:: wikimiki.org ::

Dispersion (optics)

Dispersion (optics)

In optics, dispersion is a phenomenon that causes the separation of a wave into spectral components with different frequencies, due to a dependence of the wave's speed on its frequency. It is most often described in light waves, though it may happen to any kind of wave that interacts with a medium or can be confined to a waveguide, such as sound waves. There are generally two sources of dispersion: material dispersion, which comes from a frequency-dependent response of a material to waves; and waveguide dispersion, which comes because the transverse mode solutions for waves confined laterally within a finite waveguide generally depend upon the frequency (i.e. on the relative size of the wave, the wavelength, and that of the waveguide). A related phenomenon is that of modal dispersion, which comes about if a signal consists of a superposition of multiple modes at each frequency—because different modes generally travel at different speeds, dispersion of temporal features (and thus signal degradation) results. A special case of this is polarization mode dispersion (PMD), which comes from a superposition of two modes that normally travel at the same speed due to symmetry (e.g. two orthogonal polarizations in a waveguide of circular or square cross-section), but which travel at different speeds due to random imperfections that break the symmetry.

Material dispersion in optics

polarization mode dispersion In optics, the phase velocity of a wave

v

in a given uniform medium is given by: :

v = \frac

where c is the speed of light in a vacuum and n is the refractive index of the medium. In general, the refractive index is some function of the frequency ν of the light, thus n = n(ν), or alternately, with respect to the wave's wavelength n = n(λ). The wavelength dependency of a material's refractive index is usually quantified by an empirical formula, the Cauchy or Sellmeier equations. The most commonly seen consequence of dispersion in optics is the separation of white light into a color spectrum by a prism. From Snell's law it can be seen that the angle of refraction of light in a prism depends on the refractive index of the prism material. Since that refractive index varies with wavelength, it follows that the angle that the light is refracted will also vary with wavelength, causing an angular separation of the colors known as angular dispersion. For visible light, most transparent materials (e.g. glasses) have: :

1 or alternatively:

: \frac < 0,

that is, refractive index

n decreases with increasing wavelength λ. At the interface of such a material with air or vacuum (index of ~1), Snell's law predicts that light incident at an angle θ to the normal will be refracted at an angle

\arcsin(\sin(\theta)/n)

. Thus, blue light, with a higher refractive index, will be bent more strongly than red light, resulting in the well-known rainbow pattern.

Group and phase velocity

Another consequence of dispersion manifests itself as a temporal effect. The formula above, v = c / n calculates the phase velocity of a wave; this is the velocity at which the phase of any one frequency component of the wave will propagate. This is not the same as the group velocity of the wave, which is the rate that changes in amplitude (known as the envelope of the wave) will propagate. The group velocity v_g is related to the phase velocity by, for a homogeneous medium (here

\lambda

is the wavelength in vacuum, not in the medium): :

v_g = c \left[ n - \lambda \frac \right]^

. The group velocity v_g is often thought of as the velocity at which energy or information is conveyed along the wave. In most cases this is true, and the group velocity can be thought of as the signal velocity of the waveform. In some unusual circumstances, where the wavelength of the light is close to an absorption resonance of the medium, it is possible for the group velocity to exceed the speed of light (v_g > c), leading to the conclusion that superluminal (faster than light) communication is possible. In practice, in such situations the distortion and absorption of the wave is such that the value of the group velocity essentially becomes meaningless, and does not represent the true signal velocity of the wave, which stays less than c. The group velocity itself is usually a function of the wave's frequency. This results in group velocity dispersion (GVD), which causes a short pulse of light to spread in time as a result of different frequency components of the pulse travelling at different velocities. GVD is often quantified as the group delay dispersion parameter (again, this formula is for a uniform medium only): :

D = - \frac \left( \frac \right)

. If D is less than zero, the medium is said to have normal, or positive dispersion. If D is greater than zero, the medium has anomalous, or negative dispersion. If a light pulse is propagated through a normally dispersive medium, the result is the higher frequency components travel slower than the lower frequency components. The pulse therefore becomes positively chirped, or up-chirped, increasing in frequency with time. Conversely, if a pulse travels through an anomalously dispersive medium, high frequency components travel faster than the lower ones, and the pulse becomes negatively chirped, or down-chirped, decreasing in frequency with time. The result of GVD, whether negative or positive, is ultimately temporal spreading of the pulse. This makes dispersion management extremely important in optical communications systems based on optical fiber, since if dispersion is too high, a group of pulses representing a bit-stream will spread in time and merge together, rendering the bit-stream unintelligible. This limits the length of fiber that a signal can be sent down without regeneration. One possible answer to this problem is to send signals down the optical fibre at a wavelength where the GVD is zero (e.g. around ~1.3-1.5 μm in silica fibres), so pulses at this wavelength suffer minimal spreading from dispersion—in practice, however, this approach causes more problems than it solves because zero GVD unacceptably amplifies other nonlinear effects (such as Four wave mixing). Another possible option is to use soliton pulses in the regime of anomalous dispersion, a form of optical pulse which uses a nonlinear optical effect to self-maintain its shape—solitons have the practical problem, however, that they require a certain power level to be maintained in the pulse for the nonlinear effect to be of the correct strength. Instead, the solution that is currently used in practice is to perform dispersion compensation, typically by matching the fiber with another fiber of opposite-sign dispersion so that the dispersion effects cancel; such compensation is ultimately limited by nonlinear effects such as self phase modulation, which interact with dispersion to make it very difficult to undo. Dispersion control is also important in lasers that produce short pulses. The overall dispersion of the optical resonator is a major factor in determining the duration of the pulses emitted by the laser. A pair of prisms can be arranged to produce net negative dispersion, which can be used to balance the usually positive dispersion of the laser medium. Diffraction gratings can also be used to produce dispersive effects; these are often used in high-power laser amplifier systems. Recently, an alternative to prisms and gratings has been developed: chirped dispersive mirrors. These dielectric mirrors are coated so that different wavelengths have different penetration lengths, and therefore different group delays. The coating layers can be tailored to achieve a net negative dispersion.

Dispersion in gemology

In the technical terminology of gemology, dispersion is the difference in the refractive index of a material at the B and G Fraunhofer wavelengths of 686.7 nm and 430.8 nm and is meant to express the degree to which a prism cut from the gemstone shows fire or color. Dispersion is a material property. Fire depends on the dispersion, the cut angles, the lighting environment, the refractive index, and the viewer.

External links

- [http://ioannis.virtualcomposer2000.com/spectroscope/characteristics.html Optical Characteristics of the SF10 Crystal Prism]
- [http://ioannis.virtualcomposer2000.com/spectroscope/deviationangle.html Deviation Angle for a Prism] Category:Optics

Optics

:"Optical" redirects here. For the musical artist, see Optical (artist). Optics (appearance or look in ancient Greek) is a branch of physics that describes the behavior and properties of light and the interaction of light with matter. Optics explains and is illuminated by optical phenomena. The field of optics usually describes the behavior of visible, infrared and ultraviolet light; however because light is an electromagnetic wave, analogous phenomena occur in X-rays, microwaves, radio waves, and other forms of electromagnetic radiation. Optics can thus be regarded as a sub-field of electromagnetism. Some optical phenomena depend on the quantum nature of light and as such some areas of optics are also related to quantum mechanics. In practice, the vast majority of optical phenomena can be accounted for using the electromagnetic description of light, as described by Maxwell's Equations. Optics, however, as a field is often considered largely separate from the physics community. It has its own identity, societies, and conferences. The pure science aspects of the field are often called optical science or optical physics. Applied optical sciences are often called optical engineering. Applications of optical enginering related specifically to illumination systems are called illumination engineering. Each of these disciplines tends to be quite different in its applications, technical skills, focus, and professional affiliations. Because of the wide application of the science of "light" to real-world applications, the areas of optical science and optical engineering tend to be very cross-disiplinary. Optical science is a part of many related disciplines including electrical engineering, physics, psychology, medicine, and others. Additionally, the most complete description of optical behavior, as known to physics, is unnecessarily complicated for most scenarios so particular simplified theories are used. These limited theories adequately describe subsets of optical phenomena while ignoring behavior irrelevant and/or undetectable to the system of interest.

