object. Such a distribution gives an intensity dip of 26%. The distance between the two points in this case is
\[d = \frac{0.61\lambda}{nsinu}=\frac{0.61\lambda}{NA}\]
 Sparrow resolution limit
\[d = \frac{0.5\lambda}{NA}\]  Abbe resolution limit
\[d = \frac{lambda}{NA_{objective}+NA_{condenser}}\]
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How to improve resolution
 \(\lambda\)
 \(NA\)

Principle for scanning imaging
The focused beam of a laser is scanned over the sample and the reflected intensity is displayed as a function of position to create a digital reflected light image of the sample. Scanning a focused laser beam allows the acquisition of digital images with very high resolution since the resolution is determined by the position of the beam rather than the pixel size of the detector.
Basic Principles for Microscopic Imaging 显微成像的基本原理
Geometric Optical Imaging Formula 几何光学成像公式
Rays and wavefronts
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 Rays indicate the direction of energy propagation and are normal to the wavefront surfaces.
A light ray is a line or curve that is perpendicular to the light's wavefronts (and is therefore collinear with the wave vector).  Wavefronts are surfaces of constant OPL from the source point.
 Rays are normal to wavefronts
Basic Laws
 Reflection
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\[\theta_{i}=\theta_{r}\]
 Refraction – Snell's Law
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\[n_1sin\theta_1 = n_2sin\theta_2\] \[v_1sin\theta_2 = v_2sin\theta_1\]
 Total Internal Reflection
TIR occurs when the angle of incidence of a ray propagating from a higher index medium to a lower index medium exceeds the critical angle
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Rearranging Snell's law, we get incidence,
\[sin\theta_1 = \frac{n_2}{n_1}sin\theta_2\]
The Critical angle \(\theta_c\) is the angle of incidence for which the angle of refraction is \(90^{\circ}\),
\[\theta_c = \theta_1 = arcsin(\frac{n_2}{n_1})\]
Gaussian Optics
Gaussian Optics treats imaging as a mapping from object space into image space.
 Planes perpendicular to the axis in one space are mapped to planes perpendicular to the axis in the other space.
 Lines parallel to the axis in one space map to conjugate lines in the other space that either intersect the axis at a common point (focal system), or are also parallel to the axis (afocal system).

The transverse magnification or lateral magnification is the ratio of the image point height from the axis \(h′\) to the conjugate object point height \(h\):
\[m = \frac{h'}{h}\]
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If a lens can be characterized by a single plane then the lens is “thin”. Various relations hold among the quantities shown in the figure.
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Newtonian Equations
For a focal imaging system, an object plane location is related to its conjugate image plane location through the transverse magnification associated with those planes. The Newtonian equations characterize this Gaussian mapping when the axial locations of the conjugate object and image planes are measured relative to the respective focal points. By definition, the front and rear focal lengths continue to be measured relative to the principal planes. The Newtonian equations result from the analysis of similar triangles.
\[X_1X_2 = F^2\] 
Gaussian Equations
The Gaussian equations describe the focal mapping when the respective principal planes are the references for measuring the locations of the conjugate object and image planes.
\[\frac{1}{S_1}+\frac{1}{S_2}=\frac{1}{F}\] 
Magnification
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￼ Lateral Magnification
The transverse magnification or lateral magnification(\(m\)) is the ratio of the image point height from the axis \(h′\) to the conjugate object point height \(h\):
\[m = \frac{h'}{h}\]  Longitudinal magnification
Longitudinal magnification relates the distances between pairs of conjugate planes
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\[\Delta z = z_2 – z_1\]
\[m_1 = \frac{h'_1}{h_1}\]
\[\frac{z'}{z}=\frac{f'_R}{f_F}m_1m_2\]
These equations are valid for widely separated planes. As the plane separation approaches zero, the local longitudinal magnification \(\bar{m}\) is obtained.
\[\bar{m} = \frac{n'}{n}m^2\]
\[\frac{\Delta z'/n'}{\Delta z/n} = m_1m_2\]  Angular magnification: Two additional cardinal points are the front and rear **nodal points **(N and N′) that define the location of unit angular magnification for a focal system.
