étendue
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Etendue or étendue (; ) is a property of
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– ...
in an
optical system Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultravio ...
, which characterizes how "spread out" the light is in area and angle. It corresponds to the
beam parameter product In laser science, the beam parameter product (BPP) is the product of a laser beam's divergence angle (half-angle) and the radius of the beam at its narrowest point (the beam waist). The BPP quantifies the quality of a laser beam, and how well it can ...
(BPP) in
Gaussian beam In optics, a Gaussian beam is an idealized beam of electromagnetic radiation whose amplitude envelope in the transverse plane is given by a Gaussian function; this also implies a Gaussian intensity (irradiance) profile. This fundamental (or ...
optics. Other names for etendue include acceptance, throughput, light grasp, light-gathering power, optical extent, and the AΩ product. ''Throughput'' and ''AΩ product'' are especially used in
radiometry Radiometry is a set of techniques for measurement, measuring electromagnetic radiation, including visible light. Radiometric techniques in optics characterize the distribution of the radiation's power (physics), power in space, as opposed to phot ...
and radiative transfer where it is related to the
view factor Acornsoft was the software arm of Acorn Computers, and a major publisher of software for the BBC Micro and Acorn Electron. As well as games, it also produced a large number of educational titles, extra computer languages and business and ut ...
(or shape factor). It is a central concept in
nonimaging optics Nonimaging optics (also called anidolic optics)Roland Winston et al., ''Nonimaging Optics'', Academic Press, 2004 R. John Koshel (Editor), ''Illumination Engineering: Design with Nonimaging Optics'', Wiley, 2013 is a branch of optics that is conce ...
. From the source point of view, etendue is the product of the area of the source and the
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 poin ...
that the system's
entrance pupil In an optical system, the entrance pupil is the optical image of the physical aperture stop, as 'seen' through the optical elements in front of the stop. The corresponding image of the aperture stop as seen through the optical elements behin ...
subtend In geometry, an angle subtended (from Latin for "stretched under") by a line segment at an arbitrary vertex is formed by the two rays between the vertex and each endpoint of the segment. For example, a side of a triangle ''subtends'' the op ...
s as seen from the source. Equivalently, from the system point of view, the etendue equals the area of the entrance pupil times the solid angle the source subtends as seen from the pupil. These definitions must be applied for infinitesimally small "elements" of area and solid angle, which must then be summed over both the source and the diaphragm as shown below. Etendue may be considered to be a volume in
phase space The phase space of a physical system is the set of all possible physical states of the system when described by a given parameterization. Each possible state corresponds uniquely to a point in the phase space. For mechanical systems, the p ...
. Etendue never decreases in any optical system where optical power is conserved. A perfect optical system produces an image with the same etendue as the source. The etendue is related to the Lagrange invariant and the optical invariant, which also share the property of being constant in an ideal optical system. The
radiance 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 radiati ...
of an optical system is equal to the derivative of the
radiant flux 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 ...
with respect to the etendue.


Definition

An infinitesimal surface element, , with normal is immersed in a medium of
refractive index In optics, the refractive index (or refraction index) of an optical medium is the ratio of the apparent speed of light in the air or vacuum to the speed in the medium. The refractive index determines how much the path of light is bent, or refrac ...
. The surface is crossed by (or emits) light confined to a solid angle, , at an angle with the normal . The area of projected in the direction of the light propagation is . The etendue of an infinitesimal bundle of light crossing is defined as \mathrmG = n^2\, \mathrmS \cos \theta\, \mathrm\Omega\,. Etendue is the product of geometric extent and the squared refractive index of a medium through which the beam propagates. Because angles, solid angles, and refractive indices are dimensionless quantities, etendue is often expressed in units of area (given by ). However, it can alternatively be expressed in units of area (square meters) multiplied by solid angle (steradians).


