  Magnetic ordering, as one of the most important characteristics in magnetic
materials, could have significant influence on the band structure, spin
dependent transport, and other important properties of materials. Its
measurement, especially for the case of antiferromagnetic ordering, however, is
generally difficult to be achieved. Here we demonstrate the feasibility of
magnetic ordering detection using a noncontact and nondestructive optical
method. Taking the compressive strained tetragonal BiFeO3 (BFO) as an example
and combining density functional theory calculations with the minimal one-band
tight-binding models, we find that when BFO changes from C1-type
antiferromagnetic (AFM) phase to G-type AFM phase, the top of valance band
shifts from the Z point to {\Gamma} point, which makes the original direct band
gap become indirect. This can be explained by the two-center Slater-Koster
parameters using the Harrison approach. The impact of magnetic ordering on
energy band dispersion dramatically changes the optical properties of
tetragonal BFO. For the linear ones, the energy shift of the optical band gap
could be as large as 0.4 eV. As for the nonlinear ones, the change is even
larger. The second-harmonic generation coefficient d33 of G-AFM becomes more
than 13 times smaller than that of C1-type AFM case. Finally, we propose a
practical way to distin-guish the C1- and G-type AFM of BFO using the optical
method, which might be of great importance in next-generation information
storage technologies and widens the potential application of BFO to optical
switch.
