Geometric factor

The geometric factor (also etude , from French étendue géométrique) is a physical quantity that characterizes how much light in the optical system is “expanded” in size and direction. This value corresponds to the beam quality parameter (BPP) in the physics of Gaussian beams .

From the point of view of the source, this is the product of the surface area of the source and the solid angle , which is contracted by the entrance pupil of the receiver optical system upon observation from the source. Equivalently, from the point of view of the optical system, the geometric factor is equal to the product of the area of the entrance pupil and the solid angle contracted by the source when observed from the pupil. These definitions should apply to infinitesimal elements of area and solid angle, which should then be summed over the source and aperture as shown below. The geometric factor can be considered as the volume in the phase space .

The geometric factor is an important characteristic of light, since this value never decreases in any optical system where optical power is stored. An ideal optical system creates an image with the same geometric factor as the source. The geometric factor is associated with the Lagrange invariant and the optical invariant , which are also constant in an ideal optical system. The energy brightness of the optical system is equal to the derivative of the radiation flux with respect to the geometric factor.

Content

Definition

The geometric factor for a differential surface element in 2D (left) and 3D (right).

Let an element of the surface dS with a normal n _{S be} immersed in a medium with a refractive index n . Light falls from the solid angle d Ω at the angle θ to the normal n _S onto the surface (or it radiates). The projection of the area d S in the direction of light propagation is equal to d S cos θ . The geometric factor G of the light passing through dS is defined as

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

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

Since angles, solid angles, and refractive indices are dimensionless quantities , the geometric factor has an area dimension (due to the term dS).

Saving a geometric factor

As shown below, the geometric factor is preserved during the propagation of light in free space, as well as in refractions and reflections. It also persists when light passes through optical systems, where the light undergoes ideal refractions or reflections. However, if light hits a scattering surface , the solid angle of its divergence will increase, increasing the geometric factor. The geometric factor may remain constant or increase with the passage of light through the optical system, but it cannot decrease. This is a direct consequence of the increase in entropy , which can be reversed only if a priori information is used to reverse the wavefront - for example, using phase-conjugated mirrors .

The law of conservation of the geometric factor can be derived in different contexts - from the first principles of optics, from Hamiltonian optics or from the second law of thermodynamics . ^[one]

In Free Space

Etendue in free space.

Consider the light source Σ and the detector S , both extended (and not differential elements) separated by a perfectly transparent medium with a refractive index n (see figure). To calculate the geometric factor of the system, one should consider the contribution of each point on the surface of the light source emitting rays at each point on the surface of the receiver. ^[2]

In accordance with the above definition, the geometric factor of the light crossing d Σ in the direction of d S is given by the expression:

\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}}}

{\ displaystyle \ 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 d Ω _Σ is the solid angle contracted by the area d S with respect to the surface d Σ . Similarly, the geometric factor of the light crossing d S coming from d Σ 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}}},

{\ displaystyle \ 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 d Ω _S is the solid angle contracted by dΣ. From these expressions it follows that

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

{\ displaystyle \ mathrm {d} G _ {\ Sigma} = \ mathrm {d} G_ {S},}

which means that the geometric factor is preserved during the propagation of light in free space.

The geometric factor of the whole system is then equal

G=\int _{\Sigma }\!\int _{S}\mathrm {d} G.

{\ displaystyle G = \ int _ {\ Sigma} \! \ int _ {S} \ mathrm {d} G.}

If both surfaces d Σ and d S are immersed in air (or vacuum), then n = 1 , and the expression for the geometric factor can 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 \rightarrow \mathrm {d} S},

{\ displaystyle \ 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 \ rightarrow \ mathrm {d} S },}

where F _{d Σ → d S} is the coefficient of radiation visibility between the surface elements d Σ and d S. Integration over d Σ and d S gives G = π Σ F _{Σ → S} , which allows one to obtain a geometric factor from the visibility coefficients between these surfaces, which, for example, are given in the list of visibility factors for certain geometries or in some books on heat transfer .

The preservation of the geometric factor in free space is related to the reciprocity theorem for visibility factors .

