Английская Википедия:Etendue
Шаблон:Short description Шаблон:Use dmy dates
Etendue or étendue (Шаблон:IPAc-en; Шаблон:IPA-fr) is a property of light in an optical system, which characterizes how "spread out" the light is in area and angle. It corresponds to the beam parameter product (BPP) in Gaussian beam optics. Other names for etendue include acceptance, throughput, light grasp, light-gathering power, optical extent,[1] and the AΩ product. Throughput and AΩ product are especially used in radiometry and radiative transfer where it is related to the view factor (or shape factor). It is a central concept in nonimaging optics.[2]Шаблон:Page needed[3]Шаблон:Page needed[4]Шаблон:Page needed
From the source point of view, etendue is the product of the area of the source and the solid angle that the system's entrance pupil subtends 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.
Etendue never decreases in any optical system where optical power is conserved.[5] 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 share the property of being constant in an ideal optical system. The radiance of an optical system is equal to the derivative of the radiant flux with respect to the etendue.
Definition
An infinitesimal surface element, Шаблон:Math, with normal Шаблон:Math is immersed in a medium of refractive index Шаблон:Mvar. The surface is crossed by (or emits) light confined to a solid angle, Шаблон:Math, at an angle Шаблон:Mvar with the normal Шаблон:Math. The area of Шаблон:Math projected in the direction of the light propagation is Шаблон:Math. The etendue of an infinitesimal bundle of light crossing Шаблон:Math is defined as
<math display="block">\mathrm{d}G = n^2\, \mathrm{d}S \cos \theta\, \mathrm{d}\Omega\,.</math>
Etendue is the product of geometric extent and the squared refractive index of a medium through which the beam propagates.[1] Because angles, solid angles, and refractive indices are dimensionless quantities, etendue is often expressed in units of area (given by Шаблон:Math). However, it can alternatively be expressed in units of area (square meters) multiplied by solid angle (steradians).[1][6]
In free space
Consider a light source Шаблон:Math, and a light detector Шаблон:Mvar, both of which are extended surfaces (rather than differential elements), and which are separated by a medium of refractive index Шаблон:Mvar 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.[7]Шаблон:Better source needed
According to the definition above, the etendue of the light crossing Шаблон:Math towards Шаблон:Math is given by:
<math display="block">\mathrm{d}G_\Sigma = n^2\, \mathrm{d}\Sigma \cos \theta_\Sigma\, \mathrm{d}\Omega_\Sigma = n^2\, \mathrm{d}\Sigma \cos \theta_\Sigma \frac{\mathrm{d}S \cos \theta_S}{d^2}\,,</math>
where Шаблон:Math is the solid angle defined by area Шаблон:Math at area Шаблон:Math, and Шаблон:Mvar is the distance between the two areas. Similarly, the etendue of the light crossing Шаблон:Math coming from Шаблон:Math is given by:
<math display="block">\mathrm{d}G_S = n^2\, \mathrm{d}S \cos \theta_S\, \mathrm{d}\Omega_S = n^2\, \mathrm{d}S \cos \theta_S \frac{\mathrm{d}\Sigma \cos \theta_\Sigma}{d^2}\,,</math>
where Шаблон:Math is the solid angle defined by area Шаблон:Math. These expressions result in
<math display="block">\mathrm{d}G_\Sigma = \mathrm{d}G_S\,,</math>
showing that etendue is conserved as light propagates in free space.
The etendue of the whole system is then:
<math display="block">G = \int_\Sigma\!\int_S \mathrm{d}G\,.</math>
If both surfaces Шаблон:Math and Шаблон:Math are immersed in air (or in vacuum), Шаблон:Math and the expression above for the etendue may be written as
<math display="block">\mathrm{d}G = \mathrm{d}\Sigma\, \cos \theta_\Sigma\, \frac{\mathrm{d}S\, \cos \theta_S}{d^2} = \pi\, \mathrm{d}\Sigma\,\left(\frac{\cos \theta_\Sigma \cos \theta_S}{\pi d^2}\, \mathrm{d}S\right) = \pi\, \mathrm{d}\Sigma\, F_{\mathrm{d}\Sigma \rarr \mathrm{d}S}\,,</math>
where Шаблон:Math is the view factor between differential surfaces Шаблон:Math and Шаблон:Math. Integration on Шаблон:Math and Шаблон:Math results in Шаблон:Math which allows the etendue between two surfaces to be obtained from the view factors between those surfaces, as provided in a list of view factors for specific geometry cases or in several heat transfer textbooks.
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, 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 must be constant or increasing.