Classical optics

Before Max Planck suggested that light is quantized, optics consisted mainly of the application of electromagnetism and its high frequency approximations to light. Classical optics divides into two main branches: geometric optics and physical optics. Geometric optics, or ray optics, describes light propagation in terms of "rays". Rays are bent at the interface between two dissimilar media, and may be curved in a medium in which the refractive index is a function of position. The "ray" in geometric optics is an abstract object which is perpendicular to the wavefronts of the actual optical waves. Geometric optics provides rules for propagating these rays through an optical system, which indicates how the actual wavefront will propagate. Note that this is a significant simplification of optics, and fails to account for many important optical effects such as diffraction and polarization. Geometric optics is often simplified even further by making the paraxial approximation. The mathematical behavior then becomes linear, allowing optical components and systems to be described by simple matrices. This leads to the techniques of Gaussian optics and paraxial raytracing, which are used to find first-order properties of optical systems, such as approximate image and object positions and magnifications. Gaussian beam propagation is an expansion of paraxial optics that provides a more accurate model of coherent radiation like laser beams. While still using the paraxial approximation, this technique partially accounts for diffraction, allowing accurate calculations of the rate at which a laser beam expands with distance, and the minimum size to which the beam can be focused. Gaussian beam propagation thus bridges the gap between geometric and physical optics. Physical optics models the propagation of complex wavefronts through optical systems, including both the amplitude and the phase of the wave. This technique, which is usually applied numerically on a computer, can account for diffraction, interference, and polarization effects, as well as aberrations and other complex effects. Approximations are still generally used, however, so this is not a full electromagnetic wave theory model of the propagation of light. Such a full model would (at present) be too computationally demanding to be useful for most problems, although some small-scale problems can be analyzed using complete wave models.

Topics related to classical optics

- Coherence
- Diffraction
- Dispersion
- Distortion
- Fabrication and testing (optical components)
- Fermat's principle
- Fourier optics
- Gradient index optics
- Optical lens design
- Optical resolution
- Polarization
- Ray (optics)
- Reflection
- Refraction
- Scattering
- Wave
- Geometric optics of:
- Lenses
- Mirrors
- Optical instruments
- Prisms

Modern optics

Modern optics encompasses the areas of optical science and engineering that became popular in the 20th century. These areas of optical science typically relate to the electromagnetic or quantum properties of light but do include other topics.

Topics related to modern optics

- Circular dichroism
- Crystal optics
- Diffractive optics
- Guided wave optics
- Holography
- Integrated optics
- Jones calculus
- Lasers
- Micro-optics
- Non-imaging optics
- Nonlinear optics
- Optical modeling and simulation methods
- Optical pattern recognition
- Optical processors
- Photometry
- Quantum optics
- Radiometry
- Statistical optics
- Thin-film optics

Other optical fields

- Color science
- Illumination engineering
- Image processing
- Information theory
- Linear optics
- Machine vision
- Materials science - optical properties
- Optical communication
- Optical computers
- Optical data storage
- Optical display system
- Pattern recognition
- Photography (science of)
- Thermal physics - radiative heat transfer
- Visual system

Everyday optics

Optics is part of everyday life. Rainbows and mirages are examples of optical phenomena. Many people benefit from eyeglasses or contact lenses, and optics are used in many consumer goods including cameras.

Wikibooks modules

- Optics (Physics Study Guide)
- Optics

References

-
-
-

External links

- [http://www.optics.net Optics.net] - Optical Engineering forum and resource directory
- [http://www.lightandmatter.com/area1book5.html Optics Book] - an online textbook
- [http://www.spie.org/events/op Optics & Photonics] - SPIE's Optics & Photonics Symposium, held annually, features optics and photonics research, training, and an exhibition.
- [http://www.optics2001.com Optics2001] - Optics library and community Category:Atomic, molecular, and optical physics Category:Technology ko:광학 ms:Optik ja:光学

Frequency

: Frequency is the measurement of the number of times that a repeated event occurs per unit time. It is also defined as the rate of change of phase of a sinusoidal waveform.

Measurement

To calculate the frequency of an event, the number of occurrences of the event within a fixed time interval are counted, and then divided by the length of the time interval. In SI units, the result is measured in hertz (Hz), named after the German physicist Heinrich Rudolf Hertz. 1 Hz means that an event repeats once per second, 2 Hz is twice per second, and so on. This unit was originally called a cycle per second (cps), which is still used sometimes. Other units that are used to measure frequency include revolutions per minute (rpm) and radians per second (rad/s). Heart rate and musical tempo are measured in beats per minute (BPM). An alternative method to calculate frequency is to measure the time between two consecutive occurrences of the event (the period) and then compute the frequency as the reciprocal of this time: :

f = \frac

where T is the period. A more accurate measurement takes many cycles into account and averages the period between each.

Frequency of waves

Measuring the frequency of sound, electromagnetic waves (such as radio or light), electrical signals, or other waves, the frequency in hertz is the number of cycles of the repetitive waveform per second. If the wave is a sound, frequency is what mainly characterizes its pitch. Frequency has an inverse relationship to the concept of wavelength. The frequency f is equal to the speed v of the wave divided by the wavelength λ (lambda) of the wave: :

f = \frac

In the special case of electromagnetic waves moving through a vacuum, then v = c, where c is the speed of light in a vacuum, and this expression becomes: :

f = \frac

NOTE: When waves travel from one medium to another, their frequency remains exactly the same - only their wavelength and/or speed changes.

Invariance

Apart from its being modified by Doppler effect, frequency is an invariant quantity in the universe. That is, it cannot be changed by any physical process unlike velocity of propagation or wavelength.

Examples

- The frequency of the standard pitch A above middle C is usually defined as 440 Hz, that is, 440 cycles per second () and known as concert pitch, to which an orchestra tunes.
- A baby can hear tones with oscillations up to approximately 20,000 Hz, but these frequencies become more difficult to hear as people age.
- In Europe, the frequency of the alternating current in mains is 50 Hz (close to the tone G).
- In North America, the frequency of the alternating current is 60 Hz (close to the tone B flat — that is, a minor third above the European frequency). The frequency of the 'hum' in an audio recording can show where the recording was made — in Europe or in America.

External links

- [http://www.sengpielaudio.com/calculator-wavelength.htm Conversion: frequency to wavelength and back]
- [http://www.sengpielaudio.com/calculator-period.htm Conversion: period, cycle duration, periodic time to frequency] Category:Physical quantity Category:Sound Category:Wave mechanics ko:진동수 ja:周波数 th:ความถี่

Sound

:This article is about compression waves. For other meanings, see sound (disambiguation). sound (disambiguation). Yellow: cochlea. Green: auditory receptor cells. Purple: frequency spectrum of hearing response. Orange: nerve impulse.)]] Sound is vibration, as perceived by the sense of hearing. We usually hear vibrations that travel through air, but sound can also travel through gases, liquids and solids. It cannot travel through a vacuum (such as exists in outer space). When the vibrations reach our ears, they are converted into nerve impulses that are sent to our brains, allowing us to perceive the sound. In more technical language, sound "is an alternation in pressure, particle displacement, or particle velocity propagated in an elastic material" (Olson 1957) or series of mechanical compressions and rarefactions or longitudinal waves that successively propagate through media that are at least a little compressible (solid, liquid or gas but not vacuum). In sound waves parts of matter (molecules or groups of molecules) move in a direction of the spreading of the disturbance (as opposite to transversal waves). The cause of sound waves is called the source of waves, e.g., a violin string vibrating upon being bowed or plucked. A sound wave is usually represented graphically by a wavy, horizontal line; the upper part of the wave (the crest) indicates a compression and the lower part (the trough) indicates a rarefaction.

