Pump-and-probe optical transmission phase shift as a quantitative probe of the Bogoliubov dispersion relation in a nonlinear channel waveguide

P.-É. Larré; S. Biasi; F. Ramiro-Manzano; L. Pavesi; I. Carusotto

doi:10.1140/epjd/e2017-80208-5

Research · Publication & note

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

Abstract

We theoretically investigate the dispersion relation of small-amplitude optical waves superimposing upon a beam of polarized monochromatic light propagating along a single-mode channel waveguide characterized by an instantaneous and spatially local Kerr nonlinearity. These small luminous fluctuations propagate along the waveguide as Bogoliubov elementary excitations on top of a one-dimensional dilute Bose quantum fluid evolve in time. They consequently display a strongly renormalized dispersion law, of Bogoliubov type. Analytical and numerical results are found in both the absence and the presence of one- and two-photon losses. Silicon and silicon-nitride waveguides are used as examples. We finally propose an experiment to measure this Bogoliubov dispersion relation, based on a stimulated four-wave mixing and interference spectroscopy techniques.

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Figures

Optical power and phase variation along silicon and silicon nitride waveguides — **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.

Real and imaginary parts of the normalized Bogoliubov dispersion relation — **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 k_eff(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 K_eff(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.”

Transmission phase shift — **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.

Wavevector measurement — **Figure 5.** Phase φ_L(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 φ_L(ω) as a function of the angular frequency ω; Panel B uses φ_L(ω) against k(ω) to extract the Bogoliubov dispersion relation.

Phase-shift spectra — **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.

Experimental dispersion arrangement — **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.

Research fields

Simulations & fits

Bogoliubov theory is adapted to weak optical fluctuations propagating on a Kerr-nonlinear pump in a single-mode channel waveguide. Analytical and numerical solutions include one- and two-photon losses and predict how a phase-sensitive stimulated four-wave-mixing measurement can reconstruct the renormalised dispersion.

Characterization

The proposed pump–probe interferometric protocol treats phase delay, not intensity alone, as the observable. This provides an experimentally accessible route to distinguish phonon-like and free-particle branches of the optical-fluid dispersion in an integrated waveguide.

RESEARCH TOPICS

InterferenceThe proposed measurement is a pump–probe Mach–Zehnder interferometer. A strong pump establishes the optical-fluid background inside a nonlinear waveguide, while a weak frequency-shifted probe accumulates a phase delay relative to a reference arm. Figures 5 and 6 show how that measured phase is related to the real part of the effective Bogoliubov wave vector. Phase plateaus and a residual low-frequency plateau appear when losses are included, so the dispersion cannot be recovered by assuming a simple linear phase ramp.

MultimodeAlthough the physical waveguide is single-mode, the linearised fluctuations have two coupled Bogoliubov components, conventionally written u and v, which mix positive- and negative-frequency perturbations around the pump. Their evolution produces propagating or overdamped branches depending on polarization, dispersion and loss. The analysis compares TE- and TM-like fluids of light in a 2 cm silicon-core channel. In this context “multimode” refers to the coupled excitation branches of the nonlinear optical field, not to uncontrolled transverse waveguide modes.

NonlinearKerr refraction supplies the effective photon–photon interaction that gives the optical beam its fluid-like collective dispersion. The model includes group-velocity dispersion, one-photon attenuation and two-photon absorption; the latter enters as an intensity-dependent damping term. For realistic silicon-photonics parameters near 1.55 μm, loss makes the effective wave vector complex and can turn low-frequency excitations into overdamped, nonpropagating modes. The paper therefore tests how far the conservative Bogoliubov picture survives in a driven, absorbing nonlinear waveguide.

QuantumBogoliubov theory, originally developed for weak excitations of a Bose quantum fluid, is applied to fluctuations riding on a coherent optical pump. The probe plays the role of a quasiparticle excitation and the strong beam the background fluid. The predicted dispersion contains a sound-like collective regime but is modified by propagation and two-photon losses. This is an analogue-physics proposal: it does not claim that the light is a many-body condensate, but uses a controllable photonic system to access the same linearised excitation mathematics.

SetupsThe experimental proposal sends pump and probe through the nonlinear waveguide in one Mach–Zehnder arm while the other arm supplies a phase reference. Filters separate the two frequencies at the output, shutters permit individual-arm calibration and a variable delay balances the interferometer. Scanning the pump–probe detuning maps the accumulated phase versus frequency. The chosen geometry turns a theoretical dispersion relation into an observable quantity and explicitly accounts for the finite 2 cm propagation length, coupling losses and polarization-dependent silicon parameters.