Radiation-reaction electromagnetic fields in metasurfaces, a complete description of their optical properties
Michele Merano













[]michele.merano@unipd.it
Dipartimento di Fisica e Astronomia G. Galilei, Università degli studi di Padova, via Marzolo 8, 35131 Padova, Italy












This paper derives the macroscopic electric and magnetic fields and the surface susceptibilities for a metasurface, starting from the microscopic scatterer distribution. It is assumed that these scatterers behave as electric and magnetic dipoles under the influence of the incident radiation. Interestingly not only the retarded electromagnetic fields from oscillating dipoles are relevant to pass from the microscopic to the macroscopic representation, but the advanced fields must be considered too. It is found that the macroscopic fields are the sum of the incident fields plus the radiation-reaction fields acting on a single scatterer. Both the local fields and the radiation-reaction fields are necessary to fix the electric and magnetic surface susceptibilities. 



Radiation-reaction electromagnetic fields in metasurfaces, a complete description of their optical properties
    Michele Merano
    December 30, 2023
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§ INTRODUCTION

  

A metasurface is a two-dimensional (2D) metamaterial, a material engineered to have a property that is not found in nature. As such, metasurfaces are extremely advanced, manmade, new tools. They consist of a certain density of distributed microscopic scatterers that induce unusual macroscopic reflection and transmission properties <cit.>. These scatterers are much smaller than the wavelength of the incident electromagnetic field. They are designed to have well-defined electric and magnetic polarizabilities. 

The connection between the microscopic polarizability and the macroscopic reflected and transmitted fields is not trivial. In the atomic theory for standard three dimensional (3D) materials, the procedure to pass from micro to macro runs as follow <cit.>. The contribution of the magnetic polarizability is in general neglected because it is small, and the magnetic susceptibility is set equal to zero. To estimate the difference in between the local electric field and the macroscopic electric field, we can imagine that a scatterer (an atom or a molecule) is surrounded by a sphere with a radius big compared with the interatomic distance and we can consider separately the effects produced by the matter outside and inside the sphere. It is shown that the scatterers inside the sphere do not produce any effect on the central one, hence we may regard it as being situated in a spherical region inside which there is vacuum and outside which there is a uniform polarization. This results in the Clausius-Mossotti-Lorenz-Lorentz relation that connects the microscopic atomic electrical polarizability to the macroscopic electrical susceptibility.

Up to now the relation between the microscopic and the macroscopic electromagnetic fields in a metasurface has been done in a manner analogous to this 3D procedure <cit.>. Of course in the case of a metasurface the magnetic susceptibility plays an important role, but the magnetic polarizability can be treated in a similar way to the electric one. The important difference going from the bulk to a surface is the replacement of the sphere with a disc. If for the 3D case, it is enough to choose the radius of the sphere big enough, in the 2D case the radius of the disk must be determined in some way. This leads to some arbitrariness in this choice. 

In parallel with the advent of metasurfaces, the last decade has also witnessed the advent of 2D crystals <cit.>. These single-layer atomic planes are stable under ambient conditions and appear continuous on a macroscopic scale. Macroscopic continuity is confirmed also by their optical properties that can be essentially described by a surface electric susceptibility <cit.>. Differently to a bulk material and in analogy with a metasurface, in a 2D crystal all the dipoles contribute to the local electric field, and due to the finite velocity of propagation of the dipolar potential, retardation effects manifest themselves up to the macroscopic description <cit.>. 

Starting from the expression of the local electric field, a recent paper has connected this microscopic description to the macroscopic one by deducing an expression for the macroscopic field in the 2D crystal <cit.>. Remarkably an essential role in this context is played by the radiation-reaction field acting on a single dipole, via the boundary conditions for the system. The expression of the radiation-reaction electric field coincides with the macroscopic electric field radiating from the crystal and, summed to the incident electric field, generates the total macroscopic electric field. This theory does not require the calculation of the fields of any small disk of arbitrary radius, it is based on a well-defined and relativistically invariant prescription and it is well supported by experiments <cit.>.

