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Reflection photoelasticity method
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## 1.Overview
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## 1. Overview
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In some cases, the light source of the polariscope can not be placed on the back of the granular sample. For example, when air table is used to float the particles or the driving motors of the base are placed beneath the sample. Reflective photoelasticity is utilized here using the reflective polariscope. (fig) The major difference of the polariscope comparing to that used in the transmission polariscope is that the polarization of the polarizer covering the light source and the analyzer covering the camera lens are the same.
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Reflection polariscope is used when it is hard to implement the traditional transmission polariscope. For example, the light source can not be mounted below particles when air table is used to float the particles, therefore a reflection polariscope is essential to reveal the stress inside particles [1]. In granular physics experiments, usually dark field polariscope is implemented in order to reduce the noise of the image analysis [1][2][3]. Both the polarizer and analyzer of the reflection polariscope are circular polarizers with same chirality. In the following sessions the background theory and the experimental implementation of the dark field reflection polariscope are described in detail.
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## 2. Theoretical background
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### 2.1. Reflection of circular polarized light on conductor surface
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The dark field reflection polariscope relies on the fact that when reflected by a conductor the chirality of the circular polarized light changes. This feature of conductor can be easily visualized by looking at a piece of metal through a circular polarizer (with its quarter-wave plate component towards metal). The picture (a) in figure below shows a comparison between a wood pencil and a piece of aluminum viewed by a circular polarizer. It is clear that aluminum looks almost like black while the wood pencil just becomes darker.
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![reflective_circular](uploads/38015e452e9522ec66daf0d3c200a909/reflective_circular.png)
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Picture (b) in the figure above explains the observation in (a). Consider the electric field component of a monochromatic light. After passing the linear polarizer, in region (1), $`E_{I_1} = \sqrt{I_0} * e^{i(kz-\omega t)}\hat{y} = \frac{1}{\sqrt{2}}\sqrt{I_0}e^{i(kz-\omega t)}(\hat{f}+\hat{s}) `$, where $`\hat{f}`$ and $`\hat{s}`$ are the directions of the fast and slow axis of the quarter-wave plate. After passing the quarter-wave plate the linear polarized light becomes a right hand circular polarized light in region (2). $`E_{I_2}=\frac{1}{\sqrt{2}}\sqrt{I_0}e^{i(kz-\omega t)}(-i\hat{f}+\hat{s})`$. After reflection at the mirror, which uses a layer of metal (typically silver) to reflect light, the electrodynamic boundary condition requires $`E_{I_3} = \frac{1}{\sqrt{2}}\sqrt{I_0}e^{i(-kz-\omega t)}[-i(-\hat{f})+(-\hat{s})]`$, which makes the reflected light left hand circular polarized. (See section 4) Passing the quarter-wave plate again introduce another $`\pi/2`$ phase lead to the $`\hat{f}`$ component, resulting in $`E_{I_4} = \frac{1}{\sqrt{2}}\sqrt{I_0}e^{i(-kz-\omega t)}[(-i)(-i)(-\hat{f})+(-\hat{s})] = \frac{1}{\sqrt{2}}\sqrt{I_0}e^{i(-kz-\omega t)}\hat{x} `$, which is perpendicular to the linear polarizer. So no reflection light can go through the original polarizer again, creating a dark field.
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### 2.2. Photoelasticity under 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) $`E_{I_2} = \frac{1}{\sqrt{2}}\sqrt{I_0}e^{i(kz-\omega t)}(-i\hat{f}+\hat{s}) = \frac{1}{\sqrt{2}}\sqrt{I_0}e^{i(kz-\omega t)}e^{i\phi}(-i\hat{m_1}+\hat{m_2})`$. Passing the specimen results in a $`\Delta = 2\phi(\sigma_1-\sigma_2)/f_{\sigma}`$ phase lead to the $`\hat{m_1}`$ component of light. So in region (3): $`E_{I_3} = \frac{1}{\sqrt{2}}\sqrt{I_0}e^{i(kz-\omega t)}e^{i\phi}(-ie^{-i\Delta}\hat{m_1}+\hat{m_2})`$. After reflection, the electrodynamic boundary condition requires (see section 4) $`E_{I_4} = \frac{1}{\sqrt{2}}\sqrt{I_0}e^{i(-kz-\omega t)}e^{i\phi}(+ie^{-i\Delta}\hat{m_1}-\hat{m_2})`$ in region (4). Passing the specimen again gives another $`\Delta`$ phase lead to the $`\hat{m_1}`$ light component, which makes $`E_{I_5} = \frac{1}{\sqrt{2}}\sqrt{I_0}e^{i(-kz-\omega t)}e^{i\phi}(+ie^{-2i\Delta}\hat{m_1}-\hat{m_2}) =`$ $` \frac{1}{\sqrt{2}}\sqrt{I_0}e^{i(-kz-\omega t)}e^{i\phi}[(ie^{-2i\Delta}cos\phi+sin\phi)\hat{f}+(ie^{-2i\Delta}sin\phi-cos\phi)\hat{s}] `$ in region (5). Passing the quarter-wave plate again gives a $`-i`$ factor to the $`\hat{f}`$ component: $`E_{I_6} = \frac{1}{\sqrt{2}}\sqrt{I_0}e^{i(-kz-\omega t)}e^{i\phi}[-i(ie^{-2i\Delta}cos\phi+sin\phi)\hat{f}+(ie^{-2i\Delta}sin\phi-cos\phi)\hat{s}] `$. Finally, the light intensity that can be observed by the observer behind the linear polarizer is
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```math
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I = |E_{I_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. The factor of 2 must be included.
