  Ultra-massive multiple-input multiple-output (UM-MIMO) is the enabler of
Terahertz (THz) communications in next-generation wireless networks. In THz
UM-MIMO systems, a new paradigm of cross-field communications spanning from
near-field to far-field is emerging, since the near-field range expands with
higher frequencies and larger array apertures. Precise beam alignment in
cross-field is critical but challenging. Specifically, unlike far-field beams
that rely only on the angle domain, the incorporation of dual-domain (angle and
distance) training significantly increases overhead. A natural question arises
of whether far-field beam training can be deployed for cross-field beam
alignment. In this paper, this question is answered, by demonstrating that the
far-field training enables sufficient signal-to-noise ratio (SNR) in both far-
and near-field scenarios, while exciting all channel dimensions. Based on that,
we propose a subarray-coordinated hierarchical (SCH) training with greatly
reduced overhead. To further obtain high-precision beam designs, we propose a
two-phase angle and distance beam estimator (TPBE). Extensive simulations
demonstrate the effectiveness of the proposed methods. Compared to near-field
exhaustive search, the SCH possesses 0.2\% training overhead. The TPBE achieves
0.01~degrees and 0.02~m estimation root-mean-squared errors for angle and
distance. Furthermore, with the estimated beam directions, a near-optimal SNR
with 0.11~dB deviation is attained after beam alignment.