Conservation of etendue can be derived in different contexts, such as from optical first principles, from Hamiltonian optics or from the second law of thermodynamics.[2]Шаблон:Page needed
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 particular, etendue is conserved in refractions and reflections.[2]Шаблон:Page needed Figure "etendue in refraction" shows an infinitesimal surface Шаблон:Math on the Шаблон:Mvar plane separating two media of refractive indices Шаблон:Math and Шаблон:Math.
The normal to Шаблон:Math points in the direction of the Шаблон:Mvar-axis. Incoming light is confined to a solid angle Шаблон:Math and reaches Шаблон:Math at an angle Шаблон:Math to its normal. Refracted light is confined to a solid angle Шаблон:Math and leaves Шаблон:Math at an angle Шаблон:Math to its normal. The directions of the incoming and refracted light are contained in a plane making an angle Шаблон:Mvar to the Шаблон:Mvar-axis, defining these directions in a spherical coordinate system. With these definitions, Snell's law of refraction can be written as
<math display="block">n_\Sigma \sin \theta_\Sigma = n_S \sin \theta_S\,,</math>
and its derivative relative to Шаблон:Mvar
<math display="block">n_\Sigma \cos \theta_\Sigma\, \mathrm{d}\theta_\Sigma = n_S \cos \theta_S \mathrm{d}\theta_S\,,</math>
multiplied by each other result in
<math display="block">n_\Sigma^2 \cos \theta_\Sigma\!\left(\sin \theta_\Sigma\, \mathrm{d}\theta_\Sigma\, \mathrm{d}\varphi\right) = n_S^2 \cos \theta_S\!\left(\sin \theta_S\, \mathrm{d}\theta_S\, \mathrm{d}\varphi\right)\,,</math>
where both sides of the equation were also multiplied by Шаблон:Math which does not change on refraction. This expression can now be written as
<math display="block">n_\Sigma^2 \cos \theta_\Sigma\, \mathrm{d}\Omega_\Sigma = n_S^2 \cos \theta_S\, \mathrm{d}\Omega_S\,.</math>
Multiplying both sides by Шаблон:Math we get
<math display="block">n_\Sigma^2\, \mathrm{d}S \cos \theta_\Sigma\, \mathrm{d}\Omega_\Sigma = n_S^2\, \mathrm{d}S \cos \theta_S\, \mathrm{d}\Omega_S\,;</math>
that is
<math display="block">\mathrm{d}G_\Sigma = \mathrm{d}G_S\,,</math>
showing that the etendue of the light refracted at Шаблон:Math is conserved. The same result is also valid for the case of a reflection at a surface Шаблон:Math, in which case Шаблон:Math and Шаблон:Math.
Brightness theorem
A consequence of the conservation of etendue is the brightness theorem, which states that no linear optical system can increase the brightness 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).[8]
Conservation of basic radiance
Radiance of a surface is related to etendue by:
<math display="block">L_{\mathrm{e},\Omega} = n^2 \frac{\partial \Phi_\mathrm{e}}{\partial G}\,,</math>
where
- Шаблон:Math is the radiant flux emitted, reflected, transmitted or received;
- Шаблон:Mvar is the refractive index in which that surface is immersed;
- Шаблон:Mvar 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:[9]Шаблон:Page needed
<math display="block">L_{\mathrm{e},\Omega}^* = \frac{L_{\mathrm{e},\Omega}}{n^2}</math>
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, at a point in space, a light ray may be completely defined by a point Шаблон:Math, a unit Euclidean vector Шаблон:Math indicating its direction and the refractive index Шаблон:Mvar at point Шаблон:Math. The optical momentum of the ray at that point is defined by
<math display="block">\mathbf{p} = n(\cos \alpha_X, \cos \alpha_Y, \cos \alpha_Z) = (p, q, r)\,,</math>
where Шаблон:Math. The geometry of the optical momentum vector is illustrated in figure "optical momentum".
