In spectroscopy, spectral flux density is the quantity that describes the rate at which energy is transferred by electromagnetic radiation through a real or virtual surface, per unit surface area and per unit wavelength (or, equivalently, per unit frequency). It is a radiometric rather than a photometric measure. In SI units it is measured in W m−3, although it can be more practical to use W m−2 nm−1 (1 W m−2 nm−1 = 1 GW m−3 = 1 W mm−3) or W m−2 μm−1 (1 W m−2 μm−1 = 1 MW m−3), and respectively by W·m−2·Hz−1, Jansky or solar flux units. The terms irradiance, radiant exitance, radiant emittance, and radiosity are closely related to spectral flux density.
The terms used to describe spectral flux density vary between fields, sometimes including adjectives such as "electromagnetic" or "radiative", and sometimes dropping the word "density". Applications include:
For the flux density received from a remote unresolvable "point source", the measuring instrument, usually telescopic, though not able to resolve any detail of the source itself, must be able to optically resolve enough details of the sky around the point source, so as to record radiation coming from it only, uncontaminated by radiation from other sources. In this case, [1] spectral flux density is the quantity that describes the rate at which energy transferred by electromagnetic radiation is received from that unresolved point source, per unit receiving area facing the source, per unit wavelength range.
At any given wavelength λ, the spectral flux density, Fλ, can be determined by the following procedure:
Spectral flux density is often used as the quantity on the y-axis of a graph representing the spectrum of a light-source, such as a star.
There are two main approaches to definition of the spectral flux density at a measuring point in an electromagnetic radiative field. One may be conveniently here labelled the 'vector approach', the other the 'scalar approach'. The vector definition refers to the full spherical integral of the spectral radiance (also known as the specific radiative intensity or specific intensity) at the point, while the scalar definition refers to the many possible hemispheric integrals of the spectral radiance (or specific intensity) at the point. The vector definition seems to be preferred for theoretical investigations of the physics of the radiative field. The scalar definition seems to be preferred for practical applications.
The vector approach defines flux density as a vector at a point of space and time prescribed by the investigator. To distinguish this approach, one might speak of the 'full spherical flux density'. In this case, nature tells the investigator what is the magnitude, direction, and sense of the flux density at the prescribed point. [2] [3] [4] [5] [6] [7] For the flux density vector, one may write
where denotes the spectral radiance (or specific intensity) at the point at time and frequency , denotes a variable unit vector with origin at the point , denotes an element of solid angle around , and indicates that the integration extends over the full range of solid angles of a sphere.
Mathematically, defined as an unweighted integral over the solid angle of a full sphere, the flux density is the first moment of the spectral radiance (or specific intensity) with respect to solid angle. [5] It is not common practice to make the full spherical range of measurements of the spectral radiance (or specific intensity) at the point of interest, as is needed for the mathematical spherical integration specified in the strict definition; the concept is nevertheless used in theoretical analysis of radiative transfer.
As described below, if the direction of the flux density vector is known in advance because of a symmetry, namely that the radiative field is uniformly layered and flat, then the vector flux density can be measured as the 'net flux', by algebraic summation of two oppositely sensed scalar readings in the known direction, perpendicular to the layers.
At a given point in space, in a steady-state field, the vector flux density, a radiometric quantity, is equal to the time-averaged Poynting vector, [8] an electromagnetic field quantity. [4] [7]
Within the vector approach to the definition, however, there are several specialized sub-definitions. Sometimes the investigator is interested only in a specific direction, for example the vertical direction referred to a point in a planetary or stellar atmosphere, because the atmosphere there is considered to be the same in every horizontal direction, so that only the vertical component of the flux is of interest. Then the horizontal components of flux are considered to cancel each other by symmetry, leaving only the vertical component of the flux as non-zero. In this case [4] some astrophysicists think in terms of the astrophysical flux (density), which they define as the vertical component of the flux (of the above general definition) divided by the number π. And sometimes [4] [5] the astrophysicist uses the term Eddington flux to refer to the vertical component of the flux (of the above general definition) divided by the number 4π.
