Isotropic Radiator
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An isotropic radiator is a theoretical point source of
electromagnetic In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of a ...
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 Gain or GAIN may refer to: Science and technology * Gain (electronics), an electronics and signal processing term * Antenna gain * Gain (laser), the amplification involved in laser emission * Gain (projection screens) * Information gain in d ...
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 Longitudinal waves are waves in which the vibration of the medium is parallel ("along") to the direction the wave travels and displacement of the medium is in the same (or opposite) direction of the wave propagation. Mechanical longitudinal wa ...
. The unrelated term '' isotropic radiation'' refers to radiation which has the same intensity in all directions, thus an isotropic radiator does ''not'' radiate isotropic radiation.


Physics

In physics, an isotropic radiator is a point radiation or sound source. At a distance, the sun is an isotropic radiator of electromagnetic radiation.


Antenna theory

In antenna theory, an isotropic antenna is a hypothetical antenna radiating the same intensity of
radio wave Radio waves are a type of electromagnetic radiation with the longest wavelengths in the electromagnetic spectrum, typically with frequencies of 300 gigahertz ( GHz) and below. At 300 GHz, the corresponding wavelength is 1 mm (sho ...
s in all directions. It thus is said to have a directivity of 0 dBi (dB relative to isotropic) in all directions. Since it is entirely non-directional, it serves as a hypothetical worst-case against which directional antennas may be compared. In reality, a '' coherent'' isotropic radiator of linear
polarization Polarization or polarisation may refer to: Mathematics *Polarization of an Abelian variety, in the mathematics of complex manifolds *Polarization of an algebraic form, a technique for expressing a homogeneous polynomial in a simpler fashion by ...
can be shown to be impossible. Its radiation field could not be consistent with the Helmholtz wave equation (derived from
Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. Th ...
) in all directions simultaneously. Consider a large sphere surrounding the hypothetical point source, in the far field of the radiation pattern so that at that radius the wave over a reasonable area is essentially planar. In the far field the electric (and magnetic) field of a plane wave in free space is always perpendicular to the direction of propagation of the wave. So the electric field would have to be tangent to the surface of the sphere everywhere, and continuous along that surface. However the
hairy ball theorem The hairy ball theorem of algebraic topology (sometimes called the hedgehog theorem in Europe) states that there is no nonvanishing continuous tangent vector field on even-dimensional ''n''-spheres. For the ordinary sphere, or 2‑sphere, ...
shows that a continuous vector field
tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...
to the surface of a sphere must fall to zero at one or more points on the sphere, which is inconsistent with the assumption of an isotropic radiator with linear polarization. '' Incoherent'' isotropic radiators are possible and do not violate Maxwell's equations. Acoustic isotropic radiators are possible because sound waves in a gas or liquid are
longitudinal wave Longitudinal waves are waves in which the vibration of the medium is parallel ("along") to the direction the wave travels and displacement of the medium is in the same (or opposite) direction of the wave propagation. Mechanical longitudinal wa ...
s and not transverse waves. Even though an isotropic antenna cannot exist in practice, it is used as a base of comparison to calculate the directivity of actual antennas.
Antenna gain In electromagnetics, an antenna's gain is a key performance parameter which combines the antenna's directivity and radiation efficiency. The term ''power gain'' has been deprecated by IEEE. In a transmitting antenna, the gain describes h ...
\scriptstyle G, which is equal to the antenna's directivity multiplied by the
antenna efficiency Antenna ''apertureillumination efficiency is a measure of the extent to which an antenna or array is uniformly excited or illuminated. It is typical for an antenna pertureor array to be intentionally under-illuminated or under-excited in order t ...
, is defined as the ratio of the intensity \scriptstyle I (power per unit area) of the radio power received at a given distance from the antenna (in the direction of maximum radiation) to the intensity \scriptstyle I_\text received from a perfect lossless isotropic antenna at the same distance. This is called ''isotropic gain'' :G = \, Gain is often expressed in logarithmic units called decibels (dB). When gain is calculated with respect to an isotropic antenna, these are called ''decibels isotropic'' (dBi) :G\mathrm = 10\log\, The gain of any perfectly efficient antenna averaged over all directions is unity, or 0 dBi.