In ref. <cit.> only the case for a crystal with a magnetic surface susceptibility equal to zero was addressed. Here I show how to extend the same concept to a metasurface, with scatterers that have both electric and magnetic polarizabilities.



§ CLASSICAL THEORY OF A RADIATING METASURFACE

For simplicity I model a metamaterial as scatterers on a Bravais lattice (fig. (<ref>)). I consider two types of 2D  Bravais lattices, the square and the triangular one. The unit cell surface density (N) is  

    N=1/a^2  (square lattice);     N=2/√(3)a^2   (triangular lattice)

where a is the lattice spacing. An s polarized plane wave is normally incident on a free-standing metasurface. It generates a reflected and a transmitted wave, a macroscopic electric E⃗(t) and magnetic H⃗(t) field, a surface polarization P⃗(t) and a surface magnetization M⃗(t). These quantities are connected by the following relations

    P⃗(t)=ϵ_0 χ_e E⃗(t);   M⃗(t)= χ_m H⃗(t)

where ϵ_0 is the vacuum permittivity, χ_e and χ_m are the macroscopic surface electric and magnetic susceptibilities. P⃗ and M⃗ are also related to the microscopic electric (α_e) and magnetic (α_m) polarizabilities via

    P⃗(t)=Np⃗(t)=Nϵ_0 α_e E⃗_loc(t);   M⃗(t)=Nm⃗(t)=N α_m H⃗_loc(t).



E⃗_loc(t) and H⃗_loc(t)  are the local electric and magnetic fields acting on a single scatterer of the unit cell, p⃗(t) and m⃗(t) are the electric and magnetic dipole moments established for each scatterer under the influence of the local fields.
    

Given the metasurface structure with N, a, α_e, α_m and the incident electric and magnetic fields E⃗_i(t), H⃗_i(t), a complete theory must be able to compute the microscopic fields E⃗_loc(t) and H⃗_loc(t), and then the macroscopic fields  P⃗(t), M⃗(t), E⃗(t), H⃗(t) and χ_e, χ_m. 



§ MICROSCOPIC THEORY

The scatterers, excited by the incident fields, act as electric and magnetic dipoles. For simplicity I consider isotropic scatterers, even if in the more general case, metasurfaces are created by assembling arrays of miniature, anisotropic radiation scatterers. If I assume an harmonic time dependence e^i(ω t-k z) for the incident field, an electric dipole generates an electromagnetic field given by <cit.>

    E⃗_p(r⃗, t)   =   e^i(ω t-k r)/4πϵ_0r^3([3(p⃗_0·n̂)n̂-p⃗_0](1+ikr ) +k^2 r^2( n̂×p⃗_0)  ×n̂) 
    H⃗_p((r⃗, t)   =   c k^2 e^i(ω t-k r)/4π r( n̂×p⃗_0)(1+1/ikr)

where c is the speed of light, n̂ is the unit vector in the direction of r⃗ and p⃗(t)=p⃗_0 e^iω t. I note that the magnetic field is transverse to the radius vector at all distances. The electromagnetic field generated by the magnetic dipole m⃗(t)=m⃗_0 e^iω t is 

    H⃗_m(r⃗, t)   =   e^i(ω t-k r)/4π r^3([3(m⃗_0·n̂)n̂-m⃗_0](1+ikr ) +k^2 r^2( n̂×m⃗_0)  ×n̂) 
    E⃗_m((r⃗, t)   =   -η k^2 e^i(ω t-k r)/4π r( n̂×m⃗_0)(1+1/ikr)

and it can be obtained from expressions (<ref>) with the interchange E⃗_p →ηH⃗_m, η⃗H⃗_p →  -E⃗_m and p⃗_0 →m⃗_0/c <cit.>.



 §.§ The local fields

The local field acting on a scatterer is provided by the superposition principle, being the sum of the incident field plus the electromagnetic fields due to all the other scatterers except the one under consideration.

