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UNDER CONSTRUCTION, work
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First draft.
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Reflection photoelasticity method
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=================================
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## Reflective polariscope
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## 1. Reflective polariscope
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Reflective polariscope can probe the photo-elastic fringes with light source and camera on same side of the granular sample. An example implementation is schetched in fig.1(a). The reflective polariscope contains 5 parts: (1) The light source (2) the ‘polarizer’ in front of the light source (3) a mirror or a effective mirror to reflect light (4) the analyzer and (5) the camera. Similar to transmissive polariscope, both the polarizer and analyzer are usually circular polarizer. It is important to point out that a dark field reflective polariscope uses same kind of circular polarizer for polarizer and analyzer, whereas the transmissive polariscope uses circular polarizers with different chirality.
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There are two ways to implement a effective mirror. One is to use photo-elastic particles with reflective coating on one side (see J. Picket et al. Or K. Daniels et al). The left figure below shows a sketch of the implementation of the reflective polariscope using particles with reflective coating. (From j. Pucket et al) and a typical image recorded for a jammed system (from pucket et al). The other way is to put transparent particles on a big mirror. The right lower figure shows a coquette shear cell that implement a big mirror. ()
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Reflective polariscope can probe the photo-elastic fringes with light source and camera on same side of the granular sample. An example implementation is sketched in fig.1(a). The reflective polariscope contains 5 parts: (1) The light source (2) the ‘polarizer’ in front of the light source (3) a mirror or a effective mirror to reflect light (4) the analyzer and (5) the camera. Similar to transmissive polariscope, both the polarizer and analyzer are usually circular polarizer. It is important to point out that a dark field reflective polariscope uses same kind of circular polarizer for polarizer and analyzer, 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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## 1.1. Example experimental realizations
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There are two ways to implement the mirror in real granular physics experiments:
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## Technique for coating the particles
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A typical way to coat the particle is using mirror effect painting powders. The lower left figure shows a typical painting material (RUST-OLEUM mirror effect) that provides good reflection for the light, while attaches firmly on the vanchy PSM material. To ensure uniform coating. Usually a whole sheet of photo elastic material is painted. And the particles are cutter from the sheet afterwards. The lower right figure shows a picture of the painted psm layer after cutting of the particles (see cut section to learn how to perform the cut). The figure below also shows different angle of a particle after this coating process.
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![particles](uploads/ba1713b99aba580dd47cdeb73c15cc8d/particles.png)
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![reflective_circular2](uploads/28f1e3f8a9384cb6ffc5a21e1d82fdc0/reflective_circular2.png)
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## Implementation issue
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### Polarizer direction
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It is very important to note that in reflective polariscope, the circular polarizer has to make its wave-plate side towards the granular sample. However sometimes it is hard to tell which side of a circular polarizer is wave-plate. A simple trick can be used to determine this: put the circular polarizer on a piece of metal. 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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![polarizer](uploads/5ae4ce71bf075d4694730975d4b992f0/polarizer.png)
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### Light condition
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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 (see here for details) and the nonlinear fitting contact force measurement (see here for details) depends sensitively on the back ground light intensity. The reflective polariscope has a higher chance to suffer from the light heterogeneity. First, the reflection light intensity is 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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### Particle painting or big mirror?
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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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## Theoretical background
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### Reflective photo elastic theory
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### Chirality of light on the reflection process: put particles on a metal surface?
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## References
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make a list
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(1) by using photo-elastic particles with a reflective surface (see J. Picket et al. Or K. Daniels et al). The left figure below shows a sketch of an example experimental setup that implementing the reflective polariscope using reflective particles. (From j. Pucket et al) A typical force network recorded from the same experiment is also attached below, showing same type of fringes as observed in the transmissive polariscopes. (link to transmissive polariscope)(from pucket et al).
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(2) put transparent particles on a big mirror.
