The concept of coherence. Temporal and spatial coherence
As already noted, the interference pattern can be observed only when superimposing coherent waves. Let us pay attention to the fact that in the definition of coherent waves it is noted not the existence, but the observation of interference. This means that the presence or absence of coherence depends not only on the characteristics of the waves themselves, but also on the time interval for recording the intensity. The same pair of waves can be coherent at one observation time and incoherent at another.
Two light waves produced from one by the amplitude division method or the wavefront division method do not necessarily interfere with each other. At the observation point, two waves with wave vectors and are added. There are two main reasons for the possible incoherence of such waves.
The first reason is the non-monochromatic nature of the light source (or the variability of the magnitudes of the wave vectors). Monochromatic light is light of one frequency. A strictly monochromatic wave at each point in space has a time-independent amplitude and initial phase. Both the amplitude and phase of a real light wave experience some random variation over time. If the changes in frequency are small and the changes in amplitude are sufficiently slow (their frequency is small compared to the optical frequency), then the wave is said to be quasi-monochromatic.
The second reason for the possible incoherence of light waves obtained from a single wave is the spatial extent of the actual light source (or the inconstancy of the direction of each of the wave vectors).
In reality, both reasons occur simultaneously. However, for simplicity, we will analyze each reason separately.
Temporal coherence.
Let there be spot Light source S and and , which are real or imaginary images of it (Fig. 3.6.3 or 3.6.4). Let us assume that the radiation from the source consists of two close and equally intense waves with wavelengths and (obviously the same will be true for sources and ). Let the initial phases of the sources be the same. Rays with wavelengths will arrive at a certain point on the screen in the same phases. Let's call this point the center of the interference pattern. For both waves there will be a light stripe. At another point on the screen, where the path difference ( N– integer, band number) for the wavelength, a light interference fringe will also be obtained. If it is the same, then rays with a wavelength will arrive at the same point on the screen in antiphase, and for this wavelength the interference fringe will be dark. Under this condition, at the point of the screen under consideration, the light stripe will overlap the dark stripe - the interference pattern will disappear. Thus, the condition for the disappearance of fringes is , whence the maximum number of the interference fringe
Let us now turn to the case when the light from the source is a collection of waves with lengths lying in the interval . Let us divide this spectral interval into a set of pairs of infinitely narrow spectral lines, the wavelengths of which differ by . Formula (3.7.1) is applicable to each such pair, where it must be replaced by . Therefore, the disappearance of the interference pattern will occur for the interference order
This formula gives an estimate of the maximum possible interference order. The quantity is usually called degree of monochromaticity of the wave.
Thus, to observe the interference pattern when a wave is split along the path of the beam, the difference in the paths of the two resulting waves should not exceed a value called coherence length l
The concept of coherence length can be explained as follows. Consider two points on one beam as two possible secondary light sources for observing the interference pattern. In this case, the distance from each point to the mental screen is assumed to be the same (Fig. 3.7.1).
Here and are two selected along the ray
Fig.3.7.1. points at which we mentally place translucent plates to obtain an interference pattern on the screen. Let the optical path difference for the interfering rays and be equal to . If it exceeds the value, then, as indicated above, the interference pattern is “smeared”, and, consequently, the secondary light sources at the points turn out to be incoherent. The distance between points and at which this begins to happen is called length coherence along the beam, longitudinal coherence length, or simply coherence length.
A distance equal to the coherence length the wave travels coherence time
Coherence time can be called the maximum period of time, when averaging over which the interference effect is still observed.
Based on the above estimates, we can estimate the thickness of the film, with the help of which an interference pattern can be obtained (decipher the term “thin film” used in the previous lecture). The film can be called “thin” if the difference in the paths of the waves that give the interference pattern does not exceed the coherence length of the light wave. When a wave falls on the film at a small angle (in a direction close to the normal), the path difference is equal to 2bn(formula (3.6.20)), where b– thickness, and n– refractive index of the film material. Therefore, the interference pattern can be obtained on a film for which 2bn ≤ l =. (3.7.5) Note that when a wave is incident at large angles, it is also necessary to take into account the possible incoherence of different points of the wave front.
Let us estimate the coherence length of light emitted by different sources.
1. Consider light emitted by a natural source (not a laser). If a glass filter is placed in the path of light, the bandwidth of which is ~ 50 nm, then for a wavelength of the middle of the optical spectral interval ~ 600 nm we obtain, according to (3.7.3), ~ 10 m. If there is no filter, then the coherence length will be approximately an order of magnitude less.
2. If the light source is a laser, then its radiation has a high degree of monochromaticity (~ 0.01 nm) and the coherence length of such light for the same wavelength will be about 4·10 m.
Spatial coherence.
The ability to observe the interference of coherent waves from extended sources leads to the concept spatial coherence of waves.
For simplicity of reasoning, let us imagine that sources of coherent electromagnetic waves with identical initial phases and wavelengths are located on a segment of length b, located at a distance l»b from the screen (Fig. 3.7.2), on which their interference is observed. The interference pattern observed on the screen can be represented as a superposition of interference patterns created by an infinite number of pairs of point coherent sources into which an extended source can be mentally divided.
