For quantification Radiation uses a fairly wide range of quantities, which can be divided into two systems of units: energy and light. Wherein energy quantities characterize radiation related to the entire optical region of the spectrum, and lighting quantities– to visible radiation. Energy quantities are proportional to the corresponding lighting quantities.
The main quantity in the energy system that allows us to judge the amount of radiation is Fe radiation flux, or radiation power, i.e. amount of energy W, emitted, transferred or absorbed per unit time:
The value of Fe is expressed in watts (W). – energy unit
In most cases they do not take into account quantum nature occurrence of radiation and consider it continuous.
A qualitative characteristic of radiation is the distribution of the radiation flux over the spectrum.
For radiation having a continuous spectrum, the concept is introduced spectral radiation flux density (j l)– the ratio of the radiation power falling on a certain narrow section of the spectrum to the width of this section (Fig. 2.2). For a narrow spectral range dl the radiation flux is equal to dФ l. The ordinate axis shows the spectral densities of radiation flux j l = dФ l / dl, therefore, the flow is represented by the area of an elementary section of the graph, i.e.
If the radiation spectrum lies within the range l 1 before l 2, then the magnitude of the radiation flux
Under luminous flux F, V general case, understand the radiation power assessed by its effect on the human eye. Unit of measurement luminous flux is lumen (lm). – lighting unit
The action of the light flux on the eye causes it to react in a certain way. Depending on the level of action of the light flux, one or another type of light-sensitive receptors of the eye, called rods or cones, works. In conditions low level illumination (for example, in the light of the Moon), the eye sees surrounding objects due to the rods. At high levels After illumination, the daytime vision apparatus, for which the cones are responsible, begins to work.
In addition, cones, according to their light-sensitive substance, are divided into three groups with different sensitivity in various areas spectrum Therefore, unlike rods, they react not only to the light flux, but also to its spectral composition.
In this regard, it can be said that light effect is two-dimensional.
The quantitative characteristic of the eye reaction associated with the level of illumination is called lightness. Qualitative characteristic associated with different levels reactions of three groups of cones are called chromaticity.
Luminous intensity (I). In lighting engineering, this value is taken as main. This choice has no basis in principle, but is made for reasons of convenience, since The intensity of light does not depend on distance.
The concept of luminous intensity applies only to point sources, i.e. to sources whose dimensions are small compared to the distance from them to the illuminated surface.
The power of light point source in some direction there is per unit solid angle W light flow F, emitted by this source in a given direction:
I = Ф / Ω
Energy Luminous intensity is expressed in watts per steradian ( Tue/Wed).
Behind lighting engineering unit of luminous intensity adopted candela(cd) is the luminous intensity of a point source that emits a luminous flux of 1 lm, distributed uniformly within a solid angle of 1 steradian (sr).
A solid angle is a part of space bounded by a conical surface and a closed curved contour that does not pass through the vertex of the angle (Fig. 2.3). When a conical surface is compressed, the dimensions of the spherical area o become infinitesimal. The solid angle in this case also becomes infinitesimal:
Figure 2.3 – Towards the definition of the concept “solid angle”
Illumination (E). Under energetic illumination E uh understand the radiation flux on unit of area illuminated surface Q:
The irradiance is expressed in W/m2.
Luminous illumination E expressed by luminous flux density F on the surface illuminated by it (Fig. 2.4):
The unit of luminous illumination is taken luxury, i.e. illumination of a surface receiving a luminous flux of 1 lm uniformly distributed over it over an area of 1 m2.
Among other quantities used in lighting engineering, important ones are energy radiation We or light energy W, as well as energy Ne or light N exposition.
The values of We and W are determined by the expressions
where are the functions of changes in radiation flux and light flux over time, respectively. We is measured in joules or W s, a W – in lm s.
Under energy H e or light exposure understand surface density radiation energy We or light energy W respectively on the illuminated surface.
That is light exposure H this is the product of illumination E, created by a radiation source, for a time t effects of this radiation.
The definitions of photometric quantities of the light series and the mathematical relationships between them are similar to the corresponding quantities and relationships of the energy series. That's why light flow, extending within the solid angle, is equal to . Unit of measurement of luminous flux ( lumen). For monochromatic light relationship between energy and light quantities is given by the formulas:
where is a constant called mechanical equivalent of light.
Luminous flux per wavelength interval from l before ,
, (30.8)
Where j– energy distribution function over wavelengths (see Fig. 30.1). Then the total luminous flux carried by all waves of the spectrum is
. (30.9)
Illumination
The luminous flux can also come from bodies that do not themselves glow, but reflect or scatter the light incident on them. In such cases, it is important to know what luminous flux falls on a particular area of the body surface. For this purpose it is used physical quantity, called illumination
. (30.10)
Illumination is numerically equal to the ratio of the total luminous flux incident on a surface element to the area of this element (see Fig. 30.4). For uniform light output
Illuminance unit 
(luxury). Lux equals the illumination of a surface with an area of 1 m2 when a luminous flux of 1 lm falls on it. The irradiance is determined similarly
Unit of irradiance.
Brightness
For many lighting calculations, some sources can be considered as point sources. However, in most cases the light sources are placed close enough to distinguish their shape, in other words, the angular dimensions of the source are within the ability of the eye or optical instrument to distinguish an extended object from a point. For such sources, a physical quantity called brightness is introduced. The concept of brightness is not applicable to sources whose angular dimensions are less than the resolution of the eye or optical instrument (for example, stars). Brightness characterizes the radiation luminous surface in a certain direction. The source can glow with its own or reflected light.
Let us select a luminous flux propagating in a certain direction in a solid angle from a section of the luminous surface. The beam axis forms an angle with the normal to the surface (see Fig. 30.5).
Projection of a section of the luminous surface onto an area perpendicular to the selected direction,
(30.14)
called visible surface element of the source site (see Fig. 30.6).
The value of the luminous flux depends on the area of the visible surface, on the angle and on the solid angle:
The proportionality factor is called brightness. It depends on optical properties radiating surface and may be different for different directions. From (30.5) brightness
. (30.16)
Thus, brightness is determined by the luminous flux emitted in a certain direction by a unit of visible surface per unit solid angle. Or in other words: brightness in a certain direction is numerically equal to the intensity of light created per unit area of the visible surface of the source.
