Colors of noise

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Colors_of_noise#Blue_noise

 

Colors of noise
White
Pink
Red (Brownian)
Purple
Grey

In audio engineering, electronics, physics, and many other fields, the color of noise refers to the power spectrum of a noise signal (a signal produced by a stochastic process). Different colors of noise have significantly different properties. For example, as audio signals they will sound different to human ears, and as images they will have a visibly different texture. Therefore, each application typically requires noise of a specific color. This sense of 'color' for noise signals is similar to the concept of timbre in music (which is also called "tone color"); however, the latter is almost always used for sound, and may consider very detailed features of the spectrum.

The practice of naming kinds of noise after colors started with white noise, a signal whose spectrum has equal power within any equal interval of frequencies. That name was given by analogy with white light, which was (incorrectly) assumed to have such a flat power spectrum over the visible range. Other color names, such as pink, red, and blue were then given to noise with other spectral profiles, often (but not always) in reference to the color of light with similar spectra. Some of those names have standard definitions in certain disciplines, while others are very informal and poorly defined. Many of these definitions assume a signal with components at all frequencies, with a power spectral density per unit of bandwidth proportional to 1/f ^β and hence they are examples of power-law noise. For instance, the spectral density of white noise is flat (β = 0), while flicker or pink noise has β = 1, and Brownian noise has β = 2.

Simulated power spectral densities as a function of frequency for various colors of noise (violet, blue, white, pink, brown/red). The power spectral densities are arbitrarily normalized such that the value of the spectra are approximately equivalent near 1 kHz. Note the slope of the power spectral density for each spectrum provides the context for the respective electromagnetic/color analogy.

Technical definitions

Various noise models are employed in analysis, many of which fall under the above categories. AR noise or "autoregressive noise" is such a model, and generates simple examples of the above noise types, and more. The Federal Standard 1037C Telecommunications Glossary defines white, pink, blue, and black.

The color names for these different types of sounds are derived from a loose analogy between the spectrum of frequencies of sound wave present in the sound (as shown in the blue diagrams) and the equivalent spectrum of light wave frequencies. That is, if the sound wave pattern of "blue noise" were translated into light waves, the resulting light would be blue, and so on.

White noise

Main article: White noise

 

White noise spectrum. Flat power spectrum.  (logarithmic frequency axis).

White noise is a signal (or process), named by analogy to white light, with a flat frequency spectrum when plotted as a linear function of frequency (e.g., in Hz). In other words, the signal has equal power in any band of a given bandwidth (power spectral density) when the bandwidth is measured in Hz. For example, with a white noise audio signal, the range of frequencies between 40 Hz and 60 Hz contains the same amount of sound power as the range between 400 Hz and 420 Hz, since both intervals are 20 Hz wide. Note that spectra are often plotted with a logarithmic frequency axis rather than a linear one, in which case equal physical widths on the printed or displayed plot do not all have the same bandwidth, with the same physical width covering more Hz at higher frequencies than at lower frequencies. In this case a white noise spectrum that is equally sampled in the logarithm of frequency (i.e., equally sampled on the X axis) will slope upwards at higher frequencies rather than being flat. However it is not unusual in practice for spectra to be calculated using linearly-spaced frequency samples but plotted on a logarithmic frequency axis, potentially leading to misunderstandings and confusion if the distinction between equally spaced linear frequency samples and equally spaced logarithmic frequency samples is not kept in mind.

Pink noise

Main article: Pink noise

 

Pink noise spectrum. Power density falls off at 10 dB/decade (−3.01 dB/octave).

The frequency spectrum of pink noise is linear in logarithmic scale; it has equal power in bands that are proportionally wide. This means that pink noise would have equal power in the frequency range from 40 to 60 Hz as in the band from 4000 to 6000 Hz. Since humans hear in such a proportional space, where a doubling of frequency (an octave) is perceived the same regardless of actual frequency (40–60 Hz is heard as the same interval and distance as 4000–6000 Hz), every octave contains the same amount of energy and thus pink noise is often used as a reference signal in audio engineering. The spectral power density, compared with white noise, decreases by 3.01 dB per octave (density proportional to 1/f ). For this reason, pink noise is often called "1/f noise".

Since there are an infinite number of logarithmic bands at both the low frequency (DC) and high frequency ends of the spectrum, any finite energy spectrum must have less energy than pink noise at both ends. Pink noise is the only power-law spectral density that has this property: all steeper power-law spectra are finite if integrated to the high-frequency end, and all flatter power-law spectra are finite if integrated to the DC, low-frequency limit.

Brownian noise

Main article: Brownian noise

 

Brown spectrum (−6.02 dB/octave)

Brownian noise, also called Brown noise, is noise with a power density which decreases 6.02 dB per octave with increasing frequency (frequency density proportional to 1/f²) over a frequency range excluding zero (DC).

