One of the most controversial phenomena in nonlinear dynamics is the reappearance of initial conditions. Celebrated as the Fermi-Pasta-Ulam-Tsingou problem, the attempt to understand how these recurrences form during the complex evolution that leads to equilibrium has deeply influenced the entire development of nonlinear science. The enigma is rendered even more intriguing by the fact that integrable models predict recurrence as exact solutions, but the difficulties involved in upholding integrability for a sufficiently long dynamic has not allowed a quantitative experimental validation. In natural processes, coupling with the environment rapidly leads to thermalization, and finding nonlinear multimodal systems presenting multiple returns is a long-standing open challenge. Here, we report the observation of more than three Fermi-Pasta-Ulam-Tsingou recurrences for nonlinear optical spatial waves and demonstrate the control of the recurrent behavior through the phase and amplitude of the initial field. The recurrence period and phase shift are found to be in remarkable agreement with the exact recurrent solution of the nonlinear Schrödinger equation, while the recurrent behavior disappears as integrability is lost. These results identify the origin of the recurrence in the integrability of the underlying dynamics and allow us to achieve one of the basic aspirations of nonlinear dynamics: the reconstruction, after several return cycles, of the exact initial condition of the system, ultimately proving that the complex evolution can be accurately predicted in experimental conditions.

Terahertz imaging has the potential to identify and decipher portions of ancient manuscripts, which may be unreadable at infrared and visible wavelengths. We use a scanning confocal terahertz microscope to scan a medieval parchment with music notes and pentagrams written with different inks. The microscope is based on a continuous-wave solid-state source at 0.3 THz, emitting in the free space with a horn antenna, and a high numerical-aperture ellipsoidal reflector. We present terahertz images with diffraction-limited lateral resolution of approximately 0.5 mm, where the different inks all give similar high contrast. Symbols written on the 'verso" side of the parchment, barely glimpsed in the near-infrared photograph, leave a clear imprint in the terahertz images. Artifacts due to imperfect flatness of the parchment are also briefly discussed

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Nonlinear response in a material increases with its index of refraction as n4. Commonly, n ~ 1 so that diffraction, dispersion, and chromatic walk-off limit nonlinear scattering. Ferroelectric crystals with a periodic 3D polarization structure overcome some of these constraints through versatile Cherenkov and quasi-phase-matching mechanisms. Three-dimensional self-structuring can also lead to a giant optical refraction. Here, we perform second-harmonic-generation experiments in KTN:Li in conditions of giant broadband refraction. Enhanced response causes wavelength conversion to occur in the form of bulk Cherenkov radiation without diffraction and chromatic walk-off, even in the presence of strong wave-vector mismatch and highly focused beams. The process occurs with a wide spectral acceptance of more than 100 nm in the near infrared spectrum, an ultra-wide angular acceptance of up to ±40∘, with no polarization selectivity, and can be tuned to allow bulk supercontinuum generation. Results pave the way to highly efficient and adaptable nonlinear optical devices with the promise of single-photon-to-single-photon nonlinear optics.

We study theoretically and experimentally the propagation of optical solitons in a lattice nonlinearity, a periodic pattern that both affects and is strongly affected by the wave. Observations are carried out using spatial photorefractive solitons in a volume microstructured crystal with a built-in oscillating low-frequency dielectric constant. The pattern causes an oscillating electro-optic response that induces a periodic optical nonlinearity. On-axis results in potassium-lithium-tantalate-niobate indicate the appearance of effective continuous saturated-Kerr solitons, where all spatial traces of the lattice vanish, independently of the ratio between beam width and lattice constant. Decoupling the lattice nonlinearity allows the detection of discrete delocalized and localized light distributions, demonstrating that the continuous solitons form out of the combined compensation of diffraction and of the underlying periodic volume pattern.