Attributes of sound

The characteristics of sound are frequency, wavelength, amplitude and velocity.

Frequency and wavelength

The frequency is the number of air pressure oscillations per second at a fixed point occupied by a sound wave. One single oscillatory cycle per second corresponds to 1 Hz. The wavelength is the distance between two successive crests and is the distance that a wave travels in the time of one oscillatory cycle. The wavelength of a sound wave of frequency f and travelling at speed c is given by c/f. Given a speed of 343 m/s, a 20 kHz sound wave has a wavelength of about 17 mm. For comparison, an A440 has a nominal wavelength of about 78 cm, and a 20 Hz sound wave has a wavelength of 17 m. Suppose sound is emitted as a sine wave travelling outward spherically from a point source. The pressure (above ambient, see gauge pressure) of the sound wave can be written as :

P(r,t) = P_0 \sin\left(2 \pi f \left(t-\frac\right)\right)

where P(r,t) is the pressure at distance r at time t, P₀ is the amplitude of the pressure variation (0 to peak), f is the frequency of oscillation, and c is the speed of sound.

Amplitude

The amplitude is the magnitude of sound pressure change within the wave, or basically, the maximum amount of pressure at any point in the sound wave. A sound wave is caused literally by increases in pressure at certain points (of a material) causing a "domino effect" outward, the high pressure points are the crests mentioned above, and behind them are low pressure points which tail them, those are the troughs mentioned above. Amplitude is the maximal displacement of particles of matter that is obtained in compressions, where the particles of matter move towards each other and pressure increases the most and in rarefactions, where the pressure lessens the most. See also particle displacement and particle velocity. While the pressure can be measured in pascals, the amplitude is more often referred to as sound pressure level and measured in decibels, or dBSPL, sometimes written as dBspl or dB(SPL). When the measurement is adjusted based on how the human ear perceives loudness based on frequency, it is called dBA or A-weighting. See decibels for a more thorough discussion.
- [http://www.makeitlouder.com/Decibel%20Level%20Chart.txt Chart showing decibel level of many sounds]

Velocity

Sound's propagation speed depends on the type, temperature and pressure of the medium through which it propagates. Under normal conditions, however, because air is nearly a perfect gas, the speed of sound does not depend on air pressure. In dry air at 20 °C (68 °F) the speed of sound is approximately 343 m/s (approximately 1 meter every 2.9 milliseconds). The speed of sound relates frequency to wavelength. Thus, a tone of 343 Hz (F4 minus 31 cents) traveling in air has a wavelength of 1 meter.

Types of sounds

Noises are irregular and disordered vibrations including all possible frequencies. Their wave diagram does not repeat in time. Noise is an aperiodic series of waves. Sounds that are sine waves with fixed frequency and amplitude are perceived as pure tones. While sound waves are usually visualised as sine waves, sound waves can have arbitrary shapes and frequency content, limited only by the apparatus that generates them and the medium through which they travel. In fact, most sound waves consist of multiple overtones or harmonics and any sound can be thought of as being composed of sine waves (see additive synthesis). Waveforms commonly used to approximate harmonic sounds in nature include sawtooth waves, square waves and triangle waves. While a sound may still be referred to as being of a single frequency (for example, a piano striking the A above middle C is said to be playing a note at 440 Hz), the sound perceived by a listener will be colored by all of the sound wave's frequency components and their relative amplitudes, as well as how the sound changes over time (see timbre.) For convenience in this article, however, it is best to think of sound waves as sine waves.

Perception of sound

The frequency range of sound audible to humans is approximately between 20 and 20,000 Hz. This range varies by individual and generally shrinks with age. It is also an uneven curve - sounds near 3,500 Hz are often perceived as louder than a sound with the same amplitude at a much lower or higher frequency. Above and below this range are ultrasound and infrasound, respectively. The amplitude range of sound for humans has a lower limit of 0dBSPL, called the threshold of hearing. Sound is technically at its upper limit at 194.09 dB. Above this level it should be called a shock wave. Sounds begin to do damage to ears at 85 dBSPL and sounds above approximately 130 dBSPL (called the threshold of pain) cause pain. Again, this range varies by individual and changes with age. The perception of sound is the sense of hearing. In humans and many animals this is accomplished by the ears, but loud sounds and low frequency sounds can be perceived by other parts of the body through the sense of touch. Sounds are used in several ways, most notably for communication through speech or, for example, music. Sound perception can also be used for acquiring information about the surrounding environment in properties such as spatial characteristics and presence of other animals or objects. For example, bats use one sort of echolocation, ships and submarines use sonar, and humans can determine spatial information by the way in which they perceive sounds.

Reference

- Olson (1957) cited in Roads, Curtis (2001). Microsound. MIT. ISBN 0262182157.
- Roederer, Juan C. Introduction to the Physics and Psychophysics of Music (2nd ed.). New York: Springer-Verlag, 1979.
- Charles Dodge and Thomas A. Jerse, "Computer Music". New York, Schirmer Books, 1997. ISBN 0028646827
- Grey, J. M. "An Exploration of Musical Timbre." Doctoral dissertation, Stanford University, 1975.

External links

- [http://hyperphysics.phy-astr.gsu.edu/hbase/sound/soucon.html HyperPhysics: Sound and Hearing]
- [http://www.sengpielaudio.com/Calculations03.htm Audio calculations and online acoustics conversion engine]
- [http://www.acoustics.salford.ac.uk/schools/index.htm Sounds Amazing a learning resource for sound and waves]
- [http://www.plosbiology.org/plosonline/?request=get-document&doi=10.1371%2Fjournal.pbio.0030026 Computation Provides a Virtual Recording of Auditory Signaling - PLoS Biol 2005.3(1).e26] Category:Hearing Category:Waves ko:소리 ja:音 simple:Sound th:เสียง

Waveguide

In electromagnetics and communications engineering, a waveguide is a physical structure that guides the propagation of electromagnetic waves. Waveguides can be constructed to carry waves over a wide portion of the electromagnetic spectrum, but are especially useful in the microwave and optical frequency ranges. Depending on the frequency, they can be constructed from either conductive or dielectric materials. Waveguides are used for transferring both power and communication signals.

History

The first waveguide was proposed by J. J. Thomson in 1893 and experimentally verified by O. J. Lodge in 1894; the mathematical analysis of the propagating modes within a hollow metal cylinder was first performed by Lord Rayleigh in 1897. (McLachan, 1947.)

Principles of operation

Waveguides are analyzed by solving Maxwell's equations, or their reduced form, the electromagnetic wave equation, with boundary conditions determined by the properties of the materials and their interfaces. These equations have multiple solutions, or modes, which are eigenfunctions of the equation system. Each mode is therefore characterized by an eigenvalue, which corresponds to the axial propogation velocity of the wave in the guide. Waveguide propagation modes depend on the operating wavelength and polarization and the shape and size of the guide. In hollow metallic waveguides, the fundamental modes are the transverse electric TE_1,0 mode for rectangular and TE_1,1 for circular waveguides. Image:TE10.gif|TE_1,0 mode of a rectangular hollow metallic waveguide. Image:TE11.gif|TE_1,1 mode of a circular hollow metallic waveguide.

Hollow metallic waveguides

In the microwave region of the electromagnetic spectrum, a waveguide normally consists of a hollow metallic conductor. Hollow waveguides must be one-half wavelength or more in diameter in order to support one or more transverse wave modes. In some waveguides, there may be a positive pressure internally present, allowing for the detection of potentially dangerous RF leaks. Another solution to detect RF leakage of a waveguide is to have a vacuum present inside. Then leaks can be detected in basically the same way. A slotted waveguide is generally used for radar and other similar applications. The waveguide structure has the capability of confining and supporting the energy of an electromagnetic wave to a specific relatively narrow and controllable path. In physics and electronics every Klystron has a waveguide to process and propagating the transmitted RF output. A closed waveguide is an electromagnetic waveguide (a) that is tubular, usually with a circular or rectangular cross section, (b) that has electrically conducting walls, (c) that may be hollow or filled with a dielectric material, (d) that can support a large number of discrete propagating modes, though only a few may be practical, (e) in which each discrete mode defines the propagation constant for that mode, (f) in which the field at any point is describable in terms of the supported modes, (g) in which there is no radiation field, and (h) in which discontinuities and bends cause mode conversion but not radiation.

