... | ... | @@ -8,12 +8,12 @@ Reflection photoelasticity method |
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Reflective polariscope can probe the photoelastic fringes with light source and camera on same side of the photoelastic specimen. As shown in the figure below, the reflective polariscope contains 5 basic elements:
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* The light source.
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* The *Polarizer*, which is the circular polarizer (plotted as a combination of a linear polarizer and quarter-wave plate below) between the light source and photoelastic specimen.
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* The *Polarizer*, which is usually a circular polarizer (plotted as a combination of a linear polarizer and quarter-wave plate below) between the light source and photoelastic specimen.
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* The mirror to reflect light.
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* The *Analyzer*, which is the circular polarizer (plotted as a combination of a linear polarizer and quarter-wave plate below) between the camera and the photoelastic specimen.
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* The *Analyzer*, which is usually a circular polarizer (plotted as a combination of a linear polarizer and quarter-wave plate below) between the camera and the photoelastic specimen.
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* The camera.
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Similar to the [transmissive polariscope](#https://git-xen.lmgc.univ-montp2.fr/PhotoElasticity/Main/wikis/transmission-photoelasticity), both the *Polarizer* and the *Analyzer* are usually circular polarizers. Thus the principle axis of the quarter-wave plate in figure below must form $`45^{\circ}`$ angle with the direction of polarization of the linear polarizer. It is important to point out that a dark field reflective polariscope uses same kind of circular polarizer for both the *Polarizer* and the *Analyzer* (see Sec.2 for mathematical proof), whereas the transmissive polariscope uses circular polarizers with different chirality.
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Similar to the [transmissive polariscope](#https://git-xen.lmgc.univ-montp2.fr/PhotoElasticity/Main/wikis/transmission-photoelasticity), both the *Polarizer* and the *Analyzer* are usually circular polarizers. Thus the principle axis of the quarter-wave plate in figure below must form $`45^{\circ}`$ angle with the direction of polarization of the linear polarizer. It is important to point out that a dark field reflective polariscope uses circular polarizers with same chirality for both the *Polarizer* and the *Analyzer* (see Sec.2 for mathematical proof), whereas the transmissive polariscope uses circular polarizers with different chirality.
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![reflective_circular_4](uploads/9b9996db2ba507c78fd8e16f65a6144c/reflective_circular_4.png)
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... | ... | @@ -21,14 +21,14 @@ Similar to the [transmissive polariscope](#https://git-xen.lmgc.univ-montp2.fr/P |
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There are two typical ways to implement the mirror in real granular physics experiments:
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* Using photo-elastic particles with a reflective surface. The left figure below shows a sketch of an example experimental setup using this technique. (see [*J. Puckett et al.*](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.110.058001) or [*K. E. Daniels et al.*](https://aip.scitation.org/doi/abs/10.1063/1.4983049) for more details about the experiment) A typical image recorded from this same experiment for a jammed disc packing is also attached below (from [*the PhD Thesis of J. G. Puckett*](http://nile.physics.ncsu.edu/pub/Publications/papers/Puckett-2012-thesis.pdf) ), showing same type of fringes as observed in the transmissive polariscopes.
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* Using a mirror table. Particles are then put on this table to perform experiments. This technique allows usage of transparent photoelastic particles the same way as in the transmissive polariscope case. An example implementation of the mirror table is shown in the up-right subfigure of the figure below. (see [*Y. Zhao et al.*](https://www.epj-conferences.org/articles/epjconf/abs/2017/09/epjconf162348/epjconf162348.html) for more details of this experiment)
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* Using photoelastic particles with one reflective surface. The left figure below shows a sketch of an example experimental setup using this technique. (see [*J. Puckett et al.*](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.110.058001) or [*K. E. Daniels et al.*](https://aip.scitation.org/doi/abs/10.1063/1.4983049) for more details about the experiment) A typical image recorded from this experiment for a jammed disc packing is also attached below (from [*the PhD Thesis of J. G. Puckett*](http://nile.physics.ncsu.edu/pub/Publications/papers/Puckett-2012-thesis.pdf) ), showing same type of fringes as observed in the transmissive polariscopes.
