We present a field-theoretic framework for modeling electromagnetic energy propagation in heterogeneous media by introducing the concept of electromagnetic geodesics. Unlike traditional ray optics, which assumes either a straight-line propagation or a simple bending in refractive media, our approach formulates wave propagation as geodetic motion in a curved spatial geometry induced by variations in refractive index. Building on earlier work, we move beyond scalar refractive index analogies and instead construct a local Riemannian metric characterized by an orthogonal geometric tensor derived from the Helmholtz representation. This tensor encodes spatial anisotropy and curvature, enabling a rigorous description of energy flow through complex media. We derive the electromagnetic geodesics by formulating and solving a Lagrangian system, yielding equations of motion for wavefront trajectories, group velocity, and intensity distribution. The concept of refractive tension—the vector displacement between Euclidean and transformed positions—plays a central role in defining the transformation matrix and associated metric. Numerical simulations for a spherical inhomogeneity embedded in vacuum demonstrate the emergence of curved geodesics and localized energy redistribution, illustrating the model's potential for interpreting interstellar electromagnetic phenomena and refractive effects in astrophysical environments. In particular, it shows the spatial dispersion of a the energy flow in the vicinity of the spherical inhomogeneity.

Plain Language Summary
We propose a new way to describe how light and electromagnetic energy move through complex materials. Instead of straight or bent paths, waves follow curved “geodesics” shaped by refractive index changes. This geometric approach explains energy flow, predicts intensity patterns, and helps interpret astrophysical and interstellar phenomena.

This booklet provides additional information to the work of art 'Soft Layer' located in front of the faculty of Civil Engineering and Geosciences. This piece of art is an initiative of students and alumni of the Mijnbouwkundige Vereeniging and has been revealed on the 24th of February, 2020.

Maxwell's equations are transformed from a Cartesian geometry to a Riemannian geometry. A geodetic path in the Riemannian geometry is defined as the raypath on which the electromagnetic energy efficiently travels through the medium. Consistent with the spatial behavior of the Poynting vector, the metric tensor is required to be functionally dependent on the refractive index of the medium. A symmetric nonorthogonal transformation is introduced, in which the metric is a function of an electromagnetic tension. This so-called refractional tension determines the curvature of the geodetic line. To verify the geodetic propagation paths and wavefronts, a spherical object with a refractive index not equal to one is considered. A full 3-D numerical simulation based on a contrast-source integral equation for the electric field vector is used. These experiments corroborate that the geodesics support the actual wavefronts. This result has consequences for the explanation of the light bending around the Sun. Next to Einstein's gravitational tension there is room for an additional refractional tension. In fact, the total potential interaction energy controls the bending of the light. It is shown that this extended model is in excellent agreement with historical electromagnetic deflection measurements.

Synthetic-aperture (SA) imaging is a popular method to visualize the reflectivity of an object from ultrasonic reflections. The method yields an image of the (volume) contrast in acoustic impedance with respect to the embedding. Typically, constant mass density is assumed in the underlying derivation. Due to the band-limited nature of the recorded data, the image is blurred in space, which is quantified by the associated point spread function. SA volume imaging is valid under the Born approximation, where it is assumed that the contrast is weak. When objects are large with respect to the wavelength, it is questionable whether SA volume imaging should be the method-of-choice. Herein, we propose an alternative solution that we refer to as SA interface imaging. This approach yields a vector image of the discontinuities of acoustic impedance at the tissue interfaces. Constant wave speed is assumed in the underlying derivation. The image is blurred in space by a tensor, which we refer to as the interface spread function. SA interface imaging is valid under the Kirchhoff approximation, where it is assumed that the wavelength is small compared to the spatial dimensions of the interfaces. We compare the performance of volume and interface imaging on synthetic data and on experimental data of a gelatin cylinder with a radius of 75 wavelengths, submerged in water. As expected, the interface image peaks at the gelatin-water interface, while the volume image exposes a peak and trough on opposing sides of the interface.

In acoustic reflectivity imaging, we infer the internal reflectivity of an unknown object from reflected waveforms. A common assumption is that the mass density is constant and that the recorded pressure field is related to a volume contrast in the wave speed by a nonlinear volume-integral representation. This representation is typically linearized under the Born approximation and solved for the volume contrast by iterative inversion. We propose an alternative methodology, which we refer to as interface contrast imaging. In our derivation, we assume a medium with constant wave speed, which contains discontinuities of the acoustic impedance at a collection of interfaces between piecewise-homogeneous subdomains. A linear relationship is established between the recorded data and the gradient of the acoustic impedance at the interfaces, which we refer to as an interface contrast. This contrast can be solved for by iterative inversion. With this procedure, acoustic interfaces can be delineated with superior resolution compared to volume contrast imaging. Since the convergence speed is relatively fast and a reasonable image can already be obtained after a single iteration, real-time applications seem feasible. If necessary, the acoustic impedance can also be imaged by integrating the retrieved reflectivity contrast over space.