In refractions and reflections

The geometric factor in refraction.

It was shown above that the geometric factor is preserved in the case of light propagation in free space or, in a more general case, in a medium with a constant refractive index. However, the geometric factor also persists in refractions and reflections. ^[1] The figure on the right shows an element of the surface d S in the xy plane separating two media with refractive indices n _Σ and n _S.

The normal to d S is aligned with the z axis. The incident light is bounded by a solid angle d Ω _Σ and reaches d S at an angle θ _Σ to the normal. Refracted light is bounded by the solid angle d Ω _S and proceeds from d S at an angle θ _S to the normal. The directions of the incident and refracted light are contained in a plane located at an angle φ to the x axis, which defines these directions in a spherical coordinate system . With these notation , Snell's law can be written as

n_{\Sigma }\sin \theta _{\Sigma }=n_{S}\sin \theta _{S},

{\ displaystyle n _ {\ Sigma} \ sin \ theta _ {\ Sigma} = n_ {S} \ sin \ theta _ {S},}

and, differentiating with respect to θ , we obtain

n_{\Sigma }\cos \theta _{\Sigma }\,\mathrm {d} \theta _{\Sigma }=n_{S}\cos \theta _{S}\mathrm {d} \theta _{S}.

{\ displaystyle n _ {\ Sigma} \ cos \ theta _ {\ Sigma} \, \ mathrm {d} \ theta _ {\ Sigma} = n_ {S} \ cos \ theta _ {S} \ mathrm {d} \ theta _ {S}.}

We multiply these expressions by each other and by the factor d φ , which does not change upon refraction, and we obtain

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).

{\ displaystyle 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).}

This expression can be written as

n_{\Sigma }^{2}\cos \theta _{\Sigma }\,\mathrm {d} \Omega _{\Sigma }=n_{S}^{2}\cos \theta _{S}\,\mathrm {d} \Omega _{S},

{\ displaystyle n _ {\ Sigma} ^ {2} \ cos \ theta _ {\ Sigma} \, \ mathrm {d} \ Omega _ {\ Sigma} = n_ {S} ^ {2} \ cos \ theta _ { S} \, \ mathrm {d} \ Omega _ {S},}

a, multiplying both sides of the equation by d S , we obtain

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},

{\ displaystyle 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},}

those.

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

{\ displaystyle \ mathrm {d} G _ {\ Sigma} = \ mathrm {d} G_ {S}.}

Thus, the geometric factor of the light refracted by d S is preserved. The same result is valid for the case of reflection from the surface d S , where n _Σ = n _S and θ _Σ = θ _S should be set.

Preservation of reduced energy brightness

The energy brightness of the surface is related to the geometric factor by the expression

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

{\ displaystyle L _ {\ mathrm {e}, \ Omega} = n ^ {2} {\ frac {\ partial \ Phi _ {\ mathrm {e}}} {\ partial G}},}

Where

n is the refractive index of the medium into which the surface is immersed;
G is the geometric factor of the light beam.

When light propagates through an ideal optical system, the geometric factor and radiation flux are preserved. Therefore, the reduced energy brightness defined as ^[3]

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

{\ displaystyle L _ {\ mathrm {e}, \ Omega} ^ {*} = {\ frac {L _ {\ mathrm {e}, \ Omega}} {n ^ {2}}},}

also saved. In real systems, the geometric factor may increase (for example, due to scattering), or the radiation flux (for example, due to absorption) may decrease, and therefore the reduced energy brightness may decrease. However, the geometric factor cannot decrease, and the radiation flux cannot increase, so the reduced energy brightness cannot increase either.

Geometric factor as volume in phase space

Optical pulse.

