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

Introduction

The coherence of light waves plays a big role nowadays, because... Only coherent waves can interfere. Light interference has a wide range of applications. This phenomenon is used for: surface quality control, creating light filters, antireflective coatings, measuring the length of light waves, precise distance measurements, etc. Holography is based on the phenomenon of light interference.

Coherent electromagnetic oscillations in the decimeter-millimeter wavelength range are predominantly used in areas such as radio electronics and communications. But over the past 10-15 years, their use in non-traditional fields has been increasing at an increasingly rapid pace, among which medicine and biology occupy a prominent place.

The purpose of our work is to study the problem of coherence of light waves.

The objectives of this work are:

1. Study the concept of coherence.

2. Study of sources of coherent waves.

3. Identification of areas of science in which this phenomenon is used.

Concept of coherence

Coherence is the coordinated occurrence of several oscillatory or wave processes. The degree of consistency may vary. Accordingly, we can introduce the concept of the degree of coherence of two waves. There are temporal and spatial coherence. We'll start by looking at temporal coherence. Temporal coherence. The interference process described in the previous paragraph is idealized. In reality, this process is much more complex. This is due to the fact that the monochromatic wave described by the expression

where A, and are constants, represent an abstraction. Any real light wave is formed by the superposition of oscillations of all possible frequencies (or wavelengths), contained in a more or less narrow, but finite interval of frequencies (respectively, wavelengths). Even for light that is considered monochromatic (one color), the range of frequencies C is finite. In addition, the amplitude of wave A and phase a undergo continuous random (chaotic) changes over time. Therefore, oscillations excited at a certain point in space by two overlapping light waves have the form

Moreover, chaotic changes in functions are completely independent. For simplicity, we will assume that the amplitudes and a are constant. Changes in frequency and phase can be reduced to either a change in phase alone or a change in frequency alone. Let's imagine the function

where is some average frequency value, and introduce the notation: Then formula (2) will take the form

We have obtained a function in which only the oscillation phase undergoes chaotic changes.

On the other hand, in mathematics it is proven that a non-harmonic function, for example function (2), can be represented as a sum of harmonic functions with frequencies contained in a certain interval (see formula (4)).

Thus, when considering the issue of coherence, two approaches are possible: “phase” and “frequency”. Let's start with the "phase" approach. Let us assume that the frequencies and in formulas (1) satisfy the condition: ==const, and find out what effect the change in phases and has. Under the assumptions made, the intensity of light at a given point is determined by the expression

where The last term in this formula is called the interference term. Any device with which you can observe an interference pattern (eye, photographic plate, etc.) has some inertia. In this regard, it registers a picture averaged over the period of time required for the device to “operate.” If the multiplier takes on all values ​​from -1 to +1 over time, the average value of the interference term will be zero. Therefore, the intensity recorded by the device will be equal to the sum of the intensities created at a given point by each of the waves separately - there is no interference, and we are forced to recognize the waves as incoherent.

If the value remains practically unchanged over time, the device will detect interference, and the waves must be considered coherent.

From the above it follows that the concept of coherence is relative; two waves can behave as coherent when observed with one device (with low inertia) and as incoherent when observed with another device (with greater inertia). To characterize the coherent properties of waves, the coherence time is introduced, which is defined as the time during which a random change in the wave phase (t) reaches an order value. Over time, the oscillation seems to forget its initial phase and becomes incoherent with itself.

Using the concept of coherence time, we can say that in cases where the time constant of the device is much greater than the coherence time of the superimposed waves), the device will not detect interference. If the device detects a clear interference pattern. At intermediate values, the clarity of the picture will decrease as it increases from values ​​smaller to values ​​larger.

The distance a wave travels in time is called the coherence length (or train length). The coherence length is the distance at which a random phase change reaches a value of ~n. To obtain an interference pattern by dividing a natural wave into two parts, it is necessary that the optical path difference be less than the coherence length. This requirement limits the number of visible interference fringes observed in the diagram in Fig. 1.

As the stripe number m increases, the stroke difference increases, as a result of which the accuracy of the stripes becomes worse and worse. Let us move on to clarify the role of non-monochromaticity of light waves. Let us assume that light consists of a sequence of identical trains of frequency and duration. When one train is replaced by another, the phase undergoes random changes, as a result of which the trains turn out to be mutually incoherent. Under these assumptions, the duration of the train practically coincides with the coherence time.