Transmission lines

For low frequency and RF applications, a variety of two-conductor waveguides exist, including open two-wire waveguides, coaxial cable, and microstrip transmission lines. There is no lower frequency limit for two-conductor waveguides, and they can conduct electromagnetic energy all the way down to DC. Also, their diameter is not limited by wave propagation modes , and may be far narrower than one-half wavelength. A simplified analysis technique, transmission line analysis, can be used for many of these applications. Several waveguides based on entrainment of EM waves also exist. These can take the form of single conductors with or without a dielectric coating, e.g. the Goubou line and helical waveguides. The same principle is used in dielectric rod waveguides, linear arrays of short transverse conductors, planar resistive conductors, and dielectric rods. These function by slowing the EM wave velocity below the free space velocity, which continuously bends the EM wavefronts towards the waveguide, keeping them entrained. Helical waveguides and linear arrays of short conductors are used as part of "end-fire" antennas such as the helical antenna and Yagi antenna. Planar resistive waveguides are used in Over-The-Horizon radar, where the conductive surface of the ocean serves to entrain the waves. In digital computing, the term waveguide can also be used for data buffers used as delay lines that simulate physical waveguide behavior, such as in digital waveguide synthesis.

Dielectric waveguides

digital waveguide synthesis A dielectric waveguide is a waveguide that consists of a dielectric material surrounded by another dielectric material, such as air, glass, or plastic, with a lower refractive index. An example of a dielectric waveguide is an optical fiber. Paradoxically, a metallic waveguide filled with a dielectric material is not a dielectric waveguide.

External links

- [http://www2.slac.stanford.edu/vvc/accelerators/waveguide.html Waveguides in particle accelerators incorporating Klystons]

References

This article is based in part on material from Federal Standard 1037C and from MIL-STD-188, and ATIS
- J. J. Thomson, Recent Researches (1893).
- O. J. Lodge, Proc. Roy. Inst. 14, p. 321 (1894).
- Lord Rayleigh, Phil. Mag. 43, p. 125 (1897).
- N. W. McLachlan, Theory and Applications of Mathieu Functions, p. 8 (1947) (reprinted by Dover: New York, 1964). Category:Electrodynamics Category:Wave mechanics Category:Communication engineering Category:Microwave technology

Polarization mode dispersion

Polarization mode dispersion (PMD) is a form of modal dispersion where two different polarizations of light in a waveguide, which normally travel at the same speed, travel at different speeds due to random imperfections and asymmetries, causing random spreading of optical pulses. Unless it is compensated, which is difficult, this ultimately limits the rate at which data can be transmitted over a fiber.

Overview

In an ideal optical fiber, the core has a perfectly circular cross-section. In this case, the fundamental mode has two orthogonal polarizations (orientations of the electric field) that travel at the same speed. The signal that is transmitted over the fiber is randomly polarized, i.e. a random superposition of these two polarizations, but that would not matter in an ideal fiber because the two polarizations would propagate identically (are degenerate). In a realistic fiber, however, there are random imperfections that break the circular symmetry, causing the two polarizations to propagate with different speeds. In this case, the two polarization components of a signal will slowly separate, e.g. causing pulses to spread and overlap. Because the imperfections are random, the pulse spreading effects correspond to a random walk, and thus have a mean polarization-dependent time-differential

\Delta\tau

proportional to the square root of propagation distance

L

: :

\Delta\tau = D_\textrm \sqrt

D_\textrm

is the PMD parameter of the fiber, typically measured in ps/√km, a measure of the strength and frequency of the imperfections. The symmetry-breaking random imperfections fall into several categories. First, there is geometric asymmetry, e.g. slightly elliptical cores. Second, there are stress-induced material birefringences, in which the refractive index itself depends on the polarization. Both of these effects can stem from either imperfection in manufacturing (which is never perfect or stress-free) or from thermal and mechanical stresses imposed on the fiber in the field — moreover, the latter stresses generally vary over time.

Compensating for PMD

A PMD compensation system is a device which uses a polarization controller to compensate for PMD in fibers. Essentially, one splits the output of the fiber into two principal polarizations (usually those with

d\tau/d\omega = 0

, i.e. no first-order variation of time-delay with frequency), and applies a differential delay to bring them back into synch. Because the PMD effects are random and time-dependent, this requires an active device that responds to feedback over time. Such systems are therefore expensive and complex; combined with the fact that PMD is not yet the limiting factor in the lower data rates still in common use, this means that PMD-compensation systems have seen limited deployment in largescale telecommunications systems. Another alternative would be to use a polarization-maintaining fiber (PMF), a fiber whose symmetry is so strongly broken (e.g. a highly elliptical core) that an input polarization along a principal axis is maintained all the way to the output. Since the second polarization is never excited, PMD does not occur. Such fibers currently have practical problems, however, such as higher losses than ordinary optical fiber and higher cost. An extension of this idea is a single-polarization fiber in which only a single polarization state is allowed to propagate along the fiber (the other polarization is not guided and escapes).

Related Phenomena

A related effect is polarization-dependent loss (PDL), in which two polarizations suffer different rates of loss in the fiber due, again, to asymmetries. PDL similarly degrades signal quality. Strictly speaking, a circular core is not required in order to have two degenerate polarization states. Rather, one requires a core whose symmetry group admits a two-dimensional irreducible representation. For example, a square or equilateral-triangle core would also have two equal-speed polarization solutions for the fundamental mode; such general shapes also arise in photonic-crystal fibers. Again, any random imperfections that break the symmetry would lead to PMD in such a waveguide.

References

- Rajiv Ramaswami and Kumar N. Sivarajan, Optical Networks: A Practical Perspective (Harcourt: San Diego, 1998). Category:Polarization

Speed of light

. The effect is due to electrons moving faster than the speed at which light moves in water.]] The speed of light in a vacuum is defined to be exactly 299,792,458 metres per second (or 1,079,252,848.8 km/h, which is approximately 186,282.397 miles per second, or 670,616,629.4 miles per hour). This value is denoted by the letter c, reputedly from the Latin celeritas, "speed", and also known as Einstein's constant. Note that this speed is a definition, not a measurement; in fact, the fundamental SI unit of distance, the metre, is defined in terms of the speed of light: it is the distance light travels in a vacuum in 1/299,792,458 of a second. The speed of light through a transparent medium (that is, not in vacuum) is less than c; the ratio of c to this speed is called the refractive index of the medium. "Speed of light" is sometimes abbreviated SOL.