    E⃗_loc(t)   =   E⃗_i(t)+∑_(n, s)'E⃗_p  (n,s)+∑_(n,s)'E⃗_m (n,s)
    H⃗_loc(t)   =   H⃗_i(t) +∑_(n,s)'H⃗_m (n,s)+∑_(n,s)'H⃗_p (n,s)

where the sites of the Bravais lattice are labeled by the integers (n, s) and the prime indicates that the scatterer, for which the local field is computed, does not contribute. From the expressions (<ref>, <ref>) for simmetry reasons

    ∑_(n,s)'E⃗_m (n,s)=0   V/m;   ∑_(n,s)'H⃗_p (n,s)=0 A/m.

Sums (<ref>) reduce in this way to those already computed in reference <cit.> and give the following results

    E⃗_i(t)   =   E⃗_loc(t)
    (1-α_e C_0/4π a^3+i α_e N k/2)
    H⃗_i(t)   =   H⃗_loc(t)
    (1-α_m C_0/4π a^3+i α_m N k/2)

where the values of C_0 for the square and the triangular lattice are reported in <cit.>.



§ FROM MICRO TO MACRO

The microscopic theory provides E⃗_loc(t) and H⃗_loc(t). From the local fields it is then possible (eqs. <ref>)  to compute P⃗(t) and M⃗(t). We still do not know (eqs. <ref>) both E⃗(t), χ_e and H⃗(t), χ_m. It is clear that something is missing in the theory. For a 2D crystal, reference <cit.> has shown that the missing term is the radiation-reaction electric field E⃗_R(t) acting on a single scatter. I prove below that the same is true for a metasurface.



 §.§ Radiation-reaction electromagnetic fields

E⃗_R(t) is the difference between one-half the retarded and one-half the advanced electric fields generated by all the other scatterers except the one under consideration<cit.>. In an analogous way this definition is extended to the radiation-reaction magnetic field

    E⃗_R(t)    =    ∑_(n,s)' 1/2( E⃗_ret,  (n,s)(t)-E⃗_adv,  (n,s)(t) )     
    H⃗_R(t)    =    ∑_(n,s)' 1/2( H⃗_ret,  (n,s)(t)-H⃗_adv,  (n,s)(t) ).
   
Equations (<ref>, <ref>) give the retarded electromagnetic fields generated by oscillating electric and magnetic dipoles. The advanced fields are easily related to the retarded ones <cit.>, if the retarded fields are given by 

    E⃗_ret(r⃗, t)= e^i ω tf⃗(r⃗);   H⃗_ret(r⃗, t)= e^i ω tg⃗(r⃗)

the advanced fields are

    E⃗_adv(r⃗, t)= e^i ω tf⃗^*(r⃗);   H⃗_adv(r⃗, t)= e^i ω tg⃗^*(r⃗).

Keeping in mind eqs. (<ref>, <ref>) I obtain

    E⃗_R(t)    =    -i α_e N k/2E⃗_loc(t)=- η/2Ṗ⃗̇(t)=-η/2J⃗_p(t) 
    H⃗_R(t)    =    -i α_m N k/2H⃗_loc(t)= -Ṁ⃗̇(t)/2c=-1/2ηJ⃗_m(t)
 
where J⃗_p=Ṗ⃗̇ is the surface polarization current and J⃗_m = μ_0 Ṁ⃗̇ is the surface magnetization current. From these expressions it is evident that E⃗_R(t) and H⃗_R(t) are macroscopic quantities.