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* Using a mirror table. Transparent particles are then put on this table to perform experiments. This technique allows usage of transparent photoelastic particles the same way as in the transmissive polariscope case. An example implementation of the mirror table is shown in the up-right subfigure of the figure below. (see [*Y. Zhao et al.*](https://www.epj-conferences.org/articles/epjconf/abs/2017/09/epjconf162348/epjconf162348.html) for more details of this experiment)
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![james](uploads/958f33b4b081ddc4ee59e906709a9a28/james.png)
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### 1.2. How to make the photoelastic particles reflective?
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A typical way to create the reflective surface for photoelastic particles is to coat one side of the particles with mirror effect paint. An empirical choice that works well is the [*Rust-Oleum Mirror Effect spray*](https://www.amazon.com/Rust-Oleum-267727-Specialty-Mirror-6-Ounce/dp/B00FMRXJW2/ref=sr_1_1?ie=UTF8&qid=1544796251&sr=8-1&keywords=rust-oleum+mirror+effect) (figure below). To ensure uniform coating, it is typical to first paint a sheet of photoelastic material and then cut particles from it. The lower right figure below shows a picture of the painted photoelastic sheet after [cutting of the particles](#https://git-xen.lmgc.univ-montp2.fr/PhotoElasticity/Main/wikis/cutting-sample). The figure below also shows different angles of a particle after this coating process.
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A typical way to create the reflective surface for photoelastic particles is to coat one side of the particles with mirror effect paint. An empirical choice that works well is the [*Rust-Oleum Mirror Effect spray*](https://www.amazon.com/Rust-Oleum-267727-Specialty-Mirror-6-Ounce/dp/B00FMRXJW2/ref=sr_1_1?ie=UTF8&qid=1544796251&sr=8-1&keywords=rust-oleum+mirror+effect) (figure below). To ensure uniform coating, it is typical to first paint a sheet of photoelastic material and then cut particles from it. The lower right figure below shows a picture of the painted photoelastic sheet after [cutting of the particles](#https://git-xen.lmgc.univ-montp2.fr/PhotoElasticity/Main/wikis/cutting-sample). The figure below also shows different angles of particles after this coating process.
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... | ... | @@ -37,17 +37,17 @@ A typical way to create the reflective surface for photoelastic particles is to |
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### 1.3. Determine the configuration of circular polarizers
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It is very important to note that in reflective polariscope, the circular polarizers used as both *Polarizer* and *Analyzer* must have their quarter-wave plate side towards the granular sample. However sometimes it is hard to tell which side of a circular polarizer is quarter-wave plate by simple looking at it. A simple trick can be used to solve this problem is: put the circular polarizer on a piece of metal (can be any metal in the lab, even your lab keys). If the metal becomes black then the wave-plate side is towards the metal, otherwise the linear polarizer side is towards the metal.
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It is very important to note that in reflective polariscope, the circular polarizers used as both *Polarizer* and *Analyzer* must have their quarter-wave plate side towards the granular sample. However sometimes it is hard to tell which side of a circular polarizer is quarter-wave plate. A simple trick can be used to solve this problem: put the circular polarizer on a piece of metal (can be any metal in the lab, even your lab keys). If the metal becomes black then the wave-plate side is towards the metal, otherwise the linear polarizer side is towards the metal. (see Sec.2 for why)
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![polarizer](uploads/5ae4ce71bf075d4694730975d4b992f0/polarizer.png)
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### 1.4. Ensure a uniform distribution of light intensity
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In experiments using photoelastic particles, it is important to keep the light intensity distribution uniform among the system. Because both [empirical pressure measurement](#https://git-xen.lmgc.univ-montp2.fr/PhotoElasticity/Main/wikis/gradient-analysis) and the [nonlinear fitting force measurement](#https://git-xen.lmgc.univ-montp2.fr/PhotoElasticity/Main/wikis/inverse-analysis) depend sensitively on the background light intensity. The reflective polariscope has a higher chance to suffer from the light heterogeneity, comparing to the transmissive polariscope. First, the reflection light intensity is more sensitive to the relative position between light source, particle and camera (figure below). Second, if the effective mirror is implemented using coated particles, small titing of particles may induce big change of background light intensity for that particle (figure below).
