Anderson localization of three-dimensional electromagnetic waves: the end of a 40-year mystery

What is Anderson localization

Anderson localization (AL) refers to the phenomenon in which the diffusion propagation of waves is suppressed in disordered systems. It can occur both in quantum waves (such as electron waves) and classical waves (such as electromagnetic waves, sound waves, water waves, seismic waves, etc.). Anderson localization is a universal physical phenomenon that has nothing to do with the dimension, geometry, boundary conditions, etc. of the system, but only depends on the degree of disorder and wavelength.

Anderson localization was first proposed by Anderson in 1958 to explain the mechanism of metal-insulator transition. He found that in one- or two-dimensional disordered electronic systems, electron waves would be fully localized, that is, they could not transmit current; In the three-dimensional disordered electronic system, there is a critical disorder strength, when the disordered intensity is lower than this critical value, the electron wave can diffuse and propagate, manifested as metal behavior; When the disorder strength is above this critical value, the electron waves are localized and behave as insulators. This critical disorder intensity corresponds to an energy value called the mobility edge, which separates the diffuse and local states.

Why is it difficult to achieve

Although Anderson localization has been extensively studied for more than 40 years, the localization of three-dimensional electromagnetic waves has not been experimentally observed, and some people even question whether it really exists. The reasons for this difficulty are as follows:

Anderson localization of three-dimensional electromagnetic waves requires a very strong degree of disorder. According to the Ioffe–Regel criterion, the moving edge can be reached when the effective wavenumber Keff and the scattering average free path ls satisfy the keffls ≈\u20091. This means either reducing KEFF (by introducing partial ordering or spatial correlation) or decreasing LS (by increasing scattering intensity). However, actual optical materials often have limited refractive index differences and scattering intensities, which is difficult to meet this condition.

Anderson localization of three-dimensional electromagnetic waves needs to consider vector properties. An electromagnetic wave is a transverse wave that has two polarization components, corresponding to the electric field and the magnetic field. The propagation behavior of these two polarizing components in disordered media may be different, resulting in different degrees of localization. Therefore, to observe the Anderson localization of three-dimensional electromagnetic waves, it is necessary to consider the coupling effect of the two polarization components at the same time.

Anderson localization of three-dimensional electromagnetic waves requires the exclusion of interference from other factors. For example, the absorption of metallic materials leads to a decrease in the lifetime of photons, thus masking the characteristics of localization; Inelastic scattering in the medium causes a loss of photon energy, which changes the position of the moving edge; The finite size effect, boundary conditions, detection methods, etc. in the experiment will also affect the judgment of localization.

How the new paper is implemented

To overcome these difficulties, a paper recently published in the journal Nature Physics uses a novel numerical simulation method, the finite difference time domain (FDTD) method. FDTD method is a method for directly solving Maxwell's equations, which can accurately simulate the propagation process of electromagnetic waves in any medium, including scattering, reflection, refraction, interference and other phenomena. The advantage of the FDTD method is that it does not require any approximation or simplification of the disordered system, and can directly obtain the distribution of electric and magnetic fields in space-time. The disadvantage of the FDTD method is that it consumes a lot of computational resources and time, because it needs to divide the entire calculation area into very small meshes and iteratively update the electric and magnetic fields on each mesh.

To improve the efficiency of the FDTD method, this paper uses a parallel computing technique based on a graphics processing unit (GPU). A GPU is a chip specifically designed to handle data-intensive tasks such as images and videos, and it has the ability to be highly parallelized and perform high-speed operations. By distributing the computational tasks in the FDTD method to multiple GPU cores, and utilizing the cache and memory inside the GPU, this paper achieves hundreds of times faster the FDTD method in three-dimensional disordered systems.

Using this accelerated FDTD method, the paper simulates two different types of three-dimensional disordered systems: one is a random stacker of metal balls and the other is a random stacker of dielectric spheres. Metal balls have a high refractive index and absorption coefficient, while dielectric spheres have a lower refractive index and absorption coefficient. Both systems have the same volume fraction (i.e. the proportion of the balls occupying space) and both have the same degree of overlap (i.e., there is some overlap between the balls).

By performing spatiotemporal analysis of the electric and magnetic fields of these two systems, the paper found the following important results:

In the metal ball stack, the propagation of electromagnetic waves shows obvious Anderson localization characteristics, that is, the strength of electric and magnetic fields shows exponential attenuation behavior in space, and the localized area shrinks with the increase of disorder. This shows that the metal ball stacked satisfies the conditions of Anderson's localization, i.e. keffls ≈\u20091.

In the dielectric ball stack, the propagation of electromagnetic waves exhibits different behaviors, that is, the strength of the electric field and the magnetic field exhibit a power-law attenuation behavior in space, and the attenuation index decreases with the increase of disorder12. This shows that the dielectric ball stacker did not meet the conditions of Anderson localization, i.e., keffls ≫\u20091.

In both systems, the vector nature of electromagnetic waves has little effect on localization, that is, the degree of localization of the two polarization components is comparable. This shows that under strong scattering conditions, the coupling effect between the two polarization components is negligible.

In both systems, the overlap between the spheres also has little effect on localization, that is, the degree of overlap has no significant effect on the strength and distribution of electric and magnetic fields. This shows that under strong scattering conditions, the overlap between spheres can be seen as an equivalent source of disorder.

The significance of the paper

This paper is the first time that Anderson localization of electromagnetic waves has been observed in a three-dimensional random medium, thus validating Anderson's original theoretical prediction. This paper is of great significance and value for understanding and controlling the phenomenon of electromagnetic wave transmission in three-dimensional disordered systems. It can provide guidance and inspiration for designing and manufacturing materials with special optical functions and properties. For example, the localization of three-dimensional electromagnetic waves can realize applications such as ultra-high sensitivity optical sensors, ultra-high efficiency optical luminaires, and ultra-high-density optical memory.