On loss compensation, amplification and lasing in metallic metamaterials

The design of metamaterials, which are artificial materials that can offer unique electromagnetic properties, is based on the excitation of strong resonant modes. Unfortunately, material absorption—mainly due to their metallic parts—can damp their resonances and hinder their operation. Incorporating a gain material can balance these losses, but this must be performed properly, as a reduced or even eliminated absorption does not guarantee loss compensation. Here we examine the possible regimes of interaction of a gain material with a passive metamaterial and show that background amplification and loss compensation are two extreme opposites, both of which can lead to lasing.

and σa is the coupling strength of Pa to the electric field and ΔN = N2 − N1 is the population difference that drives the polarization. Depending on the sign of ΔN, energy can be transferred from the fields to the medium (i.e. absorption for ΔN < 0) or from the medium to the fields (i.e. amplification for ΔN > 0). The latter case refers to what is widely known as 'population inversion' and is a prerequisite for a material to provide gain.
Depending on the experiment to be simulated, the pump can be either pulsed or constant and this is reflected in the evolution of populations over time; for pulsed pump (Rp = Rp(t) with max(Rp(t)) = Rp0) the populations are initially redistributed, but then relax to their initial condition (Fig. S2a), while for constant pump (Rp = const.) they reach a steady state (Fig. S2b). The pump conditions in Fig. 2 have been chosen in accordance with the discussion in Section 4, i.e. for the constant pump Rp = 3×10 8 s -1 and for the pulsed pump Rp0 = 3×10 11 s -1 and τpump = 0.15 ps. The constant pump reaches a steady state after approximately 200 ps, which is the gain used in the examples of Fig.4-6. This amount of gain is achieved with the pulsed pump, approximately 9 ps after excitation, as is shown in Fig. S2a, as the populations N2 and N1 become equal to their CW equivalents in (b). Figure S2. Evolution of populations for our gain system when the pump is (a) pulsed with 0.15 ps duration, Rp0 = 1×10 11 s -1 and (b) constant with Rp = 3×10 8 s -1 . The red dotted lines denote the asymptotic limit of the populations, which have been normalized to the total population Ntotal = 5 × 10 23 . For the considered pump conditions in (a), 9 ps after excitation, the populations N2 and N1 become equal to their CW equivalents in (b).
For a certain constant pump rate Rp, the population difference ΔN=N2-N1 provided by a certain gain material can be directly calculated from Eq. (S1) with setting the fields and the derivatives to zero, to account for t→∞. Then we obtain the result 23 : For another gain material with the exact same parameters, but with different total population *   total total N C N , as discussed in the main paper (section: Observable regimes for different gain materials), to achieve the same population inversion (gain), a different pump rate * p R has to be applied, as already shown in Fig. 10. To relate the two pump rates, Eq. (S3) can be applied for both systems and equating ΔN for both cases leads to: For C<1, the denominator of the right side of Eq. (S4) can become negative, which is an unphysical solution. This indicates that the same population inversion ΔΝ cannot be achieved with the new gain material. In our examples, this occurs for the cases with C = 0.15 and C = 0.06, which are the two cases shown to have inadequate gain to reach lasing.

FDTD self-consistent calculations
The gain material is characterized by the lifetimes τ32 = 0.05 ps, τ21 = 80 ps and τ10 = 0.05 ps and the coupling constant is σα = 10 -4 C 2 /kg (τ30 is assumed for simplicity to be very large, i.e. the nonradiative N3→N0 transition to be absent). The gain medium is assumed to have a Lorentzian response which is homogeneously broadened with linewidth Γα = 2π  20  10 12 rad/s and emission frequency ωα = 2π  200  10 12 rad/s. To avoid effects from frequency mismatch between the gain emission and the SRR resonance, we tune ωα to coincide with the resonant frequency of the SRR-gain composite system. For example, for the strongly coupled case (δz = 0 nm) we set ωα = 2π  184 10 12 rad/s and for the uncoupled case (δz = 80 nm) we set ωα = 2π  199 10 12 rad/s. Similarly, we repeat this tuning for each δz separation. In our simulations the total electron density is considered to be N0(t=0) = N0(t) + N1(t) + N2(t) + N3(t) = C×5×10 23 m -3 and the initial condition is that all electrons are in the ground state and all electric, magnetic and polarization fields are zero. For the major part of the paper we have used C = 1, while for the simulations presented in section "Observable regimes for different gain materials", we have set C to 10, 0.5, 0.15 and 0.06 to account for gain materials that provide different amounts of maximum gain. For the pump-probe simulations we first pump the system with a pulsed pump and then probe it with a weak Gaussian pulse and repeat for different pump-probe delays (for the CW pump simulations the pump-probe delay is irrelevant, because the gain is constant). For the lasing simulations (Fig. 9) we insert noise in the system, then pump with a CW pump and do not send a probe pulse, but monitor the outgoing waves instead and this procedure is repeated for several pump rates. In all cases, the system of the Maxwell equations coupled with the atomic rate equations is self-consistently solved in a Finite-Difference Time-Domain (FDTD) scheme S3 using an approach similar to the one outlined in 15 .

Retrieval of effective parameters
The retrieval of effective metamaterial parameters is a very popular technique 30,31 , according to which the effective refractive index n and impedance z are calculated from the reflection and transmission coefficients, r and t respectively. To do so, it is assumed that the metamaterial is subwavelength enough to be homogenizable and it is then replaced by a homogeneous slab of thickness d with the respective properties n and z that need to be calculated. Then, the effective permittivity and permeability of the metamaterial are simply expressed as εr = n/z and μr = nz. The inversion of r and t, however, introduces ambiguity both in the sign of z and the value of n. The sign of z can be determined by causality arguments, but lifting the ambiguity in n is a more demanding task, as multi-valued functions are involved.
The expression for n can be written as    and Arg[g] is the principal argument of g defined to lie in the interval (-π,π]. It is evident that while g(ω) is a single-valued function of ω that is directly calculated from r and t, f(ω) (and hence n) consists of a multivalued real part and a single-valued imaginary part and is therefore a multi-valued function of ω.