Encyclopedic Dictionary, 1998

coherence

COHERENCE (from the Latin cohaerens - in connection) the coordinated occurrence in time of several oscillatory or wave processes. If the phase difference between 2 oscillations remains constant over time or changes according to a strictly defined law, then the oscillations are called coherent. Oscillations in which the phase difference changes randomly and quickly compared to their period are called incoherent.

Coherence

(from the Latin cohaerens ≈ in connection), the coordinated occurrence in time of several oscillatory or wave processes, manifested when they are added. Oscillations are called coherent if the difference in their phases remains constant over time and, when the oscillations are added, determines the amplitude of the total oscillation. Two harmonic (sinusoidal) oscillations of the same frequency are always coherent. Harmonic oscillation is described by the expression: x = A cos (2pvt + j), (

where x ≈ oscillating quantity (for example, the displacement of the pendulum from the equilibrium position, the strength of the electric and magnetic fields, etc.). The frequency of a harmonic oscillation, its amplitude A and phase j are constant in time. When two harmonic oscillations with the same frequency v, but different amplitudes A1 and A2 and phases j1 and j2 are added, a harmonic oscillation of the same frequency is formed. Amplitude of the resulting oscillation:

can vary from A1 + A2 to A1 ≈ A2 depending on the phase difference j1 ≈ j2 (). The intensity of the resulting vibration, proportional to Ap2, also depends on the phase difference.

In reality, ideally harmonic oscillations are not feasible, since in real oscillatory processes the amplitude, frequency and phase of oscillations continuously change chaotically in time. The resulting amplitude Ap depends significantly on how quickly the phase difference changes. If these changes are so rapid that they cannot be detected by the instrument, then only the average amplitude of the resulting vibration can be measured. At the same time, because the average value of cos (j1≈j2) is equal to 0, the average intensity of the total oscillation is equal to the sum of the average intensities of the initial oscillations: ═and, thus, does not depend on their phases. The original oscillations are incoherent. Chaotic rapid changes in amplitude also disrupt K.

If the phases of oscillations j1 and j2 change, but their difference j1 ≈ j2 remains constant, then the intensity of the total oscillation, as in the case of ideally harmonic oscillations, is determined by the difference in the phases of the added oscillations, that is, K occurs. If the difference in the phases of two oscillations changes very slowly , then they say that the oscillations remain coherent for some time, until their phase difference has had time to change by an amount comparable to p.

You can compare the phases of the same oscillation at different times t1 and t2, separated by an interval t. If the inharmonicity of an oscillation manifests itself in a disorderly, random change in time of its phase, then for a sufficiently large t the change in the oscillation phase can exceed p. This means that after time t the harmonic oscillation “forgets” its original phase and becomes incoherent “to itself.” Time t is called the K time of a nonharmonic oscillation, or the duration of a harmonic train. After one harmonic train has passed, it is, as it were, replaced by another with the same frequency but a different phase.

When a plane monochromatic electromagnetic wave propagates in a homogeneous medium, the electric field strength E along the direction of propagation of this wave oh at time t is equal to:

where l = cT ≈ wavelength, c ≈ speed of its propagation, T ≈ oscillation period. The phase of oscillations at any specific point in space is maintained only during the time CT. During this time, the wave will propagate over a distance сt and oscillations E at points distant from each other by a distance сt, along the direction of propagation of the wave, turn out to be incoherent. The distance equal to сt along the direction of propagation of a plane wave at which random changes in the oscillation phase reach a value comparable to p is called the K length, or train length.

Visible sunlight, occupying the range from 4×1014 to 8×1014Hz on the frequency scale of electromagnetic waves, can be considered as a harmonic wave with rapidly changing amplitude, frequency and phase. In this case, the length of the train is ~ 10≈4 cm. Light emitted by a rarefied gas in the form of narrow spectral lines is closer to monochromatic. The phase of such light practically does not change at a distance of 10 cm. The length of the laser radiation train can exceed kilometers. In the radio wave range, there are more monochromatic sources of oscillation (see Quartz oscillator, Quantum frequency standards), and the wavelength l is many times longer than for visible light. The length of a radio wave train can significantly exceed the size of the solar system.

