High resolution NMR spectroscopy in organic and organometallic chemistry. Coherent waves Coherence definition
Phase coherence
Experimental confirmation (14). It follows from this equation that it is possible to increase, either by increasing the amplitude of the component B i, or by increasing the duration of the pulse. In the experiments of Fribolin with the H 2 O sample, the B i kept constant. It follows from the obtained results that the amplitude maximum corresponds to
As for the populations of the energy levels, the situation is reversed at , and there will be more nuclei at the upper energy level than at the lower one.
For , we have a more complicated situation, because M2 = 0 and both Zeeman levels are populated in the same way. This case differs from saturation, because in this situation we have M y" present, but not when saturated. The appearance of transverse magnetization in this case is explained by the fact that under the influence of B1 nuclear dipoles do not precess around the double cone uniformly, but form "rare" and "dense" fractions that precess "in phase". This phenomenon is called "phase coherence". |
coherenceь (from the Latin cohaerens - being in connection) is considered as a coordinated flow in time of several oscillatory or wave processes, which manifests itself when they are added. Oscillations are called coherent if the difference between their phases remains constant in time and, when the oscillations are added, determines the amplitude of the total oscillation. Two harmonic (sinusoidal) oscillations of the same frequency are coherent.
When adding two harmonic oscillations with the same frequency, but with different amplitudes A 1 and A 2 and phases φ 1 and φ 2 a harmonic oscillation of the same frequency ν :
where the amplitude of the resulting oscillations
and the phase shift
The amplitude of the resulting oscillations can vary from A 1 + A 2 to A 1 - A 2 depending on the phase difference φ 1 - φ 1 .
Coherence manifests itself as a property of two (or more) oscillatory processes, capable of mutually reinforcing or weakening the effect of interaction when added.
The stimulated emission of photons has significant features. First, the frequency of a quantum of light emitted under the action of an external monochromatic field coincides exactly with the frequency of the external field. Secondly, the direction of propagation and polarization of the emitted photon coincide with the direction of propagation and polarization of the external electromagnetic field causing the radiation. Thus, the radiation of individual elementary emitters under the action of a common external field will be coherent. These features of stimulated emission of light quanta are characteristic of the active medium of lasers and are effectively used to amplify and form powerful monochromatic radiation.
To explain the concept of coherence, it is convenient to use the wave representation of light. On fig. 6 the radiation is shown as "elementary waves" originating in the active medium; they are commonly referred to as tsugami. The situation in fig. 3.13a corresponds to incoherent light, and in fig. 3.13b - ideally coherent. In the latter case, all wave trains propagate in the same direction, have the same wavelength, and are in phase with each other. All this is a consequence of the forced emission of light. With stimulated emission, the secondary train exactly copies the primary train in the direction of propagation, in wavelength, in phase. On fig. 3.13b, the dashed line shows the surface of the same phase (wave front).
Figure.3.13 Scheme of propagation of incoherent (a) and coherent (b) light
The coherence of the laser beam manifests itself, in particular, in an extremely high degree of its monochromaticity, as well as in a very small divergence of the laser beam.
Orientation
Directivity is one of the main properties of laser radiation. Directional is radiation that propagates within a small solid angle.
A measure of the parallelism of radiation is the divergence of the laser beam.
Divergence of laser radiation- it's flat θ or a solid angle with a vertex coinciding with the point of intersection of the resonator axis with the waist plane.
This divergence is also called angular. The spatial parameters of the laser beam are obtained experimentally or calculated from the known parameters of the resonator. The relationship between the beam parameters and the resonator parameters is determined by the resonator type.
On fig. 3.14 shows a confocal resonator, consisting of two mirrors 1, 2 with radii r 1 and r 2, respectively. In the case of r 1 = r 2, the radiation waist will be in the center of the resonator, its diameter (for single-mode radiation) is determined by the expression:
where = 2 /λ - wave number; d is the length of the resonator.
The radiation diameter at a distance z from the waist is expressed by the formula:
.
Figure 3.14 - Diagram of a confocal resonator
Beam divergence for a uniform energy distribution, which corresponds to the multimode nature of radiation, is determined by the equality:
where 2y is the aperture size on the output mirror; k Ф - coefficient depending on the distribution of energy and the shape of the active element.