Overview

According to standard modern physical theory, all electromagnetic radiation, including visible light, propagates (or moves) at a constant speed in a vacuum, commonly known as the speed of light, which is a physical constant denoted as c. This speed c is also the speed of propagation of gravity in the theory of general relativity. One consequence of the laws of electromagnetism (such as Maxwell's equations) is that the speed c of electromagnetic radiation does not depend on the velocity of the object emitting the radiation; thus for instance the light emitted from a rapidly moving light source would travel at the same speed as the light coming from a stationary light source (although the colour, frequency, energy, and momentum of the light will be shifted, which is called the relativistic Doppler effect). If one combines this observation with the principle of relativity, one concludes that all observers will measure the speed of light in vacuum as being the same, regardless of the reference frame of the observer or the velocity of the object emitting the light. Because of this, one can view c as a fundamental physical constant. This fact can then be used as a basis for the theory of special relativity. It is worth noting that it is the constant speed c, rather than light itself, which is fundamental to special relativity; thus if light is somehow manipulated to travel at more or less than c, this will not directly affect the theory of special relativity. Observers travelling at large velocities will find that distances and times are distorted ("dilated") in accordance with the Lorentz transforms; however, the transforms distort times and distances in such a way that the speed of light remains constant. A person travelling near the speed of light would also find that colours of lights ahead were blue shifted and of those behind were red shifted. If information could travel faster than c in one reference frame, causality would be violated: in some other reference frames, the information would be received before it had been sent, so the 'cause' could be observed after the 'effect'. Due to special relativity's time dilation, the ratio between an external observer's perceived time and the time perceived by an observer moving closer and closer to the speed of light approaches zero. If something could move faster than light, this ratio would not be a real number. Such a violation of causality has never been observed. real number and those that are not.]] To put it another way, information propagates to and from a point from regions defined by a light cone. The interval AB in the diagram to the right is 'time-like' (that is, there is a frame of reference in which event A and event B occur at the same location in space, separated only by their occurring at different times, and if A precedes B in that frame then A precedes B in all frames: there is no frame of reference in which event A and event B occur simultaneously). Thus, it is hypothetically possible for matter (or information) to travel from A to B, so there can be a causal relationship (with A the 'cause' and B the 'effect'). On the other hand, the interval AC in the diagram to the right is 'space-like' (that is, there is a frame of reference in which event A and event C occur simultaneously, separated only in space; see simultaneity). However, there are also frames in which A precedes C (as shown) or in which C precedes A. Barring some way of travelling faster than light, it is not possible for any matter (or information) to travel from A to C or from C to A. Thus there is no causal connection between A and C. According to the currently prevailing definition, adopted in 1983, the speed of light is exactly 299,792,458 metres per second (approximately 3 × 10⁸ metres per second, or about thirty centimetres (one foot) per nanosecond). The value of

c

defines the permittivity of free space (

\epsilon_0

) in SI units as: :

\varepsilon_0 = 10^/4\pi c^2 \quad \mathrm

The permeability of free space (

\mu_0

) is not dependent on

c

and is defined in SI units as: :

\mu_0 = 4\,\pi\, 10^ \quad \mathrm

. These constants appear in Maxwell's equations, which describe electromagnetism, and are related by: :

c= \frac

Astronomical distances are sometimes measured in light years (the distance that light would travel in one year, roughly 9.46 × 10¹² kilometres or about 5.88 × 10¹² miles) especially in popularised texts.

Communications

The speed of light is of relevance to communications. For example, given that the equatorial circumference of the Earth is 40,075 km and

c

, the theoretical shortest amount of time for a piece of information to travel half the globe is 0.067 second. The actual transit time is longer, in part because the speed of light is slower by about 30% in an optical fibre and straight lines rarely occur in global communications situations, but also because delays are created when the signal passes through an electronic switch or signal regenerator. A typical time as of 2004 for an Australia or Japan to US computer-to-computer ping is 0.18 second. The speed of light additionally affects wireless communications design. The finite speed of light became quite apparent to everybody following the communication of Houston ground control and Neil Armstrong when he became the first man to set foot on the Moon: For every question, Houston had to wait nearly 3 seconds for the answer to arrive, and would have to do so even if the astronauts replied immediately. (See animation.) Similarly, instantaneous remote control of an interplanetary spacecraft is impossible, in the sense that the time it takes, for example, for the earth-based controllers to become aware of a problem, plus the time it takes for the spacecraft to receive their response, can be some hours. The speed of light can also be of concern on short distances. In supercomputers, the speed of light imposes a limit on how quickly data can be sent between processors. If a processor operates at 1 GHz, a signal can only travel a maximum of 300 mm in a single cycle. Processors must therefore be placed close to each other to minimise communication latencies. If clock frequencies continue to increase, the speed of light will eventually become a limiting factor for the internal design of single chips.

Physics

Constant velocity from all reference frames

It is important to realise that the speed of light is not a "speed limit" in the conventional sense. An observer chasing a beam of light will measure it moving away from him at the same speed as a stationary observer. This leads to some unusual consequences for velocities. Most individuals are accustomed to the addition rule of velocities: if two cars approach each other from opposite directions, each travelling at a speed of 50 kilometres per hour (31 miles per hour), one expects that each car will perceive the other as approaching at a combined speed of 50 + 50 = 100 km/h (62 mph) to a very high degree of accuracy. At velocities at or approaching the speed of light, however, it becomes clear from experimental results that this rule does not apply. Two spaceships approaching each other, each travelling at 90% the speed of light relative to some third observer between them, do not perceive each other as approaching at 90% + 90% = 180% the speed of light; instead they each perceive the other as approaching at slightly less than 99.5% the speed of light. This last result is given by the Einstein velocity addition formula: :

u =  \,\!

where v and w are the speeds of the spaceships as observed by the third observer, and u is the speed of either space ship as observed by the other. Contrary to one's usual intuitions, regardless of the speed at which one observer is moving relative to another observer, both will measure the speed of an incoming light beam as the same constant value, the speed of light. The above equation was derived by Albert Einstein from his theory of special relativity, which takes the principle of relativity as a main premise. This principle (originally proposed by Galileo Galilei) requires physical laws to act in the same way in all reference frames. As Maxwell's equations directly give a speed of light, it should be the same for every observer—a consequence which sounded obviously wrong to the 19th century physicists, who assumed that the speed of light given by Maxwell's theory is valid relative to the luminiferous aether. But the Michelson-Morley experiment, arguably the most famous and useful failed experiment in the history of physics, could not find this aether, suggesting instead that the speed of light is constant in all frames of reference. Although it is uncertain whether Einstein knew the results of the Michelson-Morley experiment, he took the speed of light being constant as a given fact, understood it as reaffirming Galilei's principle of relativity, and deduced the consequences, now known as the theory of special relativity which includes the counter-intuitive addition formula above.

Interaction with transparent materials

special relativity, as demonstrated by this prism (in the case of a prism splitting white light into a spectrum of colours, the refraction is known as dispersion).]] In passing through materials, light is slowed to less than c by the ratio called the refractive index of the material. The speed of light in air is only slightly less than

c

. Denser media, such as water and glass, can slow light much more, to fractions such as 3/4 and 2/3 of c. This reduction in speed is also responsible for bending of light at an interface between two materials with different indices, a phenomenon known as refraction. Since the speed of light in a material depends on the refractive index, and the refractive index depends on the frequency of the light, light at different frequencies travels at different speeds through the same material. This can cause distortion of electromagnetic waves that consist of multiple frequencies, called dispersion. Note that the speed of light referred to is the observed or measured speed in some medium and not the true speed of light (as observed in vacuum). On the microscopic scale, considering electromagnetic radiation to be like a particle, refraction is caused by continual absorption and re-emission of the photons that compose the light by the atoms or molecules through which it is passing. In some sense, the light itself travels only through the vacuum existing between these atoms, and is impeded by the atoms. The process of absorption and re-emission itself takes time thereby creating the impression that the light itself has undergone delay (i.e. loss of speed) between entry and exit from the medium in question. It may be noted, that once the light has emerged from the medium it changes back to its original speed and this is without gaining any energy. This can mean only one thing - that the light's speed itself was never altered in the first place. Alternatively, considering electromagnetic radiation to be like a wave, the charges of each atom (primarily the electrons) interfere with the electric and magnetic fields of the radiation, slowing its progress.