Everything said is true for a plane wave. However, a perfectly plane wave is just as impracticable as a perfectly harmonic oscillation (see Waves). In real wave processes, the amplitudes and phase of oscillations change not only along the direction of wave propagation, but also in a plane perpendicular to this direction. Random changes in the phase difference at two points located in this plane increase with increasing distance between them. The vibrational effect at these points weakens and at a certain distance l, when random changes in the phase difference become comparable to p, disappear. To describe the coherent properties of a wave in a plane perpendicular to the direction of its propagation, the term spatial coherence is used, in contrast to temporal coherence, which is associated with the degree of monochromaticity of the wave. The entire space occupied by the wave can be divided into regions, in each of which the wave retains a space. The volume of such a region (volume of the wave) is approximately equal to the product of the train length st and the area of ​​a circle with a diameter of / (the size of the spatial space).

Violation of spatial signalization is associated with the peculiarities of the processes of radiation and wave formation. For example, the spatial radiation of a light wave emitted by an extended heated body disappears at a distance of only a few wavelengths from its surface, because different parts of a heated body radiate independently of each other (see Spontaneous emission). As a result, instead of a single plane wave, the source emits a set of plane waves propagating in all possible directions. As it moves away from the heat source (of finite dimensions), the wave approaches more and more flat. The size of the spatial K. l increases in proportion to l ═≈ where R ≈ distance to the source, r ≈ size of the source. This makes it possible to observe the interference of light from stars, despite the fact that they are thermal sources of enormous size. By measuring / for light from nearby stars, it is possible to determine their sizes r. The value l/r is called the angle K. With distance from the source, the light intensity decreases as 1/R2. Therefore, using a heated body it is impossible to obtain intense radiation with a large spatial K.

The light wave emitted by the laser is formed as a result of coordinated stimulated emission of light throughout the entire volume of the active substance. Therefore, the spatial K. of light at the laser output aperture is preserved throughout the entire cross section of the beam. Laser radiation has enormous spatial radiation, that is, high directivity compared to radiation from a heated body. With the help of a laser, it is possible to obtain light whose volume of radiation is 1017 times greater than the volume of radiation of a light wave of the same intensity obtained from the most monochromatic non-laser light sources.

In optics, the most common way to produce two coherent waves is to split the wave emitted by one non-monochromatic source into two waves that travel along different paths, but ultimately meet at one point, where they are combined (Fig. 2). If the delay of one wave relative to another, associated with the difference in the paths they travel, is less than the duration of the train, then the oscillations at the point of addition will be coherent and interference of light will be observed. When the difference in the paths of the two waves approaches the length of the train, the radiation of the rays weakens. Fluctuations in screen illumination decrease, illumination I tends to a constant value equal to the sum of the intensities of two waves incident on the screen. In the case of a non-point (extended) heat source, two rays arriving at points A and B may turn out to be incoherent due to the spatial incoherence of the emitted wave. In this case, interference is not observed, since the interference fringes from different points of the source are shifted relative to each other by a distance greater than the fringe width.

The concept of quantum mechanics, which originally arose in the classical theory of oscillations and waves, is also applied to objects and processes described by quantum mechanics (atomic particles, solids, etc.).

Lit.: Landsberg G.S., Optics, 4th ed., M., 1957; Gorelik G.S., Oscillations and waves, 2nd ed., M., 1959; Fabrikant V.A., New about coherence, “Physics at school”, 1968, ╧ 1; Franson M., Slansky S., Coherence in optics, trans. from French, M., 1968; Martinsen V., Spiller E., What is coherence, “Nature”, 1968, ╧ 10.

