[Enlarge image]Left: Artist’s interpretation of an optical Trojan beam guided through a transparent dielectric tube. Top right: Side-view image and output intensity profile of the trapped Trojan beam (red) when helicoidally traversing a PDMS-filled tube heated by a twisted iron wire. Bottom right: Side-view image and output intensity profile of the diffracted Gaussian beam when the current in the wire is turned off.
Guided wave transmission is a cornerstone of modern photonic technologies. Traditionally, light confinement has relied on total internal reflection (TIR) induced by high-index profiles, as in optical fibers and chip-based waveguides.1 Beyond these systems, alternative guiding mechanisms have been explored, such as in periodic photonic structures, plasmonic systems and scenarios involving gain and loss. This raises a critical question: Can light be guided within a fully dielectric, passive material without relying on TIR or periodicity, but instead through a physical process that acts from afar?
We recently introduced a novel approach to guiding light in transparent dielectric structures by leveraging stable Lagrange points.2 This mechanism enables the capture and guidance of optical beams even in defocusing environments where the refractive-index profile lacks conventional guiding features. To enable guiding, this approach uses a periodically swirling index profile created through attractive or repulsive potentials along the propagation direction.
We refer to these guided waves as “optical Trojan beams,” inspired by Trojan asteroids captured at the Lagrange points of the sun–Jupiter system. In celestial mechanics,3 Lagrange points represent equilibrium positions at which the gravitational forces of two orbiting massive bodies balance the centrifugal force. Of the five known Lagrange points, only L4 and L5 are stable, capable of capturing planetesimal objects. Despite their seemingly unstable nature, these points dynamically trap a body with the aid of the Coriolis force. In our work, we harnessed this principle to guide light, as the ray dynamics resemble those in Newtonian mechanics. A wave analysis confirmed that a helical index distribution can indeed guide Trojan modes.
To observe optical Trojan beams, we used a glass cylinder filled with polydimethylsiloxane (PDMS) with a refractive index of n0 = 1.46. The large negative thermo-optic effect of PDMS was deployed to induce optical Lagrange points. By inserting a twisted iron wire into the tube and passing a current, the PDMS was heated to create a spiraling defocusing index potential, establishing a stable Lagrange point. This allowed a Trojan optical beam to be stably guided, maintaining its spot size along a helical trajectory. The Trojan beams also exhibited a distinctive X-shaped wavefront, differing from the Gaussian phase profile of the input laser beam.
We believe this guiding mechanism, based on dynamic stability at Lagrange points, offers a promising solution for environments where TIR is challenging, such as in liquids or gases. It may also have implications for guiding other types of waves, including acoustic and electron beams, extending its applications beyond photonic systems.
Researchers
Haokun Luo, Yunxuan Wei, Fan O. Wu, Georgios G. Pyrialakos, Demetrios N. Christodoulides and Mercedeh Khajavikhan, University of Southern California, Los Angeles, CA, USA
References
1. A.W. Snyder and J.D. Love. Optical Waveguide Theory (Springer, 1983).
2. H. Luo et al. Nat. Phys. 20, 95 (2024).
3. A. Pérez-Villegas et al. Astrophys. J. Lett. 840, L2 (2017).