In general, brightness depends on direction, but there are light sources for which brightness does not depend on direction. Such sources are called Lambertian or cosine, because Lambert’s law is valid for them: the intensity of light in a certain direction is proportional to the cosine of the angle between the normal to the surface of the source and this direction:
where is the light intensity in the direction of the normal to the surface, and is the angle between the normal to the surface and the selected direction. To ensure equal brightness in all directions, technical luminaires are equipped with milk glass shells. Lambertian sources that emit diffuse light include surfaces coated with magnesium oxide, unglazed porcelain, drawing paper, and freshly fallen snow.
Brightness unit 
(nit). Here are the brightness values of some light sources:
Moon – 2.5 knt,
fluorescent lamp – 7 knt,
light bulb filament – 5 MNT,
solar surface – 1.5 Gnt.
The lowest brightness perceived by the human eye is about 1 μnt, and brightness exceeding 100 μnt causes painful sensation in the eye and may damage vision. The brightness of a sheet of white paper when reading and writing should be at least 10 nits.
Energy brightness is determined similarly
. (30.18)
A unit of measurement for radiance.
Luminosity
Let us consider a light source of finite dimensions (illuminating with its own or reflected light). Luminosity source is the surface density of the luminous flux emitted by a surface in all directions within a solid angle. If a surface element emits a luminous flux, then
For uniform luminosity we can write:
A unit of measurement for luminosity.
The energetic luminosity is determined similarly
Unit energetic luminosity.
Laws of illumination
Photometric measurements are based on two laws of illumination.
1. Illumination of a surface by a point light source varies in inverse proportion to the square of the distance of the source from the illuminated surface. Consider a point source (see Fig. 30.7) emitting light in all directions. Let us describe spheres with radii and concentric with the source around the source. It is obvious that the luminous flux through the surface areas and is the same, since it propagates in the same solid angle. Then the illumination of the areas will be, respectively, and . Expressing the elements of spherical surfaces through the solid angle, we obtain:
. (30.22)
2. Illumination created on an elementary surface area by a luminous flux incident on it at a certain angle is proportional to the cosine of the angle between the direction of the rays and the normal to the surface. Let us consider a parallel beam of rays (see Fig. 29.8) incident on sections of surfaces and . The rays fall on the surface along the normal, and on the surface - at an angle to the normal. The same luminous flux passes through both sections. The illumination of the first and second sections will be, respectively, 
. But, therefore,
Combining these two laws, we can formulate basic law of illumination: illumination of a surface by a point source is directly proportional to the luminous intensity of the source, the cosine of the angle of incidence of the rays and inversely proportional to the square of the distance from the source to the surface
. (30.24)
Calculations using this formula give a fairly accurate result if the linear dimensions of the source do not exceed 1/10 of the distance to the illuminated surface. If the source is a disk with a diameter of 50 cm, then at a point normal to the center of the disk relative error in calculations for a distance of 50 cm it reaches 25%, for a distance of 2 m it does not exceed 1.5%, and for a distance of 5 m it decreases to 0.25%.
If there are several sources, then the resulting illumination is equal to the sum of the illumination created by each individual source. If the source cannot be considered as a point source, its surface is divided into elementary sections and, having determined the illumination created by each of them, according to the law 
, are then integrated over the entire surface of the source.
There are lighting standards for workplaces and premises. On the tables classrooms The illumination must be at least 150 lux; for reading books, illumination is needed, and for drawing - 200 lux. For corridors, illumination is considered sufficient, for streets - .
The most important source of light for all life on Earth is the Sun, which creates upper limit atmosphere, the energy irradiance called the solar constant - and the illuminance is 137 klx. The energy illumination created on the Earth's surface by direct rays in summer is two times less. The illumination created by direct sunlight at midday at an average latitude is 100 klx. The change of seasons on Earth is explained by a change in the angle of incidence sun rays to its surface. In the northern hemisphere, the angle of incidence of rays on the Earth's surface is greatest in winter, and the smallest in summer. The illumination in an open area under a cloudy sky is 1000 lux. Illumination in a bright room near a window is 100 lux. For comparison, we present the illumination from full moon– 0.2 lux and from the night sky on a moonless night – 0.3 mlx. The distance from the Sun to the Earth is 150 million kilometers, but due to the fact that the force sunlight equals , the illumination created by the Sun on the surface of the Earth is so great.
For sources whose luminous intensity depends on the direction, sometimes they use average spherical luminous intensity, where is the total luminous flux of the lamp. Luminous Flux Ratio electric lamp to its electrical power is called luminous efficiency lamps: . For example, a 100 W incandescent lamp has an average spherical luminous intensity of about 100 cd. The total luminous flux of such a lamp is 4 × 3.14 × 100 cd = 1260 lm, and the luminous efficiency is 12.6 lm/W. The luminous efficiency of fluorescent lamps is several times greater than that of incandescent lamps and reaches 80 lm/W. In addition, the service life of fluorescent lamps exceeds 10 thousand hours, while for incandescent lamps it is less than 1000 hours.
Over millions of years of evolution, the human eye has adapted to sunlight, and therefore it is desirable that the spectral composition of the lamp light be as close as possible to the spectral composition of sunlight. Fluorescent lamps meet this requirement to the greatest extent. That is why they are also called fluorescent lamps. The brightness of a light bulb filament causes pain in the eye. To prevent this, milk glass lampshades and lampshades are used.
With all their advantages, fluorescent lamps also have a number of disadvantages: the complexity of the switching circuit, pulsation of the light flux (with a frequency of 100 Hz), the impossibility of starting in the cold (due to mercury condensation), throttle buzzing (due to magnetostriction), environmental hazard (mercury from a broken lamp poisons environment).
In order for the spectral composition of the radiation of an incandescent lamp to be the same as that of the Sun, it would be necessary to heat its filament to the temperature of the surface of the Sun, i.e., up to 6200 K. But tungsten, the most refractory of metals, melts already at 3660 K.
