Polarization (waves)
Polarization (
An
Light or other electromagnetic radiation from many sources, such as the sun, flames, and
According to
Polarization is an important parameter in areas of science dealing with transverse waves, such as optics, seismology, radio, and microwaves. Especially impacted are technologies such as lasers, wireless and optical fiber telecommunications, and radar.
Introduction
Wave propagation and polarization
Most sources of light are classified as incoherent and unpolarized (or only "partially polarized") because they consist of a random mixture of waves having different spatial characteristics, frequencies (wavelengths), phases, and polarization states. However, for understanding electromagnetic waves and polarization in particular, it is easier to just consider coherent
Transverse electromagnetic waves
Thus the leading vectors e and h each contain up to two nonzero (complex) components describing the amplitude and phase of the wave's x and y polarization components (again, there can be no z polarization component for a transverse wave in the +z direction). For a given medium with a characteristic impedance η, h is related to e by:
In a dielectric, η is real and has the value η0/n, where n is the refractive index and η0 is the impedance of free space. The impedance will be complex in a conducting medium. Note that given that relationship, the dot product of E and H must be zero:
Knowing the propagation direction (+z in this case) and η, one can just as well specify the wave in terms of just ex and ey describing the electric field. The vector containing ex and ey (but without the z component which is necessarily zero for a transverse wave) is known as a
However, the wave's state of polarization is only dependent on the (complex) ratio of ey to ex. So let us just consider waves whose |ex|2 + |ey|2 = 1; this happens to correspond to an intensity of about 0.00133 W/m2 in free space (where η = η0). And because the absolute phase of a wave is unimportant in discussing its polarization state, let us stipulate that the phase of ex is zero; in other words ex is a real number while ey may be complex. Under these restrictions, ex and ey can be represented as follows:
Non-transverse waves
In addition to transverse waves, there are many wave motions where the oscillation is not limited to directions perpendicular to the direction of propagation. These cases are far beyond the scope of the current article which concentrates on transverse waves (such as most electromagnetic waves in bulk media), but one should be aware of cases where the polarization of a coherent wave cannot be described simply using a Jones vector, as we have just done.
Just considering electromagnetic waves, we note that the preceding discussion strictly applies to plane waves in a homogeneous isotropic non-attenuating medium, whereas in an
Even in free space, longitudinal field components can be generated in focal regions, where the plane wave approximation breaks down. An extreme example is
For
Polarization state
Polarization is best understood by initially considering only pure polarization states, and only a coherent sinusoidal wave at some optical frequency. The vector in the adjacent diagram might describe the oscillation of the electric field emitted by a single-mode laser (whose oscillation frequency would be typically 1015 times faster). The field oscillates in the x-y plane, along the page, with the wave propagating in the z direction, perpendicular to the page. The first two diagrams below trace the electric field vector over a complete cycle for linear polarization at two different orientations; these are each considered a distinct state of polarization (SOP). Note that the linear polarization at 45° can also be viewed as the addition of a horizontally linearly polarized wave (as in the leftmost figure) and a vertically polarized wave of the same amplitude in the same phase.
Now if one were to introduce a
Of course the orientation of the x and y axes used in this description is arbitrary. The choice of such a coordinate system and viewing the polarization ellipse in terms of the x and y polarization components, corresponds to the definition of the Jones vector (below) in terms of those
Moreover, one can use as basis functions any pair of orthogonal polarization states, not just linear polarizations. For instance, choosing right and left circular polarizations as basis functions simplifies the solution of problems involving circular birefringence (optical activity) or circular dichroism.
Polarization ellipse
Consider a purely polarized monochromatic wave. If one were to plot the electric field vector over one cycle of oscillation, an ellipse would generally be obtained, as is shown in the figure, corresponding to a particular state of elliptical polarization. Note that linear polarization and circular polarization can be seen as special cases of elliptical polarization.