The scalar approach defines flux density as a scalar-valued function of a direction and sense in space prescribed by the investigator at a point prescribed by the investigator. Sometimes [9] this approach is indicated by the use of the term 'hemispheric flux'. For example, an investigator of thermal radiation, emitted from the material substance of the atmosphere, received at the surface of the earth, is interested in the vertical direction, and the downward sense in that direction. This investigator thinks of a unit area in a horizontal plane, surrounding the prescribed point. The investigator wants to know the total power of all the radiation from the atmosphere above in every direction, propagating with a downward sense, received by that unit area. [10] [11] [12] [13] [14] For the flux density scalar for the prescribed direction and sense, we may write
where with the notation above, indicates that the integration extends only over the solid angles of the relevant hemisphere, and denotes the angle between and the prescribed direction. The term is needed on account of Lambert's law. [15] Mathematically, the quantity is not a vector because it is a positive scalar-valued function of the prescribed direction and sense, in this example, of the downward vertical. In this example, when the collected radiation is propagating in the downward sense, the detector is said to be "looking upwards". The measurement can be made directly with an instrument (such as a pyrgeometer) that collects the measured radiation all at once from all the directions of the imaginary hemisphere; in this case, Lambert-cosine-weighted integration of the spectral radiance (or specific intensity) is not performed mathematically after the measurement; the Lambert-cosine-weighted integration has been performed by the physical process of measurement itself.
In a flat horizontal uniformly layered radiative field, the hemispheric fluxes, upwards and downwards, at a point, can be subtracted to yield what is often called the net flux. The net flux then has a value equal to the magnitude of the full spherical flux vector at that point, as described above.
The radiometric description of the electromagnetic radiative field at a point in space and time is completely represented by the spectral radiance (or specific intensity) at that point. In a region in which the material is uniform and the radiative field is isotropic and homogeneous, let the spectral radiance (or specific intensity) be denoted by I (x, t ; r1, ν), a scalar-valued function of its arguments x, t, r1, and ν, where r1 denotes a unit vector with the direction and sense of the geometrical vector r from the source point P1 to the detection point P2, where x denotes the coordinates of P1, at time t and wave frequency ν. Then, in the region, I (x, t ; r1, ν) takes a constant scalar value, which we here denote by I. In this case, the value of the vector flux density at P1 is the zero vector, while the scalar or hemispheric flux density at P1 in every direction in both senses takes the constant scalar value πI. The reason for the value πI is that the hemispheric integral is half the full spherical integral, and the integrated effect of the angles of incidence of the radiation on the detector requires a halving of the energy flux according to Lambert's cosine law; the solid angle of a sphere is 4π.
The vector definition is suitable for the study of general radiative fields. The scalar or hemispheric spectral flux density is convenient for discussions in terms of the two-stream model of the radiative field, which is reasonable for a field that is uniformly stratified in flat layers, when the base of the hemisphere is chosen to be parallel to the layers, and one or other sense (up or down) is specified. In an inhomogeneous non-isotropic radiative field, the spectral flux density defined as a scalar-valued function of direction and sense contains much more directional information than does the spectral flux density defined as a vector, but the full radiometric information is customarily stated as the spectral radiance (or specific intensity).
For the present purposes, the light from a star, and for some particular purposes, the light of the sun, can be treated as a practically collimated beam, but apart from this, a collimated beam is rarely if ever found in nature, [16] though artificially produced beams can be very nearly collimated. [17] The spectral radiance (or specific intensity) is suitable for the description of an uncollimated radiative field. The integrals of spectral radiance (or specific intensity) with respect to solid angle, used above, are singular for exactly collimated beams, or may be viewed as Dirac delta functions. Therefore, the specific radiative intensity is unsuitable for the description of a collimated beam, while spectral flux density is suitable for that purpose. [18] At a point within a collimated beam, the spectral flux density vector has a value equal to the Poynting vector, [8] a quantity defined in the classical Maxwell theory of electromagnetic radiation. [7] [19] [20]
Sometimes it is more convenient to display graphical spectra with vertical axes that show the relative spectral flux density. In this case, the spectral flux density at a given wavelength is expressed as a fraction of some arbitrarily chosen reference value. Relative spectral flux densities are expressed as pure numbers without any units.