Temperatures close to the surface temperature of the Sun are achieved in arc discharge in mercury vapor or xenon under a pressure of about 15 atm. The power of light arc lamp can be increased to 10 Mkd. Such lamps are used in film projectors and spotlights. Lamps filled with sodium vapor are distinguished by the fact that in them a significant part of the radiation (about a third) is concentrated in visible area spectrum (two intense yellow lines 589.0 nm and 589.6 nm). Although the emission of sodium lamps is very different from the sunlight familiar to the human eye, they are used to illuminate highways, as their advantage is their high luminous efficiency, reaching 140 lm/W.
Photometers
Instruments designed to measure luminous intensity or luminous fluxes from different sources are called photometers. Based on the registration principle, photometers are of two types: subjective (visual) and objective.
The principle of operation of a subjective photometer is based on the ability of the eye to record with sufficient accuracy the sameness of illumination (more precisely, brightness) of two adjacent fields, provided that they are illuminated by light of the same color.
Photometers for comparing two sources are designed in such a way that the role of the eye is reduced to establishing the sameness of illumination of two adjacent fields illuminated by the sources being compared (see Fig. 30.9). The observer's eye examines a white triangular prism installed in the middle of a blackened pipe inside. The prism is illuminated by sources and. By changing the distances from the sources to the prism, you can equalize the illumination of the surfaces and. Then , where and are the intensities of light, respectively, the sources and . If the luminous intensity of one of the sources is known (reference source), then the luminous intensity of the other source in the selected direction can be determined. By measuring the luminous intensity of the source in different directions, the total luminous flux, illumination, etc. are found. The reference source is an incandescent lamp, the luminous intensity of which is known.
The inability to change the distance ratio within very wide limits forces the use of other methods of attenuating the flux, such as light absorption by a filter of variable thickness - a wedge (see Fig. 30.10).
One of the varieties of the visual photometry method is the extinction method, which is based on the use of a constant threshold sensitivity of the eye for each individual observer. The threshold sensitivity of the eye is the lowest brightness (about 1 micron) to which the human eye reacts. Having previously determined the sensitivity threshold of the eye, in some way (for example, a calibrated absorbing wedge) the brightness of the source under study is reduced to the sensitivity threshold. Knowing how many times the brightness is attenuated, you can determine the absolute brightness of the source without a reference source. This method is extremely sensitive.
Direct measurement of the total luminous flux of the source is carried out in integral photometers, for example, in a spherical photometer (see Fig. 30.11). The source under study is suspended in the internal cavity of a sphere whitened with a matte surface inside. As a result of multiple reflections of light inside the sphere, illumination is created, determined by medium strength light source. The illumination of the hole, protected from direct rays by the screen, is proportional to the luminous flux: , where is the constant of the device, depending on its size and color. The hole is covered with milky glass. The brightness of milk glass is also proportional to the luminous flux. It is measured using the photometer described above or by another method. In technology, automated spherical photometers with photocells are used, for example, to control incandescent lamps on the conveyor of an electric lamp plant.
Objective methods Photometry is divided into photographic and electrical. Photographic methods are based on the fact that the blackening of the photosensitive layer is, over a wide range, proportional to the density of light energy falling on the layer during its illumination, i.e. exposure (see Table 30.1). This method determines relative intensity two closely located spectral lines in one spectrum or compare the intensities of the same line in two adjacent (taken on one photographic plate) spectra based on the blackening of certain areas of the photographic plate.
Visual and photographic methods are gradually being replaced by electrical ones. The advantage of the latter is that they quite simply carry out automatic registration and processing of results, up to the use of a computer. Electric photometers make it possible to measure radiation intensity beyond the visible spectrum.
CHAPTER 31. THERMAL RADIATION
31.1. Characteristics thermal radiation
Bodies heated to sufficiently high temperatures glow. The glow of bodies caused by heating is called thermal (temperature) radiation. Thermal radiation, being the most common in nature, occurs due to energy thermal movement atoms and molecules of a substance (i.e., due to its internal energy) and is characteristic of all bodies at temperatures above 0 K. Thermal radiation is characterized by a continuous spectrum, the position of the maximum of which depends on temperature. At high temperatures short-term (visible and ultraviolet) radiation is emitted electromagnetic waves, at low ones - predominantly long (infrared).
A quantitative characteristic of thermal radiation is spectral density energetic luminosity (emissivity) of the body- radiation power per unit surface area of the body in a frequency range of unit width:
Rv,T =, (31.1)
where is energy electromagnetic radiation, emitted per unit time (radiation power) from a unit surface area of a body in the frequency range v before v+dv.
Energy luminosity spectral density unit Rv,T- joule per meter squared (J/m2).
The written formula can be represented as a function of wavelength:
=Rv,Tdv= R λ ,T dλ. (31.2)
Because с =λvυ, That dλ/ dv = - c/v 2 = - λ 2 /With,
where the minus sign indicates that with an increase in one of the quantities ( λ or v) another quantity decreases. Therefore, in what follows we will omit the minus sign.
Thus,
R υ,T =Rλ,T . (31.3)
Using formula (31.3) you can go from Rv,T To Rλ,T and vice versa.
Knowing the spectral density of energetic luminosity, we can calculate integral energy luminosity(integral emissivity), summing over all frequencies:
R T = . (31.4)
The ability of bodies to absorb radiation incident on them is characterized by spectral absorptivity
A v,T =(31.5)
showing what fraction of the energy brought per unit time per unit surface area of a body by electromagnetic waves incident on it with frequencies from v before v+dv, is absorbed by the body.
Spectral absorption capacity is a dimensionless quantity. Quantities Rv,T And A v,T depend on the nature of the body, its thermodynamic temperature and at the same time differ for radiation with different frequencies. Therefore, these values are referred to as certain T And v(or rather, to a fairly narrow frequency range from v before v+dv).
A body capable of completely absorbing at any temperature all radiation of any frequency incident on it is called black. Consequently, the spectral absorption capacity of a black body for all frequencies and temperatures is identically equal to unity ( A h v,T = 1). There are no absolutely black bodies in nature, but bodies such as soot, platinum black, black velvet and some others, in a certain frequency range, are close to them in their properties.
The ideal model the black body is a closed cavity with a small hole, inner surface which is blackened (Fig. 31.1). A ray of light entering Fig. 31.1.
such a cavity experiences multiple reflections from the walls, as a result of which the intensity of the emitted radiation is practically equal to zero. Experience shows that when the hole size is less than 0.1 of the cavity diameter, incident radiation of all frequencies is completely absorbed. Consequently open windows the houses appear black from the street, although the inside of the rooms is quite light due to the reflection of light from the walls.