A. V. Francesson.

Wikipedia

Coherence (physics)

Coherence(from - " in touch") - the correlation of several oscillatory or wave processes in time, manifested when they are added. Oscillations are coherent if their phase difference is constant over time, and when adding the oscillations, an oscillation of the same frequency is obtained.

The classic example of two coherent oscillations is two sinusoidal oscillations of the same frequency.

Coherence radius is the distance at which, when displaced along the pseudo-wave surface, a random phase change reaches an order of magnitude.

The process of decoherence is a violation of coherence caused by the interaction of particles with the environment.

Coherence (philosophical speculative strategy)

In a thought experiment proposed by the Italian probability theorist Bruno de Finetti to justify Bayesian probability, the array of bets is exactly coherent, if he does not expose the bettor to certain loss regardless of the outcome of the events on which he bets, providing his opponent with a reasonable choice.

Coherence

Coherence(from - " in touch»):

• The coherence of several oscillatory or wave processes of these processes in time, manifested when they are added.
• Coherence of an array of bets is a property of an array of bets, which means that a bettor who bets on some outcomes of some events will never lose the argument, regardless of the outcomes of these events.
• Memory coherence is a property of computer systems that allows two or more processors or cores to access the same memory area.

Examples of the use of the word coherence in literature.

Regardless of the plane of polarization of the Ghosts' radiation, we can now adjust to any and make sure that coherence really exists and is constant over time.

They also perceive the phase of the wave, but at the same time they themselves provide coherence, emitting signals at strictly defined intervals.

Coherence, but this is a coherence that does not allow the existence of my coherence, the coherence of the world and the coherence of God.

The entire Composition of the Total Number of Incarnations of the Essence of the Supreme, as well as the entire Composition of the Total Number of Represented Incarnations of the Essence of the Supreme, along with the Composition of the Total Number of Imaginary Incarnations of the Essence of the Supreme, are imprinted in the Bowl of Accumulations of the Essence of the Divine Man-Buddha in an information-energetic holographic way coherence Spirit, for He is Alpha-and-Omega - the First-and-Last One Supreme, Encompassing in His Creation all Those That Exist with the Creator.

External communications The RA-8000 has the means to effectively maintain coherence cache in multiprocessor systems.

Impressions in the Fabrics of Saraswati's Clothes occur by the Power of the Essence of the Divine Human - in an information-energetic holographic way, that is coherence psychocorrelative quantum fields, leaving the holographic information-energy code of Human Co-Existence, as a living Memory in the Eternal Unchangeable Form of the Soul of Creation.

Each Person has his own individual Composition of the Total Number of Incarnations of the Essence of the Supreme, and this Composition is imprinted in the Human Chalice in an information-energetic holographic way - high coherence radiations of psychocorrelative quantum fields that are generated by the Essence of the Divine Man in the process of his Education by the Supreme.

The Essence of Divine Man, as a result of Thinking in Images of the Supreme, gives birth to myriads of elementary particles of Matter, which are focused high coherence Spirit in the Lens of density of curvature of Space Images of the general picture of the hologram of what is happening in Saraswati from the senses.

Figure 5 -- Formation of the Theroidsphere of Influx by the creation of high density Curvature of Space coherence Spirit.

Individual electrons observed in a specific physical experiment are, according to Tsech, the result of destruction by a measuring device coherence a single electron-positron field.

The processes of self-organization of social consciousness are subject to the general laws of formation: coherence, the coherence of events of the emergence of certain social stereotypes, etc.

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, antireflection 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 wavelike 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.

Coherence b (from the Latin cohaerens - in connection) is considered as the coordinated occurrence in time of several oscillatory or wave processes, manifested when they are added. Oscillations are called coherent if the difference in their phases remains constant over time and, when adding the oscillations, 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 is formed ν :

and 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 that, when added, can mutually enhance or weaken the interaction effect.

Stimulated emission of photons has significant features. Firstly, the frequency of a quantum of light emitted under the influence of an external monochromatic field exactly coincides with the frequency of the external field. Secondly, the direction of propagation and polarization of the emitted photon coincides with the direction of propagation and polarization of the external electromagnetic field causing the radiation. Thus, the radiation of individual elementary emitters under the influence 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 generate powerful monochromatic radiation.