Spectra showing the relative spectral flux density are used when we are interested in comparing the spectral flux densities of different sources; for example, if we want to show how the spectra of blackbody sources vary with absolute temperature, it is not necessary to show the absolute values. The relative spectral flux density is also useful if we wish to compare a source's flux density at one wavelength with the same source's flux density at another wavelength; for example, if we wish to demonstrate how the Sun's spectrum peaks in the visible part of the EM spectrum, a graph of the Sun's relative spectral flux density will suffice.
The reflectance of the surface of a material is its effectiveness in reflecting radiant energy. It is the fraction of incident electromagnetic power that is reflected at the boundary. Reflectance is a component of the response of the electronic structure of the material to the electromagnetic field of light, and is in general a function of the frequency, or wavelength, of the light, its polarization, and the angle of incidence. The dependence of reflectance on the wavelength is called a reflectance spectrum or spectral reflectance curve.
Flux describes any effect that appears to pass or travel through a surface or substance. Flux is a concept in applied mathematics and vector calculus which has many applications to physics. For transport phenomena, flux is a vector quantity, describing the magnitude and direction of the flow of a substance or property. In vector calculus flux is a scalar quantity, defined as the surface integral of the perpendicular component of a vector field over a surface.
In physics, the Poynting vector represents the directional energy flux or power flow of an electromagnetic field. The SI unit of the Poynting vector is the watt per square metre (W/m2); kg/s3 in base SI units. It is named after its discoverer John Henry Poynting who first derived it in 1884. Nikolay Umov is also credited with formulating the concept. Oliver Heaviside also discovered it independently in the more general form that recognises the freedom of adding the curl of an arbitrary vector field to the definition. The Poynting vector is used throughout electromagnetics in conjunction with Poynting's theorem, the continuity equation expressing conservation of electromagnetic energy, to calculate the power flow in electromagnetic fields.
In physics, Planck's law describes the spectral density of electromagnetic radiation emitted by a black body in thermal equilibrium at a given temperature T, when there is no net flow of matter or energy between the body and its environment.
Transmittance of the surface of a material is its effectiveness in transmitting radiant energy. It is the fraction of incident electromagnetic power that is transmitted through a sample, in contrast to the transmission coefficient, which is the ratio of the transmitted to incident electric field.
In radiometry, radiance is the radiant flux emitted, reflected, transmitted or received by a given surface, per unit solid angle per unit projected area. Radiance is used to characterize diffuse emission and reflection of electromagnetic radiation, and to quantify emission of neutrinos and other particles. The SI unit of radiance is the watt per steradian per square metre. It is a directional quantity: the radiance of a surface depends on the direction from which it is being observed.
In radiometry, irradiance is the radiant flux received by a surface per unit area. The SI unit of irradiance is the watt per square metre (W⋅m−2). The CGS unit erg per square centimetre per second (erg⋅cm−2⋅s−1) is often used in astronomy. Irradiance is often called intensity, but this term is avoided in radiometry where such usage leads to confusion with radiant intensity. In astrophysics, irradiance is called radiant flux.
In classical electromagnetism, magnetic vector potential is the vector quantity defined so that its curl is equal to the magnetic field: . Together with the electric potential φ, the magnetic vector potential can be used to specify the electric field E as well. Therefore, many equations of electromagnetism can be written either in terms of the fields E and B, or equivalently in terms of the potentials φ and A. In more advanced theories such as quantum mechanics, most equations use potentials rather than fields.
In radiometry, radiant intensity is the radiant flux emitted, reflected, transmitted or received, per unit solid angle, and spectral intensity is the radiant intensity per unit frequency or wavelength, depending on whether the spectrum is taken as a function of frequency or of wavelength. These are directional quantities. The SI unit of radiant intensity is the watt per steradian, while that of spectral intensity in frequency is the watt per steradian per hertz and that of spectral intensity in wavelength is the watt per steradian per metre —commonly the watt per steradian per nanometre. Radiant intensity is distinct from irradiance and radiant exitance, which are often called intensity in branches of physics other than radiometry. In radio-frequency engineering, radiant intensity is sometimes called radiation intensity.
In computer graphics, the rendering equation is an integral equation in which the equilibrium radiance leaving a point is given as the sum of emitted plus reflected radiance under a geometric optics approximation. It was simultaneously introduced into computer graphics by David Immel et al. and James Kajiya in 1986. The various realistic rendering techniques in computer graphics attempt to solve this equation.