Brownian noise can be generated with temporal integration of white noise. "Brown" noise is not named for a power spectrum that suggests the color brown; rather, the name derives from Brownian motion. Also known as "random walk" or "drunkard's walk". "Red noise" describes the shape of the power spectrum, with pink being between red and white.

Blue noise

Blue spectrum (+3.01 dB/octave)

Blue noise is also called azure noise. Blue noise's power density increases 3.01 dB per octave with increasing frequency (density proportional to f ) over a finite frequency range. In computer graphics, the term "blue noise" is sometimes used more loosely as any noise with minimal low frequency components and no concentrated spikes in energy. This can be good noise for dithering. Retinal cells are arranged in a blue-noise-like pattern which yields good visual resolution.

Cherenkov radiation is a naturally occurring example of almost perfect blue noise, with the power density growing linearly with frequency over spectrum regions where the permeability of index of refraction of the medium are approximately constant. The exact density spectrum is given by the Frank–Tamm formula. In this case, the finiteness of the frequency range comes from the finiteness of the range over which a material can have a refractive index greater than unity. Cherenkov radiation also appears as a bright blue color, for these reasons.

Violet noise

Violet spectrum (+6.02 dB/octave)

Violet noise is also called purple noise. Violet noise's power density increases 6.02 dB per octave with increasing frequency (density proportional to f ²) over a finite frequency range. It is also known as differentiated white noise, due to its being the result of the differentiation of a white noise signal.

Due to the diminished sensitivity of the human ear to high-frequency hiss and the ease with which white noise can be electronically differentiated (high-pass filtered at first order), many early adaptations of dither to digital audio used violet noise as the dither signal.

Acoustic thermal noise of water has a violet spectrum, causing it to dominate hydrophone measurements at high frequencies.

Grey noise

Main article: Grey noise

 

Grey spectrum

Grey noise is random white noise subjected to a psychoacoustic equal loudness curve (such as an inverted A-weighting curve) over a given range of frequencies, giving the listener the perception that it is equally loud at all frequencies. This is in contrast to standard white noise which has equal strength over a linear scale of frequencies but is not perceived as being equally loud due to biases in the human equal-loudness contour.

Informal definitions

There are also many colors used without precise definitions (or as synonyms for formally defined colors), sometimes with multiple definitions.

Red noise

• A synonym for Brownian noise, as above
• Similar to pink noise, but with different spectral content and different relationships (i.e. 1/f for pink noise, while 1/f² for red noise).
• In areas where terminology is used loosely, "red noise" may refer to any system where power density decreases with increasing frequency.

Green noise

• The mid-frequency component of white noise, used in halftone dithering
• Bounded Brownian noise
• Vocal spectrum noise used for testing audio circuits

• Joseph S. Wisniewski writes that "green noise", is marketed by producers of ambient sound effects recordings as "the background noise of the world". It simulates the spectra of natural settings, without human-made noises. Pink noise is similar, but has more energy in the area of 500 Hz.

Black noise

• Silence
• Noise with a 1/f^β spectrum, where β > 2. This formula is used to model the frequency of natural disasters.
• Noise that has a frequency spectrum of predominantly zero power level over all frequencies except for a few narrow bands or spikes. Note: An example of black noise in a facsimile transmission system is the spectrum that might be obtained when scanning a black area in which there are a few random white spots. Thus, in the time domain, a few random pulses occur while scanning.
• Noise with a spectrum corresponding to the blackbody radiation (thermal noise). For temperatures higher than about 3×10⁻⁷ K the peak of the blackbody spectrum is above the upper limit of human hearing range. In those situations, for the purposes of what is heard, black noise is well approximated as violet noise. At the same time, Hawking radiation of black holes may have a peak in hearing range, so the radiation of a typical stellar black hole with a mass equal to 6 solar masses will have a maximum at a frequency of 604.5 Hz - this noise is similar to green noise. A formula is: ${\displaystyle f_{\text{max}}\approx 3627\times {{\text{M}}_{\odot } \over {\text{M}}}}$ Hz. Several examples of audio files with this spectrum can be found here.

Noisy white

In telecommunication, the term noisy white has the following meanings:

• In facsimile or display systems, such as television, a nonuniformity in the white area of the image, i.e., document or picture, caused by the presence of noise in the received signal.
• A signal or signal level that is supposed to represent a white area on the object, but has a noise content sufficient to cause the creation of noticeable black spots on the display surface or record medium.

Noisy black

In telecommunication, the term noisy black has the following meanings:

• In facsimile or display systems, such as television, a nonuniformity in the black area of the image, i.e., document or picture, caused by the presence of noise in the received signal.
• A signal or signal level that is supposed to represent a black area on the object, but has a noise content sufficient to cause the creation of noticeable non-black spots on the display surface or record medium.

