A. Evolution of the power radiated and absorbed with the level of gain

Figure 2 shows the evolution with the level of gain of the total power radiated (A) and absorbed (B) by the active–passive dimer when excited by a dipole source. Clearly, as F increases, both the radiated power and the absorbed power display a monotonic behaviour.

B. Dipole model

Within the dipolar approximation, we can model the optical response of the active–passive dimer excited by a dipole emitter at its gap as a collection of three point dipole emitters. By doing so, the differential power radiated by the system per unit of solid angle in the direction of n ^ can be written (using Gaussian units), as

d P d Ω = ω 4 2 π c 3 ​ ​ ​ ∑ i , j = 0, G , L ​ ​ ​ ​ e i k ⋅ ( r j − r i ) [ p i ⋅ p j * − ( p i ⋅ n ^ ) ( p j * ⋅ n ^ ) ] ,

where k = ω / ​ c   n ^ is the wavevector of the radiation, and r_i is the position of dipole i. Furthermore, p₀ is the dipole moment of the dipole emitter, whereas p_G=α_GG₀p₀ and p_L=α_LG₀p₀ are the dipole moment induced in the active and passive particles, respectively. We write the latter in terms of frequency-dependent polarisabilities α_G and α_L, which we assume to already include the effect of the interaction between the two particles, and G₀, which is the dipole–dipole interaction tensor that describes the field produced by p₀ at the position of the particles. Using the geometry shown in Figure 1A, we can rewrite this expression as

d P d Ω = ω 4 2 π c 3 ∑ i = 0, G , L ( | p i | 2 − | n ^ ⋅ p i | 2 ) + ω 4 π c 3 R e   { e − i k d cos θ [ p G ⋅ p L * − ( p G ⋅ n ^ ) ( p L * ⋅ n ^ ) ] } + ω 4 π c 3 R e   { e i k d 2 cos θ [ p 0 ⋅ p G * − ( p 0 ⋅ n ^ ) ( p G * ⋅ n ^ ) ] } + ω 4 π c 3 R e   { e − i k d 2 cos θ [ p 0 ⋅ p L * − ( p 0 ⋅ n ^ ) ( p L * ⋅ n ^ ) ] } ,

where d is the distance separating the gain and loss particles, and θ is the angle between the dimer axis and the unit vector n ^ . After integrating this expression over the azimuthal angle and normalising to the power radiated by the p₀ dipole in vacuum, P₀=4ω⁴|p₀|²/(3c³), we obtain (1).

C. Normalised energy transfer rate

Following previous works [61], [62], we define the normalised energy transfer rate (nETR) between two emitters placed at the gaps of a pair of active–passive dimers as

where n ^ r corresponds to the unit vector parallel to the receiver dipole, whereas E_e(r_r) and E_e,0(r_r) represent the electric field produced by the emitter dipole at the position of the receiver in presence of the active–passive dimers and in vacuum, respectively.

Assuming that the two dimers are located in the far-field region of each other, the field radiated by the emitter dipole can be written as a spherical wave [63] fe^ikr/r, with r being the distance between the emitter and the receiver dipoles, and f, the far-field amplitude. This spherical wave can be approximated, at the position of the receiver dimer, as a plane wave, and therefore, we can write (3) as

nETR ≈ | f | 2 | f 0 | 2 FE ( θ ) ,

where FE(θ) is the field intensity enhancement (Figure 4), and f 0 = k 2 [ p e − n ^ ( n ^ ⋅ p e ) ] is the far-field amplitude for the emitter dipole in vacuum. As the radiated power is proportional to the squared magnitude of the far-field amplitude, we can rewrite the factor |f|²/|f₀|², for θ=0° and θ=180°, as

| f | 2 | f 0 | 2 = 4 3 P ( θ ) P 0 ,

where P(θ) is the half-integrated power radiated in the direction θ, which is plotted in Figure 3. Combining these expressions, we obtain (2). It is worth noting that Lorentz reciprocity [60] implies that 4P(θ)/(3P₀) is equal to FE(θ) for θ=0° and 180°. This means that, for these angles, we can write the nETR as nETR ≈ 16 P 2 ( θ ) / ( 9 P 0 2 ) and therefore use (1) to obtain an analytical result within the dipolar approximation.