- Open Access
Optimal Diffractive Focusing of Matter and Light Waves
Phys. Rev. Lett. 135, 113604 – Published 12 September, 2025
DOI: https://doi.org/10.1103/h6yy-vvmf
Abstract
Following the familiar analogy between the optical paraxial wave equation and the Schrödinger equation, we derive the optimal, real-valued wave function for focusing in one- and two-space dimensions without the use of any phase component. We compare and contrast the focusing parameters of the optimal waves with those of other diffractive focusing approaches, such as Fresnel zones. Moreover, we experimentally demonstrate these focusing properties on optical beams using both reflective and transmissive liquid crystal devices. Our results provide an alternative direction for focusing waves where phase elements are challenging to implement, such as for x-rays, THz radiation, and electron beams.
Physics Subject Headings (PhySH)
Article Text
References (19)
- M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University Press, New York, 1999).
- M. Khorasaninejad and F. Capasso, Metalenses: Versatile multifunctional photonic components, Science 358, eaam8100 (2017).
- A. Arbabi and A. Faraon, Advances in optical metalenses, Nat. Photonics 17, 16 (2023).
- R. Menon and B. Sensale-Rodriguez, Inconsistencies of metalens performance and comparison with conventional diffractive optics, Nat. Photonics 17, 923 (2023).
- W. T. Chen, A. Y. Zhu, V. Sanjeev, M. Khorasaninejad, Z. Shi, E. Lee, and F. Capasso, A broadband achromatic metalens for focusing and imaging in the visible, Nat. Nanotechnol. 13, 220 (2018).
- S. Banerji, M. Meem, A. Majumder, F. G. Vasquez, B. Sensale-Rodriguez, and R. Menon, Imaging with flat optics: Metalenses or diffractive lenses?, Optica 6, 805 (2019).
- M. A. Cirone, K. Rzążewski, W. P. Schleich, F. Straub, and J. A. Wheeler, Quantum anticentrifugal force, Phys. Rev. A 65, 022101 (2001).
- I. Białynicki-Birula, M. A. Cirone, J. P. Dahl, M. Fedorov, and W. P. Schleich, In- and outbound spreading of a free-particle -wave, Phys. Rev. Lett. 89, 060404 (2002).
- W. B. Case, E. Sadurni, and W. P. Schleich, A diffractive mechanism of focusing, Opt. Express 20, 27253 (2012).
- D. Weisman, S. Fu, M. Gonçalves, L. Shemer, J. Zhou, W. P. Schleich, and A. Arie, Diffractive focusing of waves in time and in space, Phys. Rev. Lett. 118, 154301 (2017).
- D. Weisman, C. M. Carmesin, G. G. Rozenman, M. A. Efremov, L. Shemer, W. P. Schleich, and A. Arie, Diffractive guiding of waves by a periodic array of slits, Phys. Rev. Lett. 127, 014303 (2021).
- P. Boegel, M. Meister, J.-N. Siemß, N. Gaaloul, M. A. Efremov, and W. P. Schleich, Diffractive focusing of a uniform Bose–Einstein condensate, J. Phys. B 54, 185301 (2021).
- D. Attwood, Soft X-Rays and Extreme Ultraviolet Radiation: Principles and Applications (Cambridge University Press, Cambridge, England, 1999).
- M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (US Government Printing Office, 1968), Vol. 55.
- H. Larocque, J. Gagnon-Bischoff, F. Bouchard, R. Fickler, J. Upham, R. W. Boyd, and E. Karimi, Arbitrary optical wavefront shaping via spin-to-orbit coupling, J. Opt. 18, 124002 (2016).
- E. Bolduc, N. Bent, E. Santamato, E. Karimi, and R. W. Boyd, Exact solution to simultaneous intensity and phase encryption with a single phase-only hologram, Opt. Lett. 38, 3546 (2013).
- R. Dorn, S. Quabis, and G. Leuchs, Sharper focus for a radially polarized light beam, Phys. Rev. Lett. 91, 233901 (2003).
- H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, Creation of a needle of longitudinally polarized light in vacuum using binary optics, Nat. Photonics 2, 501 (2008).
- E. Karimi, B. Piccirillo, L. Marrucci, and E. Santamato, Improved focusing with Hypergeometric-Gaussian type-II optical modes, Opt. Express 16, 21069 (2008).
