Structured 3D linear space-time light bullets by nonlocal nanophotonics

The paper addresses the challenge of synthesizing propagation-invariant 3D optical wave packets (light bullets) in free space that remain localized in three spatial dimensions and in time. Such pulses require a specific space-time coupling relating temporal and spatial frequency components, typically of the form ω = v_g k + b, to eliminate diffraction and dispersion. Prior free-space methods have generated restricted classes (e.g., X-waves with b = 0, superluminal v_g, and focus wave modes with v_g = c, b ≠ 0), but a general route to 3D light bullets with arbitrary, especially subluminal, group velocities has been lacking. The authors propose using a single passive nonlocal nanophotonic surface with engineered wavevector-dependent transfer functions to imprint the required space-time coupling onto an incident Gaussian wave packet, enabling 3D linear light bullets with controllable group velocity and propagation distance, and with tunable internal structure such as SAM and OAM. This approach eliminates the need for bulky pulse-shaping and spatial-beam modulation setups.

The authors review free-space propagation-invariant wave packets: X-waves (b = 0, v_g > c) and focus wave modes (v_g = c, b ≠ 0). Recent work by Abouraddy et al. synthesized 2D space-time light sheets with arbitrary group velocity by combining spatial-beam modulation and ultrafast pulse shaping; however, this method cannot directly extend to 3D light bullets because it consumes an extra spatial dimension to spread the temporal spectrum. Li et al. proposed Gauss–Bessel pulses with conical-pulse-front pre-deformation to adjust group velocity, but the resulting wave packets vary significantly during propagation. Thus, a general, compact technique to synthesize 3D, propagation-invariant light bullets with controllable v_g, particularly for v_g < c, has remained an open problem. Nonlocal nanophotonics, with wavevector-dependent transfer functions, have enabled operations like optical differentiation, image filtering, and squeezing free space, suggesting a pathway to enforce the required space-time coupling for light bullets.

The authors formulate the generation of 3D linear light bullets using a single ultrathin periodic nanophotonic layer supporting guided resonances. A scalar optical field U(x, y, t, z) with envelope A(x, y, t, z) propagates in free space via a spatiotemporally shift-invariant linear map on its spectrum A(kx, ky, Ω, 0). Light bullets require a spatiotemporal spectrum A_s(kx, ky, Ω) constrained by a delta function Ω = v_g[k(kx, ky, Ω) − k0], ensuring rigid propagation of the envelope at group velocity v_g. In practice, finite-energy light bullets approximate this delta constraint, resulting in long but finite propagation with gradual intensity decay. In the paraxial regime (k_⊥^2 ≪ k0^2), the required space-time coupling simplifies to a quadratic relationship Ω ≈ (ω0/c)(β/k0) k_⊥^2, where β determines v_g. The central idea is to implement a nonlocal wavevector-dependent narrowband bandpass filter that enforces this quadratic dispersion using a guided-resonance band in a photonic crystal slab. Near resonance, the reflection amplitude is modeled as r(k, ω) = r_a + f(ω − ω_r(k))/γ(k), where f = r_a + t_a, ω_r(k) is the guided-resonance frequency, and γ(k) its radiative linewidth. To realize the desired space-time coupling, the device should satisfy: (1) negligible background reflection (r_a ≈ 0) to yield a Lorentzian lineshape, (2) narrow linewidth γ(k) such that the Lorentzian approximates a delta function in frequency at each wavevector, and (3) isotropic quadratic band dispersion ω_r(k_⊥) ≈ ω0 + α k_⊥^2 with α set to match the target β. An illustrative design considers a 2D photonic crystal slab (lattice constant a = 1 μm) supporting a single guided-resonance band with ω0 = 1.0 × 2πc/a, β = −2.0, and γ(k_⊥) = 1.67 × 10^−6 × 2πc/a. A Gaussian incident pulse (center wavelength λ_c = 1 μm, waist W0 = 30 μm, temporal width τ0 = 5 ps, Rayleigh range z_R = 2.83 mm) impinges on the slab. In reflection, the nonlocal, narrowband resonance filters the spectrum onto the desired conic section in (kx, ky, ω) space, imprinting the space-time coupling. The authors simulate free-space propagation of the reflected pulse at distances up to 100 mm and beyond in the moving frame r = t − z/c, verifying rigid, diffraction-free propagation at the designed v_g. They further analyze how the real part (dispersion α, via β) and imaginary part (linewidth γ) of the guided-resonance band independently control v_g and the maximum propagation distance L_max, respectively.

