Coherent synthesis of cycle-engineered terahertz (THz) field waveforms is demonstrated using two collinear femtosecond laser pulses with a tunable delay, undergoing optical rectification in a nonlinear crystal. The output THz waveforms are tailored with sub-quarter-cycle precision by varying the delay time between the laser pulses. A straightforward scaling of this concept to THz field synthesis with multiple driver pulses suggests ways toward the ultrahigh precision of THz waveform engineering and coherent enhancement to high peak powers in the THz range.

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The observation of magnetoresistance (MR) varying with the rotation of magnetization in the plane perpendicular to the electric current is an important discovery in spintronics in recent years. The famous conventional anisotropic MR (AMR) says that the resistance of a polycrystalline magnetic material must depend on magnetization component along the current direction only, thus cannot account for this newly observed unusual AMR (UAMR). This UAMR leads to the notion of the spin-Hall MR (SMR) in the famous SMR theory. However, the SMR theory may only explain UAMR observed in heavy-metal/magnetic-insulator bilayers, not other types of bilayers. Here, we present a two-vector theory that can explain not only all existing experiments on the unusual angular dependence of longitudinal and transverse resistivity when the magnetization rotates in three mutually perpendicular planes, but also how three amplitudes of MR angular oscillation are related to each other. The theory is very general and its correctness depends only on the assumption that the magnetization and interfacial field are the only vectors affecting electron transport besides of other scalar variables such as the temperatures and impurities. Experiments that can test this theory against the SMR theory are also proposed.