With a uniform distribution of energy for a circular aperture, k Ф = 1, for a Gaussian beam, k Ф = 1.22.
Without the use of additional optical systems, the divergence of gas lasers is a few arc minutes, solid-state lasers up to several tens of minutes, and semiconductor lasers up to tens of degrees.
The beam divergence can be reduced by collimating it with focusing of the laser beam (a small-diameter diaphragm is placed at the focus of the optical system - a spatial filter) and without focusing the laser beam - by passing the beam through a telescope (Fig. 3.15), which converts the parallel beam of rays entering the system , also into a parallel beam of rays at its exit with an increased aperture (diameter) of the beam.
Figure 3.15 - Beam collimation with a two-lens telescope
In this case, the divergence of laser radiation is inversely proportional to the increase β used telescope ( β =D2/D1):
where 1,2 - beam divergence at the entrance to the telescope and at the exit from it, respectively; D 1 , D 2 are the beam diameters at the entrance to the telescope and at the exit from it, respectively. In this case, the laser beam must completely fill the telescope.
The minimum achievable value of the divergence is determined by the diffraction phenomena of the optical wavefront at the output component of the collimating system.
In the technical specification (passport), the angle 2θ is usually indicated as the divergence.
Intensity
The concept of intensity is used to evaluate the photometric quantities by which laser radiation is characterized: radiation strength, brightness, flux, etc. For large values of these quantities, it is usually stated that the radiation is intense. The laser radiation, due to the high degree of directivity of the radiation, can be intense even when the radiation power is relatively low.
The strength of the laser radiation characterizes the spatial density of the radiation flux, that is, the magnitude of the radiant flux per unit solid angle in which the radiation propagates, and is determined by the formula:
where Fe is the radiation power, W; Ω=α 2 - solid angle, erased; α is the aperture angle of the cone that forms the solid angle, rad.
For single-mode laser radiation, the divergence of which is 2θ (the solid angle is respectively equal to α = 4θ 2), the radiation force in the direction characterized by the aperture angle 2θ to the axis is equal to
If we compare, for example, an incandescent lamp and a laser in terms of radiation strength, then with the same power consumption, lasers turn out to be more intense, having a lower efficiency. For example, a 66 W incandescent lamp has an average radiant power
and a laser of the LG-55 type with a power consumption of 66 W, a radiation power of 2 10 -3 W, and a divergence of 10" is characterized by the radiation strength
W/ster.
Radiation flux (laser power) Fe represents the energy of stimulated radiation (generation energy) passing through the cross section per unit time: Fe = dQe/dt. If the radiation occurs in the fundamental mode, then the magnitude of the flux Fe is determined by the ratio of the radius of the considered section r and the size of the mode spot ω:
where F 0 is the total laser flux measured at r>>ω.
The transition of the energy value of the flow (W) to light (lm) is carried out according to the formula
F=638Fe,
where 683 lm/W is the light equivalent of radiant energy at a wavelength corresponding to the maximum sensitivity of the eye (λ = 0.55 µm).
The transition from the lighting value of the flow to the energy value is carried out according to the formula
Ph=AF,
where A = 0.00146 W/lm is the mechanical equivalent of light (A = 1/683).
With pulsed radiation, the regime of a regular sequence of pulses is characterized by an average radiation flux, that is, the average value of the flux over a given period of time:
Фср=Фи∆t/T,
where fi - flow in impulse; ∆t - pulse duration; T is the pulse repetition period.
In prepress processes, when recording an image, the intensity of the laser beam is controlled according to the “yes - no” principle, in which the intensity changes from a maximum value to zero, to form printing or blank elements of the form, as well as to bring the intensity into line with the light or thermal sensitivity of the recorded materials. To control the intensity, special devices are used - radiation modulators.
