## 3 Results and Discussion
In the M-H curve in the hard axis direction, normal component of magnetization \(M_{z}\) is proportional to the applied magnetic field \(\mathbf{H}\). Then \(\sigma_{m}^{z}\) is given by the expression \(\sigma_{m}^{z}=M_{z}=M_{0}(1+\alpha\cos(\omega_{m}t))\cos(\omega_{c}t))\). On the other hand, the magnetic charge density on the cross-sectional surface of the main pole \(\sigma_{m}^{x}\) was generated by the parallel component of main-pole magnetization \(M_{x}\). In  
Figure 3: Normalized A-MFM image of the out-of-plane signal near the main pole for \(f_{c}\) = 0.1 (**a**), 1 (**b**), 10 (**c**), 100 (**d**) and 1000 (**e**) MHz currents with amplitude modulation at \(f_{m}\) = 89 Hz. The horizontal dimension of the images is 1 \(\upmu\)m. The topographical shape of the pole is marked with blue lines. (**f**) Line profile (white line in (**d**)) for \(f_{c}\) = 100 MHz. (**g**) Schematic model for the \(XZ\) cross-section of the main pole with magnetization vector M, its projections, and charge density \(\sigma\). magnetic rotation process in soft ferromagnet, the absolute value of magnetization is not changed, then \(M_{x}\) is obtained as following:  
\[M_{x}=\sqrt{M^{2}-M_{z}^{2}}=\sqrt{M^{2}-M_{0}^{2}(1+\alpha\cos(\omega_{m}t))^{2 }\cos^{2}(\omega_{c}t)} \tag{3}\]  
Magnetic field energy at the main pole surface and main pole edge is proportional to \((\sigma_{m}^{z})^{2}\) and \((\sigma_{m}^{x})^{2}\), respectively. Then, the following relationships are obtained:  
\[(\sigma_{m}^{z})^{2}=M_{0}^{2}(1+\alpha\cos(\omega_{m}t))^{2}\cos^{2}(\omega_{ c}t) \tag{4}\]  
\[(\sigma_{m}^{x})^{2}=M^{2}-M_{0}^{2}(1+\alpha\cos(\omega_{m}t))^{2}\cos^{2}( \omega_{c}t) \tag{5}\]  
The \(\omega_{m}\) frequency components from the above equation are expressed as following:  
\[(\sigma_{m}^{z})^{2}[\omega_{m}t]=\alpha M_{0}^{2}\cos(\omega_{m}t) \tag{6}\]  
\[(\sigma_{m}^{x})^{2}[\omega_{m}t]=-\alpha M_{0}^{2}\cos(\omega_{m}t)=\alpha M _{0}^{2}\cos(\omega_{m}t+\pi) \tag{7}\]  
Thus, \((\sigma_{m}^{z})^{2}\) and \((\sigma_{m}^{x})^{2}\) have opposite polarity. When the perpendicular component of the main pole magnetization increases, the parallel one, which generates magnetic charge at the cross-sectional surface of main pole, decreases. Thus, when the \(H^{2}\) increases at the main pole location, the \(H^{2}\) decreases near the pole edge. The source and the drain areas of magnetic field energy change alternately during the rotation of main pole magnetization. With the increasing carrier frequency \(f_{c}\) approaching the ferromagnetic resonance frequency of main pole materials, magnetic rotation process seems to change from in-plane motion to precessional motion. In the precessional motion, precession angle changes with low modulation frequency \(f_{m}\). When the magnetization \(M_{\parallel}\) parallel to the precession axis increases, the magnetization \(M_{\perp}\) normal to the precession axis decreases. So \(M_{\parallel}^{2}(=(\sigma_{m}^{z})^{2})\) and \(M_{\perp}^{2}(=(\sigma_{m}^{x})^{2})\) have opposite polarity in the similar way to in-plane magnetic rotation. When \(\alpha=0\) in the previous work [5], \((\sigma_{m}^{z})^{2}[2\omega_{c}t]=\left(\frac{M_{0}^{2}}{2}\right)\cos(2 \omega_{c}t)\) and \((\sigma_{m}^{x})^{2}[2\omega_{c}t]\)\(=-\left(\frac{M_{0}^{2}}{2}\right)\cos(2\omega_{c}t)\) for low \(\omega_{c}\) are obtained. To evaluate the spatial resolution of the present method, the magnetic size of the pole was compared with the topographic one, which is marked with blue lines in Figure 3a for comparison. Its size fits that of bright magnetic spot, which means that the magnetic pole is imaged close to surface and with sufficient resolution. Full width at half maximum (FWHM) stays at the level of \(\approx\)120 nm, which is larger than in the previous works [4,5] because the size of the used head is larger than that of the previous one.