... | ... | @@ -13,7 +13,7 @@ Reflective polariscope can probe the photoelastic fringes with light source and |
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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 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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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 section 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](#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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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. 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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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 section 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 (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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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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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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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 section 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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... | ... | @@ -75,7 +75,7 @@ where $`\hat{f}`$ and $`\hat{s}`$ are the fast and slow principle directions of |
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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 sec. 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 section 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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```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 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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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 section 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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\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 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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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 section 2.1. figure region (3) and region (4).
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## 4. References and further readings
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