(from lat. cohaerens - being in connection), a coordinated flow in time and in the pr-ve several. oscillating or waves. processes that appear when they are added. Fluctuations called. coherent if the difference between their phases remains constant (or regularly changes) in time and, when the oscillations are added, determines the amplitude of the total . harmonic oscillation is described by the expression:
Р(t)=Acos(wt+j), (1)
where P is a changing value (pendulum displacement, electric and magnetic fields, etc.), and the amplitude A, frequency co and j are constants. When adding two harmonics. oscillations with the same frequency w, but different amplitudes A1 and A2 and phases j1 and j2, a harmonic is formed. oscillation of the same frequency. The amplitude of the resulting oscillation
Ap =?(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 randomly change in time.
The addition of two harmonics. oscillations (dotted line) with amplitudes A1 and A2 at diff. phase differences. The resulting oscillation 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 phase difference of the added oscillations, i.e., the oscillations are coherent. If the phase difference of 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 m, then for a sufficiently large m, the random change in the phase of the oscillation can exceed n. This means that through m harmonic. oscillation "forgets" its original. phase and becomes incoherent "to itself". With growth t To. usually weakens gradually. For quantities, the characteristics of this phenomenon are introduced by the function R (t), called. correlation function. The result of adding two oscillations received from one source and delayed relative to each other for a time t can be represented using R (t) as:
Ap = ?(A21+A22+2A1A2R (t)coswt) , (3)
where w - cf. oscillation frequency. Function R(t)=1 at t=0 and usually drops to 0 at unlimited. growth t. The value of t, with R(t)=0.5, called. coherence time or harmonic duration. train. After one harmonic of a train of oscillations, it is, as it were, replaced by another with the same frequency, but with a different phase.
Har-r and sv-va fluctuate. processes depend 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 decomp. 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 the time K. Changes in the amplitude of the total wave are large: from 0, when the initial waves cancel each other, to max. values when the ratio of the phases of the initial waves favors their addition. Oscillations arising 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 randomly, but the resulting amplitude is kept constant. In a laser, all particles emit in a coordinated manner (stimulated emission), in phase with the oscillation established in the resonator. The phase ratios 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-oscillations. system and has a stable amplitude.
When spreading a flat el.-mag. waves in a homogeneous medium, the phase of oscillations in k.-n. def. point pr-va is preserved only during the time K. t0. During this time, the wave propagates over a distance ct0. In this case, oscillations at points separated by a distance greater than ct0 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, called. length K. or train length.
Ideally unrealizable, as well as 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 the 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 oscillation coefficient weakens at these points and disappears at a certain distance l, when the random changes in the phase difference become comparable with i. To describe the coherent light waves in a plane perpendicular to the direction of its propagation, the terms area and spatial waves are used, in contrast to temporal waves, which are related to the degree of monochromaticity of the wave. Number of 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 K., which may depend on the orientation of the segment l in the plane perpendicular to the direction of wave propagation. The entire production occupied by the wave can be divided into regions, in each of which the wave retains K. The volume of such an area (the volume of K.) is taken equal to the product of the train length and the area of \u200b\u200bthe figure bounded by the curve RI (l) \u003d 0.5 RI (0).
Violation of spaces. K. is associated with the features of the processes of radiation and the formation of waves. For example, a heated body radiates a set of spherical waves propagating in all directions. As we move away from the thermal source of finite dimensions, the wave approaches a plane 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 solar light size K. is 30 microns. With a decrease in angle. the size of the source, the size of K. grows. This makes it possible to determine the size of stars by the size of the area K. of the light coming from them. The value of l / r called. angle K. With distance from the source, the light intensity decreases in proportion. 1/r2. Therefore, with the help of a heated body, it is impossible to obtain an intense one that has a large space. K. The light wave emitted by a laser is formed as a result of stimulated emission in the entire volume of the active substance. Therefore spaces. The coherence of laser radiation is preserved over the entire cross section of the beam.
The concept of "K.", which originally arose in the classic. optics as a characteristic that determines the ability of light to interfere (see LIGHT INTERFERENCE), is widely used in describing oscillations and waves of any nature. Thanks Quantum. mechanics that spread the waves. representations for everything in the microcosm, the concept of "K." began to be applied to beams of electrons, protons, neutrons, and others. Here, K. is understood as ordered, coordinated and directed movements of a large number of quasi-independent ch-ts. The concept of "K." also penetrated into the theory of TV. bodies (eg, hypersonic phonons, (see HYPERSOUND)) and quantum. liquids. After the discovery of the superfluidity of liquid helium, the concept of “K” appeared, meaning that macroscopic. number of atoms of liquid superfluid helium can be described by a single wave. f-tion, which has one property. meaning, as if it were one ch-tsa, and not an ensemble of a huge number of interacting ch-ts.
