Reports have surfaced that certain species of insects and animals are able to detect polarized light, including ants, fruit flies, and certain fish, although the list may actually be much longer. For example, several insect species (primarily honeybees) are thought to employ polarized light in navigating to their destinations. It is also widely believed that some individuals are sensitive to polarized light, and are able to observe a yellow horizontal line superimposed on the blue sky when staring in a direction perpendicular to the sun's direction (a phenomenon termed Haidinger's brush). Yellow pigment proteins, termed macula lutea, which are dichroic crystals residing in the fovea of the human eye, are credited with enabling a person to view polarized light.

Monochromaticlight

If there were only a single light source, namely, S, then the interference pattern would be easy to predict. However, due to the contribution from S’, the pattern we see will be washed out and fuzzy. Part of the maxima and minima due to S will coincide with that due to S’, leading to a blurred picture. And when we start to move S’ away from S, the contrast will decrease even further. There will finally come a point at which the maxima of one will coincide with the minima of the other and vice versa, leading to an effectively blank screen. Let us fix S’ here, separated from S’ by l. Now, we try to estimate the separations S₁S’ and S₂S’.

This phenomenon is exactly what temporal coherence is. No light source can emit an infinitely continuous wave train and thus, even light from a lone source only remains coherent for a particular amount of time.

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A majority of the polarizing materials used today are derived from synthetic films invented by Dr. Edwin H. Land in 1932, which soon overtook all other materials as the medium of choice for production of plane-polarized light. To produce the films, tiny crystallites of iodoquinine sulfate, oriented in the same direction, are embedded in a transparent polymeric film to prevent migration and reorientation of the crystals. Land developed sheets containing polarizing films that are marketed under the trade name of Polaroid (a registered trademark), which has become the accepted generic term for these sheets. Any device capable of selecting plane-polarized light from natural (non-polarized) white light is now referred to as a polar or polarizer, a name first introduced in 1948 by A. F. Hallimond. Because these filters are capable of differentially transmitting light rays, depending upon their orientation with respect to the polarizer axis, they exhibit a form of dichroism, and are often termed dichroic filters.

Coherence

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For water (refractive index of 1.333), glass (refractive index of 1.515), and diamond (refractive index of 2.417), the critical (Brewster) angles are 53, 57, and 67.5 degrees, respectively. Light reflected from a highway surface at the Brewster angle often produces annoying and distracting glare, which can be demonstrated quite easily by viewing the distant part of a highway or the surface of a swimming pool on a hot, sunny day. Modern lasers commonly take advantage of Brewster's angle to produce linearly polarized light from reflections at the mirrored surfaces positioned near the ends of the laser cavity.

coherence中文

In linearly polarized light, the electric vector is vibrating in a plane that is perpendicular to the direction of propagation, as discussed above. Natural light sources, such as sunlight, and artificial sources, including incandescent and fluorescent light, all emit light with orientations of the electric vector that are random in space and time. Light of this type is termed non-polarized. In addition, there exist several states of elliptically polarized light that lie between linear and non-polarized, in which the electric field vector transcribes the shape of an ellipse in all planes perpendicular to the direction of light wave propagation.

The ordinary and extraordinary light waves generated when a beam of light traverses a birefringent crystal have plane-polarized electric vectors that are mutually perpendicular to each other. In addition, due to differences in electronic interaction that each component experiences during its journey through the crystal, a phase shift usually occurs between the two waves. Although the ordinary and extraordinary waves follow separate trajectories and are widely separated in the calcite crystal described previously, this is not usually the case for crystalline materials having an optical axis that is perpendicular to the plane of incident illumination.

Normally, the mirrors are kept equidistant from the beam splitter G¹, and what we expect is an excellent interference pattern at the bottom of the setup. But imagine what would happen once I started steadily moving the mirror M₂ away from the light source. Naturally, the contrast of the interference pattern would decrease and in fact, for ordinary light sources like a sodium lamp, as soon as the extra separation becomes a few millimeters, the fringes vanish completely.