In mathematics, the Fourier theorem is proven, according to which any finite and integrable function F (t) can be represented as the sum of an infinite number of harmonic components with a continuously varying frequency

Expression (4) is called the Fourier integral. The function A () under the integral sign represents the amplitude of the corresponding monochromatic component. According to the theory of Fourier integrals, the analytical form of the function A () is determined by the expression

where is an auxiliary integration variable. Let the function F(t) describe the light disturbance at some point at time t, caused by a single wave train.

Then it is determined by the conditions:

The graph of the real part of this function is given in Fig. 2. Outside the interval from - to +, the function F (t) is equal to zero. Therefore, expression (5), which determines the amplitudes of the harmonic components, has the form

After substituting the limits of integration and simple transformations, we arrive at the formula

The intensity I() of the harmonic component of the wave is proportional to the square of the amplitude, i.e., the expression

The graph of function (6) is shown in Fig. 3. From the figure it is clear that the intensity of the components whose frequencies are in the interval

significantly exceeds the intensity of the other components. This circumstance allows us to relate the duration of the train to the effective frequency range of the Fourier spectrum:

Having identified coherence with time, we arrive at the relation:

From relation (7) it follows that the wider the range of frequencies represented in a given light wave, the shorter the coherence time of this wave. Frequency is related to wavelength in vacuum by the relationship. Differentiating this relation, we find that

(we omitted the minus sign resulting from differentiation; in addition, we put it in). Replacing it in formula (7) with its expression in terms of and, we obtain the expression for the coherence time

This gives the following value for the coherence length:

The path difference at which the mth order maximum is obtained is determined by the relation:

When this path difference reaches a value on the order of the coherence length, the stripes become indistinguishable. Consequently, the maximum observed interference order is determined by the condition:

From (10) it follows that the number of interference fringes observed according to the scheme shown in Fig. 1 increases as the range of wavelengths represented in the light used decreases. Spatial coherence. According to the formula

the spread of frequencies corresponds to the spread of k values. We have established that temporal coherence is determined by meaning. Consequently, temporal coherence is associated with the spread of values ​​of the modulus of the wave vector k. Spatial coherence is associated with the spread of directions of the vector k, which is characterized by magnitude.

The occurrence of oscillations excited by waves with different wavelengths at a certain point in space is possible if these waves are emitted by different parts of an extended (non-point) light source. Let us assume for simplicity that the source has the shape of a disk, visible from a given point at an angle (see Fig. 4), it can be seen that the angle characterizes the interval in which the unit vectors are contained. We will consider this angle small. Let the light from the source fall on two narrow slits, behind which there is a screen (Fig. 5). We will consider the interval of frequencies emitted by the source to be very small so that the degree of temporal coherence is sufficient to obtain a clear interference pattern. The wave coming from the surface area indicated in Fig. 5 through O, creates a zero maximum M in the middle of the screen. The zero maximum M"-, created by the wave, coming from section O", will be shifted from the middle of the screen by a distance x". Due to the smallness of the angle and the ratio d/l, we can assume that x"=/2. The zero maximum M" created by the wave coming from section O" is shifted from the middle of the screen in the opposite direction by a distance x" equal to x". Zero maxima from the remaining sections of the source are located between the maxima M" and M".

Individual sections of the light source excite waves, the phases of which are in no way related to each other. Therefore, the interference pattern that appears on the screen will be a superposition of the patterns created by each of the sections separately. If the displacement x1" is much less than the width of the interference fringe x=l /d, the maxima from different sections of the source will practically overlap each other and the picture will be the same as from a point source. At x"x, the maxima from some sections will coincide with the minima from others, and no interference pattern will be observed. Thus, the interference pattern will be distinguishable provided that x"x, i.e.

When moving from (11) to (12), we omitted the factor 2. Formula (12) determines the angular dimensions of the source at which interference is observed. From this formula one can also determine the maximum distance between the slits at which interference from a source with an angular size can still be observed. Multiplying inequality (12) by d/, we arrive at the condition

A set of waves with different ones can be replaced by a resulting wave incident on a screen with slits. The absence of an interference pattern means that the oscillations excited by this wave at the locations of the first and second slits are incoherent. Consequently, oscillations in the wave itself at points located at a distance d from each other are incoherent. If the source were ideally monochromatic (this means that v = 0 and the surface passing through the slits would be wave and oscillations at all points of this surface would occur in the same phase. We have established that in the case of v0 and finite dimensions of the source () oscillations at surface points separated by a distance are incoherent.