Isotropic receiver

In EMF measurement applications, an isotropic receiver (also called isotropic antenna) is a calibrated
radio receiver In radio communications, a radio receiver, also known as a receiver, a wireless, or simply a radio, is an electronic device that receives radio waves and converts the information carried by them to a usable form. It is used with an antenna. T ...
with an antenna which approximates an isotropic reception pattern; that is, it has close to equal sensitivity to radio waves from any direction. It is used as a field measurement instrument to measure electromagnetic sources and calibrate antennas. The isotropic receiving antenna is usually approximated by three orthogonal antennas or sensing devices with a radiation pattern of the omnidirectional type \sin (\theta), such as short dipoles or small
loop antenna A loop antenna is a radio antenna consisting of a loop or coil of wire, tubing, or other electrical conductor, that is usually fed by a balanced source or feeding a balanced load. Within this physical description there are two (possibly three) ...
s. The parameter used to define accuracy in the measurements is called isotropic deviation.


Optics

In optics, an isotropic radiator is a point source of light. The sun approximates an isotropic radiator of light. Certain munitions such as flares and chaff have isotropic radiator properties. Whether a radiator is isotropic is independent of whether it obeys
Lambert's law In optics, Lambert's cosine law says that the radiant intensity or luminous intensity observed from an ideal diffusely reflecting surface or ideal diffuse radiator is directly proportional to the cosine of the angle ''θ'' between the direction ...
. As radiators, a spherical black body is both, a flat black body is Lambertian but not isotropic, a flat chrome sheet is neither, and by symmetry the Sun is isotropic, but not Lambertian on account of
limb darkening Limb darkening is an optical effect seen in stars (including the Sun), where the central part of the disk appears brighter than the edge, or ''limb''. Its understanding offered early solar astronomers an opportunity to construct models with such ...
.


Sound

An isotropic sound radiator is a theoretical
loudspeaker A loudspeaker (commonly referred to as a speaker or speaker driver) is an electroacoustic transducer that converts an electrical audio signal into a corresponding sound. A ''speaker system'', also often simply referred to as a "speaker" or ...
radiating equal sound volume in all directions. Since sound waves are
longitudinal wave Longitudinal waves are waves in which the vibration of the medium is parallel ("along") to the direction the wave travels and displacement of the medium is in the same (or opposite) direction of the wave propagation. Mechanical longitudinal wa ...
s, a coherent isotropic sound radiator is feasible; an example is a pulsing spherical membrane or diaphragm, whose surface expands and contracts radially with time, pushing on the air.