In short, temporal, and spatial coherence may be understood by using a Michelson interferometer and the Young Double Slit Experiment, respectively. They may be defined as follows:

Coherentlightsource examples

It is the coherence of light that makes patterns like the one shown below possible. If you have spent any time around the concepts of wave optics, it is a topic you must have come across. Here is a quick refresher: for two light waves to interfere and produce a maximum, they must be coherent and carry a phase difference equal to an integral multiple of the wavelength of the source. The resulting pattern is ethereal to look at and I have attached an image below:

The human eye lacks the ability to distinguish between randomly oriented and polarized light, and plane-polarized light can only be detected through an intensity or color effect, for example, by reduced glare when wearing polarized sun glasses. In effect, humans cannot differentiate between the high contrast real images observed in a polarized light microscope and identical images of the same specimens captured digitally (or on film), and then projected onto a screen with light that is not polarized. The basic concept of polarized light is illustrated in Figure 1 for a non-polarized beam of light incident on two linear polarizers. Electric field vectors are depicted in the incident light beam as sinusoidal waves vibrating in all directions (360 degrees; although only six waves, spaced at 60-degree intervals, are included in the figure). In reality, the incident light electric field vectors are vibrating perpendicular to the direction of propagation with an equal distribution in all planes before encountering the first polarizer.

The answer is where temporal coherence of light sources enters the picture. One thing you must keep in mind is the fact that no light source in this world can emit continuous, unbroken wave trains. Instead, all practical sources emit electromagnetic radiation that looks something like this:

where I is the intensity of light passing through the analyzer (and the total amount of light passed through the pair of crossed polarizers), I(o) is the intensity of light that is incident upon the polarizer, and θ is the angle between the transmission axes of the polarizer and analyzer. By examining the equation, it can be determined that when the two polarizers are crossed (θ = 90 degrees), the intensity is zero. In this case, light passed by the polarizer is completely extinguished by the analyzer. When the polarizers are partially crossed at 30 and 60 degrees, the light transmitted by the analyzer is reduced by 25 percent and 75 percent, respectively.

Other applications for polarized light include the Polaroid sunglasses discussed above, as well as the use of special polarizing filters for camera lenses. A variety of scientific instruments utilize polarized light, either emitted by lasers, or through polarization of incandescent and fluorescent sources by a host of techniques. Polarizers are sometimes used in room and stage lighting to reduce glare and produce a more even degree of illumination, and are worn as glasses to bestow an apparent sense of depth to three-dimensional movies. Crossed polarizers are even utilized in space suits to dramatically reduce the chances of light from the sun entering the astronaut's eyes during naps.

Polarization of light is very useful in many aspects of optical microscopy. The polarized light microscope is designed to observe and photograph specimens that are visible primarily due to their optically anisotropic character. Anisotropic materials have optical properties that vary with the propagation direction of light passing through them. In order to accomplish this task, the microscope must be equipped with both a polarizer, positioned in the light path somewhere before the specimen, and an analyzer (a second polarizer), placed in the optical pathway between the objective rear aperture and the observation tubes or camera port.

The principle behind Brewster's angle is illustrated Figure 3 for a single ray of light reflecting from the flat surface of a transparent medium having a higher refractive index than air. The incident ray is drawn with only two electric vector vibration planes, but is intended to represent light having vibrations in all planes perpendicular to the direction of propagation. When the beam arrives on the surface at a critical angle (Brewster's angle; represented by the variable θ in Figure 3), the polarization degree of the reflected beam is 100 percent, with the orientation of the electric vectors lying perpendicular to the plane of incidence and parallel to the reflecting surface. The incidence plane is defined by the incident, refracted, and reflected waves. The refracted ray is oriented at a 90-degree angle from the reflected ray and is only partially polarized.

Coherent illumination

The first clues to the existence of polarized light surfaced around 1669 when Erasmus Bartholin discovered that crystals of the mineral Iceland spar (a transparent, colorless variety of calcite) produce a double image when objects are viewed through the crystals in transmitted light. During his experiments, Bartholin also observed a quite unusual phenomenon. When the calcite crystals are rotated about a particular axis, one of the images moves in a circle around the other, providing strong evidence that the crystals are somehow splitting the light into two different beams.

One of the first polarizing filters was constructed in the early nineteenth century by French scientist François Arago, who was an active investigator into the nature of polarized light. Arago investigated the polarity of light originating from various sources in the sky and proposed a theory that predicted the velocity of light should decrease as it passes into a denser medium. He also worked with Augustin Fresnel to investigate interference in polarized light and discovered that two beams of light polarized with their vibration directions oriented perpendicular to each other will not undergo interference. Arago's polarizing filters, designed and built in 1812, were made from a stack of glass sheets pressed together.

