This paper, published in Optics Express (Vol. 28, 2020), develops a comprehensive numerical model for broadband single-cycle terahertz pulse propagation that simultaneously accounts for non-paraxial diffraction, group velocity dispersion, and time-varying nonlinear refraction including both instantaneous and delayed Kerr contributions. The work was carried out at the University of Ottawa in collaboration with the Boyd group at the University of Rochester.
Motivation and Physical Setting
Recent advances in high-field THz sources have made broadband single-cycle THz pulses with peak intensities exceeding 1013 W/m2 routinely accessible, enabling a new class of nonlinear THz time-domain spectroscopy experiments. These pulses are fundamentally different from optical pulses in the many-cycle regime: their spectral bandwidth is comparable to their central frequency, which invalidates the slowly varying envelope approximation (SVEA) and makes standard pulse propagation models inapplicable. At the same time, the THz frequency range sits close to the phonon resonances of many polar semiconductors, meaning that the nonlinear optical response of the medium cannot be treated as purely instantaneous. A retarded contribution to the Kerr effect, analogous to stimulated Raman scattering in silica fibers, arises from coherent phonon excitation and can enhance the overall third-order susceptibility by several orders of magnitude near the lattice resonance. The relative weight of this delayed response versus the instantaneous electronic Kerr effect, and how linear dispersion modulates their interplay during propagation, was not well understood.
Propagation Model: UPPE Beyond the Paraxial Approximation
The propagation framework is built on the unidirectional pulse propagation equation (UPPE), derived directly from Maxwell’s equations without invoking the SVEA. Starting from the combined Maxwell-Ampere and Maxwell-Gauss equations and projecting onto the polarization direction, one obtains a scalar wave equation of the form (∂z2 + ∇⊥2)E + k2(ω)E = -μ0ω2P. The paraxial Forward Maxwell Equation (FME) follows from this by assuming |k| >> |k⊥|, which neglects the angular extent of the beam’s spatial spectrum. In the THz regime this assumption fails: because the spectral bandwidth and central wavenumber are comparable, low-frequency components of the pulse have significant transverse wavevector components that must be retained. The z-propagated UPPE resolves this by replacing the paraxial diffraction operator with its exact non-paraxial counterpart:
∂E/∂z = i√(k²(ω) – k⊥²) E + i[μ0ω²/(2√(k²(ω) – k⊥²))] P
The first term on the right captures dispersion and frequency-resolved diffraction jointly, while the second term carries the nonlinear polarization. Simulations were carried out using the gUPPEcore package with input pulses spanning 0 to 3 THz, pulse duration τ = 0.5 ps, and initial beam waist w0 = 1.5 cm.
Nonlinear Response: Instantaneous and Delayed Kerr
The nonlinear polarization is modeled following the Blow-Wood framework for transient stimulated Raman scattering, adapted to the THz regime. The total response function is written as g(t) = α δ(t) + (1 – α) gR(t), where the first term is the instantaneous electronic Kerr contribution and the second is the delayed phonon-driven Raman response. The Raman response function gR(ω) is taken as a single Lorentzian centered at the phonon resonance frequency ωR with linewidth δR. The mixing parameter α interpolates continuously between a purely instantaneous response (α = 1) and a purely delayed one (α = 0). Material-specific values of ωR and δR are extracted from the χ(3) dispersion curve computed via the two-level atom approximation using measured linear refractive indices and absorption coefficients. Two contrasting materials are studied: silicon (Si), which exhibits weak anomalous group velocity dispersion (GVD) across 0 to 3.5 THz, and zinc selenide (ZnSe), which has a strong normal GVD and a pronounced phonon resonance at approximately 6.6 THz. The GVD parameter of ZnSe is roughly 200 times larger than that of Si in the relevant frequency range, making them ideal test cases for isolating the role of dispersion.