A polarization state can then be described in relation to the geometrical parameters of the ellipse, and its "handedness", that is, whether the rotation around the ellipse is clockwise or counter clockwise. One parameterization of the elliptical figure specifies the orientation angle ψ, defined as the angle between the major axis of the ellipse and the x-axis[13] along with the ellipticity ε = a/b, the ratio of the ellipse's major to minor axis.[14][15][16] (also known as the axial ratio). The ellipticity parameter is an alternative parameterization of an ellipse's eccentricity or the ellipticity angle, as is shown in the figure.[13] The angle χ is also significant in that the latitude (angle from the equator) of the polarization state as represented on the Poincaré sphere (see below) is equal to ±2χ. The special cases of linear and circular polarization correspond to an ellipticity ε of infinity and unity (or χ of zero and 45°) respectively.
Jones vector
Full information on a completely polarized state is also provided by the amplitude and phase of oscillations in two components of the electric field vector in the plane of polarization. This representation was used above to show how different states of polarization are possible. The amplitude and phase information can be conveniently represented as a two-dimensional complex vector (the Jones vector):
Here and denote the amplitude of the wave in the two components of the electric field vector, while and represent the phases. The product of a Jones vector with a complex number of unit
Coordinate frame
Regardless of whether polarization state is represented using geometric parameters or Jones vectors, implicit in the parameterization is the orientation of the coordinate frame. This permits a degree of freedom, namely rotation about the propagation direction. When considering light that is propagating parallel to the surface of the Earth, the terms "horizontal" and "vertical" polarization are often used, with the former being associated with the first component of the Jones vector, or zero azimuth angle. On the other hand, in astronomy the equatorial coordinate system is generally used instead, with the zero azimuth (or position angle, as it is more commonly called in astronomy to avoid confusion with the horizontal coordinate system) corresponding to due north.
s and p designations
Another coordinate system frequently used relates to the plane of incidence. This is the plane made by the incoming propagation direction and the vector perpendicular to the plane of an interface, in other words, the plane in which the ray travels before and after reflection or refraction. The component of the electric field parallel to this plane is termed p-like (parallel) and the component perpendicular to this plane is termed s-like (from senkrecht, German for perpendicular). Polarized light with its electric field along the plane of incidence is thus denoted p-polarized, while light whose electric field is normal to the plane of incidence is called s-polarized. P polarization is commonly referred to as transverse-magnetic (TM), and has also been termed pi-polarized or tangential plane polarized. S polarization is also called transverse-electric (TE), as well as sigma-polarized or sagittal plane polarized.
Degree of polarization
Degree of polarization (DOP) is a quantity used to describe the portion of an
DOP can be used to map the
DOP is also visualized using the
Unpolarized and partially polarized light
Unpolarized light can be produced from the incoherent combination of vertical and horizontal linearly polarized light, or right- and left-handed circularly polarized light.[17] Conversely, the two constituent linearly polarized states of unpolarized light cannot form an
A so-called depolarizer acts on a polarized beam to create one in which the polarization varies so rapidly across the beam that it may be ignored in the intended applications. Conversely, a polarizer acts on an unpolarized beam or arbitrarily polarized beam to create one which is polarized.
Unpolarized light can be described as a mixture of two independent oppositely polarized streams, each with half the intensity.Implications for reflection and propagation
Polarization in wave propagation
In a vacuum, the components of the electric field propagate at the speed of light, so that the phase of the wave varies in space and time while the polarization state does not. That is, the electric field vector e of a plane wave in the +z direction follows:
where k is the wavenumber. As noted above, the instantaneous electric field is the real part of the product of the Jones vector times the phase factor . When an electromagnetic wave interacts with matter, its propagation is altered according to the material's (complex)
In such media, an electromagnetic wave with any given state of polarization may be decomposed into two orthogonally polarized components that encounter different
The Jones matrix due to passage through a transparent material is dependent on the propagation distance as well as the birefringence. The birefringence (as well as the average refractive index) will generally be dispersive, that is, it will vary as a function of optical frequency (wavelength). In the case of non-birefringent materials, however, the 2×2 Jones matrix is the identity matrix (multiplied by a scalar phase factor and attenuation factor), implying no change in polarization during propagation.