Tachyon

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Tachyon

 

Tachyon

Because a tachyon would always travel faster than light, it would not be possible to see it approaching. After a tachyon has passed nearby, an observer would be able to see two images of it, appearing and departing in opposite directions. This double-image effect is most prominent for an observer located directly in the path of a superluminal object (in this example it is a sphere, shown in transparent grey). Because the tachyon arrives before the light, the observer sees nothing until the sphere has already passed, after which (from the observer's perspective) the image appears to split into two—one of the arriving sphere (to the right) and one of the departing sphere (to the left).
Classification	Elementary particles
Status	hypothetical
Theorized	1967

A tachyon (/ˈtækiɒn/) or tachyonic particle is a hypothetical particle that always travels faster than light. Most physicists believe that faster-than-light particles cannot exist because they are not consistent with the known laws of physics. If such particles did exist, and could send signals faster than light, then according to the theory of relativity they would violate causality, leading to logical paradoxes such as the grandfather paradox. Tachyons would also exhibit the unusual property of increasing in speed as their energy decreases, and would require infinite energy to slow down to the speed of light. No experimental evidence for the existence of such particles has been found.

In the 1967 paper that coined the term, Gerald Feinberg proposed that tachyonic particles could be made from excitations of a quantum field with imaginary mass. However, it was soon realized that Feinberg's model did not in fact allow for superluminal (faster-than-light) speeds. Nevertheless, in modern physics the term tachyon often refers to imaginary mass fields rather than to faster-than-light particles. Such fields have come to play a significant role in modern physics.

The term comes from the Greek: ταχύ, tachy, meaning swift. The complementary particle types are called luxons (which always move at the speed of light) and bradyons (which always move slower than light); both of these particle types are known to exist.

History

The term tachyon was coined by Gerald Feinberg in a 1967 paper titled "Possibility of faster-than-light particles". He had been inspired by the science-fiction story "Beep" by James Blish. Feinberg studied the kinematics of such particles according to special relativity. In his paper he also introduced fields with imaginary mass (now also referred to as tachyons) in an attempt to understand the microphysical origin such particles might have.

The first hypothesis regarding faster-than-light particles is sometimes attributed to German physicist Arnold Sommerfeld in 1904, who named them "meta-particles". More recent discussions happened in 1962 and 1969.

The possibility of existence of faster-than-light particles was also proposed by Lev Yakovlevich Shtrum in 1923 (see https://arxiv.org/abs/2107.10739 and references therein).

In September 2011, it was reported that a tau neutrino had traveled faster than the speed of light in a major release by CERN; however, later updates from CERN on the OPERA project indicate that the faster-than-light readings were due to a faulty element of the experiment's fibre optic timing system.

Tachyons in relativity

In special relativity, a faster-than-light particle would have space-like four-momentum, in contrast to ordinary particles that have time-like four-momentum. Although in some theories the mass of tachyons is regarded as imaginary, in some modern formulations the mass is considered real, the formulas for the momentum and energy being redefined to this end. Moreover, since tachyons are constrained to the spacelike portion of the energy–momentum graph, they could not slow down to subluminal (meaning slower-than-light) speeds.

Mass

Main articles: Mass § Tachyonic particles and imaginary (complex) mass, and Tachyonic field

In a Lorentz invariant theory, the same formulas that apply to ordinary slower-than-light particles (sometimes called "bradyons" in discussions of tachyons) must also apply to tachyons. In particular, the energy–momentum relation:

${\displaystyle E^{2}=(pc)^{2}+(mc^{2})^{2}\;}$

(where p is the relativistic momentum of the bradyon and m is its rest mass) should still apply, along with the formula for the total energy of a particle:

${\displaystyle E={\frac {mc^{2}}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}.}$

This equation shows that the total energy of a particle (bradyon or tachyon) contains a contribution from its rest mass (the "rest mass–energy") and a contribution from its motion, the kinetic energy. When ${\displaystyle v}$ (the particle's velocity) is larger than ${\displaystyle c}$ (the speed of light), the denominator in the equation for the energy is imaginary, as the value under the square root is negative. Because the total energy of the particle must be real (and not a complex or imaginary number) in order to have any practical meaning as a measurement, the numerator must also be imaginary: i.e. the rest mass m must be imaginary, as a pure imaginary number divided by another pure imaginary number is a real number.

In some modern formulations of the theory, the mass of tachyons is regarded as real.

Speed

One curious effect is that, unlike ordinary particles, the speed of a tachyon increases as its energy decreases. In particular, ${\displaystyle E}$ approaches zero when ${\displaystyle v}$ approaches infinity. (For ordinary bradyonic matter, ${\displaystyle E}$ increases with increasing speed, becoming arbitrarily large as ${\displaystyle v}$ approaches ${\displaystyle c}$, the speed of light). Therefore, just as bradyons are forbidden to break the light-speed barrier, so too are tachyons forbidden from slowing down to below c, because infinite energy is required to reach the barrier from either above or below.