Among the entire set of sources, let us select a source located in the middle of the segment and compare the interference patterns of two pairs, one of which is formed by the central source and some arbitrarily chosen source located close to it, and the other by the central source and a source located at one of the ends of the segment. It is obvious that the interference pattern of a pair of closely located sources will have a value close to the maximum in the center of the screen at the observation point (Fig. 3.7.2). At the same time, the interference pattern of the other pair will have a value depending on the optical difference in the path of electromagnetic waves emitted by sources in the center of the segment and at its edge
| ≈ , | (3.7.6) |
where is the angular size of the source (Fig. 3.7.2), which due to “ l is small enough so that the obvious transformations used in deriving formula (3.7.6) are valid.
It follows that waves from different points of an extended source arriving at an observation point located in the center of the screen will have an optical path difference relative to the wave from the central source, varying linearly from zero to a maximum value of 0.25. For a certain source length, waves arriving at the observation point can have a phase that differs by 180° from the phase of the wave emitted by the central point of the segment. As a result of this, waves arriving at the center of the screen from different parts of the source will reduce the intensity value compared to the maximum that would occur if all the waves had the same phase. The same reasoning is valid for other points on the screen. As a result, the intensities at the maxima and minima of the interference pattern of an extended source will have similar values and the visibility of the interference pattern will tend to zero. In the case under consideration, this occurs at in (3.7.6). The value of the shortest length of the segment (source) corresponding to this condition is determined from the relation (in this case t=1):
In optics and the theory of electromagnetic waves, half of this value determines the so-called. radius of spatial coherence electromagnetic waves emitted by an extended source:
| . | (3.7.7) |
The physical meaning of the concept of the radius of spatial coherence of an extended source is the idea of the possibility of observing the interference pattern from an extended source if it is located inside a circle of radius . From the above it follows that the spatial coherence of electromagnetic waves is determined by the angular size of their source.
Spatial coherence is the coherence of light in the direction perpendicular to the beam (across the beam). It turns out that this is the coherence of different points of the surface of equal phase. But on a surface of equal phase, the phase difference is zero. However, for extended sources this is not entirely true. The real light source is not a point, so the surface of equal phases undergoes slight rotations, remaining at each moment of time perpendicular to the direction of the currently emitting point light source, located within the real light source. Rotations of the surface of equal phase are caused by the fact that light comes to the observation point from one or another point of the source. Then, if we assume that on such a pseudo-wave surface there are secondary sources, the waves from which can give an interference pattern, then we can define the coherence radius in other words. Secondary sources on the pseudo-wave surface, which can be considered coherent, are located inside a circle whose radius is equal to the coherence radius. The coherence diameter is the maximum distance between points on the pseudowave surface that can be considered coherent.
Let's return to Jung's experience (Lecture 3.6). To obtain a clear interference pattern in this experiment, it is necessary that the distance between the two slits S and did not exceed the coherence diameter. On the other hand, as can be seen from (3.7.7), the radius (and, consequently, the diameter) of interference increases with decreasing angular size of the source. That's why d- distance between slots and and b- source size S inversely related b·d ≤ l.(3.7.8)
Encyclopedic Dictionary, 1998
coherence
COHERENCE (from the Latin cohaerens - in connection) the coordinated occurrence in time of several oscillatory or wave processes. If the phase difference between 2 oscillations remains constant over time or changes according to a strictly defined law, then the oscillations are called coherent. Oscillations in which the phase difference changes randomly and quickly compared to their period are called incoherent.
Coherence
(from the Latin cohaerens ≈ in connection), the coordinated occurrence in time of several oscillatory or wave processes, manifested when they are added. Oscillations are called coherent if the difference in their phases remains constant over time and, when the oscillations are added, determines the amplitude of the total oscillation. Two harmonic (sinusoidal) oscillations of the same frequency are always coherent. Harmonic oscillation is described by the expression: x = A cos (2pvt + j), (
where x ≈ oscillating quantity (for example, the displacement of the pendulum from the equilibrium position, the strength of the electric and magnetic fields, etc.). The frequency of a harmonic oscillation, its amplitude A and phase j are constant in time. When two harmonic oscillations with the same frequency v, but different amplitudes A1 and A2 and phases j1 and j2 are added, a harmonic oscillation of the same frequency is formed. Amplitude of the resulting oscillation:
can vary from A1 + A2 to A1 ≈ A2 depending on the phase difference j1 ≈ j2 (). The intensity of the resulting vibration, proportional to Ap2, also depends on the phase difference.
In reality, ideally harmonic oscillations are not feasible, since in real oscillatory processes the amplitude, frequency and phase of oscillations continuously change chaotically in time. The resulting amplitude Ap depends significantly on how quickly the phase difference changes. If these changes are so rapid that they cannot be detected by the instrument, then only the average amplitude of the resulting vibration can be measured. At the same time, because the average value of cos (j1≈j2) is equal to 0, the average intensity of the total oscillation is equal to the sum of the average intensities of the initial oscillations: ═and, thus, does not depend on their phases. The original oscillations are incoherent. Chaotic rapid changes in amplitude also disrupt K.