For brevity, we will call a surface that would be a wave surface if the source were monochromatic. We could satisfy condition (12) by reducing the distance between the slits d, i.e., by taking closer points of the pseudo-wave surface. Consequently, the oscillations excited by the wave at fairly close points of the pseudo-wave surface turn out to be coherent. Such coherence is called spatial coherence. So, the phase of oscillation during the transition from one point of the pseudo-wave surface to another changes in a random manner. Let us introduce the distance at which, when displaced along the pseudo-wave surface, the random phase change reaches the value ~. Oscillations at two points of the pseudo-wave surface, spaced from each other by a distance smaller, will be approximately coherent. The distance is called the spatial coherence length or coherence radius. From (13) it follows that

The angular size of the Sun is about 0.01 rad, the length of light waves is approximately 0.5 microns. Consequently, the radius of coherence of light waves coming from the Sun has a value of the order of

0.5/0.01 =50 µm = 0.05 mm. (15)

The entire space occupied by a wave can be divided into parts, in each of which the wave approximately maintains coherence. The volume of such a part of space, called the volume of coherence, is equal in order of magnitude to the product of the length of temporary coherence and the area of ​​a circle of radius. The spatial coherence of a light wave near the surface of the heated body emitting it is limited to a size of only a few wavelengths. As you move away from the source, the degree of spatial coherence increases. Laser radiation has enormous temporal and spatial coherence. At the laser output aperture, spatial coherence is observed throughout the entire cross section of the light beam.

It would seem that interference could be observed by passing light propagating from an arbitrary source through two slits in an opaque screen. However, if the spatial coherence of the wave incident on the slits is low, the light beams passing through the slits will be incoherent, and the interference pattern will be absent.

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.

(from lat. cohaerens - in connection), coordinated flow in time and in several directions. oscillate or waves. processes that manifest themselves when they are added. Oscillations are called coherent if their phase difference remains constant (or changes naturally) over time and, when adding oscillations, determines the amplitude of the total . Harmonic the oscillation is described by the expression:

Р(t)=Acos(wt+j), (1)

where P is a changing quantity (displacement of the pendulum, intensity of electric and magnetic fields, etc.), and amplitude A, frequency co and j are constants. When adding two harmonious oscillations with the same frequency but different amplitudes A1 and A2 and phases j1 and j2 form a harmonic. oscillation of the same frequency. Amplitude of the resulting oscillation

Ar =?(A21+A22+2A1A2cos(j1-j2)) (2)

can vary from A1+A2 to AI-A2 depending on the phase difference j1-j2 (Fig.).

In fact, perfectly harmonious. fluctuations are not feasible. In real fluctuations. processes, amplitude, frequency and can continuously change chaotically in time.

Addition of two harmonious oscillations (dotted line) with amplitudes A1 and A2 at different. phase differences. The resulting swing is a solid line.

If the phases of two oscillations j1 and j2 change randomly, but their difference j1-j2 remains constant, then the amplitude of the total oscillation is determined by the difference in the phases of the added oscillations, i.e. the oscillations are coherent. If the phase difference between two oscillations changes very slowly, then in this case the oscillations remain coherent only for a certain time, until their phase difference has had time to change by an amount comparable to n.

If we compare the phases of the same oscillation at different times, separated by an interval t, then with a sufficiently large t, a random change in the phase of the oscillation can exceed l. This means that through t harmonic. hesitation “forgets” its original. phase and becomes incoherent “to itself”. As the temperature increases, it usually weakens gradually. For quantities, characteristics of this phenomenon, the function R (t) is introduced, called. correlation function. The result of the addition of two oscillations received from the same source and delayed relative to each other by a time t can be represented using R (t) in the form:

Ar = ?(A21+A22+2A1A2R (t)coswt), (3)

where w - avg. oscillation frequency. Function R(t)=1 at t=0 and usually drops to 0 at unlimited. growth t. The value of t, for which R(t) = 0.5, is called. coherence time or harmonic duration. train. After one harmonic a train of oscillations, it seems to be replaced by another with the same frequency, but with a different phase.