As noted by Albert Einstein, Tolman, and others, special relativity implies that faster-than-light particles, if they existed, could be used to communicate backwards in time.

Neutrinos

In 1985, Chodos proposed that neutrinos can have a tachyonic nature. The possibility of standard model particles moving at faster-than-light speeds can be modeled using Lorentz invariance violating terms, for example in the Standard-Model Extension. In this framework, neutrinos experience Lorentz-violating oscillations and can travel faster than light at high energies. This proposal was strongly criticized.

Cherenkov radiation

A tachyon with an electric charge would lose energy as Cherenkov radiation—just as ordinary charged particles do when they exceed the local speed of light in a medium (other than a hard vacuum). A charged tachyon traveling in a vacuum, therefore, undergoes a constant proper time acceleration and, by necessity, its world line forms a hyperbola in space-time. However reducing a tachyon's energy increases its speed, so that the single hyperbola formed is of two oppositely charged tachyons with opposite momenta (same magnitude, opposite sign) which annihilate each other when they simultaneously reach infinite speed at the same place in space. (At infinite speed, the two tachyons have no energy each and finite momentum of opposite direction, so no conservation laws are violated in their mutual annihilation. The time of annihilation is frame dependent.)

Even an electrically neutral tachyon would be expected to lose energy via gravitational Cherenkov radiation (unless gravitons are themselves tachyons), because it has a gravitational mass, and therefore increases in speed as it travels, as described above. If the tachyon interacts with any other particles, it can also radiate Cherenkov energy into those particles. Neutrinos interact with the other particles of the Standard Model, and Andrew Cohen and Sheldon Glashow used this to argue that the 2011 faster-than-light neutrino anomaly cannot be explained by making neutrinos propagate faster than light, and must instead be due to an error in the experiment. Further investigation of the experiment showed that the results were indeed erroneous.

Causality

Spacetime diagram showing that moving faster than light implies time travel in the context of special relativity. A spaceship departs from Earth from A to C slower than light. At B, Earth emits a tachyon, which travels faster than light but forward in time in Earth's reference frame. It reaches the spaceship at C. The spaceship then sends another tachyon back to Earth from C to D. This tachyon also travels forward in time in the spaceship's reference frame. This effectively allows Earth to send a signal from B to D, back in time.

Causality is a fundamental principle of physics. If tachyons can transmit information faster than light, then according to relativity they violate causality, leading to logical paradoxes of the "kill your own grandfather" type. This is often illustrated with thought experiments such as the "tachyon telephone paradox" or "logically pernicious self-inhibitor."

The problem can be understood in terms of the relativity of simultaneity in special relativity, which says that different inertial reference frames will disagree on whether two events at different locations happened "at the same time" or not, and they can also disagree on the order of the two events (technically, these disagreements occur when the spacetime interval between the events is 'space-like', meaning that neither event lies in the future light cone of the other).

If one of the two events represents the sending of a signal from one location and the second event represents the reception of the same signal at another location, then as long as the signal is moving at the speed of light or slower, the mathematics of simultaneity ensures that all reference frames agree that the transmission-event happened before the reception-event. However, in the case of a hypothetical signal moving faster than light, there would always be some frames in which the signal was received before it was sent so that the signal could be said to have moved backward in time. Because one of the two fundamental postulates of special relativity says that the laws of physics should work the same way in every inertial frame, if it is possible for signals to move backward in time in any one frame, it must be possible in all frames. This means that if observer A sends a signal to observer B which moves faster than light in A's frame but backwards in time in B's frame, and then B sends a reply which moves faster than light in B's frame but backwards in time in A's frame, it could work out that A receives the reply before sending the original signal, challenging causality in every frame and opening the door to severe logical paradoxes. This is known as the tachyonic antitelephone.

Reinterpretation principle

The reinterpretation principle asserts that a tachyon sent back in time can always be reinterpreted as a tachyon traveling forward in time, because observers cannot distinguish between the emission and absorption of tachyons. The attempt to detect a tachyon from the future (and violate causality) would actually create the same tachyon and send it forward in time (which is causal).

However, this principle is not widely accepted as resolving the paradoxes. Instead, what would be required to avoid paradoxes is that unlike any known particle, tachyons do not interact in any way and can never be detected or observed, because otherwise a tachyon beam could be modulated and used to create an anti-telephone or a "logically pernicious self-inhibitor". All forms of energy are believed to interact at least gravitationally, and many authors state that superluminal propagation in Lorentz invariant theories always leads to causal paradoxes.

Fundamental models

In modern physics, all fundamental particles are regarded as excitations of quantum fields. There are several distinct ways in which tachyonic particles could be embedded into a field theory.