To explain the concept of coherence, it is convenient to use the wave representation of light. In Fig. 6 radiation is depicted in the form of “elementary waves” arising in the active medium; they are usually called trains. 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 stimulated emission of light. In stimulated emission, the secondary train exactly copies the primary train in the direction of propagation, in wavelength, and in phase. In Fig. 3.13b the dashed line shows a 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 is manifested, in particular, in its extremely high degree of monochromaticity, as well as in the very low divergence of the laser beam.

Focus

Directivity is one of the main properties of laser radiation. Directional radiation is radiation that propagates within a small solid angle.

A measure of parallelism of radiation is the divergence of the laser beam.

Laser divergence- it's flat θ or a solid angle with the 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 using the known parameters of the resonator. The relationship between the beam parameters and the resonator parameters is determined by the type of resonator.

In Fig. Figure 3.14 shows a confocal resonator consisting of two mirrors 1, 2 with radii r 1 and r 2, respectively. In the case r 1 = r 2, the radiation waist will be located in the center of the resonator, its diameter (for single-mode radiation) is determined by the expression:

Where = 2 /λ - wave number; d- 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

The beam divergence with a uniform energy distribution, which corresponds to the multimode nature of the radiation, is determined by the equality:

where 2у is the size of the aperture on the output mirror; k Ф is a coefficient depending on the energy distribution and the shape of the active element.

With uniform energy distribution for a circular diaphragm 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 minutes of arc, that of solid-state lasers is up to several tens of minutes, and that of semiconductor lasers is up to tens of degrees.

The divergence of the beam 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 a parallel beam of rays entering the system , also into a parallel beam of rays at the exit from it with an increased aperture (diameter) of the beam.

Figure 3.15 – Beam collimation using a two-lens telescope

In this case, the divergence of laser radiation is inversely proportional to the increase β telescope used ( β = D2/D1):

where 1.2 is the beam divergence at the entrance to the telescope and at the exit from it, respectively; D 1 , D 2 - the diameter of the beam 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 divergence value is determined by the diffraction phenomena of the optical wavefront at the output component of the collimating system.

In the technical specifications (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. At large values ​​of these quantities it is usually stated that the radiation is intense. Laser radiation, due to the high degree of directionality of the radiation, can be intense even when the radiation power is relatively low.

The laser radiation strength characterizes the spatial density of the radiation flux, that is, the amount of 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 by which the solid angle is formed, rad.

With single-mode laser radiation, the divergence of which is 2θ (the solid angle is correspondingly 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, the intensity of radiation between an incandescent lamp and a laser, then for the same power consumption, lasers turn out to be more intense and have a lower efficiency. For example, a 66 W incandescent lamp has an average radiation intensity

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 power

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 radiation occurs in the fundamental mode, then the magnitude of the Fe flux is determined by the ratio of the radius of the section under consideration r and the size of the mode spot ω:

Where F 0 - total laser flux measured at r>>ω.

The transition from the energy value of the flux (W) to the light value (lm) is carried out according to the formula

F=638Fe,

where 683 lm/W is the light equivalent of radiant energy at the wavelength corresponding to the maximum sensitivity of the eye (λ = 0.55 µm).

The transition from the lighting flux value to the energy value is carried out according to the formula

Fe=AF,

Where A = 0.00146 W/lm - mechanical equivalent of light (A = 1/683).

With pulsed radiation, the mode of a regular sequence of pulses is characterized by the average radiation flux, that is, the average flux value over a given period of time:

Fsr=Phi∆t/T,

Where Fi - flow in impulse; ∆t - pulse duration; T - pulse repetition period.

In pre-press 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 the maximum value to zero, to form printing or white-space elements of the form, as well as to match the intensity with the light or thermal sensitivity of the recorded materials. To control the intensity, special devices are used - radiation modulators.