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![james](uploads/958f33b4b081ddc4ee59e906709a9a28/james.png)
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## 1.2. Technique for coating the particles
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A typical way to coat the particle is using mirror effect painting powders. The lower left figure shows a typical painting material (RUST-OLEUM mirror effect) that provides good reflection for the light, while attaches firmly on the vanchy PSM material. To ensure uniform coating. Usually a whole sheet of photo elastic material is painted. And the particles are cutter from the sheet afterwards. The lower right figure shows a picture of the painted psm layer after cutting of the particles (see cut section to learn how to perform the cut). The figure below also shows different angle of a particle after this coating process.
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![particles](uploads/ba1713b99aba580dd47cdeb73c15cc8d/particles.png)
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## 1.3. Polarizer configuration
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It is very important to note that in reflective polariscope, the circular polarizer has to make its wave-plate side towards the granular sample. However sometimes it is hard to tell which side of a circular polarizer is wave-plate. A simple trick can be used to determine this: put the circular polarizer on a piece of metal. 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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![polarizer](uploads/5ae4ce71bf075d4694730975d4b992f0/polarizer.png)
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## 1.4. Light condition
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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 (see here for details) and the nonlinear fitting contact force measurement (see here for details) depends sensitively on the back ground light intensity. The reflective polariscope has a higher chance to suffer from the light heterogeneity. First, the reflection light intensity is 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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![intensity_adjust](uploads/646bf87c6e83f6eb63f2a513e8582df6/intensity_adjust.png)
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## 1. Overview
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## 1.5. Coated particle or mirror table?
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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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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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## 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),
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```math
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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})
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```
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, 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).
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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})
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```
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After reflection at the mirror, which uses a layer of metal (typically silver) to reflect light, the electrodynamic boundary condition requires
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```math
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E_{I_3} = \frac{1}{\sqrt{2}}\sqrt{I_0}e^{i(-kz-\omega t)}[-i(-\hat{f})+(-\hat{s})]
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```
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, 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
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```math
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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}
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```
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, 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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### 2.1. Photoelasticity in the reflection polariscope
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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)
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```math
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... | ... | @@ -123,40 +83,8 @@ I = |E_{I_6}\cdot \hat{y}|^2 = I_0sin^2\Delta = I_0sin^2(\frac{2\pi(\sigma_1-\si |
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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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### 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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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. Mathematic derivation that a circular polarized light changes its chirality when reflected by a mirror.
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### 2.2. Chirality change of circular polarized light by reflection
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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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... | ... | @@ -228,30 +156,10 @@ This gives the form of the reflection light: |
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```math
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\vec{B_R}(z,t) = -\frac{k_1}{\omega}\tilde{E_I}e^{i(-k_1z-\omega t+\pi)}\hat{y}
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```
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which defers from the incident light by a phase factor $`\pi`$ besides the change of the propagation direction.
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![particles](uploads/336241f612e8089abb733cf13b43c1c5/particles.png)
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which defers from the incident light by a phase factor $\pi$ besides the change of the propagation direction.
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### circular polarizer
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The incident light with circular polarization 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}+e^{i(k_1z-\omega t+\pi/2)}\hat{y})
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```
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```math
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\vec{B_I}(z,t) = -\frac{k_1}{\omega}\tilde{E_I}(e^{i(k_1z-\omega t)}\hat{y}+e^{i(k_1z-\omega t+\pi/2)}\hat{x})
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```
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Following same calculations as in the linear polarization case for components of the circular polarized light, we get the polarization for the reflective light:
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```math
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\vec{E_R}(z,t) = \tilde{E_I}(e^{i(-k_1z-\omega t+\pi)}\hat{x}+e^{i(-k_1z-\omega t+3\pi/2)}\hat{y})
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```
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```math
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\vec{B_R}(z,t) = -\frac{k_1}{\omega}\tilde{E_I}(e^{i(-k_1z-\omega t+\pi)}\hat{y}+e^{i(-k_1z-\omega t+3\pi/2)}\hat{x})
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```
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For a circular polarized light, after reflection, both its $`\hat{x}`$ and $`\hat{y}`$ components are shifted by a phase factor $`\pi`$. So the relative difference of the components remains unchanged. Therefore, the rotation direction of the $`\vec{E_R}`$ in the xy plane remains the same as $`\vec{E_I}`$. However, the direction of the propagation is changed, this results the change of chirality.
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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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## 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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