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Image contrast arises from the interaction of plane-polarized light with a birefringent (or doubly-refracting) specimen to produce two individual wave components that are polarized in mutually perpendicular planes. The velocities of these components are different and vary with the propagation direction through the specimen. After exiting the specimen, the light components are out of phase and sweep an elliptical geometry that is perpendicular to the direction of propagation, but are recombined through constructive and destructive interference when they pass through the analyzer. Polarized light microscopy is a contrast-enhancing technique that improves the quality of the image obtained with birefringent materials when compared to other techniques such as darkfield and brightfield illumination, differential interference contrast, phase contrast, Hoffman modulation contrast, and fluorescence. In addition, use of polarized light allows the measurement of optical properties of minerals and similar materials and can aid in the classification and identification of unknown substances.

Presented in Figure 5 is an illustration of the construction of a typical Nicol prism. A crystal of doubly refracting (birefringent) material, usually calcite, is cut along the plane labeled a-b-c-dand the two halves are then cemented together to reproduce the original crystal shape. A beam of non-polarized white light enters the crystal from the left and is split into two components that are polarized in mutually perpendicular directions. One of these beams (labeled the ordinary ray) is refracted to a greater degree and impacts the cemented boundary at an angle that results in its total reflection out of the prism through the uppermost crystal face. The other beam (extraordinary ray) is refracted to a lesser degree and passes through the prism to exit as a plane-polarized beam of light.

Gas and water molecules in the atmosphere scatter light from the sun in all directions, an effect that is responsible for blue skies, white clouds, red sunsets, and a phenomenon termed atmospheric polarization. The amount of light scattered (termed Rayleigh scattering) depends upon the size of the molecules (hydrogen, oxygen, water) and the wavelength of light, as demonstrated by Lord Rayleigh in 1871. Longer wavelengths, such as red, orange, and yellow, are not scattered as effectively as are the shorter wavelengths, such as violet and blue.

where n is the refractive index of the medium from which the light is reflected, θ(i) is the angle of incidence, and θ(r) is the angle of refraction. By examining the equation, it becomes obvious that the refractive index of an unknown specimen can be determined by the Brewster angle. This feature is particularly useful in the case of opaque materials that have high absorption coefficients for transmitted light, rendering the usual Snell's Law formula inapplicable. Determining the amount of polarization through reflection techniques also eases the search for the polarizing axis on a sheet of polarizing film that is not marked.

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Continued rotation of the analyzer transmission axis, to a 60-degree angle with respect to the transmission axis of the polarizer, further reduces the magnitude of the vector component that is transmitted through the analyzer (Figure 6(c)). When the analyzer and polarizer are completely crossed (90-degree angle), the vertical component becomes negligible (Figure 6(d)) and the polarizers have achieved their maximum extinction value.

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When considering the incidence of non-polarized light on a flat insulating surface, there is a unique angle at which the reflected light waves are all polarized into a single plane. This angle is commonly referred to as Brewster's angle, and can be easily calculated utilizing the following equation for a beam of light traveling through air:

Here, S and S’ are two point-sources which have no defined or predictable phase relationship. S₁ and S₂ are the two slits, separated from the plane of the light sources by a distance D, whereas T represents the screen, on which we have selected an arbitrary point O. Note that as far as our interest goes, the areas away from O where the distances S₁O and S₂O differ significantly have no value.

Several versions of prism-based polarizing devices were once widely available, and these were usually named after their designers. The most common polarizing prism (illustrated in Figure 5) was named after William Nicol, who first cleaved and cemented together two crystals of Iceland spar with Canada balsam in 1829. Nicol prisms were first used to measure the polarization angle of birefringent compounds, leading to new developments in the understanding of interactions between polarized light and crystalline substances.

In other words, if instead of two point-sources, we had a single extended source from S to S’, it would no longer remain coherent beyond a certain distance l, approximately given as λD/2d. This is an effective way to understand what spatial coherence is.

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The polarizers illustrated in Figure 1 are actually filters containing long-chain polymer molecules that are oriented in a single direction. Only the incident light that is vibrating in the same plane as the oriented polymer molecules is absorbed, while light vibrating at right angles to the polymer plane is passed through the first polarizing filter. The polarizing direction of the first polarizer is oriented vertically to the incident beam so it will pass only the waves having vertical electric field vectors. The wave passing through the first polarizer is subsequently blocked by the second polarizer, because this polarizer is oriented horizontally with respect to the electric field vector in the light wave. The concept of using two polarizers oriented at right angles with respect to each other is commonly termed crossed polarization and is fundamental to the concept of polarized light microscopy.

