By solving self-consistently an effective Hamiltonian including interactions for both antiferromagnetic spin-density wave (SDW) and d-wave superconducting (DSC) orderings, a comparison study is made for the local magnetic structure around superconducting vortices and unitary impurities. To represent the optimally doped regime of cuprates, the parameter values are chosen such that the DSC is dominant while the SDW is vanishingly small. We show that when vortices are introduced into the superconductor, an oscillating SDW is induced around them. The oscillation period of the SDW is microscopically found, consistent with experiments, to be eight lattice constants (8a0). The associated charge-density wave (CDW) oscillates with a period of one half (4a0) of the SDW. In the case of unitary impurities, we find a SDW modulation with identical periodicity, however without an associated CDW. We propose neutron scattering experiments to test this prediction.PACS numbers: 74.62.Dh The common feature of all high-transition-temperature (high-T c ) cuprates is the proximity between antiferromagnetic (AF) and d-wave superconducting (DSC) phases controlled by the doping-At low temperatures, the AF order of the parent compounds is replaced by a DSC order upon doping. The main theme of high-T c superconducivity research centers on how to establish the connection between these two kinds of orderings, i.e., whether they exclude each other, or coexist microscopically. In one of the strategies, the interplay between the AF and DSC ordering is explored by weakening the superconductivity in the optimally or slightly overdoped regime. The inelastic neutron scattering (INS) measurements on both optimally doped and underdoped La 2−x Sr x CuO 4 (LSCO) samples by Lake and co-workers provide the first evidence of the field-induced magnetic order [1,2]. The field-induced spin fluctuations have a spatial periodicity of 8a 0 with the wave vector pointing along the Cu-O bond directions. The principle magnetization oscillations in the vortex state are coherent over a distance L M > 20a 0 , substantially longer than the superconducting coherence length ξ 0 (∼ 5a 0 ). The field-induced enhancement of the Bragg peak intensity was also observed by Khaykovich et al. [3] in the elastic neutron scattering (ENS) measurement on a related material, La 2 CuO 4+y (LCO), but with L M > 100a 0 , indicating the existence of the field-induced static AF order of 8a 0 periodicity. Strong AF fluctuations have also been observed by Mitrovic et al.[4] in a high-field nuclear magnetic resonance (NMR) imaging experiment on nearoptimally doped YBa 2 Cu 3 O 7−x (YBCO). More recently, the scanning tunneling microscopy imaging by Hoffman et al.[5] has revealed the quasiparticle states around the vortex cores in slightly overdoped Bi 2 Sr 2 CaCu 2 O 8+x (BSCCO) a Cu-O bond-oriented "checkboard" pattern with 4a 0 periodicity. The periodicity (4a 0 ) of charge modulation is one half of that (8a 0 ) of the field-induced SDW modulation. Theoretically, Demler and co-workers, ...