 §.§ Boundary conditions for a metasurface

Next we consider the amplitudes of the reflected E⃗_r, H⃗_r and transmitted E⃗_t, H⃗_t fields. These relations hold

    ηH⃗_i=ŝ_i∧E⃗_i;   ηH⃗_r=ŝ_r∧E⃗_r;   ηH⃗_t=ŝ_t∧E⃗_t;

where ŝ_(i, r, t) are the unit vectors along the respective propagation directions. The boundary conditions demand that across the metasurface the tangential components of the electric and the magnetic fields are discontinuous <cit.>

    κ̂∧(E⃗|_z+-E⃗|_z-)=-J⃗_m=2ηH⃗_R;   κ̂∧(H⃗|_z+-H⃗|_z-)=J⃗_p=-2/ηE⃗_R

where κ̂ is the unit vector along the z direction. From eqs. (<ref>) I obtain

    E_i+E_r=E_t-2η H_R;    H_i-H_r=H_t-2/ηE_R.
 
From eqs. (<ref>) and (<ref>) the amplitude of the reflected and the transmitted fields are

    E_r=E_R-η H_R;    E_t=E_i+E_R+η H_R;    H_r=E_R/η- H_R  ;   H_t=H_i+E_R/η+ H_R.
 



 §.§ The macroscopic fields

In analogy with reference <cit.>, I define the macroscopic fields as the average fields at the metasurface

    E⃗=1/2(E⃗|_z++E⃗|_z-);   H⃗=1/2(H⃗|_z++H⃗|_z-).

From eqs. (<ref>) I obtain that the macroscopic electric (magnetic) field is the sum of the incident electric (magnetic) field and the radiation-reaction electric (magnetic) field

    E⃗(t)=E⃗_i(t)+E⃗_R(t);   H⃗(t)=H⃗_i(t)+H⃗_R(t).

Both the local fields and the radiation-reaction fields (eqs. (<ref>, <ref>, <ref>, <ref>)) are necessary to deduce the expressions of the electric and magnetic surface susceptibilities

    χ_e    =    P⃗(t)/ϵ_0 E⃗(t)=N α_e E⃗_loc(t)/E⃗_i(t)+E⃗_R(t)=Nα_e/1-C_0α_e/4π a^3
    χ_m    =    M⃗(t)/H⃗(t)=N α_m H⃗_loc(t)/H⃗_i(t)+H⃗_R(t)=Nα_m/1-C_0α_m/4π a^3.
 
The macroscopic electric field and the electric surface susceptibility have exactly the same expressions valid for a 2D crystal <cit.>.



 §.§ The reflection and transmission coefficients

I express now the components of the reflected and transmitted waves in term of those of the incident wave, giving

    r_s=E_r/E_i=-ikχ_e/ikχ_e+2+ikχ_m/ikχ_m+2;    t_s=E_t/E_i=1-ikχ_e/ikχ_e+2-ikχ_m/ikχ_m+2.

 The reflection and the transmission coefficients depend only on macroscopic quantities. The microscopic α_e and α_m enter in these expressions only through χ_e and χ_m.



§ CONCLUSION

A complete classical description of the optical properties of a metasurface has been presented. Starting from the metasurface structure and the electric and magnetic polarizabilities of the microscopic scatterers, this theory computes both the local fields and the macroscopic fields. It provides also the electric and magnetic surface susceptibilities and the reflected and transmitted fields. 

The approach that has been adopted is analogous to the one developed for the optical response of a dielectric 2D crystal <cit.> (with null magnetic surface susceptibility) and it is valid also for magnetic 2D crystals <cit.>. The analogy is possible because an oscillating magnetic dipole produces a magnetic field like the electric field from an oscillating electric dipole and vice-versa.

The radiation-reaction electric and magnetic fields play an essential role in this theory. They derive from the microscopic description and they are proportional respectively to the macroscopic surface polarization and surface magnetization currents. Via the boundary conditions a definition of the macroscopic fields is then possible. Then the radiation-reaction fields complement the local fields to fix χ_e and χ_m. 

For a dielectric 2D crystal the macroscopic electric field coincides with the transmitted electric field because E⃗ is continuous across the boundary. In a metasurface both the electric and magnetic fields are discontinuous across the boundary. Following <cit.> I have defined the macroscopic fields, as the average fields at the metasurfaces. As for a dielectric 2D crystal, the expression of the macroscopic fields is the sum of the incident plus the radiation-reaction fields.     