Along with the concept of a black body, the concept is used gray body- a body whose absorption capacity is less than unity, but is the same for all frequencies and depends only on the temperature, material and state of the surface of the body. Thus, for the gray body And with v,T< 1.
Kirchhoff's law
Kirchhoff's law: the ratio of the spectral density of energetic luminosity to the spectral absorptivity does not depend on the nature of the body; it is a universal function of frequency (wavelength) and temperature for all bodies:
= rv,T(31.6)
For black body A h v,T=1, therefore it follows from Kirchhoff’s law that Rv,T for a black body is equal to r v,T. Thus, the universal Kirchhoff function r v,T is nothing more than the spectral density of the energy luminosity of a black body. Therefore, according to Kirchhoff's law, for all bodies the ratio of the spectral density of energetic luminosity to the spectral absorptivity is equal to the spectral density of energetic luminosity of a black body at the same temperature and frequency.
From Kirchhoff’s law it follows that the spectral density of the energy luminosity of any body in any region of the spectrum is always less than the spectral density of the energy luminosity of a black body (for the same values T And v), because A v,T < 1, и поэтому Rv,T < r v υ,T. In addition, from (31.6) it follows that if a body at a given temperature T does not absorb electromagnetic waves in the frequency range from v, before v+dv, then it is in this frequency range at temperature T and does not emit, since when A v,T=0, Rv,T=0
Using Kirchhoff's law, the expression for the integral energy luminosity of a black body (31.4) can be written as
R T = .(31.7)
For gray body R with T = A T = A T R e, (31.8)
Where R e= -energy luminosity of a black body.
Kirchhoff's law describes only thermal radiation, being so characteristic of it that it can serve as a reliable criterion for determining the nature of radiation. Radiation that does not obey Kirchhoff's law is not thermal.
For practical purposes, it follows from Kirchhoff's law that bodies with a dark and rough surface have an absorption coefficient close to 1. For this reason, they prefer to wear dark clothes in winter, and light ones in summer. But bodies with an absorption coefficient close to unity also have a correspondingly higher energetic luminosity. If you take two identical vessels, one with a dark, rough surface, and the walls of the other are light and shiny, and pour the same amount of boiling water into them, then the first vessel will cool faster.
31.3. Stefan-Boltzmann laws and Wien displacements
It follows from Kirchhoff's law that the spectral density of the energy luminosity of a black body is a universal function, therefore finding its explicit dependence on frequency and temperature is an important task in the theory of thermal radiation.
Stefan, analyzing experimental data, and Boltzmann, using the thermodynamic method, solved this problem only partially, establishing the dependence of the energy luminosity R e on temperature. According to Stefan-Boltzmann law,
R e = σ T 4, (31.9)
that is, the energetic luminosity of a black body is proportional to quarters of the power of its thermodynamic temperature; σ
- Stefan-Boltzmann constant: her experimental value equals 5.67 × 10 -8 W/(m 2 × K 4).
Stefan-Boltzmann law, defining the dependence R e on temperature does not provide an answer regarding the spectral composition of black body radiation. From the experimental curves of the function r λ,T from wavelength λ (r λ,T =´ ´ r ν,T) at different temperatures (Fig. 30.2) Fig. 31.2.
it follows that the energy distribution in the black body spectrum is uneven. All curves have a clearly defined maximum, which shifts toward shorter wavelengths as the temperature rises. Area enclosed by the curve r λ,T from λ and x-axis, proportional to energetic luminosity R e black body and, therefore, according to the Stefan-Boltzmann law, quarter powers of temperature.
V. Vin, relying on the laws of thermo- and electrodynamics, established the dependence of the wavelength λ max corresponding to the maximum of the function r λ,T, on temperature T. According to Wien's displacement law,
λ max =b/T, (31.10)
i.e. wavelength λ
max corresponding maximum value spectral
luminosity density r λ,T of a black body is inversely proportional to its thermodynamic temperature. b - constant guilt its experimental value is 2.9×10 -3 m×K.
Expression (31.10) is called Wien’s displacement law; it shows the displacement of the position of the maximum of the function r λ,T as the temperature increases into the region of short wavelengths. Wien's law explains why, as the temperature of heated bodies decreases, long-wave radiation increasingly dominates in their spectrum (for example, the transition white heat turns red when the metal cools).
Rayleigh-Jeans and Planck formulas
From the consideration of the Stefan-Boltzmann and Wien laws it follows that the thermodynamic approach to solving the problem of finding universal function Kirchhoff did not give the desired results.
A rigorous attempt to theoretically deduce the relationship r λ,T belongs to Rayleigh and Jeans, who applied methods to thermal radiation statistical physics, who took advantage classical law uniform distribution energy by degrees of freedom.
The Rayleigh-Jeans formula for the spectral luminosity density of a black body has the form:
r ν , T = <E> = kT, (31.11)
Where <Е>= kT– average energy of the oscillator with natural frequency ν .
As experience has shown, expression (31.11) is consistent with experimental data only in the region of sufficiently low frequencies and high temperatures. In the region of high frequencies, this formula diverges from experiment, as well as from Wien’s displacement law. And obtaining the Stefan-Boltzmann law from this formula leads to absurdity. This result was called the “ultraviolet catastrophe.” Those. within classical physics failed to explain the laws of energy distribution in the spectrum of a black body.
In the high-frequency range, good agreement with experiment is given by Wien’s formula (Wien’s radiation law):
r ν, T =Сν 3 А e –Аν/Т, (31.12)
Where r ν, T- spectral density of energy luminosity of a black body, WITH And A – constants. In modern notation using
Planck's constant Wien's radiation law can be written as
r ν, T = . (31.13)
The correct expression for the spectral density of the energy luminosity of a black body, consistent with experimental data, was found by Planck. According to the put forward quantum hypothesis, atomic oscillators emit energy not continuously, but in certain portions - quanta, and the energy of the quantum is proportional to the oscillation frequency
E 0 =hν = hс/λ,
Where h=6.625×10 -34 J×s – Planck’s constant. Since radiation is emitted in portions, the oscillator energy E can only take certain discrete values , multiples of an integer number of elementary portions of energy E 0
E = nhν(n= 0,1,2…).