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.

A distinction is made between time 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 are constants, represents an abstraction. Any real light wave is formed by the superposition of oscillations of various frequencies (or wavelengths), contained in a more or less narrow, but finite interval of frequencies (respectively, wavelengths). Even for quasi-monochromatic light (see page 327), the frequency range 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, the chaotic changes in functions are completely independent.

where is some average frequency value, and introduce the notation: Then formula (120.2) 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 (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's 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 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 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 a certain quantity x change in jumps equal to and the increments are equally probable. This behavior of a quantity is called random walk. 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 probable. Let's assume that random walks are performed multiple times, starting each time, and find the average value. It is equal to (the double product disappears during averaging). Therefore, regardless of the value of N, the average value increases by Therefore. Thus, a quantity performing a random walk, on average, moves further and further 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 during which a random change in the wave phase reaches an order value is called the coherence time. During this time, the oscillation seems to forget its initial phase and becomes incoherent with itself.

As an example, we point out that quasi-monochromatic light, containing wavelengths in the interval , is characterized by the order of c. The emission 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 the random phase change reaches the 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 diagram 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 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, Fourier's theorem is proven, according to which any finite and integrable function can be represented as the sum of an infinite number of harmonic components with a continuously varying frequency:

(120.4)

Expression (120.4) is called the Fourier integral. The function 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 is determined by the expression

(120.5)

where is the auxiliary integration variable.

Let the function describe the light disturbance at some point at an instant in time caused by a single wave train. Then it is determined by the conditions:

The graph of the real part of this function is shown 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., the expression

The graph of function (120.6) is shown in Fig. 120.2. It can be seen from the figure that the intensity of the components whose frequencies are contained 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 with coherence time, we arrive at the relation

(the sign means: “equal in order of magnitude”).

From relation (120.7) it follows that the wider the range 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. Having differentiated this relation, we find that (we omitted the minus sign resulting from 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

This gives the following value for the coherence length:

From formula (119.5) it follows that 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 limiting observed interference order 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 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 the value of ). 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 the value De.

The occurrence of oscillations excited by waves with different θ 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 . From Fig. 120.3 it is clear that the angle characterizes the interval in which the unit vectors e 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. 120.4).

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. 120.4 through O, creates a zero maximum M in the middle of the screen. The zero maximum created by the wave coming 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 coming from the area is shifted from the middle of the screen in the opposite direction by a distance equal to x. Zero maxima from the remaining areas of the source are located between the maxima and

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 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. When the maxima from some areas 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 moving 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 we can also determine the greatest distance between the slits at which we can still observe interference from a source with an angular size. Multiplying inequality (120.12) by 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 the surface passing through the slits would be wavelike and oscillations at all points of this surface would occur in the same phase. We have established that in the case of finite dimensions of the source) oscillations at points of the surface separated are incoherent over a distance.

The surface, which would be wave if the source were monochromatic, will be called pseudo-wave for brevity.

We could satisfy condition (120.12) 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 pcoh, when displaced along the pseudo-wave surface, a random change in phase reaches the value Oscillations at two points of the pseudo-wave surface, spaced from each other at a distance less than that, will be approximately coherent. The distance pcoh is called the spatial coherence length or coherence radius. From (120.13) 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

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 by 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 by the size of the pcog 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. Young obtained interference from two slits in 1802, increasing the spatial coherence of light incident on the slits. Jung accomplished this increase by first passing light through a small hole in an opaque screen.

The light passing through this hole illuminated the cracks in the second opaque screen. In this way, Young first observed the interference of light waves and determined the lengths of these waves.

As already noted, the interference pattern can be observed only when superimposing coherent waves. Let us pay attention to the fact that in the definition of coherent waves it is noted not the existence, but the observation of interference. This means that the presence or absence of coherence depends not only on the characteristics of the waves themselves, but also on the time interval for recording the intensity. The same pair of waves can be coherent at one observation time and incoherent at another.

