Parabolic reflector
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A parabolic (or paraboloid or paraboloidal) reflector (or dish or mirror) is a reflective surface used to collect or project
energy Energy () is the physical quantity, quantitative physical property, property that is transferred to a physical body, body or to a physical system, recognizable in the performance of Work (thermodynamics), work and in the form of heat and l ...
such as
light Light, visible light, or visible radiation is electromagnetic radiation that can be visual perception, perceived by the human eye. Visible light spans the visible spectrum and is usually defined as having wavelengths in the range of 400– ...
,
sound In physics, sound is a vibration that propagates as an acoustic wave through a transmission medium such as a gas, liquid or solid. In human physiology and psychology, sound is the ''reception'' of such waves and their ''perception'' by the br ...
, or
radio wave Radio waves (formerly called Hertzian waves) are a type of electromagnetic radiation with the lowest frequencies and the longest wavelengths in the electromagnetic spectrum, typically with frequencies below 300 gigahertz (GHz) and wavelengths g ...
s. Its shape is part of a circular paraboloid, that is, the surface generated by a
parabola In mathematics, a parabola is a plane curve which is Reflection symmetry, mirror-symmetrical and is approximately U-shaped. It fits several superficially different Mathematics, mathematical descriptions, which can all be proved to define exactl ...
revolving around its axis. The parabolic reflector transforms an incoming
plane wave In physics Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of ...
travelling along the axis into a spherical wave converging toward the focus. Conversely, a spherical wave generated by a
point source A point source is a single identifiable ''localized'' source of something. A point source has a negligible extent, distinguishing it from other source geometries. Sources are called point sources because, in mathematical modeling, these sources ...
placed in the
focus Focus (: foci or focuses) may refer to: Arts * Focus or Focus Festival, former name of the Adelaide Fringe arts festival in East Australia Film *Focus (2001 film), ''Focus'' (2001 film), a 2001 film based on the Arthur Miller novel *Focus (2015 ...
is reflected into a plane wave propagating as a collimated beam along the axis. Parabolic reflectors are used to collect energy from a distant source (for example sound waves or incoming
star A star is a luminous spheroid of plasma (physics), plasma held together by Self-gravitation, self-gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked eye at night sk ...
light). Since the principles of reflection are reversible, parabolic reflectors can also be used to collimate radiation from an isotropic source into a parallel beam. In
optics Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of optical instruments, instruments that use or Photodetector, detect it. Optics usually describes t ...
, parabolic mirrors are used to gather light in
reflecting telescope A reflecting telescope (also called a reflector) is a telescope that uses a single or a combination of curved mirrors that reflect light and form an image. The reflecting telescope was invented in the 17th century by Isaac Newton as an alternati ...
s and
solar furnace A solar furnace is a structure that uses concentrated solar power to produce high temperatures, usually for industry. Parabolic mirrors or heliostats concentrate light ( Insolation) onto a focal point. The temperature at the focal point may ...
s, and project a beam of light in
flashlight A flashlight (US English) or electric torch (Commonwealth English), usually shortened to torch, is a portable hand-held electric lamp. Formerly, the light source typically was a miniature incandescent light bulb, but these have been displaced ...
s,
searchlight A searchlight (or spotlight) is an apparatus that combines an extremely luminosity, bright source (traditionally a carbon arc lamp) with a mirrored parabolic reflector to project a powerful beam of light of approximately parallel rays in a part ...
s, stage spotlights, and car headlights. In
radio Radio is the technology of communicating using radio waves. Radio waves are electromagnetic waves of frequency between 3  hertz (Hz) and 300  gigahertz (GHz). They are generated by an electronic device called a transmitter connec ...
,
parabolic antenna A parabolic antenna is an antenna that uses a parabolic reflector, a curved surface with the cross-sectional shape of a parabola, to direct the radio waves. The most common form is shaped like a dish and is popularly called a dish antenna or p ...
s are used to radiate a narrow beam of
radio wave Radio waves (formerly called Hertzian waves) are a type of electromagnetic radiation with the lowest frequencies and the longest wavelengths in the electromagnetic spectrum, typically with frequencies below 300 gigahertz (GHz) and wavelengths g ...
s for point-to-point communications in
satellite dish A satellite dish is a dish-shaped type of parabolic antenna designed to receive or transmit information by radio waves to or from a communication satellite. The term most commonly means a dish which receives direct-broadcast satellite televisio ...
es and microwave relay stations, and to locate aircraft, ships, and vehicles in
radar Radar is a system that uses radio waves to determine the distance ('' ranging''), direction ( azimuth and elevation angles), and radial velocity of objects relative to the site. It is a radiodetermination method used to detect and track ...
sets. In
acoustics Acoustics is a branch of physics that deals with the study of mechanical waves in gases, liquids, and solids including topics such as vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an acoustician ...
, parabolic microphones are used to record faraway sounds such as bird calls, in sports reporting, and to eavesdrop on private conversations in
espionage Espionage, spying, or intelligence gathering, as a subfield of the intelligence field, is the act of obtaining secret or confidential information ( intelligence). A person who commits espionage on a mission-specific contract is called an ...
and law enforcement.