In free space

Consider a light source , and a light detector , both of which are extended surfaces (rather than differential elements), and which are separated by a
medium Medium may refer to: Aircraft *Medium bomber, a class of warplane * Tecma Medium, a French hang glider design Arts, entertainment, and media Films * ''The Medium'' (1921 film), a German silent film * ''The Medium'' (1951 film), a film vers ...
of refractive index that is perfectly transparent (shown). To compute the etendue of the system, one must consider the contribution of each point on the surface of the light source as they cast rays to each point on the receiver. According to the definition above, the etendue of the light crossing towards is given by: \mathrmG_\Sigma = n^2\, \mathrm\Sigma \cos \theta_\Sigma\, \mathrm\Omega_\Sigma = n^2\, \mathrm\Sigma \cos \theta_\Sigma \frac\,, where is the solid angle defined by area at area , and is the distance between the two areas. Similarly, the etendue of the light crossing coming from is given by: \mathrmG_S = n^2\, \mathrmS \cos \theta_S\, \mathrm\Omega_S = n^2\, \mathrmS \cos \theta_S \frac\,, where is the solid angle defined by area . These expressions result in \mathrmG_\Sigma = \mathrmG_S\,, showing that etendue is conserved as light propagates in free space. The etendue of the whole system is then: G = \int_\Sigma\!\int_S \mathrmG\,. If both surfaces and are immersed in air (or in vacuum), and the expression above for the etendue may be written as \mathrmG = \mathrm\Sigma\, \cos \theta_\Sigma\, \frac = \pi\, \mathrm\Sigma\,\left(\frac\, \mathrmS\right) = \pi\, \mathrm\Sigma\, F_\,, where is the
view factor Acornsoft was the software arm of Acorn Computers, and a major publisher of software for the BBC Micro and Acorn Electron. As well as games, it also produced a large number of educational titles, extra computer languages and business and ut ...
between differential surfaces and . Integration on and results in which allows the etendue between two surfaces to be obtained from the view factors between those surfaces.


Conservation

The etendue of a given bundle of light is conserved: etendue can be increased, but not decreased in any optical system. This means that any system that concentrates light from some source onto a smaller area must always increase the solid angle of incidence (that is, the area of the sky that the source subtends). For example, a magnifying glass can increase the intensity of sunlight onto a small spot, but does so because, viewed from the spot that the light is concentrated onto, the apparent size of the sun is increased proportional to the concentration. As shown below, etendue is conserved as light travels through free space and at refractions or reflections. It is then also conserved as light travels through optical systems where it undergoes perfect reflections or refractions. However, if light was to hit, say, a
diffuser Diffuser may refer to: Aerodynamics * Diffuser (automotive), a shaped section of a car's underbody which improves the car's aerodynamic properties * Part of a jet engine air intake, especially when operated at supersonic speeds * The channel bet ...
, its solid angle would increase, increasing the etendue. Etendue can then remain constant or it can increase as light propagates through an optic, but it cannot decrease. This is a direct result of the fact that
entropy Entropy is a scientific concept, most commonly associated with states of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynamics, where it was first recognized, to the micros ...
must be constant or increasing. Conservation of etendue can be derived in different contexts, such as from optical first principles, from
Hamiltonian optics Hamiltonian opticsH. A. Buchdahl, ''An Introduction to Hamiltonian Optics'', Dover Publications, 1993, . and Lagrangian opticsVasudevan Lakshminarayanan et al., ''Lagrangian Optics'', Springer Netherlands, 2011, . are two formulations of geometrical ...
or from the
second law of thermodynamics The second law of thermodynamics is a physical law based on Universal (metaphysics), universal empirical observation concerning heat and Energy transformation, energy interconversions. A simple statement of the law is that heat always flows spont ...
. From the perspective of thermodynamics, etendue is a form of entropy. Specifically, the etendue of a bundle of light contributes to the entropy of it by S_ = k_B\ln \Omega. Etendue may be exponentially decreased by an increase in entropy elsewhere. For example, a material might absorb photons and emit lower-frequency photons, and emit the difference in energy as heat. This increases entropy due to heat, allowing a corresponding decrease in etendue. The conservation of etendue in free space is related to the reciprocity theorem for view factors.


