Etendue

Etendue or étendue (/ˌeɪtɒnˈduː/; French pronunciation: [etɑ̃dy]) is a property of light in an optical system, which characterizes how "spread out" the light is in area and angle.

From the source point of view, it 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 is important because it never decreases in any optical system. 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.

The term étendue comes from the French étendue géométrique, meaning "geometrical extent". Other names for this property are acceptance, throughput, light-grasp, collecting power, optical extent, 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.^[1]^[2]^[3]

Definition

Etendue for a differential surface element in 2D (left) and 3D (right).

An infinitesimal surface element, dS, with normal n_S is immersed in a medium of refractive index n. The surface is crossed by (or emits) light confined to a solid angle, dΩ, at an angle θ with the normal n_S. The area of dS projected in the direction of the light propagation is dS cos θ. The etendue of this light crossing dS is defined as

\mathrm{d}G = n^2\, \mathrm{d}S \cos \theta\, \mathrm{d}\Omega.

Because angles, solid angles, and refractive indices are dimensionless quantities, etendue has units of area (given by dS).

Conservation of etendue

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.

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.^[1]

In free space

Etendue in free space.

Consider a light source Σ, and a light detector S, both of which are extended surfaces (rather than differential elements), and which are separated by a medium of refractive index n 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.^[4]

According to the definition above, the etendue of the light crossing dΣ towards dS is given by:

\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}

where dΩ_Σ is the solid angle defined by area dS at area dΣ. Accordingly, the etendue of the light crossing dS coming from dΣ is given by:

\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},

where dΩ_S is the solid angle defined by area dΣ. These expressions result in

\mathrm{d}G_\Sigma = \mathrm{d}G_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 \mathrm{d}G.

If both surfaces dΣ and dS are immersed in air (or in vacuum), n = 1 and the expression above for the etendue may be written as

\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},

where F_dΣ→dS is the view factor between differential surfaces dΣ and dS. Integration on dΣ and dS results in G = πΣ F_Σ→S 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.

The conservation of etendue in free space is related to the reciprocity theorem for view factors.

In refractions and reflections

Etendue in refraction.

The conservation of etendue discussed above applies to the case of light propagation in free space, or more generally, in a medium in which the refractive index is constant. However, etendue is also conserved in refractions and reflections.^[1] Figure "etendue in refraction" shows an infinitesimal surface dS on the xy plane separating two media of refractive indices n_Σ and n_S.

The normal to dS points in the direction of the z axis. Incoming light is confined to a solid angle dΩ_Σ and reaches dS at an angle θ_Σ to its normal. Refracted light is confined to a solid angle dΩ_S and leaves dS at an angle θ_S to its normal. The directions of the incoming and refracted light are contained in a plane making an angle φ to the x axis, defining these directions in a spherical coordinate system. With these definitions, Snell's law of refraction can be written as

n_\Sigma \sin \theta_\Sigma = n_S \sin \theta_S,

and its derivative relative to θ

n_\Sigma \cos \theta_\Sigma\, \mathrm{d}\theta_\Sigma = n_S \cos \theta_S \mathrm{d}\theta_S,

multiplied by each other result in

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),

where both sides of the equation were also multiplied by dφ which does not change on refraction. This expression can now be written as

n_\Sigma^2 \cos \theta_\Sigma\, \mathrm{d}\Omega_\Sigma = n_S^2 \cos \theta_S\, \mathrm{d}\Omega_S,

and multiplying both sides by dS we get

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

that is

\mathrm{d}G_\Sigma = \mathrm{d}G_S,

showing that the etendue of the light refracted at dS is conserved. The same result is also valid for the case of a reflection at a surface dS, in which case n_Σ = n_S and θ_Σ = θ_S.

Conservation of basic radiance

Radiance of a surface is related to étendue by:

L_{\mathrm{e},\Omega} = n^2 \frac{\partial \Phi_\mathrm{e}}{\partial G},

where

n is the refractive index in which that surface is immersed;
G is the étendue of the light beam.

As the light travels through an ideal optical system, both the étendue and the radiant flux are conserved. Therefore, basic radiance defined as:^[5]

L_{\mathrm{e},\Omega}^* = \frac{L_{\mathrm{e},\Omega}}{n^2}

is also conserved. In real systems, the étendue 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, étendue may not decrease and radiant flux may not increase and, therefore, basic radiance may not increase.