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## 3. Experimental implementation of the reflective polariscope
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To build a granular physics experiment using the reflection polariscope, a mirror must be implemented behind the specimen (particles). Two typical ways to implement a mirro has been performed: one is using particles with reflective bottoms [1][2][3] and the other is just putting transparent particles on mirrors [3].
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![reflect4](uploads/f4a4bb70dcad746c7b68f81c811d3353/reflect4.png)
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### 3.1. Particles with reflective bottoms
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This method requires to make one side of the particles reflective. This is usually achieved by coating a layer of reflective material on the particles. Note this reflective layer can not be harder than the material so that the elastic property of the particle would not change after coating. Current technique features painting the particles using the mirror effect powders (link to be added, citation to be added). Figure (c) shows the particles after this painting. Figure (a) shows an air table experiment implementing the reflective polariscope using the reflective particles.
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Advantages:
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1. The particles can be used on a air table where a big mirror table can not be placed behind the particles.
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Disadvantages:
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1. The reflective light intensity is sensitive to the tilting angle of the particle if the light source is not very uniform. (Fig) This problem introduces non-negligible errors in the stress estimations (both for qualitative $`G^2`$ method and the quantitative inverse problem solver). This problem can be overcomed by rescaling the light intensity using a non polarized image.
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2. The reflection ratio for the powders is usually not as good as a commercial mirror, creating larger noise to signal ratio in the detection of photoelastic patterns.
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### 3.2. Mirror table
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In this method, the particles are transparent but they are put on a big mirror.
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Advantage:
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1. No need for coating the particles so the particles used for this experiment can also used in other experiments with non-reflective polariscopes.
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2. Particle tilting will not cause light inhomogeneity.
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3. The reflection ratio is better so the signal-to-noise ratio is better.
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Disadvantage:
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1. For particles that is not directly beneath camera, the mirror image of their boundaries will be recorded, which reduces the accuracy of the boundary detection and thus the center detection. (Fig)
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2. For base-driven experiments to apply internal shear (cite), the split between bottom slats or bottom rings will cause discontinuous photo elastic fringes, increasing errors in the stress estimations.
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Where to buy the materials?
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## 2.Theory [1]
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In this section we show why the polarization of the circular light changes when it is reflected by a mirror.
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### 2.1. Reflection of polarized light on a insulator-conductor surface
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### 2.1. Mathematic derivation that a circular polarized light changes its chirality when reflected by a mirror.
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In a conductor, the solution of Maxwell equation gives electromagnetic waves with complex wave numbers. Without lost of generality we consider a specific solution correspond to frequency $`\omega`$.
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... | ... | @@ -105,45 +151,9 @@ Following same calculations as in the linear polarization case for components of |
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Note the incident and reflection light have reverse direction of rotation for their $`\vec{E}`$ vector and $`\vec{B}`$ vector. So we have proved that on a conductor the direction of circular polarization is switched after reflection on a conductor.
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### 2.2. Photoelasticity under reflective light
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Note: how to think of the reflected light go through the particle has different polarization with the incoming light? How does this change the photo-elastic response?
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The whole problem has reflection symmetry so the final result would not be fundamentally different. proof?
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## 3. Implementation of the reflective photoelasticimetry
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There are two ways to implement the reflective photoelasticimetry. One is by using particles with reflective bases. The other is by putting particles on mirror.
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### 3.1. Particles with reflective base
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This method requires to make one side of the particles reflective. This is usually achieved by coating a layer of reflective material. Note the reflective layer can not be harder than the material so that the elastic property of the particle would not change after coating. Current technique involves painting the particles using the mirror effect powders (link to be added, citation to be added).
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Advantages:
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1. The particles can be used on a air table where a mirror table can not be used.
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2. This method does not have all the problems associated with a mirror table described below.
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Disadvantages:
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1. The reflective light intensity is sensitive to the tilting angle of the particle if the light source is not very uniform. (Fig) This problem introduces non-negligible errors in the stress estimations (both for qualitative $`G^2`$ method and the quantitative inverse problem solver). This problem can be overcomed by rescaling the light intensity using a non polarized image.
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2. The reflection ratio for the powders is usually not as good as a commercial mirror, creating larger noise to signal ratio in the detection of photoelastic patterns. (Calibration?)
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### 3.2. Mirror base
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In this method, the particles are transparent but they are put on a big mirror.
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Advantage:
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1. No need for coating the particles so the particles used for this experiment can also used in other experiments with non-reflective polariscopes.
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2. Particle tilting will not cause light inhomogeneity.
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3. The reflection ratio is better so the signal-to-noise ratio is better.
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Disadvantage:
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1. For particles that is not directly beneath camera, the mirror image of their boundaries will be recorded, which reduces the accuracy of the boundary detection and thus the center detection. (Fig)
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2. For base-driven experiments to apply internal shear (cite), the split between bottom slats or bottom rings will cause discontinuous photo elastic fringes, increasing errors in the stress estimations.
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### 3.3. Example implementations
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#### 3.3.1 Biaxial experiment with air-table (cite)
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#### 3.3.2 Couette shear experiment with mirror bottom. (Cite)
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## 4. References
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[1] Puckett, J.G. and Daniels, K.E., 2013. Equilibrating temperaturelike variables in jammed granular subsystems. Physical Review Letters, 110(5), p.058001.
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[2]
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[1]: Griffiths, David J. (2007), Introduction to Electrodynamics, 3rd Edition; Pearson Education
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[<go back to home](https://git-xen.lmgc.univ-montp2.fr/PhotoElasticity/Main/wikis/home) |
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