Galen Duree, Jr., PhD, is Professor of Physics and Optical Engineering at Rose-Hulman Institute of Technology in Indiana, where he is also the Director of the Center for Applied Optics Studies. Duree jointly established the Ultrashort Pulse Laser Laboratory at RHIT and continues to work with the Navy.

The index of refraction: This quantity describes the effect of atoms and molecules on the light as it travels through a transparent material. Use this basic formula for the index of refraction:

Snell’s law or the law of refraction: Snell‘s law shows the relationship between the incident angle and the transmitted angle (refracted angle) for a ray of light incident on a surface of a transparent material. You can see how Snell’s law works in the following formula:

The phase shift due to the film thickness in thin film interference: When light is incident straight onto a thin film (such as an oil slick on the surface of a pool of water), light rays reflecting from the top and the bottom of the film interfere (either constructively or destructively depending on the film thickness and the wavelength of the light). The following equations determine constructive or destructive interference depending on whether the phase shift produced by reflection needs to be shifted by half the wavelength (the first equation) or maintained (the second equation):

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.

Polarizationby reflection

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The location of the bright and dark fringes in Young’s two-slit interference arrangement: The following equations allow you to calculate the location of the bright fringes (where constructive interference occurs) and dark fringes (where destructive interference occurs):

The term used for this is electric polarization. In most simple words, electric polarization is the point of differentiation between positive and negative ...

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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 location of the dark fringes produced by diffraction through a single slit: Because a slit has a width larger than the wavelength, light rays from different parts of the slit interfere with each other, creating a fringe pattern. You can relatively easily locate the points where the light destructively interferes by using the following equation:

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Intermodal dispersion in a fiber: This characteristic measures the difference in time that different fiber modes take to reach the end of the fiber. The larger this time difference, the shorter the fiber has to be so that the information on this light doesn’t turn into junk. Here’s the equation:

Polarization directionelectric field

Locating images formed by mirrors: An object placed a certain distance away from a mirror will produce an image at a certain distance from the mirror. In some cases where the mirrors are curved, you may be given the focal length of a mirror. Use these equations:

The critical angle for total internal reflection: Total internal reflection is the situation where light hits and reflects off the surface of a transparent material without transmitting through the surface. It utilizes the critical angle (the minimum angle of incidence where total internal reflection takes place.). For total internal reflection to occur, the light must start in the material with the higher index. Here’s the formula:

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.

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.

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.

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Circularpolarization

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.

Optical interference is just the interaction of two or more light waves. Optical interference is useful in many applications, so you need to understand some basic equations related to this optical phenomenon. The following equations allow you to calculate various quantities related to optical interference in the two most common interference arrangements.

The lens maker’s formula: This equation allows you to calculate the focal length of a lens if all you know is the curvature of the two surfaces. Here’s the lens maker’s formula:

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Resolution: Resolution is the minimum angular separation between two objects such that you can tell that there are two distinct objects. Here’s the equation for determining resolution:

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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.

The maximum acceptance angle for a fiber: This angle is the largest angle of incidence at which light can enter the end of the fiber and be totally internally reflected inside the fiber. Angles of incidence larger than this angle will transmit through the sides of the fiber and not make it to the other end. The equation for this angle is

Diffraction is light’s response to having something mess with its path, so diffraction occurs only when something blocks part of the wavefront. Diffraction is the phenomenon where light bends around an obstacle (this bending is not due to refraction, because the material doesn’t change as refraction requires). The following equations cover the most common situations involving diffraction, including resolution.

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.

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.

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

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Linearpolarization

The location of the different diffraction orders from a diffraction grating: A diffraction grating has a very large number of slits spaced closely together, such that the light from each of these slits interferes with the light from the others. You can pretty easily identify where the light constructively interferes by using the following equation:

Polarisation meaning in Physics

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.

Phase retardation in a birefringent material: A birefringent material has two indexes of refraction. When you send polarized light into a birefringent material, the two components travel through the material with different speeds. This discrepancy can result in a change in the polarization state or simply rotate the polarization state. Use this equation:

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.

Besides imaging, fiber-optic networks are probably the largest application of optics. Fiber optics are very long, thin glass fibers that transfer information-bearing light from one place to another, but that may not be in direct sight of each other. You need to be aware of a few characteristics of the particular fiber you’re using so that you can ensure the information is accurately transmitted from one end of the fiber to the other. The following equations cover three of the basic parameters necessary for proper use of optical fibers.

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).

Polarizationexamples

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Imaging is a key function of optics. Specific optics equations can help you determine the basic characteristics of an image and predict where it will form. Use the following optics equations for your imaging needs:

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.

Optical polarization is the orientation of the planes of oscillation of the electric field vectors for many light waves. Optical polarization is often a major consideration in the construction of many optical systems, so equations for working with polarization come in handy. The following equations highlight some important polarization concepts. The equations listed here allow you to calculate how to make polarized light by reflection and to determine how much light passes through multiple polarizers:

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.

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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.

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Reflection and refraction are two processes that change the direction light travels. Using the equations for calculating reflection and refraction, you can predict where rays encountering a surface will go — whether they reflect or refract (bounce off the surface or bend through it) — which is an important concept in the study of optics. The following equations help you determine reflection and refraction angles:

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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.

The numerical aperture for a fiber: The numerical aperture is a measure of the light-gathering power of the fiber. It has a maximum value of 1 (all the light remains trapped inside the fiber) and a minimum value of 0 (only light incident at an angle of 0 degrees on the end of the fiber remains trapped in the fiber). Use this equation:

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.

Malus’ law: This equation allows you to calculate how much polarized light passes through a linear polarizer. The equation for Malus’ law is

Polarizationof electromagnetic waves pdf

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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 thin lens equation: An object placed a certain distance away from a lens will produce an image at a certain distance from the lens, and the thin lens equation relates the image location to the object distance and focal length. The following is the thin lens equation:

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The law of reflection: The law of reflection shows the relationship between the incident angle and the reflected angle for a ray of light incident on a surface. The angles are measured relative to the surface normal (a line that is perpendicular to the surface), not relative to the surface itself. Here’s the formula:

Polarizing angle or Brewster’s angle: This angle is the angle of incidence where the reflected light is linearly polarized. Here’s the equation:

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!

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.

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.

Location of images formed by a refracting surface: An object placed a certain distance away from a refracting surface will produce an image at a certain distance from the surface. The equation for this is

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Lateral magnification: Lateral magnification is one way you can describe how big the image is compared to the original object. Here are the equations:

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.

Polarization directionin physics

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.

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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.