"Faster-than-light" observations and experiments

It has long been known theoretically that it is possible for the group velocity of light to exceed c. One recent experiment made the group velocity of laser beams travel for extremely short distances through caesium atoms at 300 times c. However, it is not possible to use this technique to transfer information faster than c: the velocity of information transfer depends on the front velocity (the speed at which the first rise of a pulse above zero moves forward) and the product of the group velocity and the front velocity is equal to the square of the normal speed of light in the material. Exceeding the group velocity of light in this manner is comparable to exceeding the speed of sound by arranging people in a distantly spaced line, and asking them all to shout "I'm here!", one after another with short intervals, each one timing it by looking at their own wristwatch so they don't have to wait until they hear the previous person shouting. The speed of light may also appear to be exceeded in some phenomena involving evanescent waves, such as tunnelling. Experiments indicate that the phase velocity of evanescent waves may exceed c; however, it would appear that neither the group velocity nor the front velocity exceed c, so, again, it is not possible for information to be transmitted faster than c. In some interpretations of quantum mechanics, quantum effects may be transmitted at speeds greater than c (indeed, action at a distance has long been perceived as a problem with quantum mechanics: see EPR paradox). For example, the quantum states of two particles can be entangled, so the state of one particle fixes the state of the other particle (say, one must have spin +½ and the other must have spin −½). Until the particles are observed, they exist in a superposition of two quantum states, (+½, −½) and (−½, +½). If the particles are separated and one of them is observed to determine its quantum state then the quantum state of the second particle is determined automatically. If, as in some interpretations of quantum mechanics, one presumes that the information about the quantum state is local to one particle, then one must conclude that second particle takes up its quantum state instantaneously, as soon as the first observation is carried out. However, it is impossible to control which quantum state the first particle will take on when it is observed, so no information can be transmitted in this manner. The laws of physics also appear to prevent information from being transferred through more clever ways and this has led to the formulation of rules such as the no-cloning theorem. So-called superluminal motion is also seen in certain astronomical objects, such as the jets of radio galaxies and quasars. However, these jets are not actually moving at speeds in excess of the speed of light: the apparent superluminal motion is a projection effect caused by objects moving near the speed of light and at a small angle to the line of sight. Although it may sound paradoxical, it is possible for shock waves to be formed with electromagnetic radiation. As a charged particle travels through an insulating medium, it disrupts the local electromagnetic field in the medium. Electrons in the atoms of the medium will be displaced and polarised by the passing field of the charged particle, and photons are emitted as the electrons in the medium restore themselves to equilibrium after the disruption has passed. (In a conductor, the disruption can be restored without emitting a photon.) In normal circumstances, these photons destructively interfere with each other and no radiation is detected. However, if the disruption travels faster than the photons themselves travel, the photons constructively interfere and intensify the observed radiation. The result (analogous to a sonic boom) is known as Cherenkov radiation. The ability to communicate or travel faster-than-light is a popular topic in science fiction. Particles that travel faster than light, dubbed tachyons, have been proposed by particle physicists but have yet to be observed. Some physicists, notably João Magueijo and John Moffat, have proposed that in the past light travelled much faster than the current speed of light. This theory is called variable speed of light (VSL) and its supporters claim that it has the ability to explain many cosmological puzzles better than its rival, the inflation model of the universe. However, it has yet to gain wide acceptance.

Light-slowing experiments

universe, are due to the slower speed of light in a medium (water, in this case).]] In a sense, any light travelling through a medium other than a vacuum travels below

c

as a result of refraction. However, certain materials have an exceptionally high refractive index: in particular, the optical density of a Bose-Einstein condensate can be very high. In 1999, a team of scientists led by Lene Hau were able to slow the speed of a light beam to about 17 metres per second, and, in 2001, they were able to momentarily stop a beam. In 2003, Mikhail Lukin, with scientists at Harvard University and the Lebedev Institute in Moscow, succeeded in completely halting light by directing it into a mass of hot rubidium gas, the atoms of which, in Lukin's words, behaved "like tiny mirrors", due to an interference pattern in two "control" beams.

History

Until relatively recent times, the speed of light was largely a matter of conjecture. Empedocles maintained that light was something in motion, and therefore there had to be some time elapsed in travelling. Aristotle said that, on the contrary, "light is due to the presence of something, but it is not a movement". Furthermore, if light had a finite speed, it would have to be very great; Aristotle asserted "the strain upon our powers of belief is too great" to believe this. One of the ancient theories of vision is that light is emitted from the eye, instead of being reflected into the eye from another source. On this theory, Heron of Alexandria advanced the argument that the speed of light must be infinite, since distant objects such as stars appear immediately when one opens one's eyes.

Medieval and early modern theories

The Islamic philosophers Avicenna and Alhazen believed that light has a finite speed, although most philosophers agreed with Aristotle on this point. The Aryan school of philosophy in ancient India also held the speed of light to be finite. The 14th century philosopher Sayana wrote the following comment on verse 1.50 of the Rig Veda: :"Thus it is remembered: [O Sun] you who traverse 2202 yojanas in half a nimesa." According to some, this refers to the speed of light. It is not known exactly how long a yojana and a nimesa is, but this value is possibly accurate to within 1% (Kak, 1998), though by adopting other possible values of these units the accuracy of the statement can be reduced to a factor of 4. Johannes Kepler believed that the speed of light is infinite since empty space presents no obstacle to it. Francis Bacon argued that the speed of light is not necessarily infinite, since something can travel too fast to be perceived. René Descartes argued that if the speed of light were finite, the Sun, Earth, and Moon would be noticeably out of alignment during a lunar eclipse. Since such misalignment had not been observed, Descartes concluded the speed of light is infinite. In fact, Descartes was convinced that if the speed of light were finite, his whole system of philosophy would be demolished.

Measurement of the speed of light

Isaac Beeckman, a friend of Descartes, proposed an experiment (1629) in which one would observe the flash of a cannon reflecting off a mirror about one mile away. Galileo proposed an experiment (1638), with an apparent claim to have performed it some years earlier, to measure the speed of light by observing the delay between uncovering a lantern and its perception some distance away. Descartes criticised this experiment as superfluous, in that the observation of eclipses, which had more power to detect a finite speed, gave a negative result. This experiment was carried out by the Accademia del Cimento of Florence in 1667, with the lanterns separated by about one mile. No delay was observed. Robert Hooke explained the negative results as Galileo had: by pointing out that such observations did not establish the infinite speed of light, but only that the speed must be very great. The first quantitative estimate of the speed of light was made in 1676 by Ole Rømer, who was studying the motions of Jupiter's satellite Io with a telescope. It is possible to time the revolution of Io because it is entering/exiting Jupiter's shadow at regular intervals. Rømer observed that Io revolved around Jupiter once every 42.5 hours when Earth was closest to Jupiter. He also observed that, as Earth and Jupiter moved apart, Io's exit from the shadow would begin progressively later than predicted. It was clear that these exit "signals" took longer to reach Earth, as Earth and Jupiter moved further apart, as a result of the extra time it took for light to cross the extra distance between the planets, which had accumulated in the interval between one signal and the next. Similarly, about half a year later, Io's entries into the shadow happened more frequently, as Earth and Jupiter were now drawing closer together. On the basis of his observations, Rømer estimated that it would take light 22 minutes to cross the diameter of the orbit of the Earth (that is, twice the astronomical unit); the modern estimate is closer to 16 minutes and 40 seconds. Around the same time, the astronomical unit was estimated to be about 140 million kilometres. The astronomical unit and Rømer's time estimate were combined by Christiaan Huygens, who estimated the speed of light to be 1000 Earth diameters per minute. This is about 220,000 kilometres per second (136,000 miles per second), well below the currently accepted value, but still very much faster than any physical phenomenon then known. Isaac Newton also accepted the finite speed. In his book "Opticks" he, in fact, reports the more accurate value of 16 minutes per diameter, which it seems he inferred for himself (whether from Rømer's data, or otherwise, is not known). The same effect was subsequently observed by Rømer for a "spot" rotating with the surface of Jupiter. And later observations also showed the effect with the three other Galilean moons, where it was more difficult to observe, thus laying to rest some further objections that had been raised. Even if, by these observations, the finite speed of light may not have been established to everyone's satisfaction (notably Jean-Dominique Cassini's), after the observations of James Bradley (1728), the hypothesis of infinite speed was considered discredited. Bradley deduced that starlight falling on the Earth should appear to come from a slight angle, which could be calculated by comparing the speed of the Earth in its orbit to the speed of light. This "aberration of light", as it is called, was observed to be about 1/200 of a degree. Bradley calculated the speed of light as about 185,000 miles per second (298,000 kilometres per second). This is only slightly less than the currently accepted value. The aberration effect has been studied extensively over the succeeding centuries, notably by Friedrich Georg Wilhelm Struve and Magnus Nyren. Magnus Nyren The first successful measurement of the speed of light using an earthbound apparatus was carried out by Hippolyte Fizeau in 1849. Fizeau's experiment was conceptually similar to those proposed by Beeckman and Galileo. A beam of light was directed at a mirror several thousand metres away. On the way from the source to the mirror, the beam passed through a rotating cog wheel. At a certain rate of rotation, the beam could pass through one gap on the way out and another on the way back. But at slightly higher or lower rates, the beam would strike a tooth and not pass through the wheel. Knowing the distance to the mirror, the number of teeth on the wheel, and the rate of rotation, the speed of light could be calculated. Fizeau reported the speed of light as 313,000 kilometres per second. Fizeau's method was later refined by Marie Alfred Cornu (1872) and Joseph Perrotin (1900). Leon Foucault improved on Fizeau's method by replacing the cogwheel with a rotating mirror. Foucault's estimate, published in 1862, was 298,000 kilometres per second. Foucault's method was also used by Simon Newcomb and Albert A. Michelson. Michelson began his lengthy career by replicating and improving on Foucault's method. In 1926, Michelson used rotating mirrors to measure the time it took light to make a round trip from Mount Wilson to Mount San Antonio in California. The precise measurements yielded a speed of 186,285 miles per second (299,796 kilometres per second).