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In the example above, the time τ is referred to as the coherence time of the light source, and the length of each wave train l=cτ is known as the longitudinal coherence length.

The phenomenon of optical activity in certain chemicals derives from their ability to rotate the plane of polarized light. Included in this category are many sugars, amino acids, organic natural products, certain crystals, and some drugs. Rotation is measured by placing a solution of the target chemical between crossed polarizers in an instrument termed a polariscope. First observed in 1811 by French physicist Dominique Arago, optical activity plays an important role in a variety of biochemical processes where the structural geometry of molecules governs their interactions. Chemicals that rotate the vibrational plane of polarized light in a clockwise direction are termed dextrorotatory, while those that rotate the light in a counterclockwise direction are referred to as levorotatory. Two chemicals having the same molecular formula but different optical properties are termed optical isomers, which rotate the plane of polarized light in different directions.

To talk about spatial coherence, we turn our attention towards the famous and ubiquitous Young's Double Slit Experiment. In its original form, this remarkable setup was able to demonstrate the wave nature of light and later, it even gave rise to a definitive demonstration of the dual nature of matter. Above all, the probabilistic nature of quantum mechanics can also be described via this experiment, but that is not our target for now.

Coherentlight

We begin by considering a Michelson interferometer as shown below, consisting of a light source S, a beam splitter G, two mirrors M₁ and M₂, the former of which is fixed in place while the latter is moveable forwards and backwards, and finally, the observer O. If you recall, most variations of the Michelson interferometer also keep a glass plate to compensate for the optical path difference introduced while traversing the beam splitter, but that is irrelevant in this scenario. Here is the apparatus we have with us now:

Polarized light microscopy was first introduced during the nineteenth century, but instead of employing transmission-polarizing materials, light was polarized by reflection from a stack of glass plates set at a 57-degree angle to the plane of incidence. Later, more advanced instruments relied on a crystal of doubly refracting material (such as calcite) specially cut and cemented together to form a prism. A beam of white non-polarized light entering a crystal of this type is separated into two components that are polarized in mutually perpendicular (orthogonal) directions.

The angle subtended by the light source at the plane of the slit can be approximated as l/D, and we represent it by θ. Let

The amount of light passing through a crossed pair of high-quality polarizers is determined by the orientation of the analyzer with respect to the polarizer. When the polarizers are oriented perpendicular to each other, they display a maximum level of extinction. However, at other angles, varying degrees of extinction are obtained, as illustrated by the vector diagrams presented in Figure 6. The analyzer is utilized to control the amount of light passing through the crossed pair, and can be rotated in the light path to enable various amplitudes of polarized light to pass through. In Figure 6(a), the polarizer and analyzer have parallel transmission axes and the electric vectors of light passing through the polarizer and analyzer are of equal magnitude and parallel to each other.

Polarized light can be produced from the common physical processes that deviate light beams, including absorption, refraction, reflection, diffraction (or scattering), and the process known as birefringence (the property of double refraction). Light that is reflected from the flat surface of a dielectric (or insulating) material is often partially polarized, with the electric vectors of the reflected light vibrating in a plane that is parallel to the surface of the material. Common examples of surfaces that reflect polarized light are undisturbed water, glass, sheet plastics, and highways. In these instances, light waves that have the electric field vectors parallel to the surface are reflected to a greater degree than those with different orientations. The optical properties of the insulating surface determine the exact amount of reflected light that is polarized. Mirrors are not good polarizers, although a wide spectrum of transparent materials act as very good polarizers, but only if the incident light angle is oriented within certain limits. An important property of reflected polarized light is that the degree of polarization is dependent upon the incident angle of the light, with the increasing amounts of polarization being observed for decreasing incident angles.

The quantity lₛ = λ/θ is known as the lateral coherence width of the given light source. It depends inversely on θ and thus, decreases with the length of the source.

The concept of coherence plays a major role in a variety of optical phenomena and is also one of the major factors that differentiate an ideal source from a practical one. An ideal source would have infinite temporal and spatial coherence. Further, a LASER light has much better temporal and spatial coherence as compared to other sources, so much so that it doesn’t need to be passed through a pinhole to create a coherent source (unlike normal sources).