Fields with imaginary mass

Main article: Tachyonic field

In the paper that coined the term "tachyon", Gerald Feinberg studied Lorentz invariant quantum fields with imaginary mass. Because the group velocity for such a field is superluminal, naively it appears that its excitations propagate faster than light. However, it was quickly understood that the superluminal group velocity does not correspond to the speed of propagation of any localized excitation (like a particle). Instead, the negative mass represents an instability to tachyon condensation, and all excitations of the field propagate subluminally and are consistent with causality. Despite having no faster-than-light propagation, such fields are referred to simply as "tachyons" in many sources.

Tachyonic fields play an important role in modern physics. Perhaps the most famous is the Higgs boson of the Standard Model of particle physics, which has an imaginary mass in its uncondensed phase. In general, the phenomenon of spontaneous symmetry breaking, which is closely related to tachyon condensation, plays an important role in many aspects of theoretical physics, including the Ginzburg–Landau and BCS theories of superconductivity. Another example of a tachyonic field is the tachyon of bosonic string theory.

Tachyons are predicted by bosonic string theory and also the Neveu-Schwarz (NS) and NS-NS sectors, which are respectively the open bosonic sector and closed bosonic sector, of RNS Superstring theory prior to the GSO projection. However such tachyons are not possible due to the Sen conjecture, also known as tachyon condensation. This resulted in the necessity for the GSO projection.

Lorentz-violating theories

In theories that do not respect Lorentz invariance, the speed of light is not (necessarily) a barrier, and particles can travel faster than the speed of light without infinite energy or causal paradoxes. A class of field theories of that type is the so-called Standard Model extensions. However, the experimental evidence for Lorentz invariance is extremely good, so such theories are very tightly constrained.

Fields with non-canonical kinetic term

By modifying the kinetic energy of the field, it is possible to produce Lorentz invariant field theories with excitations that propagate superluminally. However, such theories, in general, do not have a well-defined Cauchy problem (for reasons related to the issues of causality discussed above), and are probably inconsistent quantum mechanically.

In fiction

Main article: Tachyons in fiction

Tachyons have appeared in many works of fiction. They have been used as a standby mechanism upon which many science fiction authors rely to establish faster-than-light communication, with or without reference to causality issues. The word tachyon has become widely recognized to such an extent that it can impart a science-fictional connotation even if the subject in question has no particular relation to superluminal travel (a form of technobabble, akin to positronic brain).

Cherenkov radiation

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Cherenkov_radiation 

 

Cherenkov radiation glowing in the core of the Advanced Test Reactor.

Cherenkov radiation (/tʃəˈrɛŋkɒf/; Russian: Черенков) is electromagnetic radiation emitted when a charged particle (such as an electron) passes through a dielectric medium at a speed greater than the phase velocity (speed of propagation of a wave in a medium) of light in that medium. A classic example of Cherenkov radiation is the characteristic blue glow of an underwater nuclear reactor. Its cause is similar to the cause of a sonic boom, the sharp sound heard when faster-than-sound movement occurs. The phenomenon is named after Soviet physicist Pavel Cherenkov, who shared the 1958 Nobel Prize in Physics for its discovery.

History

The radiation is named after the Soviet scientist Pavel Cherenkov, the 1958 Nobel Prize winner, who was the first to detect it experimentally under the supervision of Sergey Vavilov at the Lebedev Institute in 1934. Therefore, it is also known as Vavilov–Cherenkov radiation. Cherenkov saw a faint bluish light around a radioactive preparation in water during experiments. His doctorate thesis was on luminescence of uranium salt solutions that were excited by gamma rays instead of less energetic visible light, as done commonly. He discovered the anisotropy of the radiation and came to the conclusion that the bluish glow was not a fluorescent phenomenon.

A theory of this effect was later developed in 1937 within the framework of Einstein's special relativity theory by Cherenkov's colleagues Igor Tamm and Ilya Frank, who also shared the 1958 Nobel Prize.

Cherenkov radiation as conical wave front had been theoretically predicted by the English polymath Oliver Heaviside in papers published between 1888 and 1889 and by Arnold Sommerfeld in 1904, but both had been quickly forgotten following the relativity theory's restriction of super-c particles until the 1970s. Marie Curie observed a pale blue light in a highly concentrated radium solution in 1910, but did not investigate its source. In 1926, the French radiotherapist Lucien Mallet described the luminous radiation of radium irradiating water having a continuous spectrum.

In 2019, a team of researchers from Dartmouth’s and Dartmouth-Hitchcock's Norris Cotton Cancer Center discovered Cherenkov light being generated in the vitreous humor of patients undergoing radiotherapy. The light was observed using a camera imaging system called a CDose, which is specially designed to view light emissions from biological systems. For decades, patients had reported phenomena such as “flashes of bright or blue light” when receiving radiation treatments for brain cancer, but the effects had never been experimentally observed.