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\[m \equiv \frac{h'}{h}=\frac{z'_N}{z_N} \]
If the nodal points are coincident with the respective plans, refraction index are the same. The magnification relationship now holds for Gaussian object and image distance:
\[m \equiv \frac{h'}{h}=\frac{z'}{z}\] while \(n=n'\)
 Lateral Magnification
Related Links:
Magnification of a Microscope
 Magnifying power (MP)
MP is defined as the ratio between the angles subtended by an object with and without magnification. The magnifying power (as defined for a single lens) creates an enlarged virtual image of an object. The angle of an object observed with magnification is
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\[u' = \frac{h'}{z' l}=\frac{h(fz')}{f(z'l)}\]
Therefore,
\[MP = \frac{u'}{u}=\frac{d_0(fz')}{f(z'l)}\]
Where \(d_0 = 250mm\) is the minimum focus distance, which is the distance that the object (real or virtual) may be examined without discomfort for the average population.
\[MP \approx \frac{250mm}{f} – \frac{250mm}{z'}\]
If the virtual image is at infinity (observed with a relaxed eye), \(z' = –\infty\), and
\[MP \approx \frac{250mm}{f}\]
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The total magnifying power of the microscope results from the magnification of the microscope objective and the magnifying power of the eyepiece:
\[M_{objective}=\frac{OTL}{f_{objective}}\]
\[MP_{microscope}=M_{objective}MP_{eyepiece}=\frac{OTL}{f_{objective}}\frac{250mm}{f_{eyepiece}}\]  Resolution limit
 \(\lambda\)
 \(NA\)
 Medium
Resolution Limit Imposed by Wave Nature of Light for conventional microscopy
光的衍射和干涉效应对传统显微成像方法的限制
The Diffraction Barrier in Optical Microscopy 光的衍射效应对显微成像分辨率的限制
 Begin of diffraction story
Over the past three centuries, a vast number of technological developments and manufacturing breakthroughs have led to significantly advanced microscope designs featuring dramatically improved image quality with minimal aberration. However, despite the computeraided optical design and automated grinding methodology utilized to fabricate modern lens components, glassbased microscopes are still hampered by an ultimate limit in optical resolution that is imposed by the diffraction of visible light wavefronts as they pass through the circular aperture at the rear focal plane of the objective.
Wavefronts emanating from a point in the specimen plan of microscopy become diffracted at the edges of the objective aperture.
Due to diffraction of light, the image of a specimen never perfectly represents the real details present in the specimen because there is a lower limit below which the microscope optical system cannot resolve structural details.
￼  Abby
The diffractionlimited resolution theory was advanced by German physicist Ernst Abbe in 1873 (see Equation (1)) and later refined by Lord Rayleigh in 1896.
For a microscope objective, the aperture angle is described by the numerical aperture (NA), which includes the term \(sinθ\), the half angle over which the objective can gather light from the specimen. In terms of resolution, the radius of the diffraction Airy disk in the lateral (x,y) image plane is defined by the following formula:
\[Abbe Resolution_{x,y} = \lambda / 2NA\]
The diffractionlimited resolution theory was advanced by German physicist Ernst Abbe in 1873 (see Equation (1)) and later refined by Lord Rayleigh in 1896 (Equation (3)) to quantitate the measure of separation necessary between two Airy patterns in order to distinguish them as separate entities:
\[Abbe Resolution_{z} = 2 \lambda / NA^2\]
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According to Abbe's theory, images are composed from an array of diffractionlimited spots having varying intensity that overlap to produce the final result, as described above. The only mechanism for optimizing spatial resolution and image contrast is to minimize the size of the diffractionlimited spots by: decreasing the imaging wavelength
 increasing numerical aperture
 using an imaging medium having a larger refractive index
 Other Resolution criteria
As a result, most resolution criteria (for example, the Rayleigh criterion, Sparrow limit, or the full width at half maximum; FWHM) are directly related to the properties and geometry of the pointspread function.  Real and Fourier Space
After being subjected to Fourier transformation (see Figure 3), objects observed in the microscope (whether they are periodic or not) can be uniquely described as a summation of numerous sinusoidal curves having different spatial frequencies.
In real space, note that the image of a specimen, present in all conjugate image planes, exists as the Fourier transform in the corresponding aperture planes where higher frequencies represent fine specimen detail and lower frequencies represent coarse details.