If the phases of oscillations j1 and j2 change, but their difference j1 ≈ j2 remains constant, then the intensity of the total oscillation, as in the case of ideally harmonic oscillations, is determined by the difference in the phases of the added oscillations, that is, K occurs. If the difference in the phases of two oscillations changes very slowly , then they say that the oscillations remain coherent for some time, until their phase difference has had time to change by an amount comparable to p.
You can compare the phases of the same oscillation at different times t1 and t2, separated by an interval t. If the inharmonicity of an oscillation manifests itself in a disorderly, random change in time of its phase, then for a sufficiently large t the change in the oscillation phase can exceed p. This means that after time t the harmonic oscillation “forgets” its original phase and becomes incoherent “to itself.” Time t is called the K time of a nonharmonic oscillation, or the duration of a harmonic train. After one harmonic train has passed, it is, as it were, replaced by another with the same frequency but a different phase.
When a plane monochromatic electromagnetic wave propagates in a homogeneous medium, the electric field strength E along the direction of propagation of this wave oh at time t is equal to:
where l = cT ≈ wavelength, c ≈ speed of its propagation, T ≈ oscillation period. The phase of oscillations at any specific point in space is maintained only during the time CT. During this time, the wave will propagate over a distance сt and oscillations E at points distant from each other by a distance сt, along the direction of propagation of the wave, turn out to be incoherent. The distance equal to сt along the direction of propagation of a plane wave at which random changes in the oscillation phase reach a value comparable to p is called the K length, or train length.
Visible sunlight, occupying the range from 4×1014 to 8×1014Hz on the frequency scale of electromagnetic waves, can be considered as a harmonic wave with rapidly changing amplitude, frequency and phase. In this case, the length of the train is ~ 10≈4 cm. Light emitted by a rarefied gas in the form of narrow spectral lines is closer to monochromatic. The phase of such light practically does not change at a distance of 10 cm. The length of the laser radiation train can exceed kilometers. In the radio wave range, there are more monochromatic sources of oscillation (see Quartz oscillator, Quantum frequency standards), and the wavelength l is many times longer than for visible light. The length of a radio wave train can significantly exceed the size of the solar system.
Everything said is true for a plane wave. However, a perfectly plane wave is just as impracticable as a perfectly harmonic oscillation (see Waves). In real wave processes, the amplitudes and phase of oscillations change not only along the direction of wave propagation, but also in a plane perpendicular to this direction. Random changes in the phase difference at two points located in this plane increase with increasing distance between them. The vibrational effect at these points weakens and at a certain distance l, when random changes in the phase difference become comparable to p, disappear. To describe the coherent properties of a wave in a plane perpendicular to the direction of its propagation, the term spatial coherence is used, in contrast to temporal coherence, which is associated with the degree of monochromaticity of the wave. The entire space occupied by the wave can be divided into regions, in each of which the wave retains a space. The volume of such a region (volume of the wave) is approximately equal to the product of the length of the train ct and the area of a circle with a diameter of / (the size of the spatial space).
Violation of spatial signalization is associated with the peculiarities of the processes of radiation and wave formation. For example, the spatial radiation of a light wave emitted by an extended heated body disappears at a distance of only a few wavelengths from its surface, because different parts of a heated body radiate independently of each other (see Spontaneous emission). As a result, instead of a single plane wave, the source emits a set of plane waves propagating in all possible directions. As it moves away from the heat source (of finite dimensions), the wave approaches more and more flat. The size of the spatial K. l increases in proportion to l ═≈ where R ≈ distance to the source, r ≈ size of the source. This makes it possible to observe the interference of light from stars, despite the fact that they are thermal sources of enormous size. By measuring / for light from nearby stars, it is possible to determine their sizes r. The value l/r is called the angle K. With distance from the source, the light intensity decreases as 1/R2. Therefore, using a heated body it is impossible to obtain intense radiation with a large spatial K.
The light wave emitted by the laser is formed as a result of coordinated stimulated emission of light throughout the entire volume of the active substance. Therefore, the spatial K. of light at the laser output aperture is preserved throughout the entire cross section of the beam. Laser radiation has enormous spatial radiation, that is, high directivity compared to radiation from a heated body. With the help of a laser, it is possible to obtain light whose volume of radiation is 1017 times greater than the volume of radiation of a light wave of the same intensity obtained from the most monochromatic non-laser light sources.
In optics, the most common way to produce two coherent waves is to split the wave emitted by one non-monochromatic source into two waves that travel along different paths, but ultimately meet at one point, where they are combined (Fig. 2). If the delay of one wave relative to another, associated with the difference in the paths they travel, is less than the duration of the train, then the oscillations at the point of addition will be coherent and interference of light will be observed. When the difference in the paths of the two waves approaches the length of the train, the radiation of the rays weakens. Fluctuations in screen illumination decrease, illumination I tends to a constant value equal to the sum of the intensities of two waves incident on the screen. In the case of a non-point (extended) heat source, two rays arriving at points A and B may turn out to be incoherent due to the spatial incoherence of the emitted wave. In this case, interference is not observed, since the interference fringes from different points of the source are shifted relative to each other by a distance greater than the fringe width.
