Axion Mie theory for a spherical topological insulator

1. The authors have solved the problem of scattering by a sphere made of an isotropic dielectric characterized by a homogeneous relative permittivity and a homogeneous axionic constant.

1. The solution of the homogeneous counterpart of Eq. (10) with spatially constant $\epsilon$ and $\Theta$ in the spherical coordinate system was first given by Bohren [ “Light scattering by an optically active sphere,” Chem. Phys. Lett. 29, 458-462 (1974)]. Just set $\alpha=\beta$ in the 1974 paper.

2. The Mie theory provided by the authors was completely formulated by Bohren in 1974. Just set $\alpha=\beta$ in the 1974 paper. This landmark paper went unnoticed by Ge et al. [“Electromagnetic scattering by sphere of topological insulators,” Opt. Commun. 354, 225-230 (2015)] and well as by the authors of the present manuscript.

3. There is another way to solve the same problem, whereby the axionic term does not enter constitutive relations but enters boundary conditions. That way for the Mie theory was adopted by Lakhtakia and Mackay ["Electromagnetic scattering by homogeneous, isotropic, dielectric-magnetic sphere with topologically insulating surface states,” J. Opt. Soc. Am. B 33, 603-609 (2016)]. Section 2.F of the 2016 paper provides a comparison of the two ways.

4. Whereas Bohren provided expressions only a for an incident plane wave, Lakhtakia and Mackay considered any source that lies outside the sphere. This is significant because the authors mention the importance of the “incorporation of arbitrary sources of fields” to justify their work in comparison to that of Ge et al. [Ref. 22].

5. The vector spherical wave functions of Barrera et al. [Ref. 23] are more cumbersome than those of Stratton [Electromagnetic Theory, McGraw-Hill, 1941] and Morse and Feshbach [Methods of Theoretical Physics Vol. II, McGraw-Hill, 1953]. Many of the manipulations presented by the authors in this manuscript would be eased if the vector spherical wave functions of Stratton and Morse&Feshbach are employed. Their orthogonalities on a unit sphere are exceedingly simple and lead to very simple manipulations, as the authors would find on consulting their Ref. 20. Neither Appendix A nor Appendix B are needed in 2020.

6a. Instead or TE and TM classification which is strictly valid as $r\to\infty$, a classification in terms of toroidal and poloidal fields is much better. because it applies everywhere. The electric field is poloidal if the magnetic field is toroidal (in isotropic achiral medium) and vice versa. This classification follows immediately from the vector spherical wave functions of Stratton and Morse&Feshbach; see Chandrasekhar and Kendall [“On force-free magnetic fields,” Astrophys. J. 126, 457-461 (1957)].

6b. The toroidal and poloidal classification will impact the dubious straightness of arrows drawn in Fig. 2 at distance of just 0.1 radius from the surface of the sphere.

7a. Since the distance $r$ is specified in nm in Figs. 1 and 2, this reviewer was surprised at not being able to find the frequency or the free-space wavelength used for the calculations. Neither was the relative permittivity reported for the material of the sphere nor the axionic constant.

7b. The same lacuna is evident for Fig. 3.

8. The axionic admixing of TE and TM modes (as the authors put it) is trivial, all the more so because it is already known from the 1974 and the 2016 papers mentioned above. What is needed is a numerical estimate of a measurable quantity at a realizable frequency for a sphere made of a real material that has axionic properties. That is missing.

The manuscript lacks novelty.

The paper is an interesting exercise in textbook electrodynamics. The problem of scattering of a plane wave off a spherically symmetric object has of course been solved a long time ago: the original paper that the authors aim to generalize is from 1908.

What is new is that the advent of topological insulators has given us an excuse to renew studies of electrodynamics with an extra axion term in the Lagrangian, which leads to modified constitutive relations mixing quantities that are supposed to be magnetic or electric respectively. Many papers have been written redoing a lot of classic electrostatics and electrodynamics calculations with this extra term included: mirror charges, reflection off a plane, surface plasmons, ....

The modifications due to the axion term are often suppressed by two powers of the fine structure constant alpha and therefore difficult to observe. Some effects however are linear in alpha and, maybe more importantly, are corrections to quantities that would vanish without the axion term. The first, and most famous, such effect is the appearance of a magnetic monopole mirror charge at a planar interface.

The authors find, in this work, such an order alpha effect: a mixing of TE and TM modes in the scattering process off the sphere that can in principle be observed experimentally. This finding is in principle not surprising; exactly the same phenomenon has been observed previously for surface plasmons on a planar interface and it is the same mechanism that gives rise to this phenomenon here. But the careful generalization to the spherical geometry as well as the more detailed exposition of how one could experimentally confirm this effect make this work go substantially beyond the existing literature and make it worth while publishing.

One small modification I would like to see is a more careful discussion about the range of validity of the description of the TI in terms of the axion electrodynamics. Theorists love to use this description as it makes it easy to calculate. But this effective description is not always applicable (the surface modes need to be gapped and the effective description applies for energies below the gap), and given that the authors aim to connect to real experiments I would believe a somewhat more detailed discussion of the range of applicability of the action used would be desirable.