Har-r and saints hesitate. process depends significantly on the conditions of its occurrence. For example, emitted by a gas discharge in the form of a narrow . lines, may be close to monochromatic. The radiation of such a source consists of waves sent by different particles independently of each other and therefore with independent phases (spontaneous emission). As a result, the amplitude and phase of the total wave change chaotically with a characteristic time equal to time K. Changes in the amplitude of the total wave are large: from 0, when the original waves cancel each other, to max. values ​​when the phase relationship of the original waves favors their addition. Oscillations that occur in self-oscillations. system, for example in tube or transistor generators, lasers, have a different structure. In the first two, the frequency and phase of oscillations change chaotically, but the resulting amplitude is maintained constant. In a laser, all particles emit in concert (stimulated emission), in phase with the oscillation established in the resonator. The phase relationships of the constituent oscillations are always favorable for the formation of a stable amplitude of the total oscillation. The term "K." sometimes means that the oscillation is generated by self-oscillation. system and has a stable amplitude.

When propagating a flat el.-magn. waves in a homogeneous medium phase of oscillations in Ph.D. defined the production point is maintained only for time K. t0. During this time, the wave propagates over a distance ct0. In this case, oscillations at points located at a distance greater than ct0 from each other along the direction of wave propagation turn out to be incoherent. The distance equal to ct0 along the direction of propagation of a plane wave is called. length K. or train length.

Ideally unfeasible, just like ideally harmonious. hesitation. In real waves. processes, the amplitude 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 the 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 l, disappears. To describe the coherent light of a wave in a plane perpendicular to the direction of its propagation, the terms area and spatial color are used, in contrast to temporal color, which is associated with the degree of monochromaticity of the wave. Quantitative spaces. K. can also be characterized by the correlation function RI(l). The condition Rf(l) = 0.5 determines the size or radius of the wave, which may depend on the orientation of the segment l in a plane perpendicular to the direction of wave propagation. The entire space occupied by the wave can be divided into regions, in each of which the wave retains K. The volume of such a region (volume K) is taken equal to the product of the length of the train and the area of ​​the figure bounded by the curve RI(l) = 0.5RI (0).

Violation of spaces. K. is associated with the peculiarities of the processes of radiation and wave formation. For example, a heated body emits a set of spherical particles. waves propagating in all directions. As the wave moves away from a heat source of finite dimensions, it approaches a flat wave. At large distances from the source, the size of the K is equal to l.22lr/r, where r is the distance to the source, r is the size of the source. For sunny light size K. is 30 microns. With a decrease in atl. the size of the source, the size of K. increases. This makes it possible to determine the size of stars by the size of the area of ​​light coming from them. The value l/r is called angle K. With distance from the source, the light intensity decreases proportionally. 1/r2. Therefore, with the help of a heated body it is impossible to obtain an intense energy with a large space. K. The light wave emitted by the laser is formed as a result of stimulated emission throughout the entire volume of the active substance. Therefore, spaces. The radiation efficiency of laser radiation is maintained throughout the entire cross section of the beam.

The concept of “K.”, which originally arose in classical. in optics as a characteristic that determines the ability of light to interfere (see INTERFERENCE OF LIGHT), is widely used in describing oscillations and waves of any nature. Thanks quant. mechanics that propagated the waves. ideas for everything in the microcosm, the concept of “K.” began to be applied to beams of electrons, protons, neutrons, etc. Here, K. is understood as ordered, coordinated and directed movements of a large number of quasi-independent parts. The concept of "K." also penetrated into TV theory. bodies (for example, hypersonic phonons, (see HYPERSOUND)) and quantum. liquids. After the discovery of superfluidity of liquid helium, the concept of “K” appeared, meaning that it is macroscopic. The number of atoms of liquid superfluid helium can be described by a single wave. f-tion, having one of its own. meaning as if it were one h-ts, and not an ensemble of a huge number of interacting h-ts.

Physical encyclopedic dictionary. - M.: Soviet Encyclopedia. . 1983 .

(from Latin cohaerens - in connection) - correlated flow in time and space of several. random fluctuations. or wave processes, which makes it possible to obtain a clear interference when they are added. picture. Initially, the concept of K. arose in optics, but it refers to wave fields of any nature: electric-magnetic. waves of an arbitrary range, elastic waves, waves in plasma, quantum mechanical. probability amplitude waves, etc.