Physical origin

Basics

While the speed of light in a vacuum is a universal constant (c = 299,792,458 m/s), the speed in a material may be significantly less, as it is perceived to be slowed by the medium. For example, in water it is only 0.75c. Matter can accelerate beyond this speed (although still less than c, the speed of light in vacuum) during nuclear reactions and in particle accelerators. Cherenkov radiation results when a charged particle, most commonly an electron, travels through a dielectric (can be polarized electrically) medium with a speed greater than light's speed in that medium.

The effect can be intuitively described in the following way. From classical physics, it is known that charged particles emit EM waves and via Huygens' principle these waves will form spherical wavefronts which propagate with the phase velocity of that medium (i.e. the speed of light in that medium given by ${\displaystyle c/n}$, for ${\displaystyle n}$, the refractive index). When any charged particle passes through a medium, the particles of the medium will polarize around it in response. The charged particle excites the molecules in the polarizable medium and on returning to their ground state, the molecules re-emit the energy given to them to achieve excitation as photons. These photons form the spherical wavefronts which can be seen originating from the moving particle. If ${\displaystyle v_{p}<c/n}$, that is the velocity of the charged particle is less than that of the speed of light in the medium, then the polarization field which forms around the moving particle is usually symmetric. The corresponding emitted wavefronts may be bunched up but they do not coincide or cross and there are no interference effects to worry about. In the reverse situation, i.e. ${\displaystyle v_{p}>c/n}$, the polarization field is asymmetric along the direction of motion of the particle, as the particles of the medium do not have enough time to recover to their "normal" randomized states. This results in overlapping waveforms (as in the animation) and constructive interference leads to an observed cone-like light signal at a characteristic angle: Cherenkov light.

Animation of Cherenkov radiation

A common analogy is the sonic boom of a supersonic aircraft. The sound waves generated by the aircraft travel at the speed of sound, which is slower than the aircraft, and cannot propagate forward from the aircraft, instead forming a conical shock front. In a similar way, a charged particle can generate a "shock wave" of visible light as it travels through an insulator.

The velocity that must be exceeded is the phase velocity of light rather than the group velocity of light. The phase velocity can be altered dramatically by using a periodic medium, and in that case one can even achieve Cherenkov radiation with no minimum particle velocity, a phenomenon known as the Smith–Purcell effect. In a more complex periodic medium, such as a photonic crystal, one can also obtain a variety of other anomalous Cherenkov effects, such as radiation in a backwards direction (see below) whereas ordinary Cherenkov radiation forms an acute angle with the particle velocity.

Cherenkov radiation in the Reed Research Reactor.

In their original work on the theoretical foundations of Cherenkov radiation, Tamm and Frank wrote,"This peculiar radiation can evidently not be explained by any common mechanism such as the interaction of the fast electron with individual atom or as radiative scattering of electrons on atomic nuclei. On the other hand, the phenomenon can be explained both qualitatively and quantitatively if one takes into account the fact that an electron moving in a medium does radiate light even if it is moving uniformly provided that its velocity is greater than the velocity of light in the medium."

Emission angle

The geometry of the Cherenkov radiation shown for the ideal case of no dispersion.

In the figure on the geometry, the particle (red arrow) travels in a medium with speed ${\displaystyle v_{\text{p}}}$ such that

$${\displaystyle {\frac {c}{n}}<v_{\text{p}}<c,}$$

where ${\displaystyle c}$ is speed of light in vacuum, and ${\displaystyle n}$ is the refractive index of the medium. If the medium is water, the condition is ${\displaystyle 0.75c<v_{\text{p}}<c}$, since ${\displaystyle n\approx 1.33}$ for water at 20 °C.

We define the ratio between the speed of the particle and the speed of light as

$${\displaystyle \beta ={\frac {v_{\text{p}}}{c}}.}$$

The emitted light waves (denoted by blue arrows) travel at speed

$${\displaystyle v_{\text{em}}={\frac {c}{n}}.}$$

The left corner of the triangle represents the location of the superluminal particle at some initial moment (t = 0). The right corner of the triangle is the location of the particle at some later time t. In the given time t, the particle travels the distance

$${\displaystyle x_{\text{p}}=v_{\text{p}}t=\beta \,ct}$$

whereas the emitted electromagnetic waves are constricted to travel the distance

$${\displaystyle x_{\text{em}}=v_{\text{em}}t={\frac {c}{n}}t.}$$

So the emission angle results in

$${\displaystyle \cos \theta ={\frac {1}{n\beta }}}$$

Arbitrary emission angle

Cherenkov radiation can also radiate in an arbitrary direction using properly engineered one dimensional metamaterials. The latter is designed to introduce a gradient of phase retardation along the trajectory of the fast travelling particle (${\displaystyle d\phi /dx}$), reversing or steering Cherenkov emission at arbitrary angles given by the generalized relation:

$${\displaystyle \cos \theta ={\frac {1}{n\beta }}+{\frac {n}{k_{0}}}\cdot {\frac {d\phi }{dx}}}$$

Note that since this ratio is independent of time, one can take arbitrary times and achieve similar triangles. The angle stays the same, meaning that subsequent waves generated between the initial time t = 0 and final time t will form similar triangles with coinciding right endpoints to the one shown.