The concept of quantum mechanics, which originally arose in the classical theory of oscillations and waves, is also applied to objects and processes described by quantum mechanics (atomic particles, solids, etc.).
Lit.: Landsberg G.S., Optics, 4th ed., M., 1957; Gorelik G.S., Oscillations and waves, 2nd ed., M., 1959; Fabrikant V.A., New information about coherence, “Physics at school”, 1968, ╧ 1; Franson M., Slansky S., Coherence in optics, trans. from French, M., 1968; Martinsen V., Shpiller E., What is coherence, “Nature”, 1968, ╧ 10.
A. V. Francesson.
Wikipedia
Coherence (physics)
Coherence(from - " in touch") - the correlation of several oscillatory or wave processes in time, manifested when they are added. Oscillations are coherent if their phase difference is constant over time, and when adding the oscillations, an oscillation of the same frequency is obtained.
The classic example of two coherent oscillations is two sinusoidal oscillations of the same frequency.
Coherence radius is the distance at which, when displaced along the pseudo-wave surface, a random phase change reaches an order of magnitude.
The process of decoherence is a violation of coherence caused by the interaction of particles with the environment.
Coherence (philosophical speculative strategy)
In a thought experiment proposed by the Italian probability theorist Bruno de Finetti to justify Bayesian probability, the array of bets is exactly coherent, if he does not expose the bettor to certain loss regardless of the outcome of the events on which he bets, providing his opponent with a reasonable choice.
Coherence
Coherence(from - " in touch»):
- The coherence of several oscillatory or wave processes of these processes in time, manifested when they are added.
- Coherence of an array of bets is a property of an array of bets, which means that a bettor who bets on some outcomes of some events will never lose the argument, regardless of the outcomes of these events.
- Memory coherence is a property of computer systems that allows two or more processors or cores to access the same memory area.
Examples of the use of the word coherence in literature.
Regardless of the plane of polarization of the Ghosts' radiation, we can now adjust to any and make sure that coherence really exists and is constant over time.
They also perceive the phase of the wave, but at the same time they themselves provide coherence, emitting signals at strictly defined intervals.
Coherence, but this is a coherence that does not allow the existence of my coherence, the coherence of the world and the coherence of God.
The entire Composition of the Total Number of Incarnations of the Essence of the Supreme, as well as the entire Composition of the Total Number of Represented Incarnations of the Essence of the Supreme, along with the Composition of the Total Number of Imaginary Incarnations of the Essence of the Supreme, are imprinted in the Bowl of Accumulations of the Essence of the Divine Man-Buddha in an information-energetic holographic way coherence Spirit, for He is Alpha-and-Omega - the First-and-Last One Supreme, Encompassing in His Creation all Those That Exist with the Creator.
External communications The RA-8000 has the means to effectively maintain coherence cache in multiprocessor systems.
Impressions in the Fabrics of Saraswati's Clothes occur by the Power of the Essence of the Divine Human - in an information-energetic holographic way, that is coherence psychocorrelative quantum fields, leaving the holographic information-energy code of Human Co-Existence, as a living Memory in the Eternal Unchangeable Form of the Soul of Creation.
Each Person has his own individual Composition of the Total Number of Incarnations of the Essence of the Supreme, and this Composition is imprinted in the Human Chalice in an information-energetic holographic way - high coherence radiations of psychocorrelative quantum fields that are generated by the Essence of the Divine Man in the process of his Education by the Supreme.
The Essence of Divine Man, as a result of Thinking in Images of the Highest, gives birth to myriads of elementary particles of Matter, which are focused high coherence Spirit in the Lens of Space curvature density Images of the overall picture of the hologram What is happening in Saraswati from the senses.
Figure 5 -- Formation of the Theroidsphere of Influx by the creation of high density Curvature of Space coherence Spirit.
Individual electrons observed in a specific physical experiment are, according to Tsech, the result of destruction by a measuring device coherence a single electron-positron field.
The processes of self-organization of social consciousness are subject to the general laws of formation: coherence, the coherence of events of the emergence of certain social stereotypes, etc.
The result of the addition of two harmonic oscillations depends on the phase difference, which changes when moving to another spatial point. There are two options:
1) If both vibrations are not consistent with each other, i.e. If the phase difference changes over time in an arbitrary manner, then such oscillations are called incoherent. In real oscillatory processes, due to continuous chaotic (random) changes, the time-average value , i.e. the chaotic change of such instantaneous pictures is not perceived by the eye and a feeling of an even flow of light is created that does not change over time. Therefore, the amplitude of the resulting oscillation will be expressed by the formula:
The intensity of the resulting oscillation in this case is equal to the sum of the intensities created by each of the waves separately:
2) If the phase difference is constant in time, then such oscillations (waves) are called coherent (connected).
In general, waves of the same frequency that have a phase difference are called coherent.