In modern polarizers, incident light waves having electric vector vibrations that are parallel to the crystal axis of the polarizer are absorbed. Many of the incident waves will have a vector orientation that is oblique, but not perpendicular to the crystal axis, and will only be partially absorbed. The degree of absorption for oblique light waves is dependent upon the vibration angle at which they impact the polarizer. Those rays that have angles close to parallel with respect to the crystal axis will be adsorbed to a much greater degree than those having angles close to the perpendicular. The most common Polaroid filters (termed the H-series) transmit only about 25 percent of the incident light beam, but the degree of polarization of the transmitted rays exceeds 99 percent.

Finally, from equation (3), you can infer that a good interference pattern would be obtained when the following condition is satisfied:

Elliptical polarization, unlike plane-polarized and non-polarized light, has a rotational "sense" that refers to the direction of electric vector rotation around the propagation (incident) axis of the light beam. When viewed end-on, the direction of polarization can be either left-handed or right-handed, a property that is termed the handedness of the elliptical polarization. Clockwise rotational sweeps of the vector are referred to as right-handed polarization, and counterclockwise rotational sweeps represent left-handed polarization.

Asymmetric crystals can be utilized to produce polarized light when an electric field is applied to the surface. A common scientific device that employs this concept is termed a Pockels cell, which can be utilized in conjunction with polarized light to change the polarization direction by 90 degrees. Pockels cells can be switched on and off very rapidly by electrical currents and are often used as fast shutters that allow light to pass for very brief periods of time (ranging in nanoseconds). Presented in Figure 10 is a diagrammatic representation of polarized light passing through a Pockels cell (yellow wave). The green and red sinusoidal light waves emanating from the central region of the cell represent light that is polarized either vertically or horizontally. When the cell is turned off, the polarized light is unaffected as it passes through (green wave), but when activated or turned on, the electric vector of the light beam is shifted by 90-degrees (red wave). In situations where extremely large electric fields are available, molecules of certain liquids and gases can behave as anisotropic crystals and be aligned in the same manner. A Kerr cell, designed to house liquids and gases instead of crystals, also operates to change the angle of polarized light.

A number of applications, most notably polarized optical microscopy, rely on crossed polarizers to examine birefringent or doubly refracting specimens. When two polarizers are crossed, their transmission axes are oriented perpendicular to each other and light passing through the first polarizer is completely extinguished, or absorbed, by the second polarizer, which is typically termed an analyzer. The light-absorbing quality of a dichroic polarizing filter determines exactly how much random light is extinguished when the polarizer is utilized in a crossed pair, and is referred to as the extinction factor of the polarizer. Quantitatively, the extinction factor is determined by the ratio of light that is passed by a pair of polarizers when their transmission axes are oriented parallel versus the amount passed when they are positioned perpendicular to each other. In general, extinction factors between 10,000 and 100,000 are required to produce jet-black backgrounds and maximum observable specimen birefringence (and contrast) in polarized optical microscopy.

Lowcoherence light

For the fringes to disappear, the path difference between the light waves that arrive from S’ at the two slits must equal half the wavelength of the source, i.e., λ/2. Thus, we have

This concept is illustrated in Figure 8, where the resultant electric vector does not vibrate in a single plane, but progressively rotates around the axis of light wave propagation, sweeping out an elliptical trajectory that appears as a spiral when the wave is viewed at an angle. The size of the phase difference between the ordinary and extraordinary waves (of equal amplitude) determines whether the vector sweeps an elliptical or circular pathway when the wave is viewed end-on from the direction of propagation. If the phase shift is either one-quarter or three-quarters of a wavelength, then a circular spiral is scribed by the resultant vector. However, phase shifts of one-half or a full wavelength produce linearly polarized light, and all other phase shifts produce sweeps having various degrees of ellipticity.

If you enjoyed reading this article, consider dropping me a tip via the link provided in the end. And for learning more about LASERs, I recommend the excellent LASERS: Fundamentals and Applications by Thyagarajan and Ghatak.

Coherencelength

One of the light rays emerging from a birefringent crystal is termed the ordinary ray, while the other is called the extraordinary ray. The ordinary ray is refracted to a greater degree by electrostatic forces in the crystal and impacts the cemented surface at the critical angle of total internal reflection. As a result, this ray is reflected out of the prism and eliminated by absorption in the optical mount. The extraordinary ray traverses the prism and emerges as a beam of linearly-polarized light that is passed directly through the condenser and to the specimen (positioned on the microscope stage).