Results: THz Electric Field and Kerr-Raman Time Delays
In Si, the pulse propagating under the instantaneous Kerr effect alone (α = 1, n2 = 10-15 m2/W) accumulates approximately 3.4 ps of time delay relative to the linear case, corresponding to a refractive index change of at least Δn = n2I = 0.01. Introducing a small Raman contribution (α = 0.9) adds a further time shift, reaching approximately 5 ps total. This additional delay grows slightly as the phonon response time increases from 10 to 100 fs, consistent with the longer temporal memory of the medium. In ZnSe, the stronger dispersion reduces these delays considerably: the Kerr contribution alone produces only 240 fs of shift, and the combined Kerr and Raman effect produces approximately 500 fs. The reduced delays in ZnSe relative to Si arise because the strong normal GVD suppresses the effective nonlinear phase accumulation by temporally spreading the pulse.
Self-Focusing and the GVD-SPM Interplay
The relative importance of GVD and self-phase modulation (SPM) during propagation is quantified through the ratio of the dispersion length LD = τ²/|β2| to the nonlinear length LNL = 1/(γP), where γ = n2ω0/(cAeff) is the nonlinear coefficient and P = 7.07 × 109 W is the peak pulse power. For Si, LD/LNL ≈ 520, meaning SPM strongly dominates over GVD and self-focusing is expected and observed: the initial Gaussian spatial profile breaks up into a strong self-focused trailing peak and a weak leading pulse. For ZnSe, LD/LNL ≈ 1, placing the system in a regime where GVD and SPM balance, and the spatio-temporal bending associated with self-focusing is entirely suppressed. Normal dispersion in ZnSe effectively compensates the nonlinear refraction, inhibiting catastrophic self-focusing. These results directly demonstrate that GVD acts as a stabilizing mechanism against spatial beam collapse in the THz nonlinear regime.
Frequency-Resolved Diffraction and Spectral Broadening
Due to the long wavelengths involved, THz beams are significantly more diffractive than optical beams, and different spectral components diffract at measurably different rates. In Si, spatio-spectral analysis reveals that higher frequency components localize closer to the beam axis while lower frequencies spread outward, producing a characteristic curvature in the spectral distribution along the transverse direction. Under nonlinear propagation, SPM generates new spectral content across the bandwidth, and this broadening is more pronounced for quasi-longitudinal rays near the beam axis than for off-axis rays. Notably, replacing 10% of the Kerr contribution with a Raman contribution (α = 0.9) reproduces the same degree of spectral broadening as the pure Kerr case, confirming that the delayed phonon response contributes to frequency generation on equal footing with the instantaneous electronic response at these parameters. In ZnSe, linear absorption suppresses the high-frequency tail of the spectrum during propagation, and the overall spectral broadening due to SPM is considerably weaker than in Si, again consistent with the GVD-dominated regime. The Kerr and Kerr-Raman cases in ZnSe produce identical spectral broadening, further supporting the equivalence of the two nonlinear contributions to frequency generation.
Significance
This work establishes a self-consistent simulation framework for nonlinear THz pulse propagation that captures non-paraxiality, dispersion, and the full time-varying Kerr response within a single model. The key physical finding is that phonon-driven delayed nonlinearity contributes comparably to the instantaneous electronic Kerr effect in terms of both temporal delay and spectral broadening, and that linear GVD is the primary factor controlling whether self-focusing occurs. The model provides a direct path to extracting n2 values from nonlinear THz-TDS measurements and to identifying candidate materials with large nonlinear refractive indices arising from proximity to phonon resonances, which is an increasingly relevant problem as high-field THz sources become more widely available.
Propagation of broadband THz pulses: effects of dispersion, diffraction and time-varying nonlinear refraction
Payman Rasekh, Mohammad Saliminabi, Murat Yildirim, Robert W. Boyd, Jean-Michel Menard, and Ksenia Dolgaleva
Optics Express, Vol. 28, No. 3, February 2020
https://doi.org/10.1364/OE.381150