For propagation effects in two orthogonal modes, the Jones matrix can be written as
where g1 and g2 are complex numbers describing the
Birefringence
In a
In birefringent media there is no attenuation, but two modes accrue a differential phase delay. Well known manifestations of linear birefringence (that is, in which the basis polarizations are orthogonal linear polarizations) appear in optical
Circular birefringence is also termed
One can visualize the case of linear birefringence (with two orthogonal linear propagation modes) with an incoming wave linearly polarized at a 45° angle to those modes. As a differential phase starts to accrue, the polarization becomes elliptical, eventually changing to purely circular polarization (90° phase difference), then to elliptical and eventually linear polarization (180° phase) perpendicular to the original polarization, then through circular again (270° phase), then elliptical with the original azimuth angle, and finally back to the original linearly polarized state (360° phase) where the cycle begins anew. In general the situation is more complicated and can be characterized as a
Dichroism
Media in which transmission of one polarization mode is preferentially reduced are called dichroic or diattenuating. Like birefringence, diattenuation can be with respect to linear polarization modes (in a crystal) or circular polarization modes (usually in a liquid).
Devices that block nearly all of the radiation in one mode are known as polarizing filters or simply "polarizers". This corresponds to g2=0 in the above representation of the Jones matrix. The output of an ideal polarizer is a specific polarization state (usually linear polarization) with an amplitude equal to the input wave's original amplitude in that polarization mode. Power in the other polarization mode is eliminated. Thus if unpolarized light is passed through an ideal polarizer (where g1=1 and g2=0) exactly half of its initial power is retained. Practical polarizers, especially inexpensive sheet polarizers, have additional loss so that g1 < 1. However, in many instances the more relevant figure of merit is the polarizer's
Specular reflection
In addition to birefringence and dichroism in extended media, polarization effects describable using Jones matrices can also occur at (reflective) interface between two materials of different refractive index. These effects are treated by the Fresnel equations. Part of the wave is transmitted and part is reflected; for a given material those proportions (and also the phase of reflection) are dependent on the angle of incidence and are different for the s and p polarizations. Therefore, the polarization state of reflected light (even if initially unpolarized) is generally changed.
Any light striking a surface at a special angle of incidence known as
In the important special case of reflection at normal incidence (not involving anisotropic materials) there is no particular s or p polarization. Both the x and y polarization components are reflected identically, and therefore the polarization of the reflected wave is identical to that of the incident wave. However, in the case of circular (or elliptical) polarization, the handedness of the polarization state is thereby reversed, since by convention this is specified relative to the direction of propagation. The circular rotation of the electric field around the x-y axes called "right-handed" for a wave in the +z direction is "left-handed" for a wave in the -z direction. But in the general case of reflection at a nonzero angle of incidence, no such generalization can be made. For instance, right-circularly polarized light reflected from a dielectric surface at a grazing angle, will still be right-handed (but elliptically) polarized. Linear polarized light reflected from a metal at non-normal incidence will generally become elliptically polarized. These cases are handled using Jones vectors acted upon by the different Fresnel coefficients for the s and p polarization components.
Measurement techniques involving polarization
Some optical measurement techniques are based on polarization. In many other optical techniques polarization is crucial or at least must be taken into account and controlled; such examples are too numerous to mention.
Measurement of stress
In engineering, the phenomenon of stress induced birefringence allows for stresses in transparent materials to be readily observed. As noted above and seen in the accompanying photograph, the chromaticity of birefringence typically creates colored patterns when viewed in between two polarizers. As external forces are applied, internal stress induced in the material is thereby observed. Additionally, birefringence is frequently observed due to stresses "frozen in" at the time of manufacture. This is famously observed in cellophane tape whose birefringence is due to the stretching of the material during the manufacturing process.