In refractions and reflections

The conservation of etendue discussed above applies to the case of light propagation in free space, or more generally, in a medium of any
refractive index In optics, the refractive index (or refraction index) of an optical medium is the ratio of the apparent speed of light in the air or vacuum to the speed in the medium. The refractive index determines how much the path of light is bent, or refrac ...
. In particular, etendue is conserved in refractions and reflections. Figure "etendue in refraction" shows an infinitesimal surface on the plane separating two media of refractive indices and . The normal to points in the direction of the -axis. Incoming light is confined to a solid angle and reaches at an angle to its normal. Refracted light is confined to a solid angle and leaves at an angle to its normal. The directions of the incoming and refracted light are contained in a plane making an angle to the -axis, defining these directions in a
spherical coordinate system In mathematics, a spherical coordinate system specifies a given point in three-dimensional space by using a distance and two angles as its three coordinates. These are * the radial distance along the line connecting the point to a fixed point ...
. With these definitions,
Snell's law Snell's law (also known as the Snell–Descartes law, the ibn-Sahl law, and the law of refraction) is a formula used to describe the relationship between the angles of incidence and refraction, when referring to light or other waves passing th ...
of refraction can be written as n_\Sigma \sin \theta_\Sigma = n_S \sin \theta_S\,, and its derivative relative to n_\Sigma \cos \theta_\Sigma\, \mathrm\theta_\Sigma = n_S \cos \theta_S \mathrm\theta_S\,, multiplied by each other result in n_\Sigma^2 \cos \theta_\Sigma\!\left(\sin \theta_\Sigma\, \mathrm\theta_\Sigma\, \mathrm\varphi\right) = n_S^2 \cos \theta_S\!\left(\sin \theta_S\, \mathrm\theta_S\, \mathrm\varphi\right)\,, where both sides of the equation were also multiplied by which does not change on refraction. This expression can now be written as n_\Sigma^2 \cos \theta_\Sigma\, \mathrm\Omega_\Sigma = n_S^2 \cos \theta_S\, \mathrm\Omega_S\,. Multiplying both sides by we get n_\Sigma^2\, \mathrmS \cos \theta_\Sigma\, \mathrm\Omega_\Sigma = n_S^2\, \mathrmS \cos \theta_S\, \mathrm\Omega_S\,; that is \mathrmG_\Sigma = \mathrmG_S\,, showing that the etendue of the light refracted at is conserved. The same result is also valid for the case of a reflection at a surface , in which case and .


Brightness theorem

A consequence of the conservation of etendue is the ''brightness theorem'', which states that no linear optical system can increase the
brightness Brightness is an attribute of visual perception in which a source appears to be radiating/reflecting light. In other words, brightness is the perception dictated by the luminance of a visual target. The perception is not linear to luminance, and ...
of the light emitted from a source to a higher value than the brightness of the surface of that source (where "brightness" is defined as the optical power emitted per unit solid angle per unit emitting or receiving area).


Conservation of basic radiance

Radiance 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 radiati ...
of a surface is related to etendue by: L_ = n^2 \frac\,, where * is the
radiant flux 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 ...
emitted, reflected, transmitted or received; * is the refractive index in which that surface is immersed; * is the étendue of the light beam. As the light travels through an ideal optical system, both the etendue and the radiant flux are conserved. Therefore, ''basic radiance'' defined as: L_^* = \frac is also conserved. In real systems, the etendue may increase (for example due to scattering) or the radiant flux may decrease (for example due to absorption) and, therefore, basic radiance may decrease. However, etendue may not decrease and radiant flux may not increase and, therefore, basic radiance may not increase.


As a volume in phase space

In the context of
Hamiltonian optics Hamiltonian opticsH. A. Buchdahl, ''An Introduction to Hamiltonian Optics'', Dover Publications, 1993, . and Lagrangian opticsVasudevan Lakshminarayanan et al., ''Lagrangian Optics'', Springer Netherlands, 2011, . are two formulations of geometrical ...
, at a point in space, a light ray may be completely defined by a point , a unit
Euclidean vector In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Euclidean vectors can be added and scal ...
indicating its direction and the refractive index at point . The optical momentum of the ray at that point is defined by \mathbf = n(\cos \alpha_X, \cos \alpha_Y, \cos \alpha_Z) = (p, q, r)\,, where . The geometry of the optical momentum vector is illustrated in figure "optical momentum". In a
spherical coordinate system In mathematics, a spherical coordinate system specifies a given point in three-dimensional space by using a distance and two angles as its three coordinates. These are * the radial distance along the line connecting the point to a fixed point ...
may be written as \mathbf = n\!\left(\sin \theta \cos \varphi, \sin \theta \sin \varphi, \cos \theta \right)\,, from which \mathrmp\, \mathrmq = \frac \mathrm\theta\, \mathrm\varphi = \left(\frac \frac - \frac \frac\right) \mathrm\theta\, \mathrm\varphi = n^2 \cos \theta \sin \theta\, \mathrm\theta\, \mathrm\varphi = n^2 \cos \theta\, \mathrm\Omega\,, and therefore, for an infinitesimal area on the -plane immersed in a medium of refractive index , the etendue is given by \mathrmG = n^2\, \mathrmS \cos \theta\, \mathrm\Omega = \mathrmx\, \mathrmy\, \mathrmp\, \mathrmq\,, which is an infinitesimal volume in phase space . Conservation of etendue in phase space is the equivalent in optics to Liouville's theorem in classical mechanics. Etendue as volume in phase space is commonly used in
nonimaging optics Nonimaging optics (also called anidolic optics)Roland Winston et al., ''Nonimaging Optics'', Academic Press, 2004 R. John Koshel (Editor), ''Illumination Engineering: Design with Nonimaging Optics'', Wiley, 2013 is a branch of optics that is conce ...
.


