... | ... | @@ -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 on the [*Vishay PS-4*](http://www.vishaypg.com/micro-measurements/photo-stress-plus/category/coating/?subCategory=materials) material 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) (shown in the 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 [*Vishay PS-4*](http://www.vishaypg.com/micro-measurements/photo-stress-plus/category/coating/?subCategory=materials) sheet after cutting of the particles (see [here](#cuting) to learn how to cut). 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](#). The figure below also shows different angles of a particle after this coating process.
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... | ... | @@ -43,22 +43,27 @@ 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 (shown in 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. This is shown in figure below. This heterogeneity can be removed by rescale the polarized image using a image taken without the analyzer (which records the background 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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![reflective_light2](uploads/fd6f969c1e6d4944a63863c6870eed51/reflective_light2.png)
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![reflective_light](uploads/1837153441070c0232be07b2629e53cc/reflective_light.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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## 1.5. Coated particle or mirror table?
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![intensity_adjust](uploads/3bd8bfe6a1981a772f57e0efa3e1a1e5/intensity_adjust.png)
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The painted surface of particle usually can not reflect light as good as a commercial mirror. However, this technique is irreplaceable in cases where a big mirror can not be used. For example, in air table experiments. The big mirror solution also has its problem. In particular, the reflected image of the particle creates additional difficulties in boundary detection algorithm in data analysis (see for details of image analysis techniques). In most cases encountered in previous experiments painted particle solution works good enough. However the mirror table solution remains a choice when coating particles is not practical (for example if the the particles are also used in other experiments using transmissive polariscope).
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### 1.5. Coated particles or mirror table?
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The painted surface of particles usually can not reflect light as good as a commercial mirror. However, the coated particles are irreplaceable in cases where a mirror table can not be used. For example, in air table experiments (see [*J. Puckett et al.*](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.110.058001) for more details). The mirror table solution also has disadvantages. In particular, the mirror image of the particle boundaries will be recorded by camera, creating additional difficulties for boundary detection. In most cases, coated particles work good enough. However the mirror table solution remains a choice when coating particles is not practical (for example if the same particles are also used in other experiments with transmissive polariscope).
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## 2. Theoretical background
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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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![reflective_circular2](uploads/28f1e3f8a9384cb6ffc5a21e1d82fdc0/reflective_circular2.png)
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The above figure shows the idea of the photoelastic measurement for a specimen under the reflection polariscope, which requires the light to go through the specimen twice with different kind of polarization. Suppose the height (the size along $`z`$ direction) of the specimen is $`h`$, the pattern observed by the observer will be equivalent to the pattern under a transmission polariscope for a specimen with $`h/2`$ height. 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 by the observer will be equivalent to the pattern under a transmission polariscope for a specimen under same condition but with $`h/2`$ height. 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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```math
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E_{I_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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