Theory

Strictly, the three-dimensional shape of the reflector is called a ''
paraboloid In geometry, a paraboloid is a quadric surface that has exactly one axial symmetry, axis of symmetry and no central symmetry, center of symmetry. The term "paraboloid" is derived from parabola, which refers to a conic section that has a similar p ...
''. A parabola is the two-dimensional figure. (The distinction is like that between a sphere and a circle.) However, in informal language, the word ''parabola'' and its associated adjective ''parabolic'' are often used in place of ''paraboloid'' and ''paraboloidal''. If a parabola is positioned in Cartesian coordinates with its vertex at the origin and its axis of symmetry along the y-axis, so the parabola opens upward, its equation is 4fy = x^2, where f is its focal length. (See " Parabola#In a cartesian coordinate system".) Correspondingly, the dimensions of a symmetrical paraboloidal dish are related by the equation: where F is the focal length, D is the depth of the dish (measured along the axis of symmetry from the vertex to the plane of the rim), and R is the radius of the dish from the center. All units used for the radius, focal point and depth must be the same. If two of these three quantities are known, this equation can be used to calculate the third. A more complex calculation is needed to find the diameter of the dish ''measured along its surface''. This is sometimes called the "linear diameter", and equals the diameter of a flat, circular sheet of material, usually metal, which is the right size to be cut and bent to make the dish. Two intermediate results are useful in the calculation: P = 2F (or the equivalent: and where , , and are defined as above. The diameter of the dish, measured along the surface, is then given by: where \ln(x) means the
natural logarithm The natural logarithm of a number is its logarithm to the base of a logarithm, base of the e (mathematical constant), mathematical constant , which is an Irrational number, irrational and Transcendental number, transcendental number approxima ...
of , i.e. its logarithm to base " e". The volume of the dish is given by \frac \pi R^2 D , where the symbols are defined as above. This can be compared with the formulae for the volumes of a
cylinder A cylinder () has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an infinite ...
(\pi R^2 D), a
hemisphere Hemisphere may refer to: In geometry * Hemisphere (geometry), a half of a sphere As half of Earth or any spherical astronomical object * A hemisphere of Earth ** Northern Hemisphere ** Southern Hemisphere ** Eastern Hemisphere ** Western Hemi ...
(\frac \pi R^2 D, where D=R), and a
cone In geometry, a cone is a three-dimensional figure that tapers smoothly from a flat base (typically a circle) to a point not contained in the base, called the '' apex'' or '' vertex''. A cone is formed by a set of line segments, half-lines ...
( \frac \pi R^2 D ). \pi R^2 is the aperture area of the dish, the area enclosed by the rim, which is proportional to the amount of sunlight the reflector dish can intercept. The area of the concave surface of the dish can be found using the area formula for a
surface of revolution A surface of revolution is a Surface (mathematics), surface in Euclidean space created by rotating a curve (the ''generatrix'') one full revolution (unit), revolution around an ''axis of rotation'' (normally not Intersection (geometry), intersec ...
which gives A=\frac\left((R^2+4D^2)^-R^3\right). providing D \ne 0. The fraction of light reflected by the dish, from a light source in the focus, is given by 1 - \frac, where F, D, and R are defined as above. The parabolic reflector functions due to the geometric properties of the paraboloidal shape: any incoming ray that is parallel to the axis of the dish will be reflected to a central point, or "
focus Focus (: foci or focuses) may refer to: Arts * Focus or Focus Festival, former name of the Adelaide Fringe arts festival in East Australia Film *Focus (2001 film), ''Focus'' (2001 film), a 2001 film based on the Arthur Miller novel *Focus (2015 ...