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In experiments using photoelastic particles, it is important to keep the light intensity distribution uniform among the system. Because both [empirical pressure measurement](#https://git-xen.lmgc.univ-montp2.fr/PhotoElasticity/Main/wikis/gradient-analysis) and the [nonlinear fitting force measurement](#https://git-xen.lmgc.univ-montp2.fr/PhotoElasticity/Main/wikis/inverse-analysis) depend sensitively on the background light intensity. The reflective polariscope has a higher chance to suffer from the light heterogeneity, comparing to the transmissive polariscope. First, the reflection light intensity is more sensitive to the relative position between light source, particle and camera (left figure below). Second, if the effective mirror is implemented using coated particles, small tilting of particles may induce big change of background light intensity for that particle (right figure below).
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![reflective_light2](uploads/fd6f969c1e6d4944a63863c6870eed51/reflective_light2.png)
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This light heterogeneity can be minimized using the following trick. When taking photoelastic images using both *Polarizer* and *Analyzer* (left figure below), take another image without *Analyzer* while keep everything else the same (middle figure below). The second image gives information about the background light intensity $`I_0`$ (see sec.2 for details). Divide the first image using the second image results in a rescaled image with uniform light intensity (right figure below), which can be used in the data analysis afterwards.
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To reduce the light heterogeneity, the light source should have a large enough area. The light heterogeneity can also be reduced using the following trick. When taking photoelastic images using both *Polarizer* and *Analyzer* (left figure below), take another image without *Analyzer* while keeping everything else the same (middle figure below). The second image gives information about the background light intensity $`I_0`$ (see sec.2 for details). Divide the first image using the second image results in a rescaled image with uniform light intensity (right figure below), which can be used in the data analysis afterwards.
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![intensity_adjust](uploads/3bd8bfe6a1981a772f57e0efa3e1a1e5/intensity_adjust.png)
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... | ... | @@ -59,19 +59,23 @@ The painted surface of particles usually can not reflect light as good as a comm |
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In this section the theoretical background of the reflective polariscope is reviewed. In section 2.1. the relation between local stress and the light intensity recorded by camera is derived. In section 2.2. a proof for the chirality change of circular polarized light after reflection is presented.
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### 2.1. Photoelasticity in the reflection polariscope
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### 2.1. Photoelasticity in the reflective polariscope
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![reflective_circular2](uploads/28f1e3f8a9384cb6ffc5a21e1d82fdc0/reflective_circular2.png)
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As shown in the above figure, under reflective polariscope, the circular polarized light goes through the photoelastic specimen twice with different chirality of polarization. Suppose the height (the size along $`z`$ direction) of the specimen is $`h`$, the pattern observed here will be the same as the pattern formed by a $`h/2`$ height specimen using transmissive polariscope. This can be shown as following: suppose the principle direction for the stress tensor of the specimen point under consideration is $`\hat{m_1}`$ and $`\hat{m_2}`$ (corresponding to principle stress $`\sigma_1`$ and $`\sigma_2`$ respectively). Denote $`\phi = \alpha - \pi/4`$. Then in region (2)
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As shown in the above figure, under reflective polariscope, the circular polarized light goes through the photoelastic specimen twice with different chirality of polarization. Suppose the height (the size along $`z`$ direction) of the specimen is $`h`$, the pattern observed here will be the same as the pattern formed by a $`h/2`$ height specimen using transmissive polariscope. This can be shown as following: suppose the linear polarizer creates a polarized light along $\hat{y}$ direction. In region (1) denote the light vector as:
