The Morse lens: an intrinsic lens form Eaton and Luneburg lens
Most optical instruments, including a simple lens or sophisticated camera lenses, have various types of aberrations. Nevertheless, there do exist optical instruments that are free of aberrations and can provide sharp (stigmatic) images of all points in certain two-dimensional (2D) or three dimensional (3D) of space within geometrical optics. Such devices are called absolute instruments and have wide applications in cloaks, super-resolution and sub-wavelength focusing. Traditional examples of absolute instruments include Maxwell's fish-eye, Eaton, Luneburg and invisible sphere lenses and their refractive index profile are all radius dependence in 2D or 3D space. In addition, there are more intrinsic physical properties in one dimensional (1D) absolute instrument family. For instance, the well-known symmetric self-imaging Mikaelian lens can be mapped from the Maxwell's fish-eye lens by an exponential conformal mapping w=exp(z). Compared with cylindrical or spherical symmetry Maxwell fish-eye lens, the 1D Mikaelain lens is simpler and more intuitive both in theoretical analysis and experimental fabrication. And the physical images can also be easily extended to 2D or 3D by conformal mappings, even non-symmetric cases. Now there comes a question: Besides the Maxwell's fish-eye lens, whether the common Eaton and Luneburg lenses have such powerful 1D conformal lenses like Mikaelain lens?
The answer is yes. The research group led by Prof. Huanyang Chen from Xiamen University has found the intrinsic relationships between the Eaton lens and Luneburg lens that they can be realized from the Morse potentials proposed about 90 years ago by means of conformal mappings. The research results are published in Chinese Optics Letters, Vol. 18, Issue 6, 2020 (Huanyang Chen, Wen Xiao. Morse lens [Invited][J]. Chinese Optics Letters, 2020, 18(6): 062403).
The Morse potentials, named after physicist Philip M. Morse, is another simple analytic model of the potential energy among diatomic molecules in quantum mechanics. Compared with the famous quantum harmonic oscillator model, Morse potential is more realistic because it describes anharmonic effects, frequency doubling, and combination frequencies. In addition to being precisely solvable in quantum mechanics, the Morse potential has oscillating displacement solutions in classical mechanics as well. This kind of quantum-classical analogy relationship also occurs to some traditional potentials, such as Coulomb potential and harmonic potential. These two potentials are widely used in quantum mechanics and their analogical classical potentials are Newton potential and Hooke potential, which exactly corresponds to Eaton lens and Luneburg lens in geometric optics. In the letter, the researchers prove that, from a conformal mapping aspect, the Morse potential is related to Coulomb potential or harmonic potential, therefore they propose a core conformal lens from the Morse potential and generate an Eaton lens, Luneburg lens, and even the generalized form in geometric optics. The lens originated from the Morse potential is called the Morse lens , and after a logarithmic conformal mapping z=ln w(the inverse transformation of exponential mapping), the lens is called generalized Eaton/Luneburg lens with a cylindrical or spherical symmetry. It is easy to see that the Eaton lens and the Luneburg lens are two special cases when parameter a=-1and a=-2 . With the help of light ray simulations, they find that the familiar elliptical trajectories of Eaton lens and Luneburg lens are mapping to the 1D self-focusing asymmetry trajectories with different periods 2π and π. This is different from the Mikaelain lens, there is not symmetric self-focusing anymore but instead of asymmetric, due to the refractive index profile of the Morse lens. In addition, if parameter a is to take 1 or 2, the opposite case of Eaton and Luneburg lens, named anti-Eaton and anti-Luneburg lens, are proposed with other intriguing properties. They also find that no matter parameter are integers or fractions, the absolute instrument imaging functions are different but obey general rules. Another aspect worth to mention is that the comparison with the famous Lennard–Jones (LJ) potential or 6-12 potential. It is demonstrated that when a=4, even if their potential curves and refractive index profiles are very closed to each other, their imaging properties are opposite: the conformal potentials have four images, however its "close relative" LJ potential, cannot be perfect imaging.
The refractive index distribution (contour map) and trajectory (black curve) for a ray emitted from (1, 0) at 45° on a generalized Eaton/Luneburg lens with (a) a = −1; (b) a = −2; (c) a = −3; (d) a = −4; (e) a = 1; (f) a = 2; (g) a = 3; (h) a = 4.
Prof. Huanyang Chen believes that, "This work provides a classical optics insight for quantum potentials, showing the ability of lensing designs by combining quantum mechanics and transformation optics. The proposed Morse lens, will arise a hot research hit in absolute instrument designs and conformal cloaking designs. The Morse lens and the Morse potentials have more intrinsic physics and can combine Eaton and the Luneburg lens together. It is a novel discovery, because these two lenses are first independently proposed by Eaton and Luneburg. Morse potential enriches and deepens their physical significance and the core of conformal lens will have promising applications."
The lenses above studied are infinite and cover entire spaces, if the lens becomes finite, it is possible to design an omnidirectional concave lens, just like the Luneburg lens can be served as an omnidirectional convex lens. And the trajectories in the generalized Eaton /Luneburg lens and the Morse lens may also be likely to have similar properties to Maxwell's fish-eye lens and Miakelain lens. In another hand, quantum mechanics is a gold mine, there still exist other potentials such as Yukawa potential, and who will be the next to eat crabs?