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For the mentioned phase-matching scheme, which is denoted as XY oo-e, the pump wave has ordinary polarization, while the second-harmonic wave has extraordinary polarization. There are also cases with type II phase matching, e.g. the scheme XY oe-e, where the pump wave has both an ordinary and an extraordinary polarization component.

The polarization state of light often matters when light hits an optical surface under some angle. A linear polarization state is then denoted as p polarization when the polarization direction lies in the plane spanned by the incoming beam and the reflected beam. The polarization with a direction perpendicular to that is called s polarization. These indications have a German origin: s = senkrecht = perpendicular, p = parallel.

There are cases where polychromatic light can be described with a single Jones vector, since all its frequency components have essentially the same polarization state. However, the polarization state is substantially frequency-dependent in other cases.

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As explained above, a waveplate or other birefringent optical element may rotate the direction of linear polarization, but more generally one will obtain an elliptical polarization state after such an element. True polarization rotation, where a linear polarization state is always maintained (just with variable direction), can occur in the form of optical activity. Some optically active substances such as ordinary sugar (saccharose) can produce substantial rotation angles already within e.g. a few millimeters of propagation length. Optical activity can be accurately measured with polarimeters.

Of course, the polarization can have any other direction perpendicular to the beam axis. Note that a rotation of the polarization by 180° does not lead to a physically distinct state.

This function takes an array of vertices and indices, fetches the vertices of each triangle according to the indices and calculates its normal. In the first loop we only accumulate the normals into each of the three triangle vertices. For each triangle the normal is calculated as a cross product between the two edges that are coming out of the first vertex. Before accumulating the normal in the vertex we make sure we normalize it. The reaons is that the result of the cross product is not guaranteed to be of unit length. In the second loop we scan the array of vertices directly (since we don't care about the indices any more) and normalize the normal of each vertex. This operation is equivalent to averaging out the accumulated sum of normals and leaves us with a vertex normal that is of a unit length. This function is called before the vertex buffer is created in order to get the calculated vertex normals into the buffer along with the other vertex attributes.

Let's assume that the strength of both light rays is the same and the only difference is their direction. The model of diffuse light says that the surface on the left will be brighter than the surface on the right because the surface on the right is hit at a sharper angle than the surface on the left. In fact, the surface on the left will be the brightest possible because the light there hits at an angle of 90 degrees.

To calculate the intensity of light in the diffuse model we are going to simply use the cosine of the angle between the light and the surface normal (whereas Lambert's law refers to the more general concept of 'directionaly proportional'). Consider the following picture:

Photons carry momentum in the direction of their motion (away from the Sun), and some of this momentum is transferred to dust particles in collisions.

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In simple cases, one has a polarization direction along one of the axes of the crystal lattice. One may then call this c polarization, for example, if it is along the crystal's c axis. (Unfortunately, for one type of crystal, there are sometimes different ways of labeling the crystal axes.)

In many cases, the output of a laser is a linearly polarized laser beam. Different mechanisms can be responsible for that:

The only thing left to worry about is the coordinate space in which diffuse lighting calculations are going to take place. The vertices and their normals are specified in a local coordinate space and are transformed in the vertex shader all the way to clip space by the WVP matrix that we supply to the shader. However, specifying the direction of light in world space is the most logical course of action. After all, the direction of light is the result of some light source which is positioned in the world somewhere (even the sun is located in the "world", albeit many miles away) and sheds its light in a particular direction. Therefore, we will need to transform the normals to world space before the calculation.

Jones vectors can be used only for fully defined polarization states, not for unpolarized or partially polarized beams (see below) having a stochastic nature.

We offer a wide range of polarizers and polarization optics for many different uses. Choosing the right polarization optic for your application can be a bewildering task, as we offer a wide range for many different uses.

Fully polarized states can be associated with points on the so-called Poincaré sphere. Partially polarized states correspond to points inside that sphere; unpolarized light is represented by the point at its center.

From this discussion we draw an important conclusion - in order to have any effect on the brightness of a surface the light must hit the surface such that the angle between it and the surface normal will be greater or equal to zero and up to (but not including!) 90 degrees.

Linearly polarized light can be depolarized (made unpolarized) with a polarization scrambler, which applies the mentioned random polarization changes, or at least quasi-random changes.

I would have been glad to finally remove a serious mistake, but I believe my equations are correct. They agree with those in various textbooks and e.g. also in Wikipedia. Your argument concerning energy swapping back and forth between the electric and magnetic fields looks somewhat plausible but is not accurate.