Reverse Cherenkov effect

A reverse Cherenkov effect can be experienced using materials called negative-index metamaterials (materials with a subwavelength microstructure that gives them an effective "average" property very different from their constituent materials, in this case having negative permittivity and negative permeability). This means that, when a charged particle (usually electrons) passes through a medium at a speed greater than the phase velocity of light in that medium, that particle emits trailing radiation from its progress through the medium rather than in front of it (as is the case in normal materials with, both permittivity and permeability positive). One can also obtain such reverse-cone Cherenkov radiation in non-metamaterial periodic media where the periodic structure is on the same scale as the wavelength, so it cannot be treated as an effectively homogeneous metamaterial.

In a vacuum

The Cherenkov effect can occur in vacuum. In a slow-wave structure, like in a traveling-wave tube (TWT), the phase velocity decreases and the velocity of charged particles can exceed the phase velocity while remaining lower than ${\displaystyle c}$. In such a system, this effect can be derived from conservation of the energy and momentum where the momentum of a photon should be ${\displaystyle p=\hbar \beta }$ (${\displaystyle \beta }$ is phase constant) rather than the de Broglie relation ${\displaystyle p=\hbar k}$. This type of radiation (VCR) is used to generate high-power microwaves.

Characteristics

The frequency spectrum of Cherenkov radiation by a particle is given by the Frank–Tamm formula:

$${\displaystyle {\frac {d^{2}E}{dx\,d\omega }}={\frac {q^{2}}{4\pi }}\mu (\omega )\omega {\left(1-{\frac {c^{2}}{v^{2}n^{2}(\omega )}}\right)}}$$

The Frank–Tamm formula describes the amount of energy ${\displaystyle E}$ emitted from Cherenkov radiation, per unit length traveled ${\displaystyle x}$ and per frequency ${\displaystyle \omega }$. ${\displaystyle \mu (\omega )}$ is the permeability and ${\displaystyle n(\omega )}$ is the index of refraction of the material the charge particle moves through. ${\displaystyle q}$ is the electric charge of the particle, ${\displaystyle v}$ is the speed of the particle, and ${\displaystyle c}$ is the speed of light in vacuum.

Unlike fluorescence or emission spectra that have characteristic spectral peaks, Cherenkov radiation is continuous. Around the visible spectrum, the relative intensity per unit frequency is approximately proportional to the frequency. That is, higher frequencies (shorter wavelengths) are more intense in Cherenkov radiation. This is why visible Cherenkov radiation is observed to be brilliant blue. In fact, most Cherenkov radiation is in the ultraviolet spectrum—it is only with sufficiently accelerated charges that it even becomes visible; the sensitivity of the human eye peaks at green, and is very low in the violet portion of the spectrum.

There is a cut-off frequency above which the equation ${\displaystyle \cos \theta =1/(n\beta )}$ can no longer be satisfied. The refractive index ${\displaystyle n}$ varies with frequency (and hence with wavelength) in such a way that the intensity cannot continue to increase at ever shorter wavelengths, even for very relativistic particles (where v/c is close to 1). At X-ray frequencies, the refractive index becomes less than 1 (note that in media, the phase velocity may exceed c without violating relativity) and hence no X-ray emission (or shorter wavelength emissions such as gamma rays) would be observed. However, X-rays can be generated at special frequencies just below the frequencies corresponding to core electronic transitions in a material, as the index of refraction is often greater than 1 just below a resonant frequency.

As in sonic booms and bow shocks, the angle of the shock cone is directly related to the velocity of the disruption. The Cherenkov angle is zero at the threshold velocity for the emission of Cherenkov radiation. The angle takes on a maximum as the particle speed approaches the speed of light. Hence, observed angles of incidence can be used to compute the direction and speed of a Cherenkov radiation-producing charge.

Cherenkov radiation can be generated in the eye by charged particles hitting the vitreous humour, giving the impression of flashes, as in cosmic ray visual phenomena and possibly some observations of criticality accidents.

Uses

Detection of labelled biomolecules

Cherenkov radiation is widely used to facilitate the detection of small amounts and low concentrations of biomolecules. Radioactive atoms such as phosphorus-32 are readily introduced into biomolecules by enzymatic and synthetic means and subsequently may be easily detected in small quantities for the purpose of elucidating biological pathways and in characterizing the interaction of biological molecules such as affinity constants and dissociation rates.