Physical Encyclopedic Dictionary. - M.: Soviet Encyclopedia. . 1983 .
(from lat. cohaerens - in connection) - a 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: el.-magn. waves of an arbitrary range, elastic waves, waves in plasma, quantum mechanical. probability amplitude waves, etc.
The existence of interference. pictures is a direct consequence superposition principle for linear oscillations and waves. However, in real conditions, there are always chaotic. wave field, in particular the phase difference of the interacting waves, which leads to a rapid movement of interference. pictures in space. If through each point during the measurement time to repeatedly pass the maxima and minima of the interference. paintings, then registered cf. the value of the wave intensity will be in decomp. points identical and interference. the stripes will fade. To register a clear interference. picture, such a stability of random phase relationships is necessary, with a swarm of interference interference. fringes during the measurement time is only a small part of their width. Therefore, qualities. the concept of K. can be defined as the necessary stability of random phase relationships during the registration of interference. paintings.
Such qualities. the concept of K. in a number of cases turns out to be inconvenient or insufficient. For example, at different methods of registering interference. The picture 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 waveform is described by the 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 overline denotes the statistic. averaging over fluctuations of the wave field, and both the phase and amplitude of the wave can fluctuate; * means complex conjugate. Random (instantaneous) intensity (energy) of the wave is proportional. value . Her cf. the value is associated with G 2 f-loy. Wed the energy flux density vector S is also expressed in terms of Г":
For a multicomponent (eg, el.-magnet.) field, the scalar function Г 2 is replaced by a tensor of the second rank. If the total wave field at some point is the result of adding the initial fields
Then his cf. intensity is expressed through and 1 and and 2 Floy
the value
called complex degree coherently with-t and fields at space-time points
AND . From (3) it follows that
interference clarity. pattern is directly related to the value of . If the intensities of the interfering beams are the same (which can always be achieved experimentally), i.e., then, based on (2), we can write
If presented in the form 
, then = =. Usually within the interference. the picture changes much less than cos j. In this case, the maxima of the distribution correspond to those places where , and the minima correspond to the values , then , , and for relates. contrast mon-terf. paintings (her "visibility")
we get
Thus, "" interference. the picture is directly expressed through the degree of coherence, i.e., ultimately through the function G 2 . The most clear interference. the picture, in which , corresponds to the value . Completely washed out interference. picture, in which 
, corresponds
The value can be directly measured using relation (4), if we first ensure the equality cf. intensities . The value determines the interference offset. stripes.
It follows from the definition that the degree of coherence is maximum when the observation points are combined: . The characteristic scale of the decline in the function of the variable is called. coherence time. If, when wave fields are superimposed, the time between them is small compared to , then a clear interference can be generated. painting. Otherwise, interference will not be observed. The value also limits the interference measurement time. paintings, which was mentioned above. Value , 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 distinguished direction of propagation, then when the observation points are separated across this direction, the function will also decrease. The characteristic scale of the decline in this case is called. transverse coherence radius r0. This value characterizes the size of those sections of the wave front, from which a clear interference can be obtained. painting. As the wave propagates in a homogeneous medium, the quantity r0 increases due to diffraction (see Fig. Van Cittert-Zernike theorem). The product characterizes the volume of coherence, within which the random phase of the wave changes by an amount not exceeding
Wave fields can also be studied indirectly by studying the correlation of fluctuations of the instantaneous intensity I. In this case, the measurement time should be small compared to , and the transverse size of the detector should be small compared to r 0 . Correlation. f-tion intensity fluctuations -
It can be found if, along with G 2, the function of K. of the fourth order is also known:
If a u(r, t) is Gaussian (e.g., created by a heat source), and 
(but, of course, ), then Г 4 can be expressed in terms of Г 2 using f-lams valid for Gaussian random fields:
Therefore, for Gaussian wave fields, measurements of the quantity B I can give information about the module of the degree 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 decomp. points r 1 , . .., r p.