IN in this case average energy<E> oscillator cannot be taken equal kT.
In the approximation that the distribution of oscillators over possible discrete states obeys the Boltzmann distribution, the average oscillator energy is equal to
<E> = , (31.14)
and the spectral density of energetic luminosity is determined by the formula
r ν , T = . (31.15)
Planck derived the formula for the universal Kirchhoff function
r ν, T = , (31.16)
which is consistent with experimental data on the energy distribution in the spectra of black body radiation over the entire range of frequencies and temperatures.
From Planck's formula, knowing the universal constants h,k And With, we can calculate the Stefan-Boltzmann constants σ and Wine b. And vice versa. Planck's formula agrees well with experimental data, but also contains particular laws of thermal radiation, i.e. is complete solution thermal radiation problems.
Optical pyrometry
The laws of thermal radiation are used to measure the temperature of hot and self-luminous bodies (for example, stars). Methods for measuring high temperatures that use the dependence of the spectral density of energy luminosity or the integral energy luminosity of bodies on temperature are called optical pyrometry. Devices for measuring the temperature of heated bodies based on the intensity of their thermal radiation in the optical range of the spectrum are called pyrometers. Depending on which law of thermal radiation is used when measuring the temperature of bodies, radiation, color and brightness temperatures are distinguished.
1. Radiation temperature- this is the temperature of a black body at which its energetic luminosity R e equal to energetic luminosity R t the body under study. In this case, the energetic luminosity of the body under study is recorded and its radiation temperature is calculated according to the Stefan-Boltzmann law:
T r =.
Radiation temperature T r body is always less than its true temperature T.
2.Colorful temperature. For gray bodies (or bodies similar to them in properties), the spectral density of energy luminosity
R λ,Τ = Α Τ r λ,Τ,
Where A t = const < 1. Consequently, the energy distribution in the radiation spectrum of a gray body is the same as in the spectrum of a black body having the same temperature, therefore Wien’s displacement law is applicable to gray bodies. Knowing the wavelength λ m ax corresponding to the maximum spectral density of energy luminosity Rλ,Τ of the body being examined, its temperature can be determined
T c = b/ λ m ah,
which is called color temperature. For gray bodies, the color temperature coincides with the true one. For bodies that are very different from gray (for example, those with selective absorption), the concept of color temperature loses its meaning. In this way, the temperature on the surface of the Sun is determined ( T c=6500 K) and stars.
3.Brightness temperature T i, is the temperature of a black body at which, for a certain wavelength, its spectral luminosity density equal to the spectral density of the energy luminosity of the body under study, i.e.
r λ,Τ = R λ,Τ,
Where T– true body temperature, which is always higher than the brightness temperature.
A vanishing filament pyrometer is usually used as a brightness pyrometer. In this case, the image of the pyrometer filament becomes indistinguishable against the background of the surface of the hot body, i.e., the filament seems to “disappear.” Using a blackbody calibrated milliammeter, the brightness temperature can be determined.
Thermal light sources
The glow of hot bodies is used to create light sources. Black bodies should be the best thermal light sources, since their spectral luminosity density for any wavelength is greater than the spectral luminosity density of non-black bodies taken at the same temperatures. However, it turns out that for some bodies (for example, tungsten) that have selectivity of thermal radiation, the proportion of energy attributable to radiation in the visible region of the spectrum is significantly greater than for a black body heated to the same temperature. Therefore, tungsten, having also a high melting point, is the best material for the manufacture of lamp filaments.
The temperature of the tungsten filament in vacuum lamps should not exceed 2450K, since at higher temperatures it is strongly sputtered. The maximum radiation at this temperature corresponds to a wavelength of 1.1 microns, i.e., very far from the maximum sensitivity of the human eye (0.55 microns). Filling lamp cylinders inert gases(for example, a mixture of krypton and xenon with the addition of nitrogen) at a pressure of 50 kPa makes it possible to increase the temperature of the filament to 3000 K, which leads to an improvement in the spectral composition of the radiation. However, the light output does not increase, since additional energy losses occur due to heat exchange between the filament and the gas due to thermal conductivity and convection. To reduce energy losses due to heat exchange and increase the light output of gas-filled lamps, the filament is made in the form of a spiral, the individual turns of which heat each other. At high temperature a stationary layer of gas is formed around this spiral and heat transfer due to convection is eliminated. Energy efficiency incandescent lamps currently do not exceed 5%.
To assess the energy of radiation and its effect on radiation receivers, which include photoelectric devices, thermal and photochemical receivers, as well as the eye, energy and light quantities are used.
Energy quantities are characteristics optical radiation, relating to the entire optical range.
Eye for a long time was the only receiver of optical radiation. Therefore, historically it has developed that for the qualitative and quantitative assessment of the visible part of radiation, light (photometric) quantities are used that are proportional to the corresponding energy quantities.
The concept of radiation flux relating to the entire optical range was given above. The quantity that in the system of light quantities corresponds to the radiation flux,
is the luminous flux Ф, i.e. the radiation power estimated by a standard photometric observer.
Let's consider light quantities and their units, and then find the connection between these quantities and energy ones.
To evaluate two sources visible radiation their luminescence in the direction of the same surface is compared. If the glow of one source is taken as unity, then by comparing the glow of the second source with the first we obtain a value called luminous intensity.
IN International system SI units for the unit of luminous intensity is the candela, the definition of which was approved by the XVI General Conference (1979).
Candela is the luminous intensity in a given direction of a source emitting monochromatic radiation with a frequency of Hz, energetic force the light of which in this direction is
Luminous intensity, or angular density of luminous flux,
where is the luminous flux in a certain direction inside the solid angle
A solid angle is a part of space limited by an arbitrary conical surface. If we describe a sphere from the top of this surface as from the center, then the area of the sphere section cut off by the conical surface (Fig. 85) will be proportional to the square of the radius of the sphere:
The proportionality coefficient is the value of the solid angle.