Linearlypolarized light

The main difference between ambient light and diffuse light is the fact that diffuse light is dependent on the direction of the rays of light while ambient light ignores it completely. When only ambient light is present the entire scene is equally lit. Diffuse light makes the parts of objects that face the light brighter than the parts that are opposite from it.

This is the new DirectionalLight structure. There are two new members here: the direction is a 3 dimensional vector specified in world space and the intensity is a floating point number (will be used in the same way as the ambient intensity).

We see four light rays hitting the surface at different angles. The surface normal is the green arrow pointing out from the surface. Light ray A has the greatest strength. The angle between A and the normal is zero and the cosine of zero is 1. This means that after we multiply the intensity of light (three channels of 0 to 1) by the color of the surface we will multiply by 1. We can't get any better than this with diffuse light. Light ray B hits the surface at an angle between 0 and 90. This means that the angle between B and the normal is also between 0 and 90 and the cosine of that angle is between 0 and 1. We will scale the result of the multiplication above by the cosine of that angle which means the intensity of light will definitely be less than light ray A.

This is the core of the diffuse light calculation. We calculate the cosine of the angle between the light vector and the normal by doing a dot product between them. There are three things to note here: The normal passed from the vertex shader is normalized before it is used. This is because the interpolation the vector went through may have changed its length and it is no longer a unit vector. The light direction is reversed. If you think about this for a moment you will see that light that hits a surface at a right angle is actualy 180 degrees away from the surface normal (which simply points back at the light source). By reversing the direction of light in this case we get a vector which equals the normal. Therefore, the angle between them is zero, which is what we want. The light vector is not normalized. It will be a waste of GPU resources to normalize the same vector over and over again for all pixels. Instead, we make sure we normalize the vector the application passes is normalized before the draw call is made.

We see that the surface normal plays an important part in the calculation of diffuse light. The examples above were very simple - the surface was a single line and there was only one normal to consider. In the real world we have objects that are composed of multiple polygon and the normal of each polygon is a bit different than the one next to it. Here's an example:

This is the updated vertex shader. We have a new vertex attribute, the normal, that the application will need to supply. In addition, the world transformation has its own uniform variable and we will need to supply it in addition to the WVP matrix. The vertex shader transforms the normal to world space using the world matrix and passes it to the fragment shader. Note how the 3 dimensional normal is extended to a 4 dimensional vector, multiplied by the 4 dimensional world matrix and then reduced back to 3 dimensions using the notation (...).xyz. This capability of the GLSL language is called 'swizzling' and allows great flexibility in vector manipulations. For example, if you have a 3 dimensional vector v(1,2,3) you can write: vec4 n = v.zzyy and then vector n will contain (3,3,2,2). Remember that when we extend the normal from 3 to 4 dimensions we must place zero at the fourth component. This nullifies the effect of translation in the world matrix (the fourth column). The reason is that vectors cannot be moved like points. They can only be scaled or rotated.

This is the final lighting calculation. We add the ambient and diffuse terms and multiply the result by the color which is sampled from the texture. Now you can see that even if diffuse light has no effect on the surface (due to direction), the ambient light can still light it up, if it exists.

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If the oscillations of the horizontal and vertical electric field vector do not have the same strengths, one has the case of an elliptical polarization, where the electric field vector, projected to a plane perpendicular to the propagation direction, moves along an ellipse.

The trick is to use a concept known as a 'vertex normal'. A vertex normal is the average of the normals of all the triangles that share the vertex. Instead of having the vertex shader calculate the diffuse light we only pass through the vertex normal as an attribute to the fragment shader and nothing more. The rasterizer will get three different normals and will need to interpolate between them. The fragement shader will be invoked for each pixel with the specific normal for this pixel. We can then calculate the diffuse light at the pixel level using that specific normal. The result will be a lighting effect which nicely changes across the triangle face and between neighboring triangles. This technique is known as Phong Shading. Here's how the vertex normals look like after interpolation:

Note that radial or azimuthal polarization state requires a zero electric field strength and thus also a vanishing optical intensity on the beam axis; it is not compatible with a Gaussian beam, for example. Radially polarized beams frequently exhibit a kind of donut profile.

Your first plot shows the magnetic and electric field in phase – which is wrong. The magnetic field is made from the changing electric field. The two fields swap energy back and forth. Hence the magnetic field is at a maximum when the electric field has the largest rate of change, that is, at zero E field. The magnetic field zeros in strength when the electric field rate of change is zero, at it's peak. These are a simple consequence of Maxwell's Equations and is covered in most any text on E&M. The worst error I have found in years of use of your marvelous resource!