Quantum ones can significantly distort the results of interference. experience, if the total number of photons registered in the maximum interference. pictures are small. Because in the implementation of interference. experience, you can collect radiation from an area of the order of magnitude , and carry out measurements over time , then all photons from the volume , i.e., from the volume of coherence, will be used. If cf. number N photons in the volume of K., called the degeneracy parameter, is large, then the quantum fluctuations in the number of registered photons are relatively small () and do not affect the measurement result. If N small, these fluctuations will interfere with measurements.
The term "K." is also used in a broader sense. Thus, in quantum mechanics , for which the minimum is realized in ratio uncertainties, called coherent states. In decomp. areas of physics the term "K." used to describe correlatives. behavior of a large number of particles (as is the case, for example, when superfluidity). The term "coherent structures" in decomp. fields of science is used to refer to spontaneously emerging stable formations that retain certain regular properties against the background of chaotic. fluctuations.
Lit.: Wolf E., Mandel L., Coherent properties of optical fields, trans. from English, UFN, 1965, v. 87, p. 491; 1966, v. 88, p. 347, 619; O "Ne and 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; Clauder J. Sudarshan, E., Fundamentals of Quantum Optics, translated from English, Moscow, 1970; Perina, Ya., Coherence of Light, translated from English, Moscow, 1974. V. I. Tatarsky
Physical encyclopedia. In 5 volumes. - M.: Soviet Encyclopedia. Editor-in-Chief A. M. Prokhorov. 1988 .
Synonyms:
The result of adding two harmonic oscillations depends on the phase difference, which changes when moving to another spatial point. There are two options:
1) If both oscillations are not coordinated with each other, i.e. the phase difference changes over time in an arbitrary way, then such oscillations are called incoherent. In real oscillatory processes, due to continuous chaotic (random) changes and the time average value , i.e. the chaotic change of such instantaneous pictures is not perceived by the eye and a feeling is created of an even stream of light that does not change in time. Therefore, the amplitude of the resulting oscillation is 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 (coupled).
In the general case, waves of the same frequency are called coherent, for which the phase difference is .
In the case of a 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 .
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) .
The interference manifests itself more clearly when the intensities of the combined 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, the light flux is redistributed in space, as a result of which maxima appear in some places, and intensity minima in others. This phenomenon is called the interference of light waves.
Interference is characteristic of waves of any nature. Interference can be observed especially clearly, for example, for waves on the surface of water or sound waves. The interference of light waves in everyday life is not so common, since its observation requires certain conditions, since, firstly, ordinary light, natural light is not a monochromatic (fixed frequency) source. Secondly, ordinary 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, it is necessary that the superimposed waves be coherent.
Coherence is the coordinated flow in time and space of several oscillatory or wave processes, which manifests itself when they are added. 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 "descended" from one source). Then, after passing through various optical paths, these waves are somehow superimposed on each other and interference is observed.
Let two point coherent light sources and emit monochromatic light (Fig. 6.4). They must satisfy the coherence conditions:
To the point P the first beam passes through a medium with a refractive index path, the second beam passes through a medium with a refractive index - path. The distances and from the sources to the observed point are called the geometric lengths of the paths of the rays. The product of the refractive index of the 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 a point P swing:
and the second beam is oscillation
Phase difference of oscillations excited by rays at a point P, will be equal to:
Because (- 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 rays that should be taken into account, i.e. refractive indices of the media in which the rays propagate.
It can be seen from the expression for the phase difference 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 interference order. Therefore, this condition is the condition of the interference maximum.
If the optical path difference is equal to half an integer number of wavelengths in vacuum
then , so that oscillations at the point P are in antiphase. This is the interference minimum condition.
So, if an even number of half-wavelengths fits on a length equal to the optical path difference , then at a given point on the screen, an intensity maximum is observed. If an odd number of half-wavelengths fit along the length of the optical difference in the path of the rays, then a minimum of illumination is observed at a given point on the screen.
If two paths of rays are optically equivalent, they are called tautochronous, and optical systems - lenses, mirrors - satisfy the tautochronism condition.
Coherence is the coordinated flow 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.