The unit of solid angle is the steradian, which is equal to the solid angle with its vertex at the center of the sphere, cutting out the area on the surface of the sphere, equal to the area square with side equal to the radius spheres. A complete sphere forms a solid angle
Rice. 85. Solid angle
Rice. 86. Radiation in solid angle
If the radiation source is located at the vertex of a right circular cone, then the solid angle allocated in space is limited by the internal cavity of this conical surface. Knowing the value of the plane angle between the axis and the generatrix of the conical surface, we can determine the corresponding solid angle.
Let us select in the solid angle an infinitesimal angle that cuts out an infinitely narrow annular section on the sphere (Fig. 86). This case refers to the most common axisymmetric luminous intensity distribution.
The area of the annular section is where the distance from the axis of the cone to the narrow ring width
According to Fig. where is the radius of the sphere.
Therefore where
Solid angle corresponding to a plane angle
For a hemisphere, the solid angle for a sphere is -
From formula (160) it follows that the luminous flux
If the intensity of light does not change when moving from one direction to another, then
Indeed, if a light source with luminous intensity is placed at the vertex of a solid angle, then the same luminous flux arrives at any areas limited by a conical surface that distinguishes this solid angle in space. Let us take the indicated areas in the form of sections of concentric spheres with the center at the vertex of the solid angle . Then, as experience shows, the degree of illumination of these areas is inversely proportional to the squares of the radii of these spheres and directly proportional to the size of the areas.
Thus, the following equality holds: i.e., formula (165).
The given justification for formula (165) is valid only in the case when the distance between the light source and the illuminated area is sufficiently large compared to the size of the source and when the medium between the source and the illuminated area does not absorb or scatter light energy.
The unit of luminous flux is the lumen (lm), which is the flux within a solid angle when the luminous intensity of a source located at the vertex of the solid angle is equal to
The illumination of the area normal to the incident rays is determined by the ratio called illuminance E:
Formula (166), as well as formula (165), takes place under the condition that the light intensity I does not change when moving from one direction to another within a given solid angle. IN otherwise this formula will be valid only for an infinitesimal area
If the incident rays form angles with the normal to the illuminated area, then formulas (166) and (167) will change, since the illuminated area will increase. As a result we get:
When the site is illuminated by several sources, its illumination
where the number of radiation sources, i.e. the total illumination is equal to the sum of the illumination received by the site from each source.
The unit of illumination is taken to be the illumination of the site when a luminous flux falls on it (the site is normal to the incident rays). This unit is called luxury
If the dimensions of the radiation source cannot be neglected, then to solve a number of problems it is necessary to know the distribution of the light flux of this source over its surface. The ratio of the luminous flux emanating from a surface element to the area of this element is called luminosity and is measured in lumens per square meter Luminosity also characterizes the distribution of reflected light flux.
Thus, the luminosity
where is the surface area of the source.
The ratio of the intensity of light in a given direction to the area of projection of the luminous surface onto a plane perpendicular to this direction is called brightness.
Therefore, the brightness
where is the angle between the normal to the site and the direction of the light intensity
Substituting the value [see. formula (160)), we obtain that the brightness
From formula (173) it follows that the brightness is the second derivative of the flux with respect to the solid angle to the area.
The unit of brightness is candela per square meter
The surface density of light energy of incident radiation is called exposure:
In general, the illumination included in formula (174) can change over time
The exposition has a large practical significance, for example in photography and is measured in lux seconds
Formulas (160)-(174) are used to calculate both light and energy quantities, firstly, for monochromatic radiation, i.e. radiation with a certain wavelength, and secondly, in the absence of taking into account the spectral distribution of radiation, which, usually occurs in visual optical instruments.
The spectral composition of radiation - the distribution of radiation power over wavelengths has great importance for calculating energy quantities when using selective radiation receivers. For these calculations, the concept of spectral radiation flux density was introduced [see. formulas (157)-(159)].
In a limited range of wavelengths, we respectively have:
The energy quantities determined by the formulas also apply to the visible part of the spectrum.
The main photometric and energy quantities, their defining formulas and SI units are given in Table. 5.
Photometry is the branch of optics that deals with the measurement of light fluxes and quantities associated with such fluxes. The following quantities are used in photometry:
1) energy – characterize the energy parameters of optical radiation regardless of its effect on radiation receivers;
2) light – characterize the physiological effect of light and are assessed by the effect on the eye (based on the so-called average sensitivity of the eye) or other radiation receivers.
1. Energy quantities. Radiation flux Φ e – value, equal to the ratio energy W radiation by time t, during which the radiation occurred:
The unit of radiation flux is watt (W).
Energetic luminosity (emissivity) R e– value equal to the ratio of the radiation flux Φ e emitted by the surface to the area S cross section through which this flow passes:
those. represents the surface radiation flux density.
The unit of energetic luminosity is watt per square meter (W/m2).
Radiation intensity:
where Δ S – small surface, perpendicular to the direction of radiation propagation, through which the flux ΔΦ e is transferred.
The unit of measurement for radiation intensity is the same as for energetic luminosity – W/m2.
To determine subsequent values, you will need to use one geometric concept – solid angle , which is a measure of the opening of some conical surface. As is known, the measure of a plane angle is the ratio of the arc of a circle l to the radius of this circle r, i.e. (Fig. 3.1 a). Similarly, the solid angle Ω is determined (Fig. 3.1 b) as the ratio of the surface ball segment S to the square of the radius of the sphere:
The unit of measurement for solid angle is steradian (ср) is a solid angle, the vertex of which is located in the center of the sphere, and which cuts out an area on the surface of the sphere equal to the square of the radius: Ω = 1 ср, if . It is easy to verify that the total solid angle around a point is equal to 4π steradians - to do this, you need to divide the surface of the sphere by the square of its radius.
Energy intensity of light (radiation power ) Ie determined using concepts about a point light source – a source whose size compared to the distance to the observation site can be neglected. The energetic intensity of light is a value equal to the ratio of the source radiation flux to the solid angle Ω within which this radiation propagates:
The unit of luminous energy is watt per steradian (W/sr).
Energy brightness (radiance) V e– a value equal to the ratio of the energy intensity of light ΔI e element of the radiating surface to the area ΔS projection of this element onto a plane perpendicular to the direction of observation:
. (3.6)
The unit of radiance is watt per steradian meter squared (W/(sr m2)).