". (For a geometrical proof, click here.) Because many types of energy can be reflected in this way, parabolic reflectors can be used to collect and concentrate energy entering the reflector at a particular angle. Similarly, energy radiating from the focus to the dish can be transmitted outward in a beam that is parallel to the axis of the dish. In contrast with spherical reflectors, which suffer from a spherical aberration that becomes stronger as the ratio of the beam diameter to the focal distance becomes larger, parabolic reflectors can be made to accommodate beams of any width. However, if the incoming beam makes a non-zero angle with the axis (or if the emitting point source is not placed in the focus), parabolic reflectors suffer from an aberration called
coma A coma is a deep state of prolonged unconsciousness in which a person cannot be awakened, fails to Nociception, respond normally to Pain, painful stimuli, light, or sound, lacks a normal Circadian rhythm, sleep-wake cycle and does not initiate ...
. This is primarily of interest in telescopes because most other applications do not require sharp resolution off the axis of the parabola. The precision to which a parabolic dish must be made in order to focus energy well depends on the wavelength of the energy. If the dish is wrong by a quarter of a wavelength, then the reflected energy will be wrong by a half wavelength, which means that it will interfere destructively with energy that has been reflected properly from another part of the dish. To prevent this, the dish must be made correctly to within about of a wavelength. The wavelength range of visible light is between about 400 and 700 nanometres (nm), so in order to focus all visible light well, a reflector must be correct to within about 20 nm. For comparison, the diameter of a human hair is usually about 50,000 nm, so the required accuracy for a reflector to focus visible light is about 2500 times less than the diameter of a hair. For example, the flaw in the
Hubble Space Telescope The Hubble Space Telescope (HST or Hubble) is a space telescope that was launched into low Earth orbit in 1990 and remains in operation. It was not the Orbiting Solar Observatory, first space telescope, but it is one of the largest and most ...
mirror (too flat by about 2,200 nm at its perimeter) caused severe spherical aberration until corrected with COSTAR. Microwaves, such as are used for satellite-TV signals, have wavelengths of the order of ten millimetres, so dishes to focus these waves can be wrong by half a millimetre or so and still perform well.


Variations


Focus-balanced reflector

It is sometimes useful if the centre of mass of a reflector dish coincides with its
focus Focus (: foci or focuses) may refer to: Arts * Focus or Focus Festival, former name of the Adelaide Fringe arts festival in East Australia Film *Focus (2001 film), ''Focus'' (2001 film), a 2001 film based on the Arthur Miller novel *Focus (2015 ...
. This allows it to be easily turned so it can be aimed at a moving source of light, such as the Sun in the sky, while its focus, where the target is located, is stationary. The dish is rotated around
axes Axes, plural of ''axe'' and of ''axis'', may refer to * ''Axes'' (album), a 2005 rock album by the British band Electrelane * a possibly still empty plot (graphics) See also * Axis (disambiguation) An axis (: axes) may refer to: Mathematics ...
that pass through the focus and around which it is balanced. If the dish is symmetrical and made of uniform material of constant thickness, and if ''F'' represents the focal length of the paraboloid, this "focus-balanced" condition occurs if the depth of the dish, measured along the axis of the paraboloid from the vertex to the plane of the rim of the dish, is 1.8478 times ''F''. The radius of the rim is 2.7187 ''F''. The angular radius of the rim as seen from the focal point is 72.68 degrees.