In the case of superposition of coherent waves, the intensity of the resulting oscillation is determined by the formula:
where - is called the interference term, which has the greatest influence on the resulting intensity:
a) if , then the resulting intensity;
b) if , then the resulting intensity is .
This means that if the phase difference of the added oscillations remains constant over time (oscillations or waves are coherent), then the amplitude of the total oscillation, depending on, takes values from at , , to , (Fig. 6.3).
Interference manifests itself more clearly when the intensities of the added oscillations are equal:
Obviously, the maximum intensity of the resulting oscillation will be observed at and will be equal to:
The minimum intensity of the resulting oscillation will be observed at and will be equal to:
Thus, when harmonic coherent light waves are superimposed, a redistribution of the light flux in space occurs, resulting in intensity maxima in some places and intensity minima in others. This phenomenon is called interference of light waves.
Interference is typical for waves of any nature. Interference can be observed especially clearly, for example, for waves on the surface of water or sound waves. Interference of light waves does not occur so often in everyday life, since its observation requires certain conditions, since, firstly, ordinary light, natural light, is not a monochromatic (fixed frequency) source. Secondly, conventional light sources are incoherent, since when light waves from different sources are superimposed, the phase difference of light oscillations changes randomly over time, and a stable interference pattern is not observed. To obtain a clear interference pattern, the superimposed waves must be coherent.
Coherence is the coordinated occurrence in time and space of several oscillatory or wave processes, which manifests itself when they are added together. The general principle of obtaining coherent waves is as follows: a wave emitted by one light source is divided in some way into two or more secondary waves, as a result of which these waves are coherent (their phase difference is a constant value, since they “originated” from one source). Then, after passing through different optical paths, these waves are superimposed on each other in some way and interference is observed.
Let two point coherent light sources emit monochromatic light (Fig. 6.4). For them, the coherence conditions must be satisfied:
To the point P the first ray passes through a medium with a refractive index path , the second ray passes through a medium with a refractive index path . The distances from the sources to the observed point are called the geometric lengths of the ray paths. The product of the refractive index of a medium and the geometric path length is called the optical path length. and are the optical lengths of the first and second beams, respectively.
Let and be the phase velocities of the waves. The first beam will excite at the point P swing:
and the second ray is vibration
Phase difference of oscillations excited by rays at a point P, will be equal to:
Because (is the wavelength in vacuum), then the expression for the phase difference can be given the form
there is a quantity called the optical path difference. When calculating interference patterns, it is the optical difference in the path of the rays that should be taken into account, i.e. refractive indices of the media in which rays propagate.
From the expression for the phase difference it is clear that if the optical path difference is equal to an integer number of wavelengths in vacuum
then the phase difference and oscillations will occur with the same phase. The number is called the order of interference. Consequently, this condition is the condition of the interference maximum.
If the optical path difference is equal to a half-integer number of wavelengths in vacuum
then, so the oscillations at the point P are in antiphase. This is the condition of the interference minimum.
So, if at a length equal to the optical path difference of the rays, an even number of half-wavelengths fits, then a maximum intensity is observed at a given point on the screen. If an odd number of half-wavelengths fits along the length of the optical path difference of the rays, then a minimum of illumination is observed at a given point on the screen.
If two ray paths are optically equivalent, they are called tautochronic, and optical systems - lenses, mirrors - satisfy the condition of tautochronism.
Coherent waves are oscillations with a constant phase difference. Of course, the condition is not satisfied at every point in space, only in certain areas. Obviously, to satisfy the definition, the oscillation frequencies are also assumed to be equal. Other waves are coherent only in a certain region of space, and then the phase difference changes, and this definition can no longer be used.
Rationale for use
Coherent waves are considered a simplification that is not found in practice. Mathematical abstraction helps in many branches of science: space, thermonuclear and astrophysical research, acoustics, music, electronics and, of course, optics.
For real applications, simplified methods are used, among the latter the three-wave system; the basics of applicability are briefly outlined below. To analyze the interaction, it is possible to specify, for example, a hydrodynamic or kinetic model.
Solving equations for coherent waves makes it possible to predict the stability of systems operating using plasma. Theoretical calculations show that sometimes the amplitude of the result grows indefinitely in a short time. Which means creating an explosive situation. When solving equations for coherent waves, by selecting conditions, it is possible to avoid unpleasant consequences.
Definitions
First, let's introduce a number of definitions:
- A wave of a single frequency is called monochromatic. The width of its spectrum is zero. This is the only harmonic on the graph.
- The signal spectrum is a graphical representation of the amplitude of the component harmonics, where the frequency is plotted along the abscissa axis (X axis, horizontal). The spectrum of a sinusoidal oscillation (monochromatic wave) becomes a single spectrum (vertical line).
- Fourier transforms (inverse and direct) are the decomposition of a complex vibration into monochromatic harmonics and the inverse addition of the whole from disparate spectrins.
- Waveform analysis of circuits for complex signals is not performed. Instead, there is a decomposition into individual sinusoidal (monochromatic) harmonics, for each it is relatively simple to create formulas to describe the behavior. When calculating on a computer, this is enough to analyze any situations.