Energy illumination (irradiance) Her characterizes the amount of radiation flux incident on a unit of illuminated surface. The irradiance unit is the same as the luminosity unit (W/m2).
2. Light quantities. In optical measurements, various radiation detectors are used (for example, the eye, photocells, photomultipliers), which do not have the same sensitivity to the energy of different wavelengths, thus being selective (selective) . Each receiver light radiation characterized by its sensitivity curve to light of different wavelengths. That's why light measurements, being subjective, differ from objective, energetic ones, and for them are introduced light units, used for visible light only. Basic light unit in SI is the unit of luminous intensity - candela (cd), which is defined as the luminous intensity in a given direction of a source emitting monochromatic radiation with a frequency of 540·10 12 Hz, the luminous energy intensity of which in this direction is 1/683 W/sr. The definition of light units is similar to energy units.
Light flow Φ light is defined as the power of optical radiation based on the light sensation it causes (about its effect on a selective light receiver with a given spectral sensitivity).
Luminous flux unit – lumen (lm): 1 lm – luminous flux emitted by a point source with a luminous intensity of 1 cd inside a solid angle of 1 sr (with uniformity of the radiation field inside the solid angle) (1 lm = 1 cd sr).
The power of light I St. is related to the luminous flux by the relation
, (3.7)
Where dΦ St– luminous flux emitted by a source within a solid angle dΩ. If I St. does not depend on direction, the light source is called isotropic. For an isotropic source
. (3.8)
Energy flow . Φ e, measured in watts, and luminous flux Φ St., measured in lumens, are related by the relationship:
, lm, (3.9)
Where 
- constant, is a function of visibility, determined by the sensitivity of the human eye to radiation of different wavelengths. The maximum value is reached at 
. The complex uses laser radiation with a wavelength 
. In this case .
Luminosity R St is determined by the relation
. (3.10)
The unit of luminosity is lumen per square meter (lm/m2).
Brightness In φ luminous surface area S in a certain direction forming an angle φ with the normal to the surface, there is a value equal to the ratio of the intensity of light in a given direction to the area of projection of the luminous surface onto a plane perpendicular to this direction:
. (3.11)
Sources whose brightness is the same in all directions are called Lambertian (subject to Lambert's law) or cosine (the flux sent by the surface element of such a source is proportional to ). Only a completely black body strictly follows Lambert's law.
The unit of brightness is candela per meter squared (cd/m2).
Illumination E– a value equal to the ratio of the luminous flux incident on a surface to the area of this surface:
.
(3.12)
Illuminance unit – luxury (lx): 1 lx – illumination of a surface on 1 m2 of which a luminous flux of 1 lm falls (1 lm = 1 lx/m2).
Work order
 Rice. 3.2. |
Task 1. Determining the laser light intensity.
By measuring the diameter of the diverging laser beam in two of its sections, separated by a distance, we can find the small beam divergence angle and the solid angle in which the radiation propagates (Fig. 3.2):
, (3.13)
Luminous intensity in candelas is determined by the formula:
, (3.15)
Where 
- constant, the radiation power is set to the minimum - equal (the laser current adjustment knob is turned to extreme position counterclockwise), is a function of visibility determined by the sensitivity of the human eye to radiation of different wavelengths. The maximum value is reached at . The complex uses laser radiation with a wavelength . In this case .
Experiment
1. Install module 2 on the optical bench and adjust the installation according to the method described on page . After making sure that the installation is adjusted, remove module 2.
2. Place the lens attachment on the emitter (object 42). Install the condenser lens (module 5) at the end of the bench with the screen facing the emitter. Fix the coordinate of the risks of its raters. Using the condenser screen, determine the diameter of the laser beam.
3. Move the condenser to the laser 50 - 100 mm. Fix the coordinate of the mark and, accordingly, determine the beam diameter using the condenser screen.
4. Calculate linear angle beam divergence according to formula (3.13), taking . Calculate the solid angle of beam divergence using formula (3.14) and the luminous intensity using formula (3.15). Produce standard assessment errors.
5. Carry out the experiment 4 more times with other positions of the condenser.
6. Enter the measurement results into tables:
| № | , | , | ||||
| № | , % | ||||
Task 2. Intensity in a spherical wave
The laser radiation beam is transformed by a collecting lens into a spherical wave, first converging to the focus, and after the focus - diverging. It is required to trace the nature of the change in intensity with the coordinate - . The voltmeter readings are used as values without conversion to absolute values.
Experiment
1. Remove the diffuser lens attachment from the emitter. At the end of the free bench, install a microprojector (module 2) and, close in front of it, a condenser lens (module 5). Make sure that when moving module 5 away from module 2, the size of the spot on the installation screen and the radiation intensity in the center of the spot changes. Return the condenser to its original position.
2. Place a photosensor - object 38 - in the object plane of the microprojector, connect the photosensor to the multimeter, set the multimeter to measurement mode DC voltage(measurement range - up to 1 V) and remove the dependence of the voltage on the voltmeter on the coordinate of module 5 with a step of 10 mm, taking the coordinate of the marks of module 2 as the reference point. Make 20 measurements.
4. Give definitions of the main photometric quantities (energy and light) indicating units of measurement.
5. What is the basic unit of light in SI? How is it determined?
6. How are radiation flux and luminous flux related?
7. Which light source is called isotropic? How are luminous intensity and luminous flux of an isotropic source related? Why?
8. When is a light source called Lambertian? Give an example of a strictly Lambertian source.
9. How does the intensity of a light wave emitted by an isotropic point source depend on the distance to the source? Why?
Question 2. Photometric quantities and their units.
Photometry is a branch of optics dealing with measurement issues. energy characteristics optical radiation in the processes of propagation and interaction with matter. Photometry uses energy quantities that characterize the energy parameters of optical radiation, regardless of its effect on radiation receivers, and also uses light quantities that characterize the physiological effects of light and are assessed by their effect on human eyes or other receivers.
Energy quantities.
Energy flowF e – quantity numerically equal to energy W radiation passing through a section perpendicular to the direction of energy transfer per unit time
F e = W/ t, watt (W).
Energy flow is equivalent to energy power.
Energy emitted real source into the surrounding space, distributed over its surface.