Scheffler reflector

The focus-balanced configuration (see above) requires the depth of the reflector dish to be greater than its focal length, so the focus is within the dish. This can lead to the focus being difficult to access. An alternative approach is exemplified by the Scheffler reflector, named after its inventor, Wolfgang Scheffler. This is a paraboloidal mirror which is rotated about axes that pass through its centre of mass, but this does not coincide with the focus, which is outside the dish. If the reflector were a rigid paraboloid, the focus would move as the dish turns. To avoid this, the reflector is flexible, and is bent as it rotates so as to keep the focus stationary. Ideally, the reflector would be exactly paraboloidal at all times. In practice, this cannot be achieved exactly, so the Scheffler reflector is not suitable for purposes that require high accuracy. It is used in applications such as solar cooking, where sunlight has to be focused well enough to strike a cooking pot, but not to an exact point.


Off-axis reflectors

A circular paraboloid is theoretically unlimited in size. Any practical reflector uses just a segment of it. Often, the segment includes the vertex of the paraboloid, where its
curvature In mathematics, curvature is any of several strongly related concepts in geometry that intuitively measure the amount by which a curve deviates from being a straight line or by which a surface deviates from being a plane. If a curve or su ...
is greatest, and where the
axis of symmetry An axis (: axes) may refer to: Mathematics *A specific line (often a directed line) that plays an important role in some contexts. In particular: ** Coordinate axis of a coordinate system *** ''x''-axis, ''y''-axis, ''z''-axis, common names f ...
intersects the paraboloid. However, if the reflector is used to focus incoming energy onto a receiver, the shadow of the receiver falls onto the vertex of the paraboloid, which is part of the reflector, so part of the reflector is wasted. This can be avoided by making the reflector from a segment of the paraboloid which is offset from the vertex and the axis of symmetry. The whole reflector receives energy, which is then focused onto the receiver. This is frequently done, for example, in satellite-TV receiving dishes, and also in some types of astronomical telescope (''e.g.'', the Green Bank Telescope, the
James Webb Space Telescope The James Webb Space Telescope (JWST) is a space telescope designed to conduct infrared astronomy. As the largest telescope in space, it is equipped with high-resolution and high-sensitivity instruments, allowing it to view objects too old, Lis ...
). Accurate off-axis reflectors, for use in
solar furnace A solar furnace is a structure that uses concentrated solar power to produce high temperatures, usually for industry. Parabolic mirrors or heliostats concentrate light ( Insolation) onto a focal point. The temperature at the focal point may ...
s and other non-critical applications, can be made quite simply by using a rotating furnace, in which the container of molten glass is offset from the axis of rotation. To make less accurate ones, suitable as satellite dishes, the shape is designed by a computer, then multiple dishes are stamped out of sheet metal. Off-axis-reflectors heading from medium
latitude In geography, latitude is a geographic coordinate system, geographic coordinate that specifies the north-south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from −90° at t ...
s to a geostationary TV satellite somewhere above the equator stand steeper than a coaxial reflector. The effect is, that the arm to hold the dish can be shorter and snow tends less to accumulate in (the lower part of) the dish.