Relativity

Due to the works of James Clerk Maxwell, it was known that the speed of electromagnetic radiation was a constant defined by the electromagnetic properties of the vacuum (permittivity and permeability). permeability, as used for the Michelson-Morley experiment.]] In 1887, the physicists Albert Michelson and Edward Morley performed the influential Michelson-Morley experiment to measure the speed of light relative to the motion of the earth, the goal being to measure the velocity of the Earth through the "luminiferous aether", the medium that was then thought to be necessary for the transmission of light. As shown in the diagram of a Michelson interferometer, a half-silvered mirror was used to split a beam of monochromatic light into two beams travelling at right angles to one another. After leaving the splitter, each beam was reflected back and forth between mirrors several times (the same number for each beam to give a long but equal path length; the actual Michelson-Morley experiment used more mirrors than shown) then recombined to produce a pattern of constructive and destructive interference. Any slight change in speed of light along each arm of the interferometer (due to the fact that the apparatus was moving with the Earth through the proposed "aether") would change the amount of time that the beam spent in transit, which would then be observed as a change in the pattern of interference. In the event, the experiment gave a null result. Ernst Mach was among the first physicists to suggest that the experiment actually amounted to a disproof of the aether theory. Developments in theoretical physics had already begun to provide an alternate theory, Fitzgerald-Lorentz contraction, which explained the null result of the experiment. It is uncertain whether Albert Einstein knew the results of the Michelson-Morley experiment, but the null result of the experiment greatly assisted the acceptance of his theory of relativity. Einstein's theory did not require an aether and was entirely consistent with the null result of the experiment: the aether did not exist and the speed of light was the same in each direction. The constant speed of light is one of the fundamental Postulates (together with causality and the equivalence of inertial frames) of special relativity.

References

Historical references

- Ole Rømer. "Démonstration touchant le mouvement de la lumière", Journal des Sçavans, 7 Décembre 1676, pp. 223-236. Translated as "A Demonstration concerning the Motion of Light", Philosophical Transactions of the Royal Society no. 136, pp. 893-894; June 25, 1677. (Rømer's 1676 paper, in English and French, as bitmap images: [http://dbhs.wvusd.k12.ca.us/webdocs/Chem-History/Roemer-1677/Roemer-1677.html], and in French as plain text: [http://astro.campus.ecp.fr/histoire/roemer.html])
- Edmund Halley. "Monsieur Cassini, his New and Exact Tables for the Eclipses of the First Satellite of Jupiter, reduced to the Julian Stile and Meridian of London", Philosophical Transactions XVIII, No. 214, pp 237–256, Nov.–Dec., 1694.
- H.L. Fizeau. "Sur une experience relative a la vitesse de propogation de la lumiere", Comptes Rendus 29, 90–92, 132, 1849.
- J.L. Foucault. "Determination experimentale de la vitesse de la lumiere: parallaxe du Soleil", Comptes Rendus 55, 501–503, 792–796, 1862.
- A.A. Michelson. "Experimental Determination of the Velocity of Light", Proceedings of the American Association for the Advancement of Science 27, 71–77, 1878. ([http://www.gutenberg.org/etext/11753 Project Gutenberg Etext])
- Simon Newcomb. "The Velocity of Light", Nature, pp 29–32, May 13, 1886.
- Joseph Perrotin. "Sur la vitesse de la lumiere", Comptes Rendus 131, 731–734, 1900.
- A.A. Michelson, F.G. Pease, and F. Pearson. "Measurement Of The Velocity Of Light In A Partial Vacuum", Astrophysical Journal 82, 26–61, 1935.

Modern references

- Léon Brillouin. Wave propagation and group velocity. Academic Press Inc., 1960.
- John David Jackson. Classical electrodynamics. John Wiley & Sons, 2nd edition, 1975; 3rd edition, 1998. ISBN 047130932X
- Subhash Kak. The Speed of Light and Purāṇic Cosmology. In T.R.N. Rao and S. Kak, Computing Science in Ancient India, pages 80–90. USL Press, Lafayette, 1998. Available as [http://uk.arxiv.org/abs/physics/9804020 e-print physics/9804020] on the arXiv.
- R.J. MacKay and R.W. Oldford. "Scientific Method, Statistical Method and the Speed of Light", Statistical Science 15(3):254–278, 2000. (Also available on line: [http://www.stats.uwaterloo.ca/~rwoldfor/papers/sci-method/paperrev])

External links

- [http://physics.nist.gov/cgi-bin/cuu/Value?c speed of light in vacuum] (at NIST)
- [http://www.ldolphin.org/chistory.html A Brief History of c]
- [http://www.itl.nist.gov/div898/bayesian/datagall/michelso.htm Data Gallery: Michelson Speed of Light (Univariate Location Estimation)] (download data gathered by A.A. Michelson)
- [http://physicsweb.org/article/news/7/12/5 Switching light on and off] (news article on stopping light)
- [http://news.bbc.co.uk/hi/english/sci/tech/newsid_841000/841690.stm Beam smashes light barrier] (news article on group velocity experiment)
- [http://www.netspace.net.au/~gregegan/APPLETS/20/20.html Subluminal] (Java applet demonstrating group velocity information limits)
- [http://www.ertin.com/sloan_on_speed_of_light.html Light discussion on adding velocities] Category:Electromagnetic radiation Category:Units of velocity Category:Special relativity als:Lichtgeschwindigkeit ko:빛의 속도 ms:Kelajuan cahaya ja:光速度 simple:Speed of light th:อัตราเร็วของแสง

Refractive index

The refractive index of a material is the factor by which the phase velocity of electromagnetic radiation is slowed relative to vacuum. It is usually given the symbol n, and defined for a material by: :

n=\sqrt

where ε_r is the material's relative permittivity, and μ_r is its relative permeability. For a non-magnetic material, μ_r is very close to 1, therefore n is approximately

\sqrt

. The phase velocity is defined as the rate at which the crests of the waveform propagate; that is, the rate at which the phase of the waveform is moving. The group velocity is the rate that the envelope of the waveform is propagating; that is, the rate of variation of the amplitude of the waveform. It is the group velocity that (almost always) represents the rate that information (and energy) may be transmitted by the wave, for example the velocity at which a pulse of light travels down an optical fiber.