Medical imaging of radioisotopes and external beam radiotherapy

Cherenkov light emission imaged from the chest wall of a patient undergoing whole breast irradiation, using 6 MeV beam from a linear accelerator in radiotherapy.

More recently, Cherenkov light has been used to image substances in the body. These discoveries have led to intense interest around the idea of using this light signal to quantify and/or detect radiation in the body, either from internal sources such as injected radiopharmaceuticals or from external beam radiotherapy in oncology. Radioisotopes such as the positron emitters ¹⁸F and ¹³N or beta emitters ³²P or ⁹⁰Y have measurable Cherenkov emission and isotopes ¹⁸F and ¹³¹I have been imaged in humans for diagnostic value demonstration. External beam radiation therapy has been shown to induce a substantial amount of Cherenkov light in the tissue being treated, due to the photon beam energy levels used in the 6 MeV to 18 MeV ranges. The secondary electrons induced by these high energy x-rays result in the Cherenkov light emission, where the detected signal can be imaged at the entry and exit surfaces of the tissue.

Nuclear reactors

Cherenkov radiation in a TRIGA reactor pool.

Cherenkov radiation is used to detect high-energy charged particles. In open pool reactors, beta particles (high-energy electrons) are released as the fission products decay. The glow continues after the chain reaction stops, dimming as the shorter-lived products decay. Similarly, Cherenkov radiation can characterize the remaining radioactivity of spent fuel rods. This phenomenon is used to verify the presence of spent nuclear fuel in spent fuel pools for nuclear safeguards purposes.

Astrophysics experiments

When a high-energy (TeV) gamma photon or cosmic ray interacts with the Earth's atmosphere, it may produce an electron-positron pair with enormous velocities. The Cherenkov radiation emitted in the atmosphere by these charged particles is used to determine the direction and energy of the cosmic ray or gamma ray, which is used for example in the Imaging Atmospheric Cherenkov Technique (IACT), by experiments such as VERITAS, H.E.S.S., MAGIC. Cherenkov radiation emitted in tanks filled with water by those charged particles reaching earth is used for the same goal by the Extensive Air Shower experiment HAWC, the Pierre Auger Observatory and other projects. Similar methods are used in very large neutrino detectors, such as the Super-Kamiokande, the Sudbury Neutrino Observatory (SNO) and IceCube. Other projects operated in the past applying related techniques, such as STACEE, a former solar tower refurbished to work as a non-imaging Cherenkov observatory, which was located in New Mexico.

Astrophysics observatories using the Cherenkov technique to measure air showers are key to determine the properties of astronomical objects that emit Very High Energy gamma rays, such as supernova remnants and blazars.

Particle physics experiments

See also: Cherenkov detector and Ring imaging Cherenkov detector

Cherenkov radiation is commonly used in experimental particle physics for particle identification. One could measure (or put limits on) the velocity of an electrically charged elementary particle by the properties of the Cherenkov light it emits in a certain medium. If the momentum of the particle is measured independently, one could compute the mass of the particle by its momentum and velocity (see four-momentum), and hence identify the particle.

The simplest type of particle identification device based on a Cherenkov radiation technique is the threshold counter, which answers whether the velocity of a charged particle is lower or higher than a certain value (${\displaystyle v_{0}=c/n}$, where ${\displaystyle c}$ is the speed of light, and ${\displaystyle n}$ is the refractive index of the medium) by looking at whether this particle emits Cherenkov light in a certain medium. Knowing particle momentum, one can separate particles lighter than a certain threshold from those heavier than the threshold.

The most advanced type of a detector is the RICH, or ring-imaging Cherenkov detector, developed in the 1980s. In a RICH detector, a cone of Cherenkov light is produced when a high-speed charged particle traverses a suitable medium, often called radiator. This light cone is detected on a position sensitive planar photon detector, which allows reconstructing a ring or disc, whose radius is a measure for the Cherenkov emission angle. Both focusing and proximity-focusing detectors are in use. In a focusing RICH detector, the photons are collected by a spherical mirror and focused onto the photon detector placed at the focal plane. The result is a circle with a radius independent of the emission point along the particle track. This scheme is suitable for low refractive index radiators—i.e. gases—due to the larger radiator length needed to create enough photons. In the more compact proximity-focusing design, a thin radiator volume emits a cone of Cherenkov light which traverses a small distance—the proximity gap—and is detected on the photon detector plane. The image is a ring of light whose radius is defined by the Cherenkov emission angle and the proximity gap. The ring thickness is determined by the thickness of the radiator. An example of a proximity gap RICH detector is the High Momentum Particle Identification Detector (HMPID), a detector currently under construction for ALICE (A Large Ion Collider Experiment), one of the six experiments at the LHC (Large Hadron Collider) at CERN.