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Sunlight and almost every other form of natural and artificial illumination produces light waves whose electric field vectors vibrate in all planes that are perpendicular with respect to the direction of propagation. If the electric field vectors are restricted to a single plane by filtration of the beam with specialized materials, then the light is referred to as plane or linearly polarized with respect to the direction of propagation, and all waves vibrating in a single plane are termed plane parallel or plane-polarized.

The aim of this article is not to regurgitate the pure theory developed over the years about the coherence of light. Rather, my focus will be more experimental in nature, and I am hoping to provide you with a practical means of understanding how these concepts affect the interference pattern produced by a light source.

If you recall, the original experiment utilized a single point-source. We will use two point-sources instead, viz. S and S’. Here is a diagram:

Other prism configurations were suggested and constructed during the nineteenth and early twentieth centuries, but are currently no longer utilized for producing polarized light in modern applications. Nicol prisms are very expensive and bulky, and have a very limited aperture, which restricts their use at high magnifications. Instead, polarized light is now most commonly produced by absorption of light having a set of specific vibration directions in a filter medium (such as polarizing sheets) where the transmission axis of the filter is perpendicular to the orientation of the linear polymers and crystals that comprise the polarizing material.

When the ordinary and extraordinary waves emerge from a birefringent crystal, they are vibrating in mutually perpendicular planes having a total intensity that is the sum of their individual intensities. Because the polarized waves have electric vectors that vibrate in perpendicular planes, the waves are not capable of undergoing interference. This fact has consequences in the ability of birefringent substances to produce an image. Interference can only occur when the electric vectors of two waves vibrate in the same plane during intersection to produce a change in amplitude of the resultant wave (a requirement for image formation). Therefore, transparent specimens that are birefringent will remain invisible unless they are examined between crossed polarizers, which pass only the components of the elliptically and circularly polarized waves that are parallel to the axis of the polarizer closest to the observer. These components are able to produce amplitude fluctuations to generate contrast and emerge from the polarizer as linearly polarized light.

As discussed above, bright reflections originating from horizontal surfaces, such as the highway or the water in a pool, are partially polarized with the electric field vectors vibrating in a direction that is parallel to the ground. This light can be blocked by polarizing filters oriented in a vertical direction, as illustrated in Figure 4, with a pair of polarized sunglasses. The lenses of the sunglasses have polarizing filters that are oriented vertically with respect to the frames. In the figure, the blue light waves have their electric field vectors oriented in the same direction as the polarizing lenses and, thus, are passed through. In contrast, the red light wave vibration orientation is perpendicular to the filter orientation and is blocked by the lenses. Polarizing sunglasses are very useful when driving in the sun or at the beach where sunlight is reflected from the surface of the road or water, leading to glare that can be almost blinding. Polarizing filters are also quite useful in photography, where they can be attached to the front of a camera lens to reduce glare and increase overall image contrast in photographs or digital images. Polarizers utilized on cameras are generally designed with a mounting ring that allows them to be rotated in use to achieve the desired effect under various lighting conditions.

An excellent example of the basic application of liquid crystals to display devices can be found in the seven-segment liquid crystal numerical display (illustrated in Figure 9). Here, the liquid crystalline phase is sandwiched between two glass plates that have electrodes attached, similar to those depicted in the illustration. In Figure 9, the glass plates are configured with seven black electrodes that can be individually charged (these electrodes are transparent to light in real devices). Light passing through polarizer 1 is polarized in the vertical direction and, when no current is applied to the electrodes, the liquid crystalline phase induces a 90 degree "twist" of the light that enables it to pass through polarizer 2, which is polarized horizontally and is oriented perpendicular to polarizer 1. This light can then form one of the seven segments on the display.

The light waves emitted from various parts of an extended light source will not be coherent and spatial coherence is a measure of the distance up to which, appreciable coherence may be achieved in such a source.

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In cases where the major and minor vectorial axes of the polarization ellipse are equal, then the light wave falls into the category of circularly polarized light, and can be either right-handed or left-handed in sense. Another case often occurs in which the minor axis of the electric vector component in elliptically polarized light goes to zero, and the light becomes linearly polarized. Although each of these polarization motifs can be achieved in the laboratory with the appropriate optical instrumentation, they also occur (to varying, but minor, degrees) in natural non-polarized light.