In the context of Hamiltonian optics , at a given point in space, a ray of light can be completely described by a point r = ( x , y , z ) , a unit vector v = (cos α _X , cos α _Y , cos α _Z ) , indicating the direction ray, and the refractive index n at the point r . The optical pulse beam at this point is determined by the expression:

\mathbf {p} =n(\cos \alpha _{X},\cos \alpha _{Y},\cos \alpha _{Z})=(p,q,r),

{\ displaystyle \ mathbf {p} = n (\ cos \ alpha _ {X}, \ cos \ alpha _ {Y}, \ cos \ alpha _ {Z}) = (p, q, r),}

Where $\|\mathbf {p} \|=n$ ${\ displaystyle \ | \ mathbf {p} \ | = n}$ . The geometry of the optical pulse vector is shown in the figure to the right.

In a spherical coordinate system, p can be written as

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

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

where from

\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 ,

{\ displaystyle \ 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 element of area d S = d x d y on the xy plane immersed in a medium with a refractive index n , the geometric factor is defined as

\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,

{\ displaystyle \ 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 element of volume in the phase space x , y , p , q . The conservation of the geometric factor in the phase space in optics is equivalent to the Liouville theorem in classical mechanics. ^{[1] The} geometric factor as volume in phase space is often used in non-image optics .

Maximum Concentration Rate

Etendue for a large solid angle.

Consider a surface element d S immersed in a medium with a refractive index n onto which light (which emits) is incident, bounded by a cone with a solution angle α . The geometric factor of this light is given by the formula

\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 .

{\ displaystyle \ 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 light beam, we can rewrite this expression as follows:

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

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

Note that d Ω is expressed in a spherical coordinate system . Now, if light, also bounded by a cone with a solution angle α , is incident (or emits) onto an extended surface S , then the geometric factor of the light passing through S will be

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

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

Geometric factor and ideal concentration.

The limit of the maximum value of the concentration coefficient (see the figure) is reached by a device with an entrance pupil S in the air ( n _i = 1 ) that collects light from the solid angle 2 α (its reception angle ) and directs it to the surface Σ located in medium with a refractive index n , while the points of the receiving surface are illuminated by light emanating from a solid angle of 2 β . From the expression given above, the geometric factor of the incident light is

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

{\ displaystyle G _ {\ mathrm {i}} = \ pi S \ sin ^ {2} \ alpha,}

and for light reaching the receiving surface

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

{\ displaystyle G _ {\ mathrm {r}} = \ pi n ^ {2} \ Sigma \ sin ^ {2} \ beta.}

Then from the conservation of the geometric factor G _i = G _r it follows that

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

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

where C is the concentration coefficient of the optical device. For a given angular aperture α of the incident radiation, the concentration coefficient will be maximum for the maximum value of sin β , i.e. β = π / 2. Then the maximum possible concentration coefficient is ^[1] ^[4]

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

{\ displaystyle C _ {\ mathrm {max}} = {\ frac {n ^ {2}} {\ sin ^ {2} \ alpha}}.}

In the case when the refractive index of the incident light is not equal to 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 ,

{\ displaystyle G _ {\ mathrm {i}} = \ pi n _ {\ mathrm {i}} S \ sin ^ {2} \ alpha = G _ {\ mathrm {r}} = \ pi n _ {\ mathrm {r} } \ Sigma \ sin ^ {2} \ beta,}

where from

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

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

and in the limit β = π / 2, it turns out

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

{\ displaystyle C _ {\ mathrm {max}} = {\ frac {n _ {\ mathrm {r}} ^ {2}} {\ mathrm {NA} _ {\ mathrm {i}} ^ {2}}}. }

If the optical device is a collimator and not a concentrator, then the direction of light is inverted, and the conservation of the geometric factor gives the minimum aperture value S for a given angle of divergence 2 α of the output radiation.

Literature

↑ ¹ ² ³ ⁴ Chaves, Julio. Introduction to Nonimaging Optics, Second Edition . - CRC Press , 2015 .-- ISBN 978-1482206739 .
↑ Wikilivre de Photographie , Notion d'étendue géométrique (in French). Accessed Jan 27, 2009.
↑ William Ross McCluney, Introduction to Radiometry and Photometry , Artech House, Boston, MA, 1994 ISBN 978-0890066782
↑ Roland Winston et al. ,, Nonimaging Optics , Academic Press, 2004 ISBN 978-0127597515