Ellipsometry
Ellipsometry is a powerful technique for the measurement of the optical properties of a uniform surface. It involves measuring the polarization state of light following specular reflection from such a surface. This is typically done as a function of incidence angle or wavelength (or both). Since ellipsometry relies on reflection, it is not required for the sample to be transparent to light or for its back side to be accessible.
Ellipsometry can be used to model the (complex) refractive index of a surface of a bulk material. It is also very useful in determining parameters of one or more thin film layers deposited on a substrate. Due to their reflection properties, not only are the predicted magnitude of the p and s polarization components, but their relative phase shifts upon reflection, compared to measurements using an ellipsometer. A normal ellipsometer does not measure the actual reflection coefficient (which requires careful photometric calibration of the illuminating beam) but the ratio of the p and s reflections, as well as change of polarization ellipticity (hence the name) induced upon reflection by the surface being studied. In addition to use in science and research, ellipsometers are used in situ to control production processes for instance.[24]: 585ff [25]: 632
Geology
The property of (linear) birefringence is widespread in crystalline minerals, and indeed was pivotal in the initial discovery of polarization. In mineralogy, this property is frequently exploited using polarization microscopes, for the purpose of identifying minerals. See optical mineralogy for more details.[26]: 163–164
Sound waves in solid materials exhibit polarization. Differential propagation of the three polarizations through the earth is a crucial in the field of
: 56–57Chemistry
We have seen (above) that the birefringence of a type of crystal is useful in identifying it, and thus detection of linear birefringence is especially useful in
Astronomy
In many areas of
Applications and examples
Polarized sunglasses
Unpolarized light, after being reflected by a specular (shiny) surface, generally obtains a degree of polarization. This phenomenon was observed in the early 1800s by the mathematician
Wearers of polarized sunglasses will occasionally observe inadvertent polarization effects such as color-dependent birefringent effects, for example in
Sky polarization and photography
Polarization is observed in the light of the
Sky polarization has been used for orientation in navigation. The
Display technologies
The principle of
In a totally different sense, polarization encoding has become the leading (but not sole) method for delivering separate images to the left and right eye in
Although now obsolete, CRT computer displays suffered from reflection by the glass envelope, causing glare from room lights and consequently poor contrast. Several anti-reflection solutions were employed to ameliorate this problem. One solution utilized the principle of reflection of circularly polarized light. A circular polarizing filter in front of the screen allows for the transmission of (say) only right circularly polarized room light. Now, right circularly polarized light (depending on the convention used) has its electric (and magnetic) field direction rotating clockwise while propagating in the +z direction. Upon reflection, the field still has the same direction of rotation, but now propagation is in the −z direction making the reflected wave left circularly polarized. With the right circular polarization filter placed in front of the reflecting glass, the unwanted light reflected from the glass will thus be in very polarization state that is blocked by that filter, eliminating the reflection problem. The reversal of circular polarization on reflection and elimination of reflections in this manner can be easily observed by looking in a mirror while wearing 3-D movie glasses which employ left- and right-handed circular polarization in the two lenses. Closing one eye, the other eye will see a reflection in which it cannot see itself; that lens appears black. However, the other lens (of the closed eye) will have the correct circular polarization allowing the closed eye to be easily seen by the open one.
Radio transmission and reception
All radio (and microwave) antennas used for transmitting or receiving are intrinsically polarized. They transmit in (or receive signals from) a particular polarization, being totally insensitive to the opposite polarization; in certain cases that polarization is a function of direction. Most antennas are nominally linearly polarized, but elliptical and circular polarization is a possibility. As is the convention in optics, the "polarization" of a radio wave is understood to refer to the polarization of its electric field, with the magnetic field being at a 90 degree rotation with respect to it for a linearly polarized wave.
