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June 9, 2017 September 9, 2026

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

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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

Published 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.

Figures

Optical power and phase variation along silicon and silicon nitride waveguides
Figure 1. Summary. Power P0(z) = 1/2 cε0n0ρ0(z) (Panel A; Eq. (8)) and phase θ0(z) − θ0(0) (Panel B; Eqs.

Source: P.-É. Larré et al., European Physical Journal D 71, 146 (2017). © EDP Sciences, SIF and Springer-Verlag.

Real and imaginary parts of the normalized Bogoliubov dispersion relation
Figure 2. Summary. The real and imaginary parts of the lossless Bogoliubov dispersion are compared for the two possible sign combinations of group-velocity dispersion and Kerr nonlinearity. When both have the same sign (a), the dispersion is real and evolves from a linear, sound-like branch at low frequency to a quadratic, particle-like branch at high frequency. Opposite signs (b) produce a complex low-frequency branch, identifying the dynamically unstable regime and separating propagating from amplifying or decaying fluctuations.

Source: P.-É. Larré et al., European Physical Journal D 71, 146 (2017). © EDP Sciences, SIF and Springer-Verlag.

Bogoliubov dispersion
Figure 3. Summary. The plots compare the exact output Bogoliubov dispersion of transverse-magnetic and transverse-electric modes in a 2 cm silicon channel waveguide with the adiabatic prediction shown in red. Real parts appear in the upper row and loss-related imaginary parts in the lower row for a 1.55 µm, 100 mW input. Both polarizations exhibit an overdamped, nonpropagating low-frequency region, while the different curvature of their imaginary branches records the sign of group-velocity dispersion and the effect of two-photon absorption.

Source: P.-É. Larré et al., European Physical Journal D 71, 146 (2017). © EDP Sciences, SIF and Springer-Verlag.

Transmission phase shift
Figure 4. Summary. 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.

Source: P.-É. Larré et al., European Physical Journal D 71, 146 (2017). © EDP Sciences, SIF and Springer-Verlag.

Wavevector measurement
Figure 5. Summary. Panel a gives the phase accumulated by positive-branch Bogoliubov fluctuations in a lossless waveguide for four normalized propagation lengths, and panel b replots that phase against the corresponding wave number. At short length the phase follows the dispersion smoothly; at longer lengths it develops pronounced plateaux and inflection points through mode mixing. The comparison shows when a coarse-grained phase measurement can recover the Bogoliubov dispersion and where the low-frequency staircase must be included in the interpretation.

Source: P.-É. Larré et al., European Physical Journal D 71, 146 (2017). © EDP Sciences, SIF and Springer-Verlag.

Phase-shift spectra
Figure 6. Summary. 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.

Source: P.-É. Larré et al., European Physical Journal D 71, 146 (2017). © EDP Sciences, SIF and Springer-Verlag.

Experimental dispersion arrangement
Figure 7. Summary. The proposed balanced Mach-Zehnder pump-and-probe arrangement sends a strong pump and a weak, frequency-shifted probe through the channel waveguide in one arm while the second arm supplies a phase reference. Filters separate the transmitted probe from the pump and four-wave-mixing idler before detection. Scanning the pump-probe detuning measures the phase accumulated in the nonlinear arm, from which the Bogoliubov dispersion relation can be reconstructed.

Source: P.-É. Larré et al., European Physical Journal D 71, 146 (2017). © EDP Sciences, SIF and Springer-Verlag.

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Related publications

Connections within F. Ramiro-Manzano's publication record, selected from shared research topics and evidence reviewed in the source papers.

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What the related paper adds

This later paper extends the research direction developed in Pump-and-probe optical transmission phase shift as a quantitative probe of the Bogoliubov dispersion relation in a nonlinear channel waveguide. This work supplies the material constants needed to design silicon-oxynitride photonic circuits. Racetrack resonators yield both the thermo-optic coefficient and nonlinear refractive index, showing that SiON combines low loss, broad transparency and negligible telecom two-photon absorption with useful thermal and nonlinear response.

