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Only half of the original amount of light makes it through a polarizing filter, this means at full polarization you will lose 1 Stop of light, in addition the quarter wave plate loses another stop of light totaling 2 stops. Exposure can be compensated by

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.

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.

Circularlypolarized light

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.

Elliptical polarization

Imagine putting a quarter into a parking meter, the single plane slot only allows the quarter to pass through when turned to the allotted angle.

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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.

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.

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.

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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.

Electric polarization

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

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.

Polarization oflight

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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.

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.

Since typical light vibrates at all angles, only the light traveling at an angle within the threshold on the single plane of the Polarizer makes it through. The Quarter Wave Plate turns the single plane-polarized light, back into circularly polarized light which allows modern cameras to accurately focus and meter the light. Then the resulting light enters your lens and contacts your camera sensor recording the image. As the Polarizing filter is rotated, different vibrations of light are allowed into your camera producing varying amounts of perceived reflective light reduction. Basically as you turn the filter you graduate from 'on' to 'off' and all the stages in between.

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 of the polarization state, and also a significant variation in the polarization state across the beam profile.

Linearpolarization example

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).

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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.

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.

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.

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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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 electric and magnetic fields looks somewhat plausible but is not accurate.

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Linearpolarization

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Polarization

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.

Simply put a circular polarizer is a magic reflection reduction dial. VERY COOL NO? Prices vary, this is an item that expensive is better, but I recommend getting  decent cheap one like the TIFFEN Circular Polarizing Glass Filter to see how you like it before committing to a higher end $50 - $150 circular polarizing filter.

Understanding the question of "how do circular polarizers work?" can be an essential tool in elevating aspects of your photography to the next level. Learn the method, master the technique. Circular Polarizing Filters attach to the thread of your lens and reduce reflections emitting off the subjects you're photographing. The effect this can give is, darker deeper sky color and ability to see through windows and water that otherwise portray only surface reflections.

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.

It is very advantageous to be able to quickly estimate the radius of curvature of a spherical surface without having to set the subject lens or mirror up in a ...

Unpolarizedlight

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.

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.

Circular polarizers consists of two elements, a frontal polarizing filter, with what's called a quarter wave plate behind it. When the front element is rotated and gives a varied amount of reflection suppression by controlling what type of light enters the lens. Light waves emitted from typical objects we see are vibrating in all directions and are considered unpolarized. The filter is chemically composed of long chain molecules of the surface creating a 'blind-like' structure. The Polarizing filter allows light only to pass through it on a single axis parallel to the chain molecule pattern.

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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.

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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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.

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