Derivation of aperture of an isotropic antenna

The
aperture In optics, an aperture is a hole or an opening through which light travels. More specifically, the aperture and focal length of an optical system determine the cone angle of a bundle of rays that come to a focus in the image plane. An ...
of an isotropic antenna can be derived by a thermodynamic argument. Suppose an ideal (lossless) isotropic antenna ''A'' located within a
thermal cavity Black-body radiation is the thermal electromagnetic radiation within, or surrounding, a body in thermodynamic equilibrium with its environment, emitted by a black body (an idealized opaque, non-reflective body). It has a specific, continuous s ...
''CA'', is connected via a lossless
transmission line In electrical engineering, a transmission line is a specialized cable or other structure designed to conduct electromagnetic waves in a contained manner. The term applies when the conductors are long enough that the wave nature of the transmi ...
through a
band-pass filter A band-pass filter or bandpass filter (BPF) is a device that passes frequencies within a certain range and rejects ( attenuates) frequencies outside that range. Description In electronics and signal processing, a filter is usually a two- ...
''Fν'' to a matched resistor ''R'' in another thermal cavity ''CR'' (the
characteristic impedance The characteristic impedance or surge impedance (usually written Z0) of a uniform transmission line is the ratio of the amplitudes of voltage and current of a single wave propagating along the line; that is, a wave travelling in one direction ...
of the antenna, line and filter are all matched). Both cavities are at the same temperature T. The filter ''Fν'' only allows through a narrow band of
frequencies Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from '' angular frequency''. Frequency is measured in hertz (Hz) which is e ...
from \nu to \nu +\Delta\nu. Both cavities are filled with blackbody radiation in equilibrium with the antenna and resistor. Some of this radiation is received by the antenna. The amount of this power P_\text within the band of frequencies \Delta\nu passes through the transmission line and filter ''Fν'' and is dissipated as heat in the resistor. The rest is reflected by the filter back to the antenna and is reradiated into the cavity. The resistor also produces Johnson–Nyquist noise current due to the random motion of its molecules at the temperature T. The amount of this power P_\text within the frequency band \Delta\nu passes through the filter and is radiated by the antenna. Since the entire system is at the same temperature it is in
thermodynamic equilibrium Thermodynamic equilibrium is an axiomatic concept of thermodynamics. It is an internal state of a single thermodynamic system, or a relation between several thermodynamic systems connected by more or less permeable or impermeable walls. In ther ...
; there can be no net transfer of power between the cavities, otherwise one cavity would heat up and the other would cool down in violation of the
second law of thermodynamics The second law of thermodynamics is a physical law based on universal experience concerning heat and energy interconversions. One simple statement of the law is that heat always moves from hotter objects to colder objects (or "downhill"), unles ...
. Therefore the power flows in both directions must be equal :P_\text = P_\text The radio noise in the cavity is unpolarized, containing an equal mixture of
polarization Polarization or polarisation may refer to: Mathematics *Polarization of an Abelian variety, in the mathematics of complex manifolds *Polarization of an algebraic form, a technique for expressing a homogeneous polynomial in a simpler fashion by ...
states. However any antenna with a single output is polarized, and can only receive one of two orthogonal polarization states. For example, a
linearly polarized In electrodynamics, linear polarization or plane polarization of electromagnetic radiation is a confinement of the electric field vector or magnetic field vector to a given plane along the direction of propagation. The term ''linear polarizati ...
antenna cannot receive components of radio waves with electric field perpendicular to the antenna's linear elements; similarly a right circularly polarized antenna cannot receive left circularly polarized waves. Therefore the antenna only receives the component of power density ''S'' in the cavity matched to its polarization, which is half of the total power density :S_\text = S Suppose B_\nu is the spectral radiance per hertz in the cavity; the power of black body radiation per unit area (meter2) per unit
solid angle In geometry, a solid angle (symbol: ) is a measure of the amount of the field of view from some particular point that a given object covers. That is, it is a measure of how large the object appears to an observer looking from that point. The po ...
(
steradian The steradian (symbol: sr) or square radian is the unit of solid angle in the International System of Units (SI). It is used in three-dimensional geometry, and is analogous to the radian, which quantifies planar angles. Whereas an angle in radi ...
) per unit frequency (
hertz The hertz (symbol: Hz) is the unit of frequency in the International System of Units (SI), equivalent to one event (or cycle) per second. The hertz is an SI derived unit whose expression in terms of SI base units is s−1, meaning that one her ...
) at frequency \nu and temperature T in the cavity. If A_\text(\theta,\phi) is the antenna's aperture, the amount of power in the frequency range \Delta\nu the antenna receives from an increment of solid angle d\Omega = d\theta d\phi in the direction \theta,\phi is :dP_\text(\theta,\phi) = A_\text(\theta,\phi) S_\text \Delta\nu d\Omega = A_\text(\theta,\phi) B_\nu \Delta\nu d\Omega To find the total power in the frequency range \Delta\nu the antenna receives, this is integrated over all directions (a solid angle of 4\pi) :P_\text = \int\limits_ A_\text(\theta,\phi) B_\nu \Delta\nu d\Omega Since the antenna is isotropic, it has the same aperture A_\text(\theta,\phi) = A_\text in any direction. So the aperture can be moved outside the integral. Similarly the radiance B_\nu in the cavity is the same in any direction :P_\text = A_\text B_\nu \Delta\nu\int\limits_ d\Omega :P_\text = 2\pi A_\text B_\nu \Delta\nu Radio waves are low enough in frequency so the Rayleigh–Jeans formula gives a very close approximation of the blackbody spectral radianceThe Rayleigh-Jeans formula is a good approximation as long as the energy in a radio photon is small compared with the thermal energy per degree of freedom: h\nu << kT. This is true throughout the radio spectrum at all ordinary temperatures. :B_\nu = = Therefore :P_\text = \Delta\nu The Johnson–Nyquist noise power produced by a resistor at temperature T over a frequency range \Delta\nu is :P_\text = kT\Delta\nu Since the cavities are in thermodynamic equilibrium P_\text = P_\text, so :\Delta\nu = kT\Delta\nu


See also

*
Radiation pattern In the field of antenna design the term radiation pattern (or antenna pattern or far-field pattern) refers to the ''directional'' (angular) dependence of the strength of the radio waves from the antenna or other source.Constantine A. Balanis: †...
* E-plane and H-plane


References


External links


''Isotropic Radiators'', Matzner and McDonald, arXiv
Antennas

D.Jefferies
isotropic radiator
AMS Glossary

* ttp://www.airpower.maxwell.af.mil/airchronicles/apj/apj94/dil.html Non Lethal Concepts - Implications for Air Force IntelligencePublished Aerospace Power Journal, Winter 1994
GlossaryIsotropic Radiators
Holon Academic Institute of Technology {{Antenna_Types Radiation Radio frequency antenna types Antennas (radio)