###### Abstract  
Recent progress in the development of high-frequency (HF) magnetic materials and devices requires the HF imaging of magnetic field and magnetic field response with high spatial resolution. In this work, we proposed HF Alternating Magnetic Force Microscopy (A-MFM) with a superparamagnetic tip to image magnetic field energy near the sample surface. Magnetic field with a frequency from 100 kHz to 1 GHz was emanated by a writing head used for perpendicular magnetic recording. Applied HF magnetic field is amplitude modulated, and magnetic energy determines the frequency modulation of cantilever oscillation in the framework of A-MFM. The frequency modulated oscillation was caused by low frequency alternating magnetic force  
superposition of all nanoparticle contributions. In the SP materials, the blocking resonance frequency can exceed the ferromagnetic resonance frequency, which is caused by the intrinsic magnetocrystalline anisotropy. So, with the particle size decrease, SP susceptibility may be kept in GHz range [7, 8, 9, 10, 11]. HF-imaging is required to characterize HF magnetic materials and devices, such as perpendicular magnetic recording (PMR) systems. They operate in high frequencies up to GHz range to enable fast data processing [12], as well as to decrease written track width [13] for bit miniaturization. The writing currents also need to be low enough to decrease power consumption, so, PMR heads were recently augmented by heat-assisted (HAMR) [14] and microwave-assisted (MAMR) [15, 16] technologies. Thus, magnetic fields induced by the modern PMR writing heads, are highly localized spatially, and become weaker apart from the head main pole. Thus, the imaging of spatial magnetic field distribution near magnetic pole requires methods with high resolution and sensitivity. For magneto-optical Kerr microscopy [17], as well as for magnetoresistive scanning [18, 19], the spatial resolution is limited by optical wavelength or sensor dimension, respectively. Electron holography [20, 21] and x-ray magnetic circular dichroism (XMCD) [22] require complex instrumentation and provide spatial resolution of 100 nm and higher, while analysis of pole footprint on magnetic media [23] is an indirect method. Conventional magnetic force microscopy (MFM) and related methods have high potential in magnetic head imaging [24, 25, 26, 27, 28]. HF-MFM was successfully applied to PMR heads to measure stray fields emanated from magnetic pole [26, 27, 28]. Moreover, MFM also imaged the localized HF field up to 40 GHz, generated by spin Hall nano oscillator (SHNO) [29]. However, in HF applications it is difficult to separate magnetic and non-magnetic forces. Therefore, the MFM scan is performed significantly above sample surface at the distance comparable to pole dimension, which distorts the observed magnetic spot when compared to the topographical size of the pole. In the present study, we examined the possibility of HF magnetic imaging by A-MFM using SP tip from the viewpoint of high-frequency magnetic response of the SP tip. The scheme is shown in Figure 1. The external AC field causes periodical change of effective spring constant of cantilever, leading to frequency modulation of cantilever oscillation, which is demodulated and then detected by lock-in amplifier. This allows to perform scan near surface and eliminate non-magnetic interactions. The amplitude modulated AC magnetic field \(\mathbf{H}\left(=\mathbf{H}_{0}(1+\alpha\cos(\omega_{m}t))\cos(\omega_{c}t)\right)\) was induced in PMR writing head by applying amplitude modulated AC voltage. Magnetic  
Figure 1: The schematic diagram of high frequency \(\omega_{c}\) magnetic field imaging by using amplitude modulation with \(f_{m}(=\omega_{m}/2\pi)\) = 89 Hz, in HF MFM system with SP tip. The difference from the works [1, 4, 5] is the use of amplitude modulated HF current in the PMR head.