Spatiotemporal continuum generation in polariton waveguides

We demonstrate the generation of a spatiotemporal optical continuum in a highly nonlinear exciton–polariton waveguide using extremely low excitation powers (2-ps, 100-W peak power pulses) and a submillimeter device suitable for integrated optics applications. We observe contributions from several mechanisms over a range of powers and demonstrate that the strong light–matter coupling significantly modifies the physics involved in all of them. The experimental data are well understood in combination with theoretical modeling. The results are applicable to a wide range of systems with linear coupling between nonlinear oscillators and particularly to emerging polariton devices that incorporate materials, such as gallium nitride and transition metal dichalcogenide monolayers that exhibit large light–matter coupling at room temperature. These open the door to low-power experimental studies of spatiotemporal nonlinear optics in submillimeter waveguide devices.

envelope approximations with the slowly varying fields A(x, z, t) and Ψ(x, z, t) such that E = A exp(ik e z − iω e t), P = Ψ exp(ik e z − iω e t). The evolution equations for the system are then given by the coupled equations (1).
The variables ω e , k e , v g , γ p , γ e , Ω, and g are all real and positive. They respectively describe the exciton frequency, the photon wavenumber and group velocity at ω e , the photon and exciton loss rates, the strength of light matter coupling and the nonlinear renormalisa- * p.m.walker@sheffield.ac.uk tion of the exciton frequency per unit exciton density. The variables A and Ψ are in general complex.

II. CW SOLUTION
We insert the ansatz Here k p , δ are taken to be real and A 0 (z) and Ψ 0 (z) are complex functions of z only. Inserting Equations (2) into Equation (1b) gives Thus we have a relation giving the photon field for a given exciton field. Inserting Equations (2) and Equation (3) into Equation (1a) also gives The value of |Ψ 0 | 2 needed in Equation (4)  In the lossless case γ p = γ e = 0 it can be shown that |A 0 | 2 and hence |Ψ 0 | 2 are independent (4), and its conjugate).

Then Equations (3) and (4) are simply solved by
given by

III. EVOLUTION EQUATIONS FOR PERTURBATIONS
We now introduce perturbations a(x, z, t) and ψ(x, z, t) which are taken to be much smaller in amplitude than A 0 and Ψ 0 . The new ansatz for the fields is Inserting this ansatz into Equations (1) and retaining only terms linear in (or independent of) a and ψ (because they are small) we obtain evolution equations for the perturbations Here ω NL (z) = g |Ψ 0 (z)| 2 is the nonlinear renormalisation of the exciton oscillator frequency due to the exciton density and δ R (z) = δ − ω NL (z) is the renormalised detuning between the exciton and pump frequencies.
In deriving these we made use of Equation (3) to eliminate A 0 from (7b). Equations (4) and (3) were both used (via the useful intermediate relations in Equations (8)) to eliminate We have, without loss of generality, chosen Ψ 0 (0) to be real and thus fixed the reference phase for the system.

IV. INSTABILITY GROWTH RATE
Since Equation (7b) couples the perturbations with their complex conjugates, and hence couples positive and negative frequency components in the Fourier transformed fields, we insert the following ansatz for the perturbations Substituting this ansatz into the two evolution equations Equations (7) and collecting coefficients of exp (±i(kx ± ∆t)) leads to eight equations. There are four equations involving a I− , a S− , ψ I− , ψ S− and a separate four equations for the coefficients with '+' in the subscripts.
These two sets are uncoupled and identical apart from the substitution k → −k and so they may be solved independently. In the following we will consider only the set for the '−' coefficients and will drop the '−' part of the subscript. The four equations are given in We have made use of the following definitions for the sake of brevity Note also that we have taken the complex conjugate of the equations coming from collecting the coefficients of terms containing exp (+i∆t).
The algebraic equations (10c) and (10d) may be used to eliminate the exciton fields.
We then obtain a Hamiltonian describing the evolution of the perturbations (a I , a S ).
Here we have defined We note that the coefficients of the Hamiltonian are functions of z (although not of a I , a S ) and it must in general be solved numerically. However, provided that the losses are sufficiently small so that we can consider evolution of the perturbations over a length small compared to the loss length, the matrix can be approximated as constant with z. Then the system can be solved by diagonalizing the matrix.
The physical interpretation of ω T is the kinetic energy associated with transverse wavenumber relative to the pump. Meanwhile 1 + η 0 may be interpreted as an effective group index which reduces the polariton velocity at the pump frequency relative to the uncoupled photon velocity and 1 + η is the same but averaged over signal and idler states on the polariton branches, renormalized by the pump density. Equivalently, they are related to the ratios of exciton and photon densities.
The gain for the two solutions is then given by G = −2ℑ (q 1 ± q 2 ). We will now consider the form of the gain in more detail. To better obtain physical insight we will consider the lossless case. Then v g q 1 is real while v g q 2 is either purely real or imaginary. In that case only the u 1 perturbation can grow exponentially. The gain is then given by Equation (16).
For a detuning less than some critical value, ∆ 2 ≤ ∆ 2 s , (see Equation (17)) the peak gain occurs at k = 0 e.g. for waves travelling with the same transverse momentum as the pump.
When k = 0 then there is gain provided that ∆ 2 < 2∆ 2 s . The peak gain occurs at ∆ 2 m,k=0 and has value G m,k=0 given by Equations (18a) and (18b) respectively.
For ∆ 2 > ∆ 2 s the peak gain occurs at k = k m defined below.
It is interesting to note that ∆ 2 m,k=0 is always larger than ∆ 2 s so that the peak gain along k=0 is always lower than the gain at some finite k for the same perturbation frequency.