- The spectrum of any non-periodic signal is infinite. Its boundaries are trimmed to reasonable limits before analysis.
- Diffraction is the deviation of a beam (wave) from a straight path due to interaction with the propagation medium. For example, it manifests itself when the front overcomes a gap in an obstacle.
- Interference is the phenomenon of wave addition. Because of this, a very bizarre picture of alternating stripes of light and shadow is observed.
- Refraction is the refraction of a wave at the interface between two media with different parameters.
Concept of coherence
The Soviet encyclopedia says that waves of the same frequency are invariably coherent. This is true exclusively for individual fixed points in space. The phase determines the result of the addition of oscillations. For example, antiphase waves of the same amplitude produce a straight line. Such vibrations cancel each other out. The largest amplitude is for in-phase waves (the phase difference is zero). The operating principle of lasers, the mirror and focusing system of light beams, and the peculiarities of receiving radiation make it possible to transmit information over enormous distances are based on this fact.
According to the theory of interaction of oscillations, coherent waves form an interference pattern. A beginner has a question: the light of the light bulb does not seem striped. For the simple reason that the radiation is not of one frequency, but lies within a segment of the spectrum. And the plot, moreover, is of decent width. Due to the heterogeneity of frequencies, the waves are disordered and do not demonstrate their theoretically and experimentally substantiated and proven properties in laboratories.
The laser beam has good coherence. It is used for long-distance communications with line of sight and other purposes. Coherent waves propagate further in space and reinforce each other at the receiver. In a beam of light of disparate frequencies, effects can be subtracted. It is possible to select the conditions that the radiation comes from the source, but is not registered at the receiver.
Regular light bulbs also do not work at full power. It is not possible to achieve 100% efficiency at the present stage of technology development. For example, gas-discharge lamps suffer from strong frequency dispersion. As for LEDs, the founders of the nanotechnology concept promised to create an element base for the production of semiconductor lasers, but in vain. A significant part of the developments is classified and inaccessible to the average person.
Only coherent waves exhibit wave qualities. They act in concert, like the branches of a broom: one at a time is easy to break, but taken together they sweep away debris. Wave properties - diffraction, interference and refraction - are characteristic of all vibrations. It's just harder to register the effect because of the messiness of the process.
Coherent waves do not exhibit dispersion. They show the same frequency and are deflected equally by the prism. All examples of wave processes in physics are given, as a rule, for coherent oscillations. In practice, one has to take into account the small spectral width present. Which imposes special features on the calculation process. Numerous textbooks and scattered publications with intricate titles try to answer how the real result depends on the relative coherence of the wave! There is no single answer; it depends greatly on the individual situation.
Wave packets
To facilitate the solution of a practical problem, you can introduce, for example, the definition of a wave packet. Each of them is further broken down into smaller pieces. And these subsections interact coherently between similar frequencies of the other packet. This analytical method is widely used in radio engineering and electronics. In particular, the concept of spectrum was initially introduced in order to provide engineers with a reliable tool that allows them to evaluate the behavior of a complex signal in specific cases. A small fraction of the impact of each harmonic oscillation on the system is estimated, then the final effect is found by their complete addition.
Consequently, when assessing real processes that are not even closely coherent, it is permissible to break the object of analysis into its simplest components in order to evaluate the result of the process. The calculation is simplified with the use of computer technology. Machine experiments show the reliability of the formulas for the existing situation.
At the initial stage of the analysis, it is believed that packets with a small spectrum width can be conditionally replaced by harmonic oscillations and then use the inverse and direct Fourier transform to evaluate the result. Experiments have shown that the phase spread between selected packets gradually increases (fluctuates with a gradual increase in the spread). But for three waves the difference gradually smooths out, consistent with the theory presented. A number of restrictions apply:
- The space must be infinite and homogeneous (k-space).
- The amplitude of the wave does not decay with increasing range, but changes over time.
It has been proven that in such an environment each wave manages to select a final spectrum, which automatically makes machine analysis possible, and when the packets interact, the spectrum of the resulting wave broadens. The oscillations are not considered essentially coherent, but are described by the superposition equation presented below. Where the wave vector ω(k) is determined by the dispersion equation; Ek is recognized as the harmonic amplitude of the packet under consideration; k – wave number; r – spatial coordinate, the presented equation is solved for the indicator; t – time.
Coherence time
In a real situation, heterogeneous packets are coherent only over a separate interval. And then the phase discrepancy becomes too great to apply the equation described above. To derive conditions for the possibility of computation, the concept of coherence time is introduced.
It is assumed that at the initial moment the phases of all packets are the same. The selected elementary wave fractions are coherent. Then the required time is found as the ratio of Pi to the packet spectrum width. If the time has exceeded the coherent time, in this section it is no longer possible to use the superposition formula for adding oscillations - the phases are too different from each other. The wave is no longer coherent.
It is possible to treat a packet as if it were characterized by a random phase. In this case, the interaction of waves follows a different pattern. Then the Fourier components are found using the specified formula for further calculations. Moreover, the other two components taken for calculation are taken from three packages. This is the case of agreement with theory mentioned above. Therefore, the equation shows the dependency of all packages. More precisely, the result of addition.