In differential geometry, the four-gradient is the four-vector analogue of the gradient from vector calculus.
An isotropic radiator is a theoretical point source of electromagnetic or sound waves which radiates the same intensity of radiation in all directions. It has no preferred direction of radiation. It radiates uniformly in all directions over a sphere centred on the source. Isotropic radiators are used as reference radiators with which other sources are compared, for example in determining the gain of antennas. A coherent isotropic radiator of electromagnetic waves is theoretically impossible, but incoherent radiators can be built. An isotropic sound radiator is possible because sound is a longitudinal wave.
In classical electromagnetism, reciprocity refers to a variety of related theorems involving the interchange of time-harmonic electric current densities (sources) and the resulting electromagnetic fields in Maxwell's equations for time-invariant linear media under certain constraints. Reciprocity is closely related to the concept of symmetric operators from linear algebra, applied to electromagnetism.
Radiative transfer is the physical phenomenon of energy transfer in the form of electromagnetic radiation. The propagation of radiation through a medium is affected by absorption, emission, and scattering processes. The equation of radiative transfer describes these interactions mathematically. Equations of radiative transfer have application in a wide variety of subjects including optics, astrophysics, atmospheric science, and remote sensing. Analytic solutions to the radiative transfer equation (RTE) exist for simple cases but for more realistic media, with complex multiple scattering effects, numerical methods are required. The present article is largely focused on the condition of radiative equilibrium.
In radiometry, radiant flux or radiant power is the radiant energy emitted, reflected, transmitted, or received per unit time, and spectral flux or spectral power is the radiant flux per unit frequency or wavelength, depending on whether the spectrum is taken as a function of frequency or of wavelength. The SI unit of radiant flux is the watt (W), one joule per second, while that of spectral flux in frequency is the watt per hertz and that of spectral flux in wavelength is the watt per metre —commonly the watt per nanometre.
The linear attenuation coefficient, attenuation coefficient, or narrow-beam attenuation coefficient characterizes how easily a volume of material can be penetrated by a beam of light, sound, particles, or other energy or matter. A coefficient value that is large represents a beam becoming 'attenuated' as it passes through a given medium, while a small value represents that the medium had little effect on loss. The SI unit of attenuation coefficient is the reciprocal metre (m−1). Extinction coefficient is another term for this quantity, often used in meteorology and climatology. Most commonly, the quantity measures the exponential decay of intensity, that is, the value of downward e-folding distance of the original intensity as the energy of the intensity passes through a unit thickness of material, so that an attenuation coefficient of 1 m−1 means that after passing through 1 metre, the radiation will be reduced by a factor of e, and for material with a coefficient of 2 m−1, it will be reduced twice by e, or e2. Other measures may use a different factor than e, such as the decadic attenuation coefficient below. The broad-beam attenuation coefficient counts forward-scattered radiation as transmitted rather than attenuated, and is more applicable to radiation shielding.
Absorptance of the surface of a material is its effectiveness in absorbing radiant energy. It is the ratio of the absorbed to the incident radiant power. This should not be confused with absorbance and absorption coefficient.
Radiative equilibrium is the condition where the total thermal radiation leaving an object is equal to the total thermal radiation entering it. It is one of the several requirements for thermodynamic equilibrium, but it can occur in the absence of thermodynamic equilibrium. There are various types of radiative equilibrium, which is itself a kind of dynamic equilibrium.
In radiometry, spectral radiance or specific intensity is the radiance of a surface per unit frequency or wavelength, depending on whether the spectrum is taken as a function of frequency or of wavelength. The SI unit of spectral radiance in frequency is the watt per steradian per square metre per hertz and that of spectral radiance in wavelength is the watt per steradian per square metre per metre —commonly the watt per steradian per square metre per nanometre. The microflick is also used to measure spectral radiance in some fields.
In spectroscopy and radiometry, vector radiative transfer (VRT) is a method of modelling the propagation of polarized electromagnetic radiation in low density media. In contrast to scalar radiative transfer (RT), which models only the first Stokes component, the intensity, VRT models all four components through vector methods.