## 3 Results and Discussion  
The effect of the application of amplitude modulated AC magnetic field to the SP tip is shown in the spectrum of cantilever oscillation in Figure 2. Here the carrier frequency \(f_{c}\) is 100 MHz. In non-modulated magnetic field (Figure 2a), only a peak \(I(f_{0})\) is visible at \(f_{0}\), which is the oscillating frequency of the cantilever near its resonant frequency \(f_{r}(=\omega_{r}/2\pi)\). On the other hand, in amplitude-modulated magnetic field (Figure 2b), two sidebands \(I(f_{0}\pm f_{m})\), which are caused by the frequency modulation of cantilever oscillation, appear at \(f_{0}\pm f_{m}\) (Figure 2b). The intensity of the \(f_{0}\) oscillation does not change with and without amplitude modulation. In the figure, there is intensity difference between two sidebands. The asymmetry of sideband intensity comes from the difference of \(f_{0}\) and \(f_{r}\). To control the tip-sample distance during the imaging, we set the \(f_{0}\) as the frequency at which the change of oscillation amplitude against frequency has maximum and the order of the frequencies is \(f_{0}-f_{m}\) < \(f_{0}\) < \(f_{0}+f_{m}\) < \(f_{r}\). In this case, the intensity at \(f_{0}-f_{m}\) becomes smaller than the intensity at \(f_{0}+f_{m}\) (See inset in Figure 2). In this case, there is a mixture of frequency modulation and amplitude modulation in cantilever oscillation. Contribution ratio of the sideband spectra to frequency modulation and amplitude modulation are estimated as \(\frac{I(f_{0}+f_{m})+I(f_{0}-f_{m})}{2}\) and \(\frac{I(f_{0}+f_{m})-I(f_{0}-f_{m})}{2}\), respectively. In this experiment, frequency modulation mainly occurs in the cantilever oscillation. In the Figure 3, we can see the magnetic field energy images of PMR head from 100 kHz to 1 GHz. Here the lock-in signals (\(X+iY=R\exp(i\theta)\)) were optimized by the adjustment of lock-in reference phase offset \(\theta_{\text{offset}}\) to obtain maximal out-of-plane signal value \(Y_{\text{max}}=R\sin(\theta+\theta_{\text{offset}}=\pi/2)=R\) and to get only lock-in Y images [2]. When \(f_{c}\) approaches to 1 GHz, the magnetic field from the head decreases mainly due to the transmission line loss of applied current in the experimental setup. So, the intensity images in Figure 3a-e were normalized by the maximum signal corresponding to the central bright  
Figure 2: Spectra of cantilever oscillation in the 100 MHz magnetic field generated by writing head without (**a**) and with (**b**) amplitude modulation. Inset shows the scheme of the cantilever oscillation \(f_{0}\), resonance \(f_{r}\) and magnetic sideband \(f_{0}\pm f_{m}\) frequencies on amplitude frequency curve. spot, which imaged magnetic pole. The pole dimension in Figure 3a-e does not depend on the carrier frequency \(f_{c}\). Typical line profile is shown in a Figure 3f, where we can see the large positive magnetic signal peak from the main pole, while regions of negative signal near the peak sides correspond to dark areas around it in Figure 3a-e. The phenomena are explained by the rotation of in-plane magnetization near the PMR head gap [33,34]. The phenomenon is similar to the case of the magnetic energy of low frequency sinusoidal magnetic field \(\mathbf{H}=\mathbf{H}_{0}\cos(\omega_{m}t)\)[5]. Here the sign of magnetic signal indicates the increase or decrease in the magnetic field energy, which is proportional to \(H^{2}\), in the period of \(1/f_{m}\). Figure 3g is the schematic image of main pole area. Here we assumed that the head magnetization rotated in the (\(XZ\)) plane, which is normal to head surface (the \(XY\) plane). On the head surface, the applied amplitude modulated magnetic field \(\mathbf{H}\) generates the magnetic charge density \(\sigma_{m}^{z}\), from the normal component of main-pole magnetization \(M_{z}(=\mathbf{M}\cdot\mathbf{n})\). Here we assume the magnetic hard axis is parallel to the z direction.