In Fourier space, the OTF defines the extent to which spatial frequencies containing information about the specimen are lost, retained, attenuated, or phaseshifted during the imaging process.
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Spatial frequency information that is lost during imaging cannot be recovered, so one of the primary goals for all forms of microscopy is to acquire the highest frequency range as possible for the specimen.  Conclusion
In conclusion, a traditional widefield microscope generates an image of a point source by capturing the light in various locations in the objective and further processing the wavefronts as the pass through the optical train to finally interfere at the image plane.
Laser scanning confocal and multiphoton microscopy have been widely used to moderately enhance spatial resolution along both the lateral and axial axes, but the techniques remain limited in terms of achieving substantial improvement.
Partial coherence limit 部分相干性对显微成像分辨率的限制
Basic concepts
 Correlation
 Mutual coherence function
\[\Gamma_{12} = \langle U_1U^*_2 \rangle = \langle U_1(t'+t)U^*_2(t') \rangle\]  Mutual intensity
 Complex coherence factor of the light
\[\mu_{12} = \Gamma _{12}(0)/\sqrt{(I_1I_2)} = J_{12}/\sqrt{(I_1I_2)}\]
归一化的互强度  Complex degree of mutual coherence
\[\gamma_{12}(\tau) = \Gamma_{12}(\tau)/\sqrt{\Gamma_{11}(0)\Gamma_{22}(0)} = \Gamma_{12}(\tau)/\sqrt{I_1I_2}\]
归一化的互相干函数  Interference
 Van Citter Zernike
扩展光源的复相干因子的模等于光源强度的归一化傅里叶变换
The influence of the condenser on resolution in a microscope
 The degree of coherence in the image of an extended incoherence quasimonochromatic source
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The degree of coherence for any two points in the exit pupil is equal to the degree of coherence for the conjugate points in the entrance pupil; and the phases of the corresponding values of the complex degree of coherence for corresponding point pairs differ by an amount \(\Phi_{11} – \Phi_{22}\)
 Illumination in microscopy: Critical illumination and Köhler's illumination.
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The equal time complex degree of coherence \(j_{12}\) for any pair of points in the object plane of the objective is the same as that due to an incoherent source filling the condenser aperture.
The aberrations of the condenser have no influence on the resolving power of a microscope
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The complex degree of coherence of the light incident upon the object plane of a microscope is the same whether critical or Köhler's illumination is employed.
Interference limit
 Speckle Noise
 Speckle noise in OCT
Basic Principles for Modern Microscopic Imaging 现代显微成像的原理
现代显微成像术主要起源于上一世纪70年代发明的共焦显微术。人们利用信息论的结论：既然一个光学系统所能传递的信息总量是确定的，那么如果我们不需要某些信息，例如，我们已经知道被观察物体平面方向上的尺寸，我们就可以不需要探测平面方向上的信息，而可以获得更多深度方向上的信息，即系统具有深度分辨率。在具体实现上，在极限情况下，只考虑一个点，这个点可以利用空间滤波的方法实现，即物体上的照明点、入射物点和探测点互为共轭像，此即共焦的思想。理论和实验均证明，该方法可以实现深度分辨成像，从而开启了现代显微光学成像的时代。
在以下的讨论中，我们分两个方面进行讨论：一是如何实现高分辨率深度分辨光学断层成像技术，另一个是如何在横向方向上实现超过衍射极限分辨率的成像方法。
高深度分辨光学断层成像原理
现代显微成像方法是在人们不断寻求改进光学显微成像系统的平面和深度分辨率的过程中不断发展起来的，且现在仍处在不断的发展过程中。对于平面分辨率，主要集中在如何获取高于衍射受限分辨率的图像；对于深度分辨率，主要集中在如何非侵入地观测样品表面下的结构，获取接近或超过衍射受限分辨率的图像或获取与切片观察获得的图像质量相同的光学断层图像。
由于非侵入光学断层成像的许多优点，包括:
本章我们先讨论实现高深度分辨光学断层成像所依据的各种现象或性质，下一章将讨论实现平面超分辨成像的各中原理。
基于共焦现象的显微成像
https://www.microscopyu.com/techniques/confocal