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```math
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E_1 = \sqrt{I_0}e^{i(kz-\omega t)}\hat{y}
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```
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where $`I_0`=|E_1|^2$ is a constant called the background light intensity, $`k`$ is the wave number and $`\omega`$ is the frequency of the light. Suppose the principle direction for the stress tensor of the specimen point under consideration is $`\hat{m_1}`$ and $`\hat{m_2}`$ (corresponding to principle stress $`\sigma_1`$ and $`\sigma_2`$ respectively). Denote $`\phi = \alpha - \pi/4`$. Then in region (2)
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```math
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E_{2} = \frac{1}{\sqrt{2}}\sqrt{I_0}e^{i(kz-\omega t)}(-i\hat{f}+\hat{s}) = \frac{1}{\sqrt{2}}\sqrt{I_0}e^{i(kz-\omega t)}e^{i\phi}(-i\hat{m_1}+\hat{m_2})
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```
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where $`I_0`$ is a constant as is called the background light intensity, $`k`$ is the wave number and $`\omega`$ is the frequency of the light, $`\hat{f}`$ and $`\hat{s}`$ is the fast and slow principle direction of the quarter-wave plate. Passing the specimen results in a $`\Delta = 2\pi(\sigma_1-\sigma_2)/f_{\sigma}`$ phase lead [5] to the $`\hat{m_1}`$ component of light. So in region (3):
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where $`\hat{f}`$ and $`\hat{s}`$ are the fast and slow principle directions of the quarter-wave plate. Passing the specimen results in a $`\Delta = 2\pi(\sigma_1-\sigma_2)/f_{\sigma}`$ phase lead [5] to the $`\hat{m_1}`$ component of light. So in region (3):
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```math
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E_{3} = \frac{1}{\sqrt{2}}\sqrt{I_0}e^{i(kz-\omega t)}e^{i\phi}(-ie^{-i\Delta}\hat{m_1}+\hat{m_2})
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```
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After reflection, the electrodynamic boundary condition requires a $`\pi`$ phase shift for both components of the light (see section 2.2.). So the reflected light in region (4) is:
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After reflection, the electrodynamic boundary condition requires a $`\pi`$ phase shift for both components of the light (see sec. 2.2.). So the reflected light in region (4) is:
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```math
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E_{4} = \frac{1}{\sqrt{2}}\sqrt{I_0}e^{i(-kz-\omega t)}e^{i\phi}(+ie^{-i\Delta}\hat{m_1}-\hat{m_2})
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```
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... | ... | @@ -90,7 +94,7 @@ Finally, the light intensity that can be observed by the observer behind the lin |
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```math
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I = |E_{6}\cdot \hat{y}|^2 = I_0sin^2\Delta = I_0sin^2(\frac{2\pi(\sigma_1-\sigma_2)}{f_{\sigma}}h)
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```
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Note for a transmission polariscope the corresponding expression is $`I_0sin^2\frac{\Delta}{2}`$. This different needs to be taken care of when solving the contact forces by nonlinear fitting -- the stress-optic relation used in transmission polariscope can not be used directly to fit the patterns recorded by the reflection polariscope.
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Note for a transmission polariscope the corresponding expression is $`I_0sin^2\frac{\Delta}{2}`$. This different needs to be taken care of when solving the contact forces by nonlinear fitting -- the stress-optic relation used in transmission polariscope can not be used directly to fit the patterns recorded by the reflection polariscope. Also note when $\Delta=0$, i.e., for a stress-free specimen or no specimen, $I=0$. This is why metal looks black under a circular polarizer with wave plate side towards the metal.
