A ray at some reference plane is specified by:

• transverse position \((x,y)\)
• direction, e.g. \((\theta_x,\theta_y)\) or direction cosines

For a medium of refractive index \(n\), the differential étendue is

\[ dG = n^2 \cos\theta \, dA\, d\Omega \]

where

• \(dA\) is an area element,
• \(d\Omega\) is a solid-angle element,
• \(\theta\) is angle to the surface normal.

In the paraxial limit,

\[ dG \approx n^2\, dx\,dy\, d\theta_x\, d\theta_y. \]

So étendue is essentially a 4D volume in ray phase space.

—

In optics, rays follow Hamiltonian mechanics. A convenient canonical set is

\[ q_1=x,\quad q_2=y, \] \[ p_1=n\,u_x,\quad p_2=n\,u_y, \]

where \(\mathbf u\) is the unit direction vector of the ray.

Then the elementary phase-space volume is

\[ dq_1\,dq_2\,dp_1\,dp_2 = dx\,dy\,d(nu_x)\,d(nu_y). \]

Liouville’s theorem says that under Hamiltonian evolution,

\[ dx\,dy\,d(nu_x)\,d(nu_y) \]

is invariant.

That is the core mathematical statement behind étendue conservation.

—

Let the direction be written as

\[ \mathbf u = (\sin\theta\cos\phi,\; \sin\theta\sin\phi,\; \cos\theta). \]

Then

\[ u_x = \sin\theta\cos\phi,\qquad u_y = \sin\theta\sin\phi. \]

We compute the Jacobian from angular coordinates \((\theta,\phi)\) to \((u_x,u_y)\):

\[ \frac{\partial(u_x,u_y)}{\partial(\theta,\phi)} =

\begin{vmatrix} \cos\theta\cos\phi & -\sin\theta\sin\phi\\ \cos\theta\sin\phi & \sin\theta\cos\phi \end{vmatrix}

= cosθ\,sinθ. \]

Hence

\[ du_x\,du_y = \cos\theta\,\sin\theta\, d\theta\, d\phi. \]

But

\[ d\Omega = \sin\theta\, d\theta\, d\phi, \]

so

\[ du_x\,du_y = \cos\theta\, d\Omega. \]

Multiplying by \(n^2\),

\[ d(nu_x)\,d(nu_y) = n^2 \cos\theta\, d\Omega. \]

Therefore

\[ dx\,dy\,d(nu_x)\,d(nu_y) = n^2 \cos\theta\, dA\, d\Omega = dG. \]

So étendue is exactly the invariant phase-space measure.

—

In geometrical optics, rays satisfy Hamilton’s equations:

\[ \dot q_i = \frac{\partial H}{\partial p_i},\qquad \dot p_i = -\frac{\partial H}{\partial q_i}. \]

The flow in phase space has zero divergence:

\[ ∑_i \left( \frac{\partial \dot q_i}{\partial q_i}

\frac{\partial \dot p_i}{\partial p_i} \right) = ∑_i \left( \frac{\partial^2 H}{\partial p_i \partial q_i}

\frac{\partial^2 H}{\partial q_i \partial p_i} \right) =0. \]

So the phase-space volume element is preserved. That is Liouville’s theorem.

Thus for any bundle of rays transported through an ideal optical system,

\[ G = \int n^2 \cos\theta\, dA\, d\Omega \]

is constant.

—

In 1 transverse dimension, étendue reduces to

\[ dG \approx n\, dx\, d\theta. \]

Using \(p = n\theta\) in paraxial optics, this is just

\[ dG = dx\,dp. \]

A first-order optical system is represented by an ABCD matrix:

\[

\begin{pmatrix} x_2\\ p_2 \end{pmatrix}

=

\begin{pmatrix} A & B\\ C & D \end{pmatrix}

\begin{pmatrix} x₁
p₁ \end{pmatrix}. \]

For a lossless optical system,

\[ AD-BC = 1. \]

The Jacobian determinant is therefore 1, so area in \((x,p)\) space is preserved:

\[ dx_2\,dp_2 = dx_1\,dp_1. \]

That is étendue conservation in the simplest form.

—

The derivation above assumes:

• geometric optics,
• lossless propagation,
• deterministic reversible ray mapping.

If you introduce scattering, absorption/re-emission, diffraction with truncation, or other irreversible processes, the mapping is no longer a clean one-to-one Hamiltonian flow for the occupied ray set. The occupied phase-space region can spread or mix, so effective étendue increases.

—

Mathematically,

\[ dG = n^2 \cos\theta\, dA\, d\Omega \]

is conserved because it is the canonical phase-space volume element for ray optics:

\[ dG = dx\,dy\,dp_x\,dp_y, \]

and Hamiltonian evolution preserves phase-space volume by Liouville’s theorem.

If you want, I can next derive the optical Hamiltonian itself and show explicitly how Snell’s law and free-space propagation fit into this framework.