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On the other hand, the polarization state of the laser output can be disturbed e.g. by random (and temperature-dependent) birefringence, such as occurs e.g. in optical fibers (if they are not polarization-maintaining or single-polarization fibers) and also in laser crystals or glasses as a result of thermal effects (→ depolarization loss). If the laser gain is not polarization-dependent, small drifts of the birefringence may lead to large changes in the polarization state, and also a significant variation in the polarization state across the beam profile.

Since the normal is the same across the face of a polygon, it is enough to calculate the diffuse light in the vertex shader. All the three vertices in a triangle would have the same color and this will be the color of the entire triangle. However, this won't look too good. We will have a bunch of polygons where each one has a particular color which is slightly different than the one next to it and we will see how color breaks at the edges. This can definitely be improved.

Things become different with light rays C and D. C hits the surface directly from the side, at an angle of 0. The angle between C and the normal is exactly 90 degrees and the cosine is 0. This results in C having no effect on lighting the surface at all! The angle between D and the normal is obtuse which means the cosine is some negative number which is smaller than 0 and larger or equal to -1. The end result is the same as C - no effect on the surface brightness.

Linear polarization

Frequently, however, one requires a propagation direction which is not aligned with one of the crystal axes – for example, in the context of critical phase matching. The article on critical phase matching contains an example case for type I phase matching of frequency doubling in LBO, where the pump and second-harmonic beams propagate within the XY plane with an angle ($\varphi$) against the X axis). In this situation, polarization can be either ordinary or extraordinary:

In the simplest case, a light beam is linearly polarized, which means that the electric field oscillates in a certain linear direction perpendicular to the beam axis, and the magnetic field oscillates in a direction which is perpendicular both to the propagation axis and the electric field direction. The direction of polarization is taken to be the direction of the electric field oscillations (i.e., not the magnetic ones). For example, a laser beam propagating in ($z$) direction may have the electric field oscillations in the vertical (($y$)) direction and the magnetic field oscillations in the horizontal (($x$)) direction (see Figure 1); it can be called vertically polarized or ($y$)-polarized. In a different perspective, this is also shown in the second part of Figure 2.

When light propagates in a non-isotropic medium, such as a nonlinear crystal, the direction of polarization relative to the crystal axes is relevant. This is not the direction of beam propagation, but rather perpendicular to that.

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Polarization

In many respects, light can be described as a wave phenomenon (→ wave optics). More specifically, light waves are recognized as electromagnetic transverse waves, i.e., with transverse oscillations of the electric and magnetic field.

A circular polarization state can mathematically be obtained as a superposition of electric field oscillations in the vertical and horizontal direction, both with equal strength but a relative phase change of 90°. Effectively, this leads to a rapid rotation of the electric field vector – once per optical cycle – which maintains a constant magnitude.

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These are the main changes to the render loop. The pipeline class has a new function that provides the world transformation matrix (in addition to the WVP matrix). The world matrix is calculated as the multiplication of the scaling matrix by the rotation matrix and finally by the translation matrix. We enable and disable the third vertex attribute array and specify the offset of the normal within each vertex in the vertex buffer. The offset is 20 because the normal is preceded by the position (12 bytes) and the texture coordinates (8 bytes).

You may find the pyramid model that we have been using in the last few tutorials a bit strange looking with those vertex normals and decide to stick with the original normals. This is OK. However, as models become more complex (and we will see that in the future) and their surfaces become smoother I think you will find the vertex normals more appropriate.

Shalom EO offers various birefringent materials, including: MgF2 crystals, LiNbO3 and alpha-BBO crystals, quartz crystals, calcite crystals, and YVO4 crystals. The crystals exhibit excellent properties when used to make waveplates, Glan and Thompson polarizers, and other optical components. Crystal ingots, blanks, and polished and coated optical elements made of birefringent crystals are offered according to your request.

Here we calculate the diffuse term which depends on the color of light, the diffuse intensity and the direction of light. If the diffuse factor is negative or equals to zero it means that light strikes the surface at an obtuse angle (either "from the side" or "from behind"). In that case the diffuse light has no effect and the DiffuseColor vector is initialized to zero. If the angle is greater than zero we calculate the diffuse color by multiplying the basic light color by the constant diffuse intensity and then scaling the result by the diffuse factor. If the angle between the light and the normal is 0 the diffuse factor will be 1 which will provide the maximum light strength.

Circularlypolarized light

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EKSMA Optics has various kinds of birefringent crystal materials including various nonlinear crystals, some of our laser crystals and Raman crystals and polarizing optics crystals.

