Pump-and-probe optical transmission phase shift as a quantitative probe of the Bogoliubov dispersion relation in a nonlinear channel waveguide
Research article
P.-É. Larré, S. Biasi, F. Ramiro-Manzano, L. Pavesi and I. Carusotto
European Physical Journal D 71, 146 (2017) · Published online 9 June 2017
Article summary
A pump-and-probe transmission phase-shift method is developed as a quantitative probe of the Bogoliubov dispersion relation in a nonlinear channel waveguide.
Reference
P.-É. Larré et al., “Pump-and-probe optical transmission phase shift as a quantitative probe of the Bogoliubov dispersion relation in a nonlinear channel waveguide,” Eur. Phys. J. D 71, 146 (2017). https://doi.org/10.1140/epjd/e2017-80208-5
Figure 1. Power P0(z) = 1/2 cε0n0ρ0(z) (Panel A; Eq. (8)) and phase θ0(z) − θ0(0) (Panel B; Eqs. (9)) of the beam of monochromatic light as a function of the propagation distance z ∈ [0, 20 cm]. The plain (dashed) curves are obtained for a TM mode at 1.55 μm propagating along a channel waveguide with a silicon (silicon-nitride) core, the optical parameters of which are listed in the middle (right) column of Table 1.
Figure 2. Real (plain curves) and imaginary (dashed curves) parts of the normalized Bogoliubov dispersion relation k(ω)/(|γ|ρ0) against ωτ ≥ 0 in the absence of one- and two-photon losses, as given by equation (19b). The plots are symmetric with respect to the horizontal k(ω) = 0 line: the branches above (below) this line correspond to the “+” (“−”) sign in equation (19b) and are called “positive (negative) branches.” Panel A: “Dynamically” stable case where the group-velocity-dispersion parameter β2 and the Kerr-nonlinearity coefficient γ have the same sign. Panel B: “Dynamically” unstable case where β2 and γ have opposite signs.
Figure 3. Real (upper row; black plain style as in Figure 2) and imaginary (lower row; black dashed style as in Figure 2) parts of the Bogoliubov dispersion relation keff(ω ≥ 0, z = L) of “TM” (left column) and “TE” (right column) fluids of light exiting a L = 2 cm-long silicon-core single-mode channel waveguide. The plots result from the numerical diagonalization of Keff(ω,L) defined in equation (30b) and the red curves indicate the adiabatic predictions of Section 4.3.1. The operating wavelength equals 1.55 μm, the incident power is 100 mW, and the corresponding silicon parameters are given in the middle column of Table 1. The dispersions are horizontally symmetric: the upper (lower) branches correspond to the “+” (“−”) sign in the second row of equation (27b) and are called “positive (negative) branches.”
Figure 4. Same as Figure 3 for a L = 20 cm-long silicon-nitride-core single-mode channel waveguide, the parameters of which are given in the right column of Table 1.
Figure 5. Phase φL(ω) accumulated by positive-branch (“+” sign in Eq. (19b)) Bogoliubov fluctuations of the amplitude of the electric field in the course of propagation along a lossless, α0, α2 = 0, single-mode channel waveguide with β2, γ > 0 and normalized length ℓ = γρ0L = 0.75 (black densely dashed curves), 5 (dashed), 10 (weakly dashed), and 17.5 (plain). Panel A traces φL(ω) as a function of the angular frequency ω; Panel B uses φL(ω) against k(ω) to extract the Bogoliubov dispersion relation.
Figure 6. Same as Figure 5 for a TE mode propagating along a L = 20 cm-long single-mode channel waveguide with a silicon-nitride core. The operating wavelength equals 1.55 μm, the incident power is 100 mW, and the corresponding waveguide parameters are given in the right column of Table 1.
Figure 7. Schematic representation of the Mach-Zehnder-interferometry pump-and-probe experiment making it possible to measure (42), (75) and then, as explained in Section 5.1, the Bogoliubov dispersion relation of the fluid of light propagating along the channel waveguide encompassed between z = 0 and z = L.