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This paper provides an earlier basis for the later development reported in Thermo-optic coefficient and nonlinear refractive index of silicon oxynitride waveguides. This theoretical paper interprets small optical fluctuations on a nonlinear guided beam as Bogoliubov excitations of a one-dimensional photon fluid. It predicts how Kerr nonlinearity and losses reshape their dispersion and proposes a stimulated four-wave-mixing and interferometric experiment to measure that dispersion directly.

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What the related paper adds

This earlier paper establishes a scientific basis for Pump-and-probe optical transmission phase shift as a quantitative probe of the Bogoliubov dispersion relation in a nonlinear channel waveguide. This study demonstrates stimulated degenerate four-wave mixing in silicon-nanocrystal waveguides at telecom wavelengths. It measures a large nonlinear refractive index and a broad conversion bandwidth, while also identifying two-photon absorption as the main limitation and the mid infrared as the more favourable operating region.

What this paper adds to the related work

Building on the earlier contribution Stimulated Degenerate Four-Wave Mixing in Si Nanocrystal Waveguides, this paper advances the research line. This theoretical paper interprets small optical fluctuations on a nonlinear guided beam as Bogoliubov excitations of a one-dimensional photon fluid. It predicts how Kerr nonlinearity and losses reshape their dispersion and proposes a stimulated four-wave-mixing and interferometric experiment to measure that dispersion directly.

Featured image for Multi-Mode Interference Revealed by Two-Photon Absorption in Silicon-Rich SiO₂ Waveguides
Journal article
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Multi-Mode Interference Revealed by Two-Photon Absorption in Silicon-Rich SiO₂ Waveguides
S. Manna, F. Ramiro-Manzano, M. Ghulinyan, M. Mancinelli, F. Turri, G. Pucker and L. Pavesi

What the related paper adds

This earlier paper establishes a scientific basis for Pump-and-probe optical transmission phase shift as a quantitative probe of the Bogoliubov dispersion relation in a nonlinear channel waveguide. This work turns two-photon-excited luminescence into a direct image of otherwise invisible infrared modes in silicon-nanocrystal waveguides. The measured spatial beat patterns expose multimode interference, agree with finite-element simulations and simultaneously quantify the unusually large two-photon absorption of the nanocrystal material.

What this paper adds to the related work

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Complete multipanel composition showing pump and probe time traces, simulated temperature map, optical energy density and intrinsic quality-factor contributions
Journal article
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Thermo-Optical Bistability with Si Nanocrystals in a Whispering-Gallery-Mode Resonator
F. Ramiro-Manzano, N. Prtljaga, L. Pavesi, G. Pucker and M. Ghulinyan

What the related paper adds

This earlier paper establishes a scientific basis for Pump-and-probe optical transmission phase shift as a quantitative probe of the Bogoliubov dispersion relation in a nonlinear channel waveguide. This paper demonstrates thermo-optical bistability in a fully integrated silicon-nanocrystal microresonator. Time-resolved measurements and modelling separate refractive-index modulation from heat dissipation, show that the silica host carries away the heat, and preserve the cavity quality factor even at comparatively high pump powers.

What this paper adds to the related work

Building on the earlier contribution Thermo-Optical Bistability with Si Nanocrystals in a Whispering-Gallery-Mode Resonator, this paper advances the research line. This theoretical paper interprets small optical fluctuations on a nonlinear guided beam as Bogoliubov excitations of a one-dimensional photon fluid. It predicts how Kerr nonlinearity and losses reshape their dispersion and proposes a stimulated four-wave-mixing and interferometric experiment to measure that dispersion directly.

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

InterferenceA 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.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. 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 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 predicted dispersion contains a sound-like collective regime but is modified by propagation and two-photon losses.SetupsThe experimental proposal sends pump and probe through the nonlinear waveguide in one Mach-Zehnder arm while the other arm supplies a phase reference. 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.
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