Unpolarizedlight

Diffuse light also adds a twist where the angle by which the light strikes the surface determines the brightness of that surface. This concept is demonstrated by the following picture:

There are also partially polarized states of light. These can be described with Stokes vectors (but not with Jones vectors). Further, one can define a degree of polarization which can be calculated from the Stokes vector and can vary between 0 (unpolarized) and 1 (fully polarized).

polarized中文

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The model of diffuse light is actually based on Lambert's cosine law that says that the intensity of light reflected from a surface is directly proportional to the cosine of the angle between the observer's line of sight and the surface normal. Note that we changed this a bit by using the direction of light instead of the observer's line of sight (which we will use in specular light).

One distinguishes left and right circular polarization (see Figure 2). For example, left circular polarization means that the electric (and magnetic) field vector rotates in the left direction, seen in the direction of propagation. For an observer looking against the beam, the rotation of course has the opposite direction.

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The polarization state of monochromatic light is often described with a Jones vector, having complex electric field amplitudes for ($x$) and ($y$) direction, if propagation occurs in ($z$) direction. That Jones vector may be constant over some area across the beam, or it may vary, for example for a radially polarized beam (see above). The effect of optical elements such as waveplates, polarizers and Faraday rotators can be described with Jones matrices, with which the Jones vectors can be transformed by multiplication. (One assumes a linear relationship between input and output amplitudes.) A whole sequence of such optical elements can be described with a single Jones matrix, which is obtained as the product of the matrices corresponding to the components.

Mathematical note There are many sources online that tell you that you need the transpose of the inverse of the world matrix in order to transform the normal vector. This is correct, however, we usually don't need to go that far. Our world matrices are always orthogonal (their vectors are always orthogonal). Since the inverse of an orthogonal matrix is equal to its transpose, the transpose of the inverse is actually the transpose of the transpose, so we end up with the original matrix. As long as we avoid doing distortions (scaling one axis differently than the rest) we are fine with the approach I presented above.

There is no change in the calculation of the ambient color factor. We calculate and store it here and use it in the final formula below.

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To complete the demo that we see in this tutorial's picture we must also specify the diffuse intensity and the light direction. This is done in the constructor of the Tutorial18 class. The diffuse intensity is set to 0.8 and the direction of light is from left to right. The ambient intensity was decreased all the way down to zero to amplify the effect of diffuse light. You can play with the keys 'z' and 'x' to control the diffuse intensity (as well as 'a' and 's' from the previous tutorial that governs ambient intensity).

Here is the beginning of the fragment shader. It now receives the interpolated vertex normal that was transformed by the vertex shader to world space. The DirectionalLight structure was extended to match the one in the C++ code and contains the new light attributes.

This function sets the parameters of the directional light into the shader. It was extended to cover the direction vector and the diffuse intensity. Note that the direction vector is normalized before it is set. The LightingTechnique class also fetches the direction and diffuse intensity uniform locations from the shader as well as the world matrix uniform location. There is also a function to set the world transformation matrix. All this stuff is pretty routine by now and the code is not quoted here. Check the source for more details.

Note that a very small gain or loss difference for the two polarization directions can be sufficient for obtaining a stable linear polarization, provided that there is no significant coupling of polarization modes within the laser resonator.

The degree of linear polarization is often quantified with the polarization extinction ratio (PER), defined as the ratio of optical powers in the two polarization directions. It is often specified in decibels, and measured by recording the orientation-dependent power transmission of a polarizer. Of course, the extinction ratio of the polarizer itself must be higher than that of the laser beam.

There are also azimuthally polarized beams, where the electric field direction at any point is tangential, i.e., perpendicular to a line through the point and the beam axis.

A radially polarized laser beam may be generated from a linearly polarized beam with some optical element, but it is also possible to obtain radially polarized emission directly from a laser. The advantage of this approach, applied in a solid-state bulk laser, is that depolarization loss may be avoided [4]. Furthermore, there are applications benefiting from radially polarized light.

A light beam is called unpolarized when the analysis with a polarizer results in 50% of the power to be in each polarization state, regardless of the rotational orientation. Microscopically, this usually means that the polarization state is randomly fluctuating, so that on average no polarization is detected. Note that such fluctuations are not possible for strictly monochromatic light.

While optical activity usually results from the presence of chiral molecules, with a concentration difference between the two possible enantiometers, it can also be induced by a magnetic field in a substance which is not naturally optically active. That is called the Faraday effect, and is exploited in Faraday rotators and Faraday isolators.

The updated Vertex structure now includes the normal. It is initialized automatically to zero by the constructor and we have a dedicated function that scans all the vertices and calculates the normals.

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In the previous cases, the direction of the electric field vector was assumed to be constant over the full beam profile. However, there are light beams where that is not the case. For example, there are beams with radial polarization, where the polarization at any point on the beam profile is oriented in the radial direction, i.e., away from the beam axis.