Two light waves produced from one by the amplitude division method or the wavefront division method do not necessarily interfere with each other. At the observation point, two waves with wave vectors and are added. There are two main reasons for the possible incoherence of such waves.

The first reason is the non-monochromatic nature of the light source (or the variability of the magnitudes of the wave vectors). Monochromatic light is light of one frequency. A strictly monochromatic wave at each point in space has a time-independent amplitude and initial phase. Both the amplitude and phase of a real light wave experience some random variation over time. If the changes in frequency are small and the changes in amplitude are sufficiently slow (their frequency is small compared to the optical frequency), then the wave is said to be quasi-monochromatic.

The second reason for the possible incoherence of light waves obtained from a single wave is the spatial extent of the actual light source (or the inconstancy of the direction of each of the wave vectors).

In reality, both reasons occur simultaneously. However, for simplicity, we will analyze each reason separately.

Temporal coherence.

Let there be spot Light source S and and , which are real or imaginary images of it (Fig. 3.6.3 or 3.6.4). Let us assume that the radiation from the source consists of two close and equally intense waves with wavelengths and (obviously the same will be true for sources and ). Let the initial phases of the sources be the same. Rays with wavelengths will arrive at a certain point on the screen in the same phases. Let's call this point the center of the interference pattern. For both waves there will be a light stripe. At another point on the screen, where the path difference ( N– integer, band number) for the wavelength, a light interference fringe will also be obtained. If it is the same, then rays with a wavelength will arrive at the same point on the screen in antiphase, and for this wavelength the interference fringe will be dark. Under this condition, at the point of the screen under consideration, the light stripe will overlap the dark stripe - the interference pattern will disappear. Thus, the condition for the disappearance of fringes is , whence the maximum number of the interference fringe

Let us now move on to the case when the light from the source is a collection of waves with lengths lying in the interval . Let us divide this spectral interval into a set of pairs of infinitely narrow spectral lines, the wavelengths of which differ by . Formula (3.7.1) is applicable to each such pair, where it must be replaced by . Therefore, the disappearance of the interference pattern will occur for the interference order

This formula gives an estimate of the maximum possible interference order. The quantity is usually called degree of monochromaticity of the wave.

Thus, to observe the interference pattern when a wave is split along the path of the beam, the difference in the paths of the two resulting waves should not exceed a value called coherence length l

The concept of coherence length can be explained as follows. Consider two points on one beam as two possible secondary light sources for observing the interference pattern. In this case, the distance from each point to the mental screen is assumed to be the same (Fig. 3.7.1).

Here and are two selected along the ray

Fig.3.7.1. points at which we mentally place translucent plates to obtain an interference pattern on the screen. Let the optical path difference for the interfering rays and be equal to . If it exceeds the value, then, as indicated above, the interference pattern is “smeared”, and, consequently, the secondary light sources at the points turn out to be incoherent. The distance between points and at which this begins to happen is called length coherence along the beam, longitudinal coherence length, or simply coherence length.

A distance equal to the coherence length the wave travels coherence time

Coherence time can be called the maximum period of time, when averaging over which the interference effect is still observed.

Based on the above estimates, we can estimate the thickness of the film, with the help of which an interference pattern can be obtained (decipher the term “thin film” used in the previous lecture). The film can be called “thin” if the difference in the paths of the waves that give the interference pattern does not exceed the coherence length of the light wave. When a wave falls on the film at a small angle (in a direction close to the normal), the path difference is equal to 2bn(formula (3.6.20)), where b– thickness, and n– refractive index of the film material. Therefore, the interference pattern can be obtained on a film for which 2bn ≤ l =. (3.7.5) Note that when a wave is incident at large angles, it is also necessary to take into account the possible incoherence of different points of the wave front.

Let us estimate the coherence length of light emitted by different sources.