Energetic luminosity(emissivity) R e – radiation power per unit surface area in all directions:
R e = F e/ S, (W/m 2),
those. represents the surface radiation flux density.
Energy power of light (radiation strength) I e is determined using the concept of a point light source - a source whose dimensions, compared to the distance to the observation site, can be neglected. Energy power of light I e value equal to the ratio of the radiation flux F e source to solid angle ω , within which this radiation propagates:
I e = F e/ ω , (W/Wed) - watt per steradian.
A solid angle is a part of space limited by a certain conical surface. Special cases of solid angles are trihedral and polyhedral angles. Solid angle ω measured by area ratio S that part of the sphere with its center at the vertex of the conical surface, which is cut by this solid angle, to the square of the radius of the sphere, i.e. ω = S/r 2. A complete sphere forms a solid angle equal to 4π steradians, i.e. ω = 4π r 2 /r 2 = 4π Wed.
The luminous intensity of a source often depends on the direction of the radiation. If it does not depend on the direction of radiation, then such a source is called isotropic. For an isotropic source, the luminous intensity is
I e = F e/4π.
In the case of an extended source, we can talk about the luminous intensity of an element of its surface dS.
Energy brightness (radiance) IN e – value equal to the ratio of the energy intensity of light Δ I e element of the radiating surface to the area ΔS projection of this element onto a plane perpendicular to the direction of observation:
IN e = Δ I e/Δ S. [(W/(avg.m 2)].
Energy illuminance (irradiance) E e characterizes the degree of illumination of the surface and is equal to the amount of radiation flux from all directions incident on a unit of illuminated surface ( W/m 2).
In photometry the law is used inverse squares(Kepler's law): illumination of a plane from a perpendicular direction from a point source with force I e at a distance r it is equal to:
E e = I e/ r 2 .
Deviation of a beam of optical radiation from the perpendicular to the surface by an angle α leads to a decrease in illumination (Lambert’s law):
E e = I ecos α /r 2 .
Important role When measuring the energy characteristics of radiation, the temporal and spectral distribution of its power play a role. If the duration of optical radiation is less than the observation time, then the radiation is considered pulsed, and if longer, it is considered continuous. Sources can emit radiation of different wavelengths. Therefore, in practice, the concept of radiation spectrum is used - the distribution of radiation power along the wavelength scale λ (or frequencies). Almost all sources emit differently at different areas spectrum
For an infinitesimal wavelength interval dλ the value of any photometric quantity can be specified using its spectral density. For example, the spectral density of energy luminosity
R eλ = dW/dλ,
Where dW– energy emitted from a unit surface area per unit time in the wavelength range from λ before λ + dλ.
Light quantities. For optical measurements, various radiation receivers are used, spectral characteristics whose sensitivity to light of different wavelengths is different. The spectral sensitivity of an optical radiation photodetector is the ratio of the quantity characterizing the level of the receiver's response to the flux or energy of monochromatic radiation causing this reaction. There are absolute spectral sensitivity expressed in named units (for example, A/W, if the receiver response is measured in A), and dimensionless relative spectral sensitivity - the ratio of the spectral sensitivity at a given wavelength of radiation to the maximum value of spectral sensitivity or to the spectral sensitivity at a certain wavelength.
The spectral sensitivity of a photodetector depends only on its properties; it is different for different receivers. Relative spectral sensitivity of the human eye V(λ ) is shown in Fig. 5.3.
The eye is most sensitive to radiation with a wavelength λ =555 nm. Function V(λ ) for this wavelength is taken equal to unity.
With the same energy flux, the visually assessed light intensity for other wavelengths turns out to be less. The relative spectral sensitivity of the human eye for these wavelengths is less than unity. For example, the value of the function means that light of a given wavelength must have an energy flux density 2 times greater than the light for which , in order for the visual sensations to be the same.
The system of light quantities is introduced taking into account the relative spectral sensitivity of the human eye. Therefore, light measurements, being subjective, differ from objective, energy ones, and light units are introduced for them, used only for visible light. The basic unit of light in the SI system is luminous intensity - candela (cd), which is equal to the intensity of light in a given direction of a source emitting monochromatic radiation with a frequency of 5.4 10 14 Hz, whose luminous energy in this direction is 1/683 W/sr. All other luminous quantities are expressed in candela.
The definition of light units is similar to energy units. To measure light quantities use special techniques and instruments – photometers.
Light flow . The unit of luminous flux is lumen (lm). It is equal to the luminous flux emitted by an isotropic light source with an intensity of 1 cd within a solid angle of one steradian (with uniformity of the radiation field within the solid angle):
1 lm = 1 cd·1 Wed.
It has been experimentally established that a luminous flux of 1 lm generated by radiation with a wavelength λ = 555nm corresponds to an energy flow of 0.00146 W. Luminous flux in 1 lm, formed by radiation with a different wavelength λ , corresponds to the energy flow
F e = 0.00146/ V(λ ), W,
those. 1 lm = 0,00146 W.
Illumination E- a value relative to the luminous flux ratio F falling on the surface, to the area S this surface:
E = F/S, luxury (OK).
1 OK– surface illumination, per 1 m 2 of which the luminous flux falls at 1 lm (1OK = 1 lm/m 2). To measure illumination, instruments are used that measure the flux of optical radiation from all directions - lux meters.
Brightness R C (luminosity) of a luminous surface in a certain direction φ is a quantity equal to the ratio of luminous intensity I in this direction to the square S projection of the luminous surface onto a plane perpendicular to a given direction:
R C= I/(S cos φ ), (cd/m 2).
In general, the brightness of light sources is different for different directions. Sources whose brightness is the same in all directions are called Lambertian or cosine, since the luminous flux emitted by the surface element of such a source is proportional to cosφ. Only an absolutely black body strictly satisfies this condition.
Any photometer with a limited viewing angle is essentially a brightness meter. Measuring the spectral and spatial distribution of brightness and irradiance allows all other photometric quantities to be calculated by integration.
1. What is it? physical meaning absolute indicator
refraction of the medium?
2. What is relative indicator refraction?
3. Under what conditions is it observed? total reflection?
4. What is the operating principle of light guides?
5. What is Fermat's principle?
6. How do energy and light quantities differ in photometry?