Over a century later, French physicist Etienne Malus examined images made with light reflected through calcite crystals and noticed that, under certain circumstances, one of the images will disappear. He incorrectly speculated that ordinary daylight is composed of two different light forms that were passed through the calcite crystal in separate paths. It was later determined that the difference occurs due to the polarity of the light passing through the crystal. Daylight is composed of light vibrating in all planes, whereas reflected light is often restricted to a single plane that is parallel to the surface from which the light is reflected.

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Let us suppose that the average duration of the wave train that our source emits is given as τ seconds². It is evident that there will be no predictable phase relationship between each of these wave trains. If the time taken by our light to traverse the distances from G to M₁ and M₂ is significantly smaller than τ, then our resultant pattern will have excellent contrast since it is produced by the interference of two wave fronts emanating out of the same wave train. But once the temporal distance³ from G to M₁ and M₂ becomes larger than τ, the contrast will start to decrease and eventually, our pattern will vanish. This occurs when the interference starts to occur between two entirely separate wave trains with no predictable phase relation.

The amount of light passing through a pair of polarizers can be quantitatively described by applying Malus' cosine-squared law, as a function of the angles between the polarizer transmission axes, utilizing the equation:

When current is applied to the electrodes, the liquid crystalline phase aligns with the current and loses the cholesteric spiral pattern. Light passing through a charged electrode is not twisted and is blocked by polarizer 2. By coordinating the voltage on the seven positive and negative electrodes, the display is capable of rendering the numbers 0 through 9. In this example the upper right and lower left electrodes are charged and block light passing through them, allowing formation of the number "2" by the display device (seen reversed in the figure).

As you progress in your studies from secondary school to graduate level classes, you will come across the division of coherence into temporal and spatial coherence. This post will discuss these two concepts in detail.

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Rotating the analyzer transmission axis by 30-degrees with respect to that of the polarizer reduces the amplitude of a light wave passing through the pair, as illustrated in Figure 6(b). In this case, the polarized light transmitted through the polarizer can be resolved into horizontal and vertical components by vector mathematics to determine the amplitude of polarized light that is able to pass through the analyzer. The amplitude of the ray transmitted through the analyzer is equal to the vertical vector component (illustrated as the yellow arrow in Figure 6(b)).

One of the most common and practical applications of polarization is the liquid crystal display (LCD) used in numerous devices including wristwatches, computer screens, timers, clocks, and a host of others. These display systems are based upon the interaction of rod-like liquid crystalline molecules with an electric field and polarized light waves. The liquid crystalline phase exists in a ground state that is termed cholesteric, in which the molecules are oriented in layers, and each successive layer is slightly twisted to form a spiral pattern (Figure 9). When polarized light waves interact with the liquid crystalline phase the wave is "twisted" by an angle of approximately 90 degrees with respect to the incident wave. The exact magnitude of this angle is a function of the helical pitch of the cholesteric liquid crystalline phase, which is dependent upon the chemical composition of the molecules (it can be fine-tuned by small changes to the molecular structure).

A special class of materials, known as compensation or retardation plates, are quite useful in producing elliptically and circularly polarized light for a number of applications, including polarized optical microscopy. These birefringent substances are chosen because, when their optical axis is positioned perpendicular to the incident light beam, the ordinary and extraordinary light rays follow identical trajectories and exhibit a phase difference that is dependent upon the degree of birefringence. Because the pair of orthogonal waves is superimposed, it can be considered a single wave having mutually perpendicular electrical vector components separated by a small difference in phase. When the vectors are combined by simple addition in three-dimensional space, the resulting wave becomes elliptically polarized.

Atmospheric polarization is a direct result of the Rayleigh scattering of sunlight by gas molecules in the atmosphere. Upon impact between a photon from the sun and a gas molecule, the electric field from the photon induces a vibration and subsequent re-radiation of polarized light from the molecule (illustrated in Figure 7). The radiated light is scattered at right angles to the direction of sunlight propagation, and is polarized either vertically or horizontally, depending upon the direction of scatter. A majority of the polarized light impacting the Earth is polarized horizontally (over 50 percent), a fact that can be confirmed by viewing the sky through a Polaroid filter.