Etendue as a volume in phase space

Optical momentum.

In the context of Hamiltonian optics, at a point in space, a light ray may be completely defined by a point r = (x, y, z), a unit Euclidean vector v = (cos α_X, cos α_Y, cos α_Z) indicating its direction and the refractive index n at point r. The optical momentum of the ray at that point is defined by

\mathbf{p} = n(\cos \alpha_X, \cos \alpha_Y, \cos \alpha_Z) = (p, q, r),

where ||p|| = n. The geometry of the optical momentum vector is illustrated in figure "optical momentum".

In a spherical coordinate system p may be written as

\mathbf{p} = n\!\left(\sin \theta \cos \varphi, \sin \theta \sin \varphi, \cos \theta \right),

from which

\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,

and therefore, for an infinitesimal area dS = dx dy on the xy plane immersed in a medium of refractive index n, the etendue is given by

\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,

which is an infinitesimal volume in phase space x, y, p, q. Conservation of etendue in phase space is the equivalent in optics to Liouville's theorem in classical mechanics.^[1] Etendue as volume in phase space is commonly used in nonimaging optics.

Maximum concentration

Etendue for a large solid angle.

Consider an infinitesimal surface dS, immersed in a medium of refractive index n crossed by (or emitting) light inside a cone of angle α. The etendue of this light is given by

\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.

Noting that n sin α is the numerical aperture NA, of the beam of light, this can also be expressed as

\mathrm{d}G = \pi\, \mathrm{d}S \mathrm{NA}^2.

Note that dΩ is expressed in a spherical coordinate system. Now, if a large surface S is crossed by (or emits) light also confined to a cone of angle α, the etendue of the light crossing S is

G = \pi n^2 \sin^2 \alpha \int \mathrm{d}S = \pi n^2 S \sin^2 \alpha = \pi S \mathrm{NA}^2.

Etendue and ideal concentration.

The limit on maximum concentration (shown) is an optic with an entrance aperture S, in air (n_i = 1) collecting light within a solid angle of angle 2α (its acceptance angle) and sending it to a smaller area receiver Σ immersed in a medium of refractive index n, whose points are illuminated within a solid angle of angle 2β. From the above expression, the etendue of the incoming light is

G_\mathrm{i} = \pi S \sin^2 \alpha

and the etendue of the light reaching the receiver is

G_\mathrm{r} = \pi n^2 \Sigma \sin^2 \beta.

Conservation of etendue G_i = G_r then gives

C = \frac{S}{\Sigma} = n^2 \frac{\sin^2 \beta}{\sin^2 \alpha},

where C is the concentration of the optic. For a given angular aperture α, of the incoming light, this concentration will be maximum for the maximum value of β, that is β = π/2. The maximum possible concentration is then^[1]^[2]

C_\mathrm{max} = \frac{n^2}{\sin^2 \alpha}.

In the case that the incident index is not unity, we have

G_\mathrm{i} = \pi n_\mathrm{i} S \sin^2 \alpha = G_\mathrm{r} = \pi n_\mathrm{r} \Sigma \sin^2 \beta,

and so

C = \left(\frac{\mathrm{NA}_\mathrm{r}}{\mathrm{NA}_\mathrm{i}}\right)^2,

and in the best-case limit of β = π/2, this becomes

C_\mathrm{max} = \frac{n_\mathrm{r}^2}{\mathrm{NA}_\mathrm{i}^2}.

If the optic were a collimator instead of a concentrator, the light direction is reversed and conservation of etendue gives us the minimum aperture, S, for a given output full angle 2α.

References

1 2 3 4 5 Chaves, Julio (2015). Introduction to Nonimaging Optics, Second Edition. CRC Press. ISBN 978-1482206739.
1 2 Roland Winston et al.,, Nonimaging Optics, Academic Press, 2004 [ISBN 978-0127597515]
↑ Matthew S. Brennesholtz, Edward H. Stupp, Projection Displays, John Wiley & Sons Ltd, 2008 [ISBN 978-0470518038]
↑ Wikilivre de Photographie, Notion d'étendue géométrique (in French). Accessed 27 Jan 2009.
↑ William Ross McCluney, Introduction to Radiometry and Photometry, Artech House, Boston, MA, 1994 [ISBN 978-0890066782]