Editing Parabolic reflector (section)

== Theory ==
{{unreferenced section|date=November 2012}}

Strictly, the three-dimensional shape of the reflector is called a ''[[paraboloid]]''. 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''.<ref>{{Cite web |title=3D Printing Using a 60 GHz Millimeter Wave Segmented Parabolic Reflective Curved Antenna |url=https://www.researchgate.net/publication/331042915_3D_Printing_Using_a_60_GHz_Millimeter_Wave_Segmented_Parabolic_Reflective_Curved_Antenna#pf3}}</ref>

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 <math display="inline">4fy = x^2</math>, where <math display="inline"> f</math> is its focal length. (See "[[Parabola#In a cartesian coordinate system]]".) Correspondingly, the dimensions of a symmetrical paraboloidal dish are related by the equation: {{nowrap|<math display="inline"> 4FD = R^2</math>,}} where <math display="inline"> F</math> is the focal length, <math display="inline"> D</math> is the depth of the dish (measured along the axis of symmetry from the vertex to the plane of the rim), and <math display="inline"> R</math> 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: <math display="inline"> P = 2F</math> (or the equivalent: {{nowrap|<math display="inline"> P = \frac{R^2}{2D}</math>)}} and {{nowrap|<math display="inline"> Q = \sqrt {P^2+R^2}</math>,}} where {{mvar|F}}, {{mvar|D}}, and {{mvar|R}} are defined as above. The diameter of the dish, measured along the surface, is then given by: {{nowrap|<math display="inline"> \frac {RQ} {P} + P \ln \left ( \frac {R+Q} {P} \right )</math>,}} where <math display="inline">\ln(x)</math> means the [[natural logarithm]] of {{mvar|x}}, i.e. its logarithm to base "[[e (mathematical constant)|e]]".

The volume of the dish is given by <math display="inline">\frac {1} {2} \pi R^2 D ,</math> where the symbols are defined as above. This can be compared with the formulae for the volumes of a [[Cylinder (geometry)|cylinder]] <math display="inline">(\pi R^2 D),</math> a [[sphere|hemisphere]] <math display="inline"> (\frac {2}{3} \pi R^2 D,</math> where <math display="inline"> D=R),</math> and a [[Cone (geometry)|cone]] <math display="inline"> ( \frac {1} {3} \pi R^2 D ).</math>  <math display="inline"> \pi R^2 </math> 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#Area formula|surface of revolution]] which gives <math display="inline">A=\frac{\pi R}{6 D^2}\left((R^2+4D^2)^{3/2}-R^3\right)</math>. providing <math display="inline"> D \ne 0</math>. The fraction of light reflected by the dish, from a light source in the focus, is given by  <math display="inline"> 1 - \frac{\arctan\frac{R}{D-F}}{\pi}</math>, where <math>F,</math> <math>D,</math> and <math>R</math> are defined as above.

[[File:Parabola with focus and arbitrary line.svg|thumb|300px|Parallel rays coming into a parabolic mirror are focused at a point F. The vertex is V, and the axis of symmetry passes through V and F. For off-axis reflectors (with just the part of the paraboloid between the points P<sub>1</sub> and P<sub>3</sub>), the receiver is still placed at the focus of the paraboloid, but it does not cast a shadow onto the reflector.]]

The parabolic reflector functions due to the geometric properties of the paraboloidal shape: any incoming [[ray (optics)|ray]] that is parallel to the axis of the dish will be reflected to a central point, or "[[Focus (optics)|focus]]". (For a geometrical proof, click [[Parabola#Proof of the reflective property|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 reflector]]s, 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 in optical systems|aberration]] called [[Coma (optics)|coma]]. 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 {{sfrac|20}} 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&nbsp;nm. For comparison, the diameter of a human hair is usually about 50,000&nbsp;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]] mirror (too flat by about 2,200&nbsp;nm at its perimeter) caused severe [[spherical aberration]] until corrected with [[Corrective Optics Space Telescope Axial Replacement|COSTAR]].<ref name="Servicing Mission 1">{{cite web |url=http://hubble.nasa.gov/missions/sm1.php |title=Servicing Mission 1 |publisher=NASA |access-date=April 26, 2008 |url-status=dead |archive-url=https://web.archive.org/web/20080420202154/http://hubble.nasa.gov/missions/sm1.php <!--Added by H3llBot--> |archive-date=April 20, 2008}}</ref>

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.