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### 2.2. Chirality change of circular polarized light by reflection
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... | ... | @@ -107,7 +111,7 @@ where $`\epsilon`$ is the electric permittivity, $`\mu`$ is the magnetic permeab |
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```math
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\vec{J}_f=\sigma \vec{E}
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```
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Note, when $`\sigma=0`$, $\vec{E}(z,t)$ is the solution of Maxwell in vacuum (and is a good approximation in air), whose form has already been used in section 2.1.. Therefore, the incident light, which is in air, writes:
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Note, when $`\sigma=0`$, $`\vec{E}(z,t)`$ is the solution of Maxwell equations in vacuum (and is a good approximation in air), whose form has already been used in sec. 2.1.. Using subscript 1 to denote quantities in air and subscript 2 to denote quantities in metal, the incident light, which is in air, writes:
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```math
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\vec{E_I}(z,t) = \tilde{E_I}e^{i(k_1z-\omega t)}\hat{x}
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```
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```math
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\vec{B_T}(z,t) = \frac{\tilde{k_2}}{\omega}\tilde{E_T}e^{i(\tilde{k_2}z-\omega t)}\hat{y}
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```
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Inside the metal, from the continuity equation for free charge one can solve the free charge density $`\rho_f(t) = e^{-t/\tau}\rho_f(0)`$. For a metal that is a good conductor ($`\tau`$ is much larger than any interested time scale), $`\rho_f(t)=0`$ effectively. As a result, the boundary condition at the air-metal surface becomes (recall that quantities has subscript 1 means in air and subscript 2 means in the metal):
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Inside the metal, from the continuity equation for free charge one can solve the free charge density $`\rho_f(t) = e^{-t/\tau}\rho_f(0)`$. For a metal that is a good conductor ($`\tau`$ is much larger than any interested time scale), $`\rho_f(t)=0`$ effectively. As a result, the boundary condition at the air-metal surface becomes:
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```math
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\epsilon_1\vec{E_1^{\bot}}-\epsilon_2\vec{E_2^{\bot}} = \sigma_f = 0
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```
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```math
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\frac{1}{\mu_1}\vec{B_1^{\parallel}}-\frac{1}{\mu_2}\vec{B_2^{\parallel}}=\vec{K_f}\times \hat{n} = 0
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```
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In the above equations, $`\vec{K_f}`$ is the surface current of free charges, $`\vec{E_1}=\vec{E_I}+\vec{E_R}`$, $`\vec{E_2}=\vec{E_T}`$, $`\vec{B_1}=\vec{B_I}+\vec{B_R}`$, $`\vec{B_2}=\vec{B_T}`$. The boundary condition solves to be:
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In the above equations, $`\vec{K_f}`$ is the surface current of free charges, $`\vec{E_1}=\vec{E_I}+\vec{E_R}`$, $`\vec{E_2}=\vec{E_T}`$, $`\vec{B_1}=\vec{B_I}+\vec{B_R}`$, $`\vec{B_2}=\vec{B_T}`$. Solving the boundary conditions gives:
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```math
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\tilde{E_R}=(\frac{1-\tilde{\beta}}{1+\tilde{\beta}})\tilde{E_I},\tilde{E_T}=(\frac{2}{1+\tilde{\beta}})\tilde{E_I}
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```
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where $`\tilde{\beta}=\tilde{k_2}\mu_1v_1/\mu_2\omega`$. For an ideal conductor, $`\sigma=\infty`$, which results in $`|\tilde{k_2}|=\infty`$, and therefore $`|\tilde{\beta}|=\infty`$. So
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where $`\tilde{\beta}=\tilde{k_2}\mu_1/k_1\mu_2`$. For an ideal conductor, $`\sigma=\infty`$, which results in $`|\tilde{k_2}|=\infty`$, and therefore $`|\tilde{\beta}|=\infty`$. So
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```math
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\tilde{E_R}=-\tilde{E_I}, \tilde{E_T}=0
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```
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This means a $`\pi`$ phase shift is given by the reflection on an ideal mirror. Note the above calculation is for a linear polarized light. For a circular polarized light, after reflection, both its $`\hat{x}`$ and $`\hat{y}`$ components (can be regarded as linear polarized light themselves) are shifted by a phase factor $`\pi`$. So the relative difference of the components remains unchanged. Therefore, the rotation direction of $`\vec{E_R}`$ in the xy plane remains the same as $`\vec{E_I}`$. However, the direction of the propagation is changed, so the chirality of the light must be reversed. This effect is shown in the section 2.1. figure region (3) and region (4).
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This means a $`\pi`$ phase shift is given by the reflection on an ideal mirror. Note the above calculation is for a linear polarized light. For a circular polarized light, after reflection, both its $`\hat{x}`$ and $`\hat{y}`$ components (can be regarded as linear polarized light themselves) are shifted by a phase factor $`\pi`$. So the relative phase difference between the components of light remains unchanged. Therefore, the rotation direction of $`\vec{E_R}`$ in the xy plane remains the same as $`\vec{E_I}`$. However, the direction of the propagation is changed, so the chirality of the light must be reversed. This effect is shown in the sec. 2.1. figure region (3) and region (4).
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## 4. References and further readings
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