... | ... | @@ -13,7 +13,7 @@ Reflective polariscope can probe the photoelastic fringes with light source and |
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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 camera.
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Similar to the [transmissive polariscope](#), 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 [Section 2](#2.-theoretical-background) 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 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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![reflective_circular_4](uploads/9b9996db2ba507c78fd8e16f65a6144c/reflective_circular_4.png)
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... | ... | @@ -28,7 +28,7 @@ There are two typical ways to implement the mirror in real granular physics expe |
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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](#). 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 a particle after this coating process.
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... | ... | @@ -43,11 +43,11 @@ It is very important to note that in reflective polariscope, the circular polari |
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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](#) and the [nonlinear fitting force measurement](#) 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 (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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![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.](#) 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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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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![intensity_adjust](uploads/3bd8bfe6a1981a772f57e0efa3e1a1e5/intensity_adjust.png)
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... | ... | @@ -130,18 +130,18 @@ The light transmitted into metal will have complex wave number: |
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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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```math
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\epsilon_1\vec{E_1^{}}-\epsilon_2\vec{E_2^{}} = \sigma_f = 0
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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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\vec{B_1^{}}-\vec{B_2^{}} = 0
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\vec{B_1^{\bot}}-\vec{B_2^{\bot}} = 0
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```
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```math
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\vec{E_1^{||}}-\vec{E_2^{||}}=0
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\vec{E_1^{\parallel}}-\vec{E_2^{\parallel}}=0
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```
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```math
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\frac{1}{\mu_1}\vec{B_1}^{||}-\frac{1}{\mu_2}\vec{B_2}^{||}=\vec{K_f}\times \hat{n} = 0
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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 expression, $`\vec{E_1}=\vec{E_I}+\vec{E_R}`$, $`\vec{E_2}=\vec{E_T}`$, $`\vec{B_1}=\vec{B_I}+\vec{B_R}`$, $`\vec{B_2}=\vec{B_T}`$. The boundary condition solves to be:
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In the above equations, $`\vec{K_f}`$ is the surface current of free charges, $`\vec{E_1}=\vec{E_I}+\vec{E_R}`$, $`\vec{E_2}=\vec{E_T}`$, $`\vec{B_1}=\vec{B_I}+\vec{B_R}`$, $`\vec{B_2}=\vec{B_T}`$. The boundary condition solves to be:
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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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