• A single passive nonlocal nanophotonic surface (photonic crystal slab with guided resonance) can generate 3D linear light bullets in free space by enforcing the required space-time coupling through a wavevector-dependent, narrowband transfer function.
• Example device parameters: lattice constant a = 1 μm; guided-resonance center frequency ω0 = 1.0 × 2πc/a; linewidth γ = 1.67 × 10^−6 × 2πc/a; designed β = −2.0 (paraxial quadratic dispersion), yielding v_g = 0.8 c.
• Incident Gaussian pulse: center wavelength λ_c = 1 μm; waist W0 = 30 μm; temporal width τ0 = 5 ps (τ0 = 1500 a/c); Rayleigh range z_R = 2.83 mm.
• Simulations show the reflected pulse remains shape-invariant at z = 50 mm and z = 100 mm, with rigid propagation at v_g = 0.8 c, and gradual intensity decay due to finite-energy approximation.
• Tunable group velocity: by varying β (through band curvature α), achieved v_g/c = 0.9, 0.8, 0.7 for β = −4.50, −2.00, −1.17, respectively, while maintaining propagation invariance.
• Propagation distance is set by the resonance linewidth: the peak-intensity decay follows an exponential with maximum propagation distance (to 1/e^2 drop) L_max = γ/γ_c (per referenced derivation). For the example (v_g = 0.8 c, γ as above, a = 1 μm), L_max = 382 mm ≈ 135 z_R, confirmed by numerical simulation via linear fit of log-peak intensity vs distance.
• The approach offers independent control of external degrees of freedom (v_g via dispersion α; L_max via linewidth γ) and promises control of internal degrees (SAM, OAM) via wavevector-dependent modal field distributions of the guided resonance.

The study demonstrates that nonlocal nanophotonics provide a compact and versatile route to realize the specific frequency–wavevector correlation required for 3D linear light bullets. By engineering the guided-resonance dispersion to be isotropic and quadratic, the device imprints the paraxial space-time coupling onto an incident Gaussian pulse, enabling diffraction- and dispersion-free propagation at a designed group velocity, including subluminal values. The narrowband character of the resonance approximates the ideal delta-function constraint sufficiently to support long-distance propagation, with the decay rate governed by the linewidth. These findings directly address the challenge of synthesizing general 3D light bullets without bulky pulse shapers or consuming extra spatial dimensions, extending beyond prior 2D light-sheet and special-case wave packets. The demonstrated independent control over group velocity and propagation distance, alongside the potential to tailor internal SAM/OAM through mode profiles, underscores the method’s relevance for applications in imaging, communications, and quantum technologies.

The paper introduces a single-layer, passive nonlocal nanophotonic platform to generate 3D linear space-time light bullets in free space. Using guided resonances in a photonic crystal slab with engineered isotropic quadratic dispersion, the device enforces the necessary space-time coupling, producing shape-invariant pulses with controllable group velocity and long propagation distances. Numerical examples show v_g tunable between 0.7 c and 0.9 c and L_max on the order of hundreds of millimeters, far exceeding the Rayleigh range of comparable Gaussian beams. The approach is compact, versatile, and compatible with additional control of internal degrees of freedom (SAM, OAM). Future work could include experimental realization, dynamic tuning of dispersion and linewidth, extension beyond the paraxial regime, and explicit demonstrations of SAM/OAM sculpting in generated light bullets.

• Ideal light bullets with exact delta-function space-time coupling are unattainable in finite systems; practical implementations yield approximate coupling, leading to finite energy and exponential decay of peak intensity over distance.
• The presented results are theoretical and numerical; experimental validation is not included in the provided text.
• The method relies on narrowband guided resonances with isotropic quadratic dispersion and negligible background reflection; fabrication tolerances and material losses may limit achievable linewidths, isotropy, and background suppression.
• The demonstrated control over internal degrees of freedom (SAM, OAM) is proposed in principle but not detailed with quantitative examples in the provided excerpt.