The Complex Maxwell Stress Tensor Theorem: A Novel Scenery Underlying Electromagnetic Optical Forces

The optical manipulation of matter through the mechanical action of light is one of the liveliest topics in micro- and nanoscience. In 2018 Arthur Ashkin was awarded the Nobel Prize in Physics for the invention and development of optical tweezers.

The Maxwell stress tensor (MST), which expresses the conservation of linear and angular momentum, is the cornerstone of electromagnetic forces.

If the wave fields are characterized by complex functions, the MST conservation law is obtained from the real parts that give the currently observed time-averaged Lorentz forces (RLFs) and torques. In this context it is known that the RLF on a volume V of charges and currents is given by the Poynting momentum flow whose density is the real part of the Maxwell stress tensor (RMST) over any contour enclosing V. The RLF can be described as that characterized by the RMST Flux into the surface of a sphere in the far field, i.e. in the radiative zone of V, and as such to be considered a “radiant force”.

In a new article published in Light: Science & ApplicationsManuel Nieto-Vesperinas from the Instituto de Ciencia de Materiales de Madrid, CSIC, Spain and Xiaohao Xu from the State Key Laboratory of Transient Optics and Photonics and Xi’an Institute of Optics and Precision Mechanics, Chinese Academy of Sciences, China have shown that the theory through the RMST describes only half of the physics of the electromagnetic optical force.

The other half, hitherto ignored, is characterized by the imaginary part of the complex Maxwell stress tensor related to the exchange of reactive (i.e. imaginary Poynting) momentum, and gains importance as the optical manipulation of matter advances and its scope expanded by the inclusion of reactive concepts. The Maxwell imaginary stress tensor (IMST) builds in and around V, which they propose to be the orbital (or canonical) momentum (ROM) reactive strength; so this storage of ROM contributes to what they found to be the imaginary Lorentz force (ILF) and imaginary torque on V, which can also be viewed as the reactive strength of the Poynting impulse.

Such a reaction force, ILF, is not observable over time since its net value is zero, but it exists instantaneously due to the exchange of reaction momentum that alternates with time between the wave and the body. Therefore, the ILF is a fundamentally fundamental dynamic phenomenon inherent in the generation of electromagnetic and optical forces, and is also associated with the occurrence of reactive power, reactive work, and blind helicity. The former has been a well-known workhorse in RF antenna design for many years and has recently turned to micro and nano antennas.

Therefore, as in the design of RF antennas, one aims to reduce reactive power and reactive work to increase radiation efficiency. The theory presented in this article represents a tool to act on ROM and ILF to optimize a desired radiation pressure in optics manipulation. Consequently, it is shown that ILF and ROM as such play an antagonistic role with respect to the standard RLF, such that strong ILF, and hence large ROM storage, results in loss of radiant power, RLF, and vice-verse. This makes ROM and ILF indirectly observable.

The authors note that it is somewhat striking that the complex Poynting theorem and its associated reactive quantities, the IPM, reactive work and energy, the complex Maxwell stress tensor theorem, and the reactive entities it conveys, have appeared for decades never existed have been founded. Perhaps this is due to the practical difficulties involved in precisely controlling optical manipulation. However, the rapid advances and current maturity of optical manipulation of matter now justify its formulation.

From their point of view, this novel scenario completes an interpretive panorama of the forces in the science of light and classical electrodynamics, e.g. in the design of particles and structured beam illumination, which, as well as their radiant power and helicity of the field emitted, cause the efficiency of the time-averaged force, ie the RLF acting on them, can be optimized by either increasing or decreasing them.