In a spherical coordinate system Шаблон:Math may be written as
<math display="block">\mathbf{p} = n\!\left(\sin \theta \cos \varphi, \sin \theta \sin \varphi, \cos \theta \right)\,,</math>
from which
<math display="block">\mathrm{d}p\, \mathrm{d}q = \frac{\partial(p, q)}{\partial(\theta, \varphi)} \mathrm{d}\theta\, \mathrm{d}\varphi = \left(\frac{\partial p}{\partial \theta} \frac{\partial q}{\partial \varphi} - \frac{\partial p}{\partial \varphi} \frac{\partial q}{\partial \theta}\right) \mathrm{d}\theta\, \mathrm{d}\varphi = n^2 \cos \theta \sin \theta\, \mathrm{d}\theta\, \mathrm{d}\varphi = n^2 \cos \theta\, \mathrm{d}\Omega\,,</math>
and therefore, for an infinitesimal area Шаблон:Math on the Шаблон:Mvar-plane immersed in a medium of refractive index Шаблон:Mvar, the etendue is given by
<math display="block">\mathrm{d}G = n^2\, \mathrm{d}S \cos \theta\, \mathrm{d}\Omega = \mathrm{d}x\, \mathrm{d}y\, \mathrm{d}p\, \mathrm{d}q\,,</math>
which is an infinitesimal volume in phase space Шаблон:Math. Conservation of etendue in phase space is the equivalent in optics to Liouville's theorem in classical mechanics.[2]Шаблон:Page needed Etendue as volume in phase space is commonly used in nonimaging optics.
Maximum concentration
Consider an infinitesimal surface Шаблон:Math, immersed in a medium of refractive index Шаблон:Mvar crossed by (or emitting) light inside a cone of angle Шаблон:Mvar. The etendue of this light is given by
<math display="block">\mathrm{d}G = n^2\, \mathrm{d}S \int \cos \theta\, \mathrm{d}\Omega = n^2 dS \int_0^{2\pi}\!\int_0^\alpha \cos \theta \sin \theta\, \mathrm{d}\theta\, \mathrm{d}\varphi = \pi n^2 \mathrm{d}S \sin^2 \alpha\,.</math>
Noting that Шаблон:Math is the numerical aperture NA, of the beam of light, this can also be expressed as
<math display="block">\mathrm{d}G = \pi\, \mathrm{d}S\, \mathrm{NA}^2\,.</math>
Note that Шаблон:Math is expressed in a spherical coordinate system. Now, if a large surface Шаблон:Mvar is crossed by (or emits) light also confined to a cone of angle Шаблон:Mvar, the etendue of the light crossing Шаблон:Mvar is
<math display="block">G = \pi n^2 \sin^2 \alpha \int \mathrm{d}S = \pi n^2 S \sin^2 \alpha = \pi S \,\mathrm{NA}^2\,.</math>
The limit on maximum concentration (shown) is an optic with an entrance aperture Шаблон:Mvar, in air (Шаблон:Nobreak) collecting light within a solid angle of angle Шаблон:Math (its acceptance angle) and sending it to a smaller area receiver Шаблон:Math immersed in a medium of refractive index Шаблон:Mvar, whose points are illuminated within a solid angle of angle Шаблон:Math. From the above expression, the etendue of the incoming light is
<math display="block">G_\mathrm{i} = \pi S \sin^2 \alpha</math>
and the etendue of the light reaching the receiver is
<math display="block">G_\mathrm{r} = \pi n^2 \Sigma \sin^2 \beta\,.</math>
Conservation of etendue Шаблон:Math then gives
<math display="block">C = \frac{S}{\Sigma} = n^2 \frac{\sin^2 \beta}{\sin^2 \alpha}\,,</math>
where Шаблон:Mvar is the concentration of the optic. For a given angular aperture Шаблон:Mvar, of the incoming light, this concentration will be maximum for the maximum value of Шаблон:Math, that is Шаблон:Math. The maximum possible concentration is then[2]Шаблон:Page needed[3]
<math display="block">C_\mathrm{max} = \frac{n^2}{\sin^2 \alpha}\,.</math>
In the case that the incident index is not unity, we have
<math display="block">G_\mathrm{i} = \pi n_\mathrm{i} S \sin^2 \alpha = G_\mathrm{r} = \pi n_\mathrm{r} \Sigma \sin^2 \beta\,,</math>
and so
<math display="block">C = \left(\frac{\mathrm{NA}_\mathrm{r}}{\mathrm{NA}_\mathrm{i}}\right)^2\,,</math>
and in the best-case limit of Шаблон:Math, this becomes
<math display="block">C_\mathrm{max} = \frac{n_\mathrm{r}^2}{\mathrm{NA}_\mathrm{i}^2}\,.</math>
If the optic were a collimator instead of a concentrator, the light direction is reversed and conservation of etendue gives us the minimum aperture, Шаблон:Mvar, for a given output full angle Шаблон:Math.
See also
References
Further reading
Шаблон:Commons category Шаблон:Wikibooks Шаблон:Wiktionary
- Шаблон:Cite book
- Шаблон:Cite web xkcd–author Randall Munroe explains why it's impossible to light a fire with concentrated moonlight using an etendue-conservation argument.
- Шаблон:Cite journal