The vast majority of antennas are linearly polarized. In fact it can be shown from considerations of symmetry that an antenna that lies entirely in a plane which also includes the observer, can only have its polarization in the direction of that plane. This applies to many cases, allowing one to easily infer such an antenna's polarization at an intended direction of propagation. So a typical rooftop
Polarization is important in radio communications because, for instance, if one attempts to use a horizontally polarized antenna to receive a vertically polarized transmission, the signal strength will be substantially reduced (or under very controlled conditions, reduced to nothing). This principle is used in satellite television in order to double the channel capacity over a fixed frequency band. The same frequency channel can be used for two signals broadcast in opposite polarizations. By adjusting the receiving antenna for one or the other polarization, either signal can be selected without interference from the other.
Especially due to the presence of the ground, there are some differences in propagation (and also in reflections responsible for TV ghosting) between horizontal and vertical polarizations. AM and FM broadcast radio usually use vertical polarization, while television uses horizontal polarization. At low frequencies especially, horizontal polarization is avoided. That is because the phase of a horizontally polarized wave is reversed upon reflection by the ground. A distant station in the horizontal direction will receive both the direct and reflected wave, which thus tend to cancel each other. This problem is avoided with vertical polarization. Polarization is also important in the transmission of radar pulses and reception of radar reflections by the same or a different antenna. For instance, back scattering of radar pulses by rain drops can be avoided by using circular polarization. Just as specular reflection of circularly polarized light reverses the handedness of the polarization, as discussed above, the same principle applies to scattering by objects much smaller than a wavelength such as rain drops. On the other hand, reflection of that wave by an irregular metal object (such as an airplane) will typically introduce a change in polarization and (partial) reception of the return wave by the same antenna.
The effect of
Polarization and vision
Many
The naked human eye is weakly sensitive to polarization, without the need for intervening filters. Polarized light creates a very faint pattern near the center of the visual field, called Haidinger's brush. This pattern is very difficult to see, but with practice one can learn to detect polarized light with the naked eye.[36]: 118
Angular momentum using circular polarization
It is well known that electromagnetic radiation carries a certain linear momentum in the direction of propagation. In addition, however, light carries a certain angular momentum if it is circularly polarized (or partially so). In comparison with lower frequencies such as microwaves, the amount of angular momentum in light, even of pure circular polarization, compared to the same wave's linear momentum (or radiation pressure) is very small and difficult to even measure. However, it was utilized in an experiment to achieve speeds of up to 600 million revolutions per minute.[39][40]
See also
Quantum Physics
Optics
References
Cited references
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General references
- Azzam, R. M. A.; Bashara, N. M. (1977). Ellipsometry and Polarized Light. North-Holland. ISBN 0-444-87016-4.
- Born, M.; Wolf, E. (1999). ISBN 0-521-64222-1.
- Brosseau, C. (1998). Fundamentals of Polarized Light: A Statistical Optics Approach. Wiley. ISBN 0-471-14302-2.
- Collett, Edward (2005). Field Guide to Polarization. SPIE Field Guides. Vol. FG05. SPIE. ISBN 0-8194-5868-6.
- Damask, Jay N. (2004). Polarization Optics in Telecommunications. Springer. ISBN 0-387-22493-9.
- Goldstein, Dennis; Dekker, Marcel (2003). Polarized Light (2nd ed.). Taylor & Francis. ISBN 0-8247-4053-X.
- Karlsen, Leif (2003). Secrets of the Viking Navigators: How the Vikings used their amazing sunstones and other techniques to cross the open oceans. One Earth Press.
- Können, G. P. (1985). Polarized Light in Nature. Translated by Beerling, G. A. Cambridge University. ISBN 0-521-25862-6.
- ISBN 0-7503-0673-4.
- Shurcliff, William A. (1962). Polarized Light, Production and Use. Harvard University.
External links
- Feynman's lecture on polarization
- Polarized Light Digital Image Gallery: Microscopic images made using polarization effects
- MathPages: The relationship between photon spin and polarization
- A virtual polarization microscope
- Polarization angle in satellite dishes.
- Molecular Expressions: Science, Optics and You — Polarization of Light: Interactive Java tutorial
- Antenna Polarization
- Animations of Linear, Circular and Elliptical Polarizations on YouTube