History

The principle of parabolic reflectors has been known since
classical antiquity Classical antiquity, also known as the classical era, classical period, classical age, or simply antiquity, is the period of cultural History of Europe, European history between the 8th century BC and the 5th century AD comprising the inter ...
, when the mathematician Diocles described them in his book ''On Burning Mirrors'' and proved that they focus a parallel beam to a point.
Archimedes Archimedes of Syracuse ( ; ) was an Ancient Greece, Ancient Greek Greek mathematics, mathematician, physicist, engineer, astronomer, and Invention, inventor from the ancient city of Syracuse, Sicily, Syracuse in History of Greek and Hellenis ...
in the third century BCE studied paraboloids as part of his study of
hydrostatic equilibrium In fluid mechanics, hydrostatic equilibrium, also called hydrostatic balance and hydrostasy, is the condition of a fluid or plastic solid at rest, which occurs when external forces, such as gravity, are balanced by a pressure-gradient force. I ...
, and it has been
claimed "Claimed" is the eleventh episode of the The Walking Dead season 4, fourth season of the Apocalyptic and post-apocalyptic fiction, post-apocalyptic Horror fiction, horror television series ''The Walking Dead (TV series), The Walking Dead'', wh ...
that he used reflectors to set the Roman fleet alight during the Siege of Syracuse. This seems unlikely to be true, however, as the claim does not appear in sources before the 2nd century CE, and Diocles does not mention it in his book. Parabolic mirrors and reflectors were also studied extensively by the
physicist A physicist is a scientist who specializes in the field of physics, which encompasses the interactions of matter and energy at all length and time scales in the physical universe. Physicists generally are interested in the root or ultimate cau ...
Roger Bacon Roger Bacon (; or ', also '' Rogerus''; ), also known by the Scholastic accolades, scholastic accolade ''Doctor Mirabilis'', was a medieval English polymath, philosopher, scientist, theologian and Franciscans, Franciscan friar who placed co ...
in the 13th century AD. James Gregory, in his 1663 book ''Optica Promota'' (1663), pointed out that a
reflecting telescope A reflecting telescope (also called a reflector) is a telescope that uses a single or a combination of curved mirrors that reflect light and form an image. The reflecting telescope was invented in the 17th century by Isaac Newton as an alternati ...
with a mirror that was parabolic would correct spherical aberration as well as the
chromatic aberration In optics, chromatic aberration (CA), also called chromatic distortion, color aberration, color fringing, or purple fringing, is a failure of a lens to focus all colors to the same point. It is caused by dispersion: the refractive index of the ...
seen in
refracting telescope A refracting telescope (also called a refractor) is a type of optical telescope that uses a lens (optics), lens as its objective (optics), objective to form an image (also referred to a dioptrics, dioptric telescope). The refracting telescope d ...
s. The design he came up with bears his name: the " Gregorian telescope"; but according to his own confession, Gregory had no practical skill and he could find no optician capable of actually constructing one.
Isaac Newton Sir Isaac Newton () was an English polymath active as a mathematician, physicist, astronomer, alchemist, theologian, and author. Newton was a key figure in the Scientific Revolution and the Age of Enlightenment, Enlightenment that followed ...
knew about the properties of parabolic mirrors but chose a spherical shape for his
Newtonian telescope The Newtonian telescope, also called the Newtonian reflector or just a Newtonian, is a type of reflecting telescope invented by the English scientist Sir Isaac Newton, using a concave primary mirror and a flat diagonal secondary mirror. Newto ...
mirror to simplify construction.
Lighthouse A lighthouse is a tower, building, or other type of physical structure designed to emit light from a system of lamps and lens (optics), lenses and to serve as a beacon for navigational aid for maritime pilots at sea or on inland waterways. Ligh ...
s also commonly used parabolic mirrors to collimate a point of light from a lantern into a beam, before being replaced by more efficient
Fresnel lens A Fresnel lens ( ; ; or ) is a type of composite compact lens (optics), lens which reduces the amount of material required compared to a conventional lens by dividing the lens into a set of concentric annular sections. The simpler Dioptrics, d ...
es in the 19th century. In 1888,
Heinrich Hertz Heinrich Rudolf Hertz (; ; 22 February 1857 – 1 January 1894) was a German physicist who first conclusively proved the existence of the electromagnetic waves predicted by James Clerk Maxwell's equations of electromagnetism. Biography Heinri ...
, a German physicist, constructed the world's first parabolic reflector antenna.