Distinguish between time y and spatial coherence. We 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 complicated. This is due to the fact that the monochromatic wave described by the expression
where are constants, is an abstraction. Any real light wave is formed by the superposition of oscillations of various frequencies (or wavelengths), enclosed in a more or less narrow, but finite frequency interval (respectively, wavelengths). Even for quasi-monochromatic light (see p. 327) the frequency interval is finite. In addition, the amplitude of wave A and phase a undergo continuous random (chaotic) changes over time. Therefore, oscillations excited at some point in space by two superimposed light waves have the form
moreover, the chaotic changes of the functions are completely independent.
where is some average value of the frequency, and we introduce the notation: Then the formula (120.2) takes the form
We have obtained a function in which only the oscillation phase undergoes chaotic changes.
On the other hand, in mathematics, it is proved that a non-harmonic function, for example, function (120.2), can be represented as a sum of harmonic functions with frequencies contained in a certain Leo interval (see formula (120.4)).
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 in formulas (120.1) satisfy the condition: , and find out what effect the phase change has. In accordance with formula (119.2), the light intensity 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 the interference pattern (eye, photographic plate, etc.) has a certain inertia. In this regard, it registers a picture averaged over a certain period of time. If for Time the multiplier takes all values from -1 to the average value of the interference term will be equal to 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. If the value changes little over time, the device will detect interference.
Let some quantity x change in jumps, equal, and the increments are equiprobable. This behavior of the quantity is called random walks. Let's set the initial value to zero. If after N steps the value is equal, then after the step it will be equal, and both signs are equally likely. Assume that random walks are performed many times, starting each time and find the average value It is equal (the doubled product disappears when averaging). Therefore, regardless of the value of N, the mean value increases by Therefore. Thus, a quantity that performs random walks, on average, is increasingly moving away from its original value.
The phase of a wave formed by the superposition of a huge number of trains generated by individual atoms cannot make large jumps. It changes randomly in small steps, i.e., it performs random walks. The time it takes for a random change in the phase of a wave to reach an order of magnitude is called the coherence time. During this time, the oscillation, as it were, forgets its initial phase and becomes incoherent with respect to itself.
As an example, we indicate that quasi-monochromatic light containing wavelengths in the interval , is characterized by the order of c. The radiation of a helium-neon laser is of the order of c.
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. To obtain an interference pattern by dividing a natural wave into two parts, it is necessary that the optical path difference A be less than the coherence length. This requirement limits the number of visible interference fringes observed in the scheme shown in Fig. 119.2. As the band number increases, the path difference increases, as a result of which the clarity of the bands becomes worse and worse.
Let us proceed to the elucidation of the role of the nonmonochromaticity 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 train duration practically coincides with the coherence time.
In mathematics, the Fourier theorem is proved, according to which any finite and integrable function can be represented as the sum of an infinite number of harmonic components with a continuously changing frequency:
(120.4)
Expression (120.4) is called the Fourier integral. The function under the integral sign is the amplitude of the corresponding monochromatic component. According to the theory of Fourier integrals, the analytic form of a function is determined by the expression
(120.5)
where is an auxiliary integration variable.
Let the function describe a light perturbation at some point at a moment of time 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. 120.1.
Outside the interval from to the function is equal to zero. Therefore, expression (120.5), 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 of the harmonic component of the wave is proportional to the square of the amplitude, i.e., to the expression
The graph of the function (120.6) is shown in fig. 120.2. It can be seen from the figure that the intensity of the components, the frequencies of which are included in the interval, significantly exceeds the intensity of the other components.
This circumstance makes it possible to relate the train duration to the effective frequency range of the Fourier spectrum:
Identifying with the coherence time, we arrive at the relation
(the sign means: "in order of magnitude equals").
It follows from relation (120.7) that the wider the interval of frequencies represented in a given light wave, the shorter the coherence time of this wave.
The frequency is related to the wavelength in vacuum by the relation. Differentiating this relation, we find that (we omitted the minus sign obtained by differentiation, in addition, we put ). Replacing it in formula (120.7) with its expression in terms of X and , we obtain the expression for the coherence time
From here, the following value is obtained for the coherence length:
From formula (119.5) it follows that the path difference, at which the maximum of the mth order is obtained, is determined by the relation
When this path difference reaches a value on the order of the coherence length, the fringes become indistinguishable. Therefore, the limiting observed order of interference is determined by the condition
(120.10)
From (120.10) it follows that the number of interference fringes observed according to the scheme shown in Fig. 119.2, increases with decreasing range of wavelengths represented in the light used.