To obtain the best result, it is necessary that the width of the spectrum of the packet does not exceed the number Pi divided by the time to solve the problem of superposition of coherent waves. When the frequency is detuned, the amplitudes of the harmonics begin to oscillate, making it difficult to obtain an accurate result. And vice versa, for two coherent oscillations the addition formula is simplified as much as possible. The amplitude is found as the square root of the sum of the original harmonics, squared and added with its own double product, multiplied by the cosine of the phase difference. For coherent quantities, the angle is zero, the result, as indicated above, is maximum.
Along with time and coherence length, the term “train length” is used, which is an analogue of the second term. For sunlight, this distance is one micron. The spectrum of our star is extremely wide, which explains such a tiny distance where the radiation is considered coherent with itself. For comparison, the length of a gas discharge train reaches 10 cm (100,000 times longer), while laser radiation retains its properties even at kilometer distances.
It's much easier with radio waves. Quartz resonators make it possible to achieve high wave coherence, which explains the spots of reliable reception in the area bordering on silence zones. A similar thing occurs when the existing picture changes over the course of the day, the movement of clouds and other factors. The conditions for propagation of the coherent wave change, and the interference superposition has a full effect. In the radio range at low frequencies, the coherence length can exceed the diameter of the Solar System.
The conditions of addition strongly depend on the shape of the front. The problem is solved most simply for a plane wave. In reality the front is usually spherical. The points of in-phase are located on the surface of the ball. In an area infinitely distant from the source, the plane condition can be taken as an axiom, and further calculations can be carried out in accordance with the adopted postulate. The lower the frequency, the easier it is to create the conditions for performing the calculation. Conversely, light sources with a spherical front (remember the Sun) are difficult to fit into a harmonious theory written in textbooks.
But we should not think that this model will ensure the rigor of our conclusions. The real situation is much more complicated. We do not consider the influence of pulses on the relative populations of levels of coupled spin systems and their phase coherence. We have already considered methods for calculating the population of levels after exposure to a pulse in Section. 4.2.6, but this is only part of the overall picture; in this way, the phase relationships of different states cannot be modeled. However, we have reached the limit accessible by using our theoretical apparatus, and it will be quite sufficient for discussing the foundations of many experiments.
A 180° selective pulse should be used to excite a selected carbon atom, as it is easy to calibrate and does not require phase coherence with other hard carbon pulses.
At a sufficiently long time, a stationary state should be achieved for all types of resonance. The nature of the stationary state and the rate at which it is reached are determined by the Bloch equations. In his consideration, Bloch accepted that for individual processes a proportional relationship is observed between the magnetization component and the rate of its spontaneous loss, i.e., the spontaneous disappearance of first-order magnetization. The proportionality constants are inversely proportional to the two so-called relaxation times T1 - the time of longitudinal, or spin-lattice, relaxation, which is associated with changes in magnetization in the 2-direction along the constant field Ho, and Tg - the time of transverse, or spin-spin, relaxation associated with loss of phase coherence of precession in the x and y directions in a radio frequency field. In the case of ideal resonance, the linewidth is simply 1/Gr (with the appropriate definition of linewidth). simply related to signal saturation in very strong RF fields
We always consider not a single nuclear moment, but an ensemble containing a large number of identical nuclei. In Fig. 1.2, b shows the precession of nuclear moments with I - /2. All moments precess at the same frequency, since the xy directions are no different, there is no reason why the phase coherence of the moments in the xy plane would be preserved. However, the system has a dedicated direction - the z-axis, specified by the direction
After the 90° pulse and before the first gradient pulse is applied, only a slight dephasing of M occurs. As long as the gradient remains on, it naturally causes M dephasing. After g is turned off, the phase coherence again decreases very little. If the kernels are not di(un-
The basic theoretical principles of operation of phase-coherent communication systems are outlined, which are currently widely used in information transmission equipment used for communication with artificial Earth satellites and spacecraft. The book examines three groups of questions that, although independent, are closely related to the general provisions of the statistical theory of communication. The theory of operation of phase-coherent receivers of communication equipment, methods for optimizing coherent demodulators used in equipment operating on both analog and digital (discrete) principles are outlined, and a comparative analysis of coherent and incoherent demodulators is also carried out. A significant part of the book is devoted to the issues of ensuring phase coherence in the presence of interference of various types.
The book outlines the theory of phase-coherent communication systems taking into account thermal noise. It is devoted to the consideration from a single point of view of three different, but at the same time mutually related issues of statistical communication theory, the theory of operation of a phase-coherent receiver or phase-locked loop, optimization of coherent demodulators for both analog and digital modulation systems, a comparative analysis of the quality of coherent and conventional incoherent demodulators. Although the theory of phase coherence has found wide application in communication systems for space research, for communication with satellites and for military purposes, and although there is a large literature on this issue and its ramifications, there is still no manual that would consider more than just some specific aspects this theory. This is partly explained by the fact that until recently textbooks were devoted to presenting only one of the three defined branches of statistical communication theory (filtering, detection and information theory), and all three parts are needed to study coherent communication systems.