Applications

The most common modern applications of the parabolic reflector are in
satellite dish A satellite dish is a dish-shaped type of parabolic antenna designed to receive or transmit information by radio waves to or from a communication satellite. The term most commonly means a dish which receives direct-broadcast satellite televisio ...
es,
reflecting telescope A reflecting telescope (also called a reflector) is a telescope that uses a single or a combination of curved mirrors that reflect light and form an image. The reflecting telescope was invented in the 17th century by Isaac Newton as an alternati ...
s,
radio telescope A radio telescope is a specialized antenna (radio), antenna and radio receiver used to detect radio waves from astronomical radio sources in the sky. Radio telescopes are the main observing instrument used in radio astronomy, which studies the r ...
s, parabolic microphones, solar cookers, and many
lighting Lighting or illumination is the deliberate use of light to achieve practical or aesthetic effects. Lighting includes the use of both artificial light sources like lamps and light fixtures, as well as natural illumination by capturing daylight. ...
devices such as spotlights, car headlights, PAR lamps and LED housings. The
Olympic Flame The Olympic flame is a Olympic symbols, symbol used in the Olympic movement. It is also a symbol of continuity between ancient and modern games. The Olympic flame is lit at Olympia, Greece, several months before the Olympic Games. This ceremony s ...
is traditionally lit at
Olympia, Greece Olympia ( ; ), officially Archaia Olympia ( ), is a small town in Elis (regional unit), Elis on the Peloponnese peninsula in Greece, famous for the nearby archaeological site of the same name. The site was a major Panhellenic sanctuary, Panhell ...
, using a parabolic reflector concentrating
sunlight Sunlight is the portion of the electromagnetic radiation which is emitted by the Sun (i.e. solar radiation) and received by the Earth, in particular the visible spectrum, visible light perceptible to the human eye as well as invisible infrare ...
, and is then transported to the venue of the Games. Parabolic mirrors are one of many shapes for a burning glass. Parabolic reflectors are popular for use in creating
optical illusion In visual perception, an optical illusion (also called a visual illusion) is an illusion caused by the visual system and characterized by a visual perception, percept that arguably appears to differ from reality. Illusions come in a wide varie ...
s. These consist of two opposing parabolic mirrors, with an opening in the center of the top mirror. When an object is placed on the bottom mirror, the mirrors create a
real image {{citations needed, date=June 2019 In optics, an ''image'' is defined as the collection of focus points of light rays coming from an object. A real image is the collection of focus points actually made by converging/diverging rays, while a ...
, which is a virtually identical copy of the original that appears in the opening. The quality of the image is dependent upon the precision of the optics. Some such illusions are manufactured to tolerances of millionths of an inch. A parabolic reflector pointing upward can be formed by rotating a reflective liquid, like mercury, around a vertical axis. This makes the liquid-mirror telescope possible. The same technique is used in rotating furnaces to make solid reflectors. Parabolic reflectors are also a popular alternative for increasing wireless signal strength. Even with simple ones, users have reported 3 dB or more gains.


See also

* John D. Kraus * Liquid-mirror telescope, paraboloids produced by rotation *
Parabolic antenna A parabolic antenna is an antenna that uses a parabolic reflector, a curved surface with the cross-sectional shape of a parabola, to direct the radio waves. The most common form is shaped like a dish and is popularly called a dish antenna or p ...
*
Parabolic trough A parabolic trough collector (PTC) is a type of solar thermal collector that is straight in one dimension and curved as a parabola in the other two, lined with a polished metal mirror. The sunlight which enters the mirror parallel to its plane of ...
*
Solar furnace A solar furnace is a structure that uses concentrated solar power to produce high temperatures, usually for industry. Parabolic mirrors or heliostats concentrate light ( Insolation) onto a focal point. The temperature at the focal point may ...
* Toroidal reflector


Footnotes


References


External links


Java demonstration of a parabolic reflector

Parabolic Reflector Antennas
www.antenna-theory.com
Animations demonstrating parabola mirror
by QED

{{Authority control Mirrors Parabolas fr:Miroir (optique)#Miroir parabolique