  Existing research on extremely large-scale intelligent reflecting surface
(XL-IRS) beam training has assumed the far-field channel model for base station
(BS)-IRS link. However, this approach may cause degraded beam training
performance in practice due to the near-field channel model of the BS-IRS link.
To address this issue, we propose two efficient schemes to optimize BS
beamforming for improving the XL-IRS beam training performance. Specifically,
the first scheme aims to maximize total received signal power on the XL-IRS,
which generalizes the existing angle based BS beamforming design and can be
resolved using the singular value decomposition (SVD) method. The second scheme
aims to maximize the $\ell_1$-norm of incident signals on the XL-IRS, which is
shown to achieve the maximum received power at the user. To solve the
non-convex $\ell_1$-norm maximization problem, we propose an eficient algorithm
by using the alternating optimization (AO) technique. Numerical results show
that the proposed AO based BS beamforming design outperforms the SVD/angle
based BS beamforming in terms of training accuracy and achievable received
signal-to-noise ratio (SNR).