The existence of interference. the painting is a direct consequence superposition principle for linear oscillations and waves. However, in real conditions there are always chaotic conditions. wave field, in particular the phase difference of interacting waves, which leads to the rapid movement of interference. paintings in space. If through each point during the measurement time the maxima and minima of the interference manage to pass multiple times. paintings, then registered Wed. the value of the wave intensity will be different. points are the same and interference. the stripes will blur. To register clear interference. picture, such stability of random phase relationships is necessary, with a cut displacement interference. stripes during the measurement time is only a small part of their width. Therefore, quality. the concept of K. can be defined as the necessary stability of random phase relationships during the recording of interference. paintings.

Such qualities. the concept of K. in a number of cases turns out to be inconvenient or insufficient. For example, when different methods of recording interference. In the picture, it may turn out that the time required for this is different, so that a wave that is coherent according to the results of one experiment is incoherent according to the results of another. In this regard, it is convenient to have quantities. a measure of the degree of coherence, independent of the method of measuring interference. paintings.

If the wave is described using a complex amplitude , so it can be, for example, analytical signal], then the second-order mutual coherence function Г 2 is defined as cf. meaning:

The bar on top indicates statistical data. averaging over fluctuations of the wave field, and both the phase and amplitude of the wave can fluctuate; * means complex conjugation. Random (instantaneous) intensity (energy) of a wave proportional. size. Her cf. the meaning is associated with G 2 f-loy. Wed. the energy flux density vector S is also expressed through Г":

For a multicomponent (eg, electric-magnetic) field, the scalar function Г 2 is replaced by a tensor of the second rank. If the total wave field at a certain point is the result of the addition of the original fields

Then his cf. intensity is expressed through and 1 And and 2 Floy

Size

called complex degree of coherence of s-t and fields at space-time points

AND . From (3) it follows that

Interference clarity paintings are directly related to magnitude. If the intensities of the interfering beams are the same (which can always be achieved in experiment), i.e., then based on (2) we can write

If represented in the form , then = =. Usually within the interference limits. the picture changes much less than cos j. In this case, the maximums of the distribution correspond to those places where , and the minimums correspond to the values ​​of , then , , and for relates. Mon-terferential contrast paintings (its "appearances")

we get

Thus, "" interference. the picture is directly expressed through the degree of coherence, that is, ultimately through the function G 2. Maximum clear interference. the picture, in the cut, corresponds to the meaning. Completely washed out interference. a picture in which , corresponds

The value can be directly measured using relation (4), if we first ensure the equality of cf. intensities. The value determines the interference offset. stripes

From the definition it follows that the degree of coherence is maximum when the observation points are combined: . The characteristic scale of decline in the variable function is called. coherence time. If, when wave fields are superimposed, the time between them is small compared to , then clear interference can be obtained. painting. Otherwise, interference will not be observed. The value also limits the interference measurement time. paintings, which were mentioned above. Quantity where With - the speed of propagation of the wave of the type under consideration, called. longitudinal coherence radius (coherence length).

If we consider a wave beam with a clearly defined direction of propagation, then when the observation points are spaced across this direction, the function will also decrease. The characteristic scale of decline in this case is called. transverse coherence radius r 0. This value characterizes the size of those sections of the wave front from which clear interference can be obtained. painting. As the wave propagates in a homogeneous medium, the value r 0 increases due to diffraction (see. Van Zittert-Zernike theorem). The product characterizes the volume of coherence, within which the random phase of the wave changes by an amount not exceeding

The effects of wave fields can also be studied indirectly by studying the correlation of fluctuations in instantaneous intensity I. In this case, the measurement time should be short compared to , and the transverse size of the detector should be small compared to r 0 . Correlative intensity fluctuation function -

Can be found if, along with G 2, the fourth-order K function is also known:

If u(r, t) is Gaussian (for example, created by a heat source), and (but, of course,), then G 4 can be expressed through G 2 according to the formulas valid for Gaussian random fields:

Therefore, for Gaussian wave fields, measurements of the quantity B I can provide information about the degree module K. (see. Intensity interferometer). In the general case of measuring the intensity of the wave field in P points, to describe the results of the experiment, it is enough to know the function of K. order 2p:

The same functions describe the results of experiments on the statistics of photocounts, when the correlations of the numbers of photons registered in different types are measured. points r 1 , . .., r p.