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 the value of ). Consequently, temporal coherence is related to the spread of the modulus of the wave vector k. Spatial coherence is related to the spread of directions of the vector k, which is characterized by the value De.
The occurrence at some point in space of oscillations excited by waves with different , is possible if these waves are emitted by different parts of an extended (non-point) light source. Assume for simplicity that the source has the shape of a disk, visible from a given point at an angle . From fig. 120.3 it can be seen that the angle characterizes the interval in which the unit vectors e are enclosed. We will consider this angle small.
Let the light from the source fall on two narrow slits, behind which there is a screen (Fig. 120.4).
The frequency interval emitted by the source will be considered very small so that the degree of temporal coherence is sufficient to obtain a clear interference pattern. The wave that came from the surface area indicated in Fig. 120.4 through O, creates a zero maximum M in the middle of the screen. The zero maximum created by the wave that came from section O will be displaced from the middle of the screen by a distance x. Due to the smallness of the angle and ratio, we can assume that
The zero maximum created by the wave that came from the site is shifted from the middle of the screen in the opposite direction by a distance equal to x. The zero maxima from the remaining parts of the source are located between the maxima and
Separate sections of the light source excite waves whose phases are in no way connected with each other. Therefore, the interference pattern that appears on the screen will be an overlay of patterns created by each of the sections separately. If the shift x is much less than the width of the interference fringe (see formula (119.10)), the maxima from different parts of the source will practically overlap each other and the picture will be the same as from a point source. At , the maxima from some sections will fall on the minima from others, and the interference pattern will not be observed. Thus, the interference pattern will be distinguishable provided that i.e.
When passing from (120.11) to (120.12), we omitted the factor 2.
Formula (120.12) determines the angular dimensions of the source at which interference is observed. From this formula, one can also determine the largest distance between the slits, at which one can still observe interference from a source with an angular size. Multiplying inequality (120.12) by we arrive at the condition
The set of waves with different ones can be replaced by the 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 slots 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 the surface passing through the slots would be a wave and the oscillations at all points of this surface would occur in the same phase. We have established that in the case of finite source sizes ) oscillations at points of the surface spaced are incoherent over a distance.
The surface, which would be wave if the source is monochromatic, will be called pseudo-wave for brevity.
We could satisfy condition (120.12) by reducing the distance between the slots d, i.e. by taking closer points of the pseudo-wave surface. Consequently, the oscillations excited by the wave at sufficiently close points of the pseudowave surface turn out to be coherent. Such coherence is called spatial.
So, the phase of the oscillation during the transition from one point of the pseudo-wave surface to another changes randomly. Let us introduce the distance pkor, at a displacement along which along the pseudo-wave surface the random phase change reaches the value of Oscillations at two points of the pseudo-wave surface spaced from each other by a distance less than will be approximately coherent. The distance pkr is called the spatial coherence length or the coherence radius. From (120.13) it follows that
The angular size of the Sun is about 0.01 rad, the wavelength of light waves is about 0.5 microns. Consequently, the radius of coherence of light waves coming from the Sun has a value of the order
The entire space occupied by the wave can be divided into parts, in each of which the wave approximately retains 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 temporal coherence length and the area of a circle of radius .
The spatial coherence of a light wave near the surface of a heated body radiating it is limited by the size of pcr to only a few wavelengths. As the distance from the source increases, the degree of spatial coherence increases. Laser radiation has a huge temporal and spatial coherence. At the laser outlet, spatial coherence is observed over 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 small, the beams of light passing through the slits will turn out to be incoherent, and there will be no interference pattern. Young obtained in 1802 interference from two slits, increasing the spatial coherence of the light incident on the slits. Jung achieved this magnification by first passing light through a small hole in an opaque screen.
The light passing through this hole illuminated the slits in the second opaque screen. In this way, Jung first observed the interference of light waves and determined the wavelengths of these waves.