Maximum concentration

Consider an infinitesimal surface , immersed in a medium of refractive index crossed by (or emitting) light inside a cone of angle . The etendue of this light is given by \mathrmG = n^2\, \mathrmS \int \cos \theta\, \mathrm\Omega = n^2 \mathrmS \int_0^\!\int_0^\alpha \cos \theta \sin \theta\, \mathrm\theta\, \mathrm\varphi = \pi n^2 \mathrmS \sin^2 \alpha\,. Noting that is the
numerical aperture In optics, the numerical aperture (NA) of an optical system is a dimensionless number that characterizes the range of angles over which the system can accept or emit light. By incorporating index of refraction in its definition, has the property ...
''NA'', of the beam of light, this can also be expressed as \mathrmG = \pi\, \mathrmS\, \mathrm^2\,. Note that is expressed in a
spherical coordinate system In mathematics, a spherical coordinate system specifies a given point in three-dimensional space by using a distance and two angles as its three coordinates. These are * the radial distance along the line connecting the point to a fixed point ...
. Now, if a large surface is crossed by (or emits) light also confined to a cone of angle , the etendue of the light crossing is G = \pi n^2 \sin^2 \alpha \int \mathrmS = \pi n^2 S \sin^2 \alpha = \pi S \,\mathrm^2\,. The limit on maximum concentration (shown) is an optic with an entrance aperture , in air () collecting light within a solid angle of angle (its acceptance angle) and sending it to a smaller area receiver immersed in a medium of refractive index , whose points are illuminated within a solid angle of angle . From the above expression, the etendue of the incoming light is G_\mathrm = \pi S \sin^2 \alpha and the etendue of the light reaching the receiver is G_\mathrm = \pi n^2 \Sigma \sin^2 \beta\,. Conservation of etendue then gives C = \frac = n^2 \frac\,, where is the concentration of the optic. For a given angular aperture , of the incoming light, this concentration will be maximum for the maximum value of , that is . The maximum possible concentration is then C_\mathrm = \frac\,. In the case that the incident index is not unity, we have G_\mathrm = \pi n_\mathrm S \sin^2 \alpha = G_\mathrm = \pi n_\mathrm \Sigma \sin^2 \beta\,, and so C = \left(\frac\right)^2\,, and in the best-case limit of , this becomes C_\mathrm = \frac\,. If the optic were a
collimator A collimator is a device which narrows a beam of particles or waves. To narrow can mean either to cause the directions of motion to become more aligned in a specific direction (i.e., make collimated light or parallel rays), or to cause the spat ...
instead of a concentrator, the light direction is reversed and conservation of etendue gives us the minimum aperture, , for a given output full angle .


See also

* Beam emittance *
Beam parameter product In laser science, the beam parameter product (BPP) is the product of a laser beam's divergence angle (half-angle) and the radius of the beam at its narrowest point (the beam waist). The BPP quantifies the quality of a laser beam, and how well it can ...
*
Light field A light field, or lightfield, is a vector-valued function, vector function that describes the amount of light flowing in every direction through every point in a space. The space of all possible ''light rays'' is given by the Five-dimensional space ...
*
Noether's theorem Noether's theorem states that every continuous symmetry of the action of a physical system with conservative forces has a corresponding conservation law. This is the first of two theorems (see Noether's second theorem) published by the mat ...
*
Symplectic geometry Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form. Symplectic geometry has its origins in the ...


References


Further reading

* *
xkcd ''xkcd'' is a serial webcomic created in 2005 by American author Randall Munroe. Sometimes styled ''XKCD'', the comic's tagline describes it as "a webcomic of romance, sarcasm, math, and language". Munroe states on the comic's website that the ...
–author Randall Munroe explains why it's impossible to light a fire with concentrated moonlight using an etendue-conservation argument. * {{cite journal , last1= Sun , first1= Xutao , last2= Zheng , first2= Zhenrong , last3= Liu , first3= Xu , last4= Gu , first4= Peifu , date= March 2006 , title= Etendue Analysis and Measurement of Light Source with Elliptical Reflector , journal= Displays , volume= 27 , issue= 2 , pages= 56–61 Optical quantities