What exactly is a (analytical) ray tracer in a physical sense?

Question

I have programmed a simple 2D ray tracer for a radar signal and am now trying to understand it in physical terms. Basically, the ray tracer shoots a "shotgun" of rays from a transmitter in the general direction of a receiver, with some object(s) in between (but not necessarily on the line of sight). Every object involved is assumed to be homogeneous (in terms of the refractive index and attenuation coefficient) and given as a polygon; outside of the object(s), the rays travel through air. At each iteration of the tracer, I look for the first edge a given ray intersects (or if it hits the receiver or vanishes into infinity) and then apply specular reflection and refraction, i.e. the ray is split into a reflected and a transmitted ray, which are then also followed. The rays reaching the receiver are added up, with each ray (indexed as $k$) carrying an electric field given as follows: \begin{align} E_k = \prod_{l=1}^{m_k-1} \rho_l \prod_{i=1}^{m_k} \exp(- (\mu_i + j k_i) \, d_i) \text{,} \end{align} where $m_k$ is the number of segments that make up the ray, $\rho_l$ is the remaining fraction of the signal amplitude after an interface and $\mu_i$, $k_i$ and $d_i$ are the attenuation coefficient, wave number and length of the $i$-th segment, meaning $k_i = k n_i$, with $k$ the wavenumber in a vacuum. To sum up, the ray tracer allows for multipath propagation and correctly implements Snell's law of refraction and specular reflections, as well as (simplified) reflection/transmission coefficients.

After some diving into the literature, I found that the Eikonal equation is commonly used for ray tracing purposes, but it is unclear to me whether what I implemented can be considered a "simplified Eikonal solver" since I often found the Eikonal equation in the context of first arrival times. I also found that ray tracing equations could be derived from the Eikonal equation, which describe the paths of the rays.

So, while this is a broad question: how exactly can my ray tracer be considered in this context? Does it solve the Eikonal equation? If not, is there some other equation it (effectively) solves?

Answer 1

The eikonal is the wavefront of the field. It can be shown that in the high frequency (short wavelength) asymptotic approximation the electric and magnetic field intensities can be written as $$\mathbf {E(r)} \approx \mathbf {e(r)} e^{\mathfrak j \kappa_0\mathcal S (\mathbf{r})}\\\mathbf {H(r)} \approx \mathbf {h(r)} e^{\mathfrak j \kappa_0\mathcal S (\mathbf{r})} \tag{0}\label{0}$$ where $\kappa_0=\omega/c \to \infty$ is the free-space wavenumber so that the "phase" $\mathcal S$ satisfies the eikonal equation in medium of refractive index $\nu$ $$\vert \nabla \mathcal S\vert = \nu(\mathbf{r}) \tag{1}\label{1}$$ with the orthogonality conditions $$\mathbf {e(r)}\cdot\nabla \mathcal S = 0\\\mathbf {h(r)}\cdot\nabla \mathcal S = 0 \tag{2}\label{2}.$$

The rays themselves are the orthogonal trajectories to the equal phase surfaces of $\mathcal S$, in other words, the unit tangent to the rays $\mathbf t $ satisfies the differential equation: $$ \nu \mathbf t = \nabla \mathcal S \tag{3}\label{3}$$

In a piecewise homogeneous medium the rays are always straight lines. Since you in your formula you are assuming a straight line propagation in that sense you implicitly are solving the eikonal equation but there caveats. Even if you are staring with a plane wave after refractions you do not have a plane wave anymore although each orthogonal trajectory is still a straight line. Malus-Dupin law assures you that an eikonal surface will still exists and it will result in your polygonal approximation piece-wise planar wavefronts.

Upon encountering a discontinuity a plane wave will break into reflected and transmitted parts per Fresnel's equations and I assume you are using those to calculate your terms in the product formula. You can always handle a sufficiently narrow pencil of rays as a plane wave propagation and then Fresnel's refraction formula for that infinitesimal bundle is a good approximation if the curvature of the discontinuous surface is much smaller than the wavelength and the asymptotic behavior assumed in $\eqref{0}$ is valid. It will surely fail in the neighborhood of sharp edges or, in your 2D case, at the vertices of the polygonal discontinuities. t Depending on what you are trying to achieve you may or may not ignore that failure.

In a real life radar simulation you also have to take into account the vector nature of reflection and refraction, especially if you have both dielectrics and metals present. More importantly, a real multipath simulation must take into account the coherent nature of the radar signal and therefore when combining multiple rays at a single point you must take into account their individual phase evolution according to their respective optical path lengths.