1. Consider light emitted by a natural source (not a laser). If a glass filter is placed in the path of light, the bandwidth of which is ~ 50 nm, then for a wavelength of the middle of the optical spectral interval ~ 600 nm we obtain, according to (3.7.3), ~ 10 m. If there is no filter, then the coherence length will be approximately an order of magnitude less.

2. If the light source is a laser, then its radiation has a high degree of monochromaticity (~ 0.01 nm) and the coherence length of such light for the same wavelength will be about 4·10 m.

Spatial coherence.

The ability to observe the interference of coherent waves from extended sources leads to the concept spatial coherence of waves.

For simplicity of reasoning, let us imagine that sources of coherent electromagnetic waves with identical initial phases and wavelengths are located on a segment of length b, located at a distance l»b from the screen (Fig. 3.7.2), on which their interference is observed. The interference pattern observed on the screen can be represented as a superposition of interference patterns created by an infinite number of pairs of point coherent sources into which an extended source can be mentally divided.

Among the entire set of sources, let us select a source located in the middle of the segment and compare the interference patterns of two pairs, one of which is formed by the central source and some arbitrarily chosen source located close to it, and the other by the central source and a source located at one of the ends of the segment. It is obvious that the interference pattern of a pair of closely located sources will have a value close to the maximum in the center of the screen at the observation point (Fig. 3.7.2). At the same time, the interference pattern of the other pair will have a value depending on the optical difference in the path of electromagnetic waves emitted by sources in the center of the segment and at its edge

≈ ,

(3.7.6)

where is the angular size of the source (Fig. 3.7.2), which due to “ l is small enough so that the obvious transformations used in deriving formula (3.7.6) are valid.

It follows that waves from different points of an extended source arriving at an observation point located in the center of the screen will have an optical path difference relative to the wave from the central source, varying linearly from zero to a maximum value of 0.25. For a certain source length, waves arriving at the observation point can have a phase that differs by 180° from the phase of the wave emitted by the central point of the segment. As a result of this, waves arriving at the center of the screen from different parts of the source will reduce the intensity value compared to the maximum that would occur if all the waves had the same phase. The same reasoning is valid for other points on the screen. As a result, the intensities at the maxima and minima of the interference pattern of an extended source will have similar values ​​and the visibility of the interference pattern will tend to zero. In the case under consideration, this occurs at in (3.7.6). The value of the shortest length of the segment (source) corresponding to this condition is determined from the relation (in this case t=1):

In optics and the theory of electromagnetic waves, half of this value determines the so-called. radius of spatial coherence electromagnetic waves emitted by an extended source:

.

(3.7.7)

The physical meaning of the concept of the radius of spatial coherence of an extended source is the idea of ​​​​the possibility of observing the interference pattern from an extended source if it is located inside a circle of radius . From the above it follows that the spatial coherence of electromagnetic waves is determined by the angular size of their source.

Spatial coherence is the coherence of light in the direction perpendicular to the beam (across the beam). It turns out that this is the coherence of different points of the surface of equal phase. But on a surface of equal phase, the phase difference is zero. However, for extended sources this is not entirely true. The real light source is not a point, so the surface of equal phases undergoes slight rotations, remaining at each moment of time perpendicular to the direction of the currently emitting point light source, located within the real light source. Rotations of the surface of equal phase are caused by the fact that light comes to the observation point from one or another point of the source. Then, if we assume that on such a pseudo-wave surface there are secondary sources, the waves from which can give an interference pattern, then we can define the coherence radius in other words. Secondary sources on the pseudo-wave surface, which can be considered coherent, are located inside a circle whose radius is equal to the coherence radius. The coherence diameter is the maximum distance between points on the pseudo-wave surface that can be considered coherent.

Let's return to Jung's experience (Lecture 3.6). To obtain a clear interference pattern in this experiment, it is necessary that the distance between the two slits S and did not exceed the coherence diameter. On the other hand, as can be seen from (3.7.7), the radius (and, consequently, the diameter) of interference increases with decreasing angular size of the source. That's why d- distance between slots and and b- source size S inversely related b·d ≤ l.(3.7.8)