Quantum ones can significantly distort the results of interference. experience, if the total number of photons recorded at the maximum interference. paintings, small. Because when implementing interference. experiment, you can collect radiation from an area of ​​the order of magnitude and carry out measurements over time, then all photons from the volume will be used, i.e. from the volume of coherence. If wed. number N photons in the volume of quantum, called the degeneracy parameter, is large, then quantum fluctuations in the number of recorded photons are relatively small () and do not have any significant influence on the measurement result. If N is small, then these fluctuations will interfere with measurements.

The term "K." is also used in a broader sense. Thus, in quantum mechanics, for which a minimum is realized in uncertainty in the relationship, called coherent states. In decomposition areas of physics the term "K." used to describe correlates. behavior of a large number of particles (as occurs, for example, when superfluidity). The term "coherent structures" in various. fields of science is used to designate spontaneously emerging stable formations that retain certain regular properties against a chaotic background. fluctuations.

Lit.: Wolf E., Mandel L., Coherent properties of optical fields, trans. from English, "UFN", 1965, vol. 87, p. 491; 1966, vol. 88, p. 347, 619; O" N e i l E., Introduction to statistical optics, translated from English, M., 1966; Born M., Wolf E. Fundamentals of optics, translated from English, 2nd ed., M., 1973; Klauder J. Sudarshan E., Fundamentals of Quantum Optics, translated from English M., 1970; Perina Y., Coherence of Light, translated from English M., 1974. V. I. Tatarsky

Physical encyclopedia. In 5 volumes. - M.: Soviet Encyclopedia. Editor-in-chief A. M. Prokhorov. 1988 .

Synonyms:

COHERENCE(from Latin cohaerentio - connection, cohesion) - the coordinated occurrence in space and time of several oscillatory or wave processes, in which the difference in their phases remains constant. This means that waves (sound, light, waves on the surface of water, etc.) propagate synchronously, lagging one another by a very certain amount. When adding coherent oscillations, a interference; the amplitude of the total oscillations is determined by the phase difference.

Harmonic oscillations are described by the expression

A(t) = A 0cos( w t + j),

Where A 0 – initial vibration amplitude, A(t) – amplitude at the moment of time t, w– oscillation frequency, j – its phase.

Oscillations are coherent if their phases j 1, j 2 ... change randomly, but their difference is D j = j 1 – j 2 ... remains constant. If the phase difference changes, the oscillations remain coherent until it becomes comparable in magnitude to p.

Propagating from the source of oscillations, the wave after some time t can “forget” the original meaning of its phase and become incoherent with itself. The phase change usually occurs gradually, and time t 0, during which the value of D j there is less left p, is called temporal coherence. Its value is directly related to the reliability of the oscillation source: the more stable it operates, the greater the temporal coherence of the oscillation.

During t 0 wave, moving at speed With, travels the distance l = t 0c, which is called the coherence length, or train length, that is, a wave segment that has a constant phase. In a real plane wave, the phase of oscillations changes not only along the direction of wave propagation, but also in a plane perpendicular to it. In this case, they speak of spatial coherence of the wave.

The first definition of coherence was given by Thomas Young in 1801 when describing the laws of interference of light passing through two slits: “two parts of the same light interfere.” The essence of this definition is as follows.

Conventional optical radiation sources consist of many atoms, ions, or molecules that spontaneously emit photons. Each act of emission lasts 10 –5 – 10 –8 seconds; they follow randomly and with randomly distributed phases both in space and time. Such radiation is incoherent; the average sum of all oscillations is observed on the screen illuminated by it, and there is no interference pattern. Therefore, to obtain interference from a conventional light source, its beam is bifurcated using a pair of slits, a biprism or mirrors placed at a slight angle to one another, and then both parts are brought together. In fact, here we are talking about consistency, coherence of two rays of one act of radiation occurring randomly.

The coherence of laser radiation has a different nature. Atoms (ions, molecules) of the active substance of the laser emit stimulated radiation caused by the passage of an extraneous photon, “in time”, with identical phases equal to the phase of the primary, forcing radiation ( cm. LASER).

In the broadest interpretation, coherence today is understood as the joint occurrence of two or more random processes in quantum mechanics, acoustics, radiophysics, etc.

Sergei Trankovsiy