The book is intended as a presentation from a unified point of view of the theory of modulation for phase-coherent communication systems. The modulation technique dates back to the first attempts of prehistoric man to transmit information over a distance. The basic methods and theory of modulation are outlined by several authors. They paid special attention to the design and theory of conventional modulators and demodulators used in some modulation systems. Since the mid-forties, when statistical theory was first used to study communication problems, a number of important studies of modulation systems have been carried out, some of them are presented in textbooks on statistical communication theory. The work of Shannon, Wiener, and Woodward provided the theoretical basis for the design of optimal modulation systems for a variety of radio communication systems. Our book will outline the fundamentals of statistical communication theory, leading to the study and optimal construction of modulation systems for phase-coherent systems operating in the presence of thermal noise. (See also
Although the previous paragraphs discussed binary communication systems at any degree of phase coherence using phase-locked loop to isolate the reference phase, there is an important case intermediate between coherent and incoherent reception that has received considerable attention in practical applications. This method is most often called the difference coherent method, and sometimes the phase comparison method. It was developed and used for several years before it was sufficiently analyzed and is now widely used in practice.
Experiments on population transfer seem to provide the key to solving the problem, provided that there is a mechanism for the propagation of population disturbances along the entire chain. In addition, they have some characteristic practical advantages. Pulse distortions lead to the appearance of unwanted transverse magnetization components, but they can be suppressed by phase cycling, pulsed constant field gradients, or the introduction of short random delays. Since only RF pulses are required to create the inverted population, there is no need for phase coherence of the pulses to selectively excite individual transitions. The question comes down to what type of selective excitation of the population is practically available.
After the initial selective 90° pulse, the magnetization of water quickly decays due to its short time Tj, which can be artificially reduced by chemical exchange of the HjO signal with protons of a specially introduced substance, for example, ammonium chloride. If the value of t (see Fig. 13) is longer than Tj, then the magnetization of the solvent quickly loses phase coherence and cannot be refocused by a selective 180° pulse. However, if the value of m is significantly larger, then the magnetization is sufficiently restored along the 2 axis during this time due to spin-lattice relaxation. In this case, the selective 180th pulse inverts the recovering magnetization, and during the second interval t, the magnetization along axis 2 is restored again. The value of m is chosen so that the 2-magnetization of water passes through zero by the end of the second interval X. The degree of solvent signal suppression can be increased by repeating a simple postion (t-180°-t) several times, and then sampling the magnetization of the dissolved spins using compound pulses.
In this case, we can assume that the noise is white, i.e. contains all frequencies, the noise intensity at all these frequencies is the same. However, for biological molecules this condition is not always fulfilled. The value of Tg is always less than Ti, except in a few special cases. This is due to the fact that all processes occurring through the Ti-relaxation mechanism (due to a change in spin orientation during the transition from one energy state to another), accompanied by the transfer or absorption of energy as a result of the interaction of the spin with the lattice, always violate the phase coherence between neighboring spins , and this leads to the emergence of another relaxation channel according to the Tr relaxation mechanism. In this case, the worse the relation (1.36) is satisfied, the more the values of Ti and Tg will differ, and the better the inequality T > Tg will be satisfied. In subsequent sections of the book, we will limit ourselves to considering cases when inequality (1.36) is true (the case of maximum narrowing of lines and T T2).
The shape and width of nuclear resonance lines are significantly influenced by the movements of molecules and atoms, which often occur in solids. With sufficient speed, such movements lead to a narrowing of the resonant absorption line and, if the movements are sufficiently isotropic in space, to a Lorentzian line shape. Below we call this effect kinetic contraction. If the average rotation time or time between transitions of the nuclear spin is less than the phase memory time T, then the nucleus will experience the influence of a whole set of different local fields in a shorter time than T, which is required for the nucleus to break out of phase coherence with other nuclei. This will average the local fields acting on the nuclei in a time shorter than Gg, and, therefore, will narrow the resonance line. Graphically, one can imagine that the nuclei move from one position on the original resonance curve to another in a period shorter than required to pass through the original resonance line.
The least squares method, proposed by Diamond, is based on the accepted idea that coals consist of graphite-like, parallel, but randomly oriented layers with a homogeneous internal structure, connected by disorganized carbon, giving gas dispersion. In the absence of phase coherence between different HHien nBHO Tb scattering units, the scattering from such a system is a linear combination of the intensity functions given by each layer size. The intensity function for a given layer size can be expressed as follows:
Due to the large size of electron pairs, several orders of magnitude larger than the period of the metal crystal lattice, a process of pair synchronization occurs, i.e., phase coherence arises, spreading over the entire volume of the superconductor. A consequence of phase coherence is the properties of a superconductor.
Free spin precession often decays very slowly and can continue for several seconds after the H field is turned off. However, eventually the phase coherence of individual spin vectors is lost for various reasons and the oscillations die out. Many brilliant experiments were built on these effects, in which various spin echoes due to