The speed of light

The speed of all electromagnetic radiation in vacuum is the same, approximately 3×10⁸ meters per second, and is denoted by c. Therefore, if v is the phase velocity of radiation of a specific frequency in a specific material, the refractive index is given by :

n =\frac

This number is typically greater than one: the higher the index of the material, the more the light is slowed down. However, at certain frequencies (e.g. near absorption resonances, and for x-rays), n will actually be smaller than one. This does not contradict the theory of relativity, which holds that no information-carrying signal can ever propagate faster than c, because the phase velocity is not the same as the group velocity or the signal velocity. Sometimes, a "group velocity refractive index", usually called the group index is defined: :

n_g=\frac

, where v_g is the group velocity. This value should not be confused with n, which is always defined with respect to the phase velocity. At the microscale, an electromagnetic wave's phase velocity is slowed in a material because the electric field creates a disturbance in the charges of each atom (primarily the electrons) proportional to the permittivity. The charges will, in general, oscillate slightly out of phase with respect to the driving electric field. The charges thus radiate their own electromagnetic wave that is at the same frequency but with a phase delay. The macroscopic sum of all such contributions in the material is a wave with the same frequency but shorter wavelength than the original, leading to a slowing of the wave's phase velocity. Most of the radiation from oscillating material charges will modify the incoming wave, changing its velocity. However, some net energy will be radiated in other directions (see scattering). If the refractive indices of two materials are known for a given frequency, then one can compute the angle by which radiation of that frequency will be refracted as it moves from the first into the second material from Snell's law. Recent research has also demonstrated the existence of negative refractive index which can occur if ε and μ are simultaneously negative. Not thought to occur naturally, this can be achieved with so called metamaterials and offers the possibility of perfect lenses and other exotic phenomena such as a reversal of Snell's law.

Dispersion and Absorption

In real materials, the polarization does not respond instantaneously to an applied field. This causes dielectric loss, which can be expressed by a permittivity that is both complex and frequency dependent. Real materials are not perfect insulators either, i.e. they have non-zero direct current conductivity. Taking both aspects into consideration, we can define a complex index of refraction: :

\tilde=n-i\kappa

Here, n is the refractive index indicating the phase velocity as above, while κ is called the extinction coefficient, which indicates the amount of absorption loss when the electromagnetic wave propagates through the material. Both n and κ are dependent on the frequency (wavelength). The effect that n varies with frequency (except in vacuum, where all frequencies travel at the same speed, c) is known as dispersion, and it is what causes a prism to divide white light into its constituent spectral colors, explains rainbows, and is the cause of chromatic aberration in lenses. In regions of the spectrum where the material does not absorb, the of the refractive index tends to increase with frequency. Near absorption peaks, the curve of the refractive index is a complex form given by the Kramers-Kronig relations, and can decreases with frequency. Since the refractive index of a material varies with the frequency (and thus wavelength) of light, it is usual to specify the corresponding vacuum wavelength at which the refractive index is measured. Typically, this is done at various well-defined spectral emission lines; for example, n_D is the refractive index at the Fraunhofer "D" line, the centre of the yellow sodium doublet emission at 589.29 nm wavelength. The Sellmeier equation is an empirical formula that works well in describing dispersion, and Sellmeier coefficients are often quoted instead of the refractive index in tables. For some representative refractive indices at different wavelengths, see list of indices of refraction. As shown above, dielectric loss and non-zero DC conductivity in materials cause absorption. Good dielectric materials such as glass have extremely low DC conductivity, and at low frequencies the dielectric loss is also negligible, resulting in almost no absorption (κ ≈ 0). However, at higher frequencies (such as visible light), dielectric loss may increase absorption significantly, reducing the material's transparency to these frequencies. The real and imaginary parts of the complex refractive index are related through use of the Kramers-Kronig relations. For example, one can determine a material's full complex refractive index as a function of wavelength from an absorption spectrum of the material.

Anisotropy

The refractive index of certain media may be different depending on the polarization and direction of propagation of the light through the medium. This is known as birefringence or anisotropy and is described by the field of crystal optics. In the most general case, the dielectric constant is a rank-2 tensor (a 3 by 3 matrix), which cannot simply be described by refractive indices except for polarizations along principal axes. In magneto-optic (gyro-magnetic) and optically active materials, the principal axes are complex (corresponding to elliptical polarizations), and the dielectric tensor is complex-Hermitian (for lossless media); such materials break time-reversal symmetry and are used e.g. to construct Faraday isolators.

Nonlinearity

The strong electric field of high intensity light (such as output of a laser) may cause a medium's refractive index to vary as the light passes through it, giving rise to nonlinear optics. If the index varies quadratically with the field (linearly with the intensity), it is called the optical Kerr effect and causes phenomena such as self-focusing and self phase modulation. If the index varies linearly with the field (which is only possible in materials that do not possess inversion symmetry), it is known as the Pockels effect.

Inhomogeneity

If the refractive index of a medium is not constant, but varies gradually with position, the material is known as a gradient-index medium and is described by gradient index optics. Light travelling through such a medium can be bent or focussed, and this effect can be exploited to produce lenses, some optical fibers and other devices. Some common mirages are caused by a spatially-varying refractive index of air.

Applications

The refractive index of a material is the most important property of any optical system that uses refraction. It is used to calculate the focusing power of lenses, and the dispersive power of prisms. For a solution of sugar, the refractive index can be used to determine the sugar content (see Brix). The refractive index is also used in chemistry to determine the purity of chemicals.

External link

- [http://www.tf.uni-kiel.de/matwis/amat/elmat_en/index.html Dielectric materials]

Wavelength

:For the album by Van Morrison, see Wavelength (album). The wavelength is the distance between repeating units of a wave pattern. It is commonly designated by the Greek letter lambda (λ). In a sine wave, the wavelength is the distance between the midpoints of the wave: Image:Wavelength.png The x axis represents distance, and I would be some varying quantity at a given point in time as a function of x, for instance air pressure for a sound wave or strength of the electric or magnetic field for light. Wavelength λ has an inverse relationship to frequency f, the number of peaks to pass a point in a given time. The wavelength is equal to the speed of the wave type divided by the frequency of the wave. When dealing with electromagnetic radiation in a vacuum, this speed is the speed of light c, for signals (waves) in air, this is the speed of sound in air. The relationship is given by: :

\lambda = \frac

where: :λ = wavelength of a sound wave or electromagnetic wave :c = speed of light in vacuum = 299,792.458 km/s ~ 300,000 km/s = 300,000,000 m/s or :c = speed of sound in air = 343 m/s at 20 °C (68 °F) :f = frequency of the wave in 1/s = Hz For radio waves this relationship is approximated with the formula: wavelength λ (in metres) = 300 / frequency (in megahertz).
For sound waves this relationship is approximated with the formula: wavelength λ (in metres) = 333 / frequency (in hertz). When light waves (and other electromagnetic waves) enter a medium, their wavelength is reduced by a factor equal to the refractive index n of the medium but the frequency of the wave is unchanged. The wavelength of the wave in the medium, λ' is given by: :

\lambda^\prime = \frac

where: :λ₀ is the vacuum wavelength of the wave Wavelengths of electromagnetic radiation, no matter what mediu

Dispersion (optics)

Material dispersion in optics

Group and phase velocity

Dispersion in gemology

Related topics

External links

Optics

Classical optics

Topics related to classical optics

Modern optics

Topics related to modern optics

Other optical fields

Everyday optics

Wikibooks modules

See also

References

External links

Frequency

Measurement

Frequency of waves

Invariance

Examples

See also

External links

Sound

Attributes of sound

Frequency and wavelength

Amplitude

Velocity

Types of sounds

Perception of sound

See also

Sound measurement

Reference

External links

Waveguide

History

Principles of operation

Hollow metallic waveguides

Transmission lines

Dielectric waveguides

See also

External links

References

Polarization mode dispersion

Overview

Compensating for PMD

Related Phenomena

References

Speed of light

Overview

Communications

Physics

Constant velocity from all reference frames

Interaction with transparent materials

"Faster-than-light" observations and experiments

Light-slowing experiments

History

Medieval and early modern theories

Measurement of the speed of light

Relativity

See also

References

Historical references

Modern references

External links

Refractive index

The speed of light

Dispersion and Absorption

Anisotropy

Nonlinearity

Inhomogeneity

Applications

External link

See also

Wavelength