Optical spin-wave detection beyond the diffraction limit

Within magnetism, spin waves are ubiquitous in their presence and applications. Spin waves are fundamental excitations in the magnetization of a material and their behavior is governed by fundamental magnetic interactions, such as the anisotropy or exchange interaction. Based on this dependence, spin waves are often used to probe these fundamental magnetic interactions.^1–4 In recent years, it has been suggested that spin waves can also be used for novel computing methods. Spin-wave propagation occurs without charge transport, which allows for computation without Ohmic losses.^5,6 Moreover, spin waves also make excellent candidates for interference based logic devices and non-linear wave computing.^5,6

For applications, short-wavelength (⁠$<100nm$⁠) spin waves are preferred to reduce the device footprint and increase the group velocity.⁷ The conventional method of exciting and detecting spin waves is based on micrometer-sized microwave antennas through Oersted fields and magnetic induction.⁵ Scaling these antennas down such that short wavelength spin waves can be excited and detected (⁠$<100nm$⁠), however, is extremely challenging because of the impedance mismatch.

On the excitation side, several alternatives have been proposed to excite short-wavelength spin waves,⁸ such as spin-transfer torque based methods,⁹ grating-like couplers,^10,11 and using the resonance of antiferromagnetically coupled vortex states.¹² However, when it comes to the detection of short-wavelength spin waves, there are considerably fewer alternatives. The aforementioned grating couplers can also be used to detect the spin waves. Spin-pumping¹³ and spin-caloritronics¹⁴ based techniques do allow for short wavelength detection, but they are not wavelength-selective. When using a spectroscopic light-scattering approach, such as Brillouin light scattering, the minimum detectable spin-wave wavelength is dependent on the laser wavelength, unless near-field techniques are used, or samples with reduced translational symmetry are used.^15–17 Other optical detection techniques suffer from the diffraction limit, while x-ray based techniques have the required resolution but require large scale facilities.¹² In this article, we, therefore, demonstrate a grating-based magneto-optical (MO) method to circumvent the diffraction limit when using optical detection of spin waves. It is very much related to the field of near-field optics (see Ref. 18, and references therein) but simpler to implement than the techniques commonly used to reach super resolution for MO measurements.¹⁷ 

In Fig. 1(a), we illustrate the method that we demonstrate in this article. A metallic grating is placed on top of a magnetic strip with a spin wave. The incident laser light is partially reflected by this grating. However, the light that is transmitted—and is subsequently reflected off the magnetic strip—carries information on the direction of the magnetization through a rotation of its polarization as a result of the magneto-optical Kerr effect (MOKE). Any spin wave whose periodicity matches that of the grating can be detected because the grating acts as a Fourier filter for the magnetization components in the strip. This also allows us to do studies as a function of position by moving the laser spot. By adjusting the relative phase between the probing laser pulses and the excitation current we can achieve phase sensitivity as well. Moreover, this technique will work even when the grating periodicity is smaller than the diffraction limit, although complex scattering effects might need to be taken into account. In the remainder of this article, we start by demonstrating spin-wave excitation through conventional meandered microwave antennas.¹⁹ We then move on to actual optical propagating spin-wave transmission measurements using metallic grating. In the last part, we focus on an understanding of the measured spectra using a simple analytical model.

We fabricate devices such as the one shown in Fig. 1(b). Here, we show a meandering spin-wave antenna located between two gratings. This device allows us to excite spin waves with specific wavevectors belonging to the main periodicities of the spin-wave antenna (⁠$km$ and a small secondary periodicity $ks=0.36×km$⁠),¹⁹ and measure the spin waves by focusing a laser on the grating. The magnetic strip underneath the grating consists of $20nm$ of Ni$80$Fe$20$ (Py). Spin waves are measured in the Damon–Eshbach geometry, with the magnetic field and wave vector perpendicular and in the plane. The electrical characteristics of the spin-wave excitation were measured using a vector network analyzer following a procedure described elsewhere.²⁰ The optical detection of spin waves was performed using a pulsed laser (⁠$80MHz$⁠, pulse length $150fs$ at a wavelength of $780nm$⁠) with a diffraction-limited spot size of $10μm∼10μm$ measuring the Kerr rotation in combination with continuous wave excitation of the spin waves such that measured magneto-optical (MO) signal is proportional to the spin-wave amplitude at the given phase (the details on the experimental setup and sample fabrication can be found in supplementary material note I). Different devices were measured with differing combinations of gratings and spin-wave antennas resonant to spin waves with wavevectors $k=5.0,7.0,9.0μm−1$⁠, which have wavelengths smaller than our laser spot size and which have long enough attenuation lengths to measure propagating spin waves away from the antenna.

To demonstrate spin-wave excitation, we show the self-induction $L$ of a $km=9μm−1$ antenna in Fig. 2(a). In this spectrum, two resonances are observed at $60$ and $100mT$⁠, which correspond to the excitation of the spin waves with wavelengths equal to the two main periodicities of the antenna (⁠$km,s=9.0,3.2μm−1$⁠).^19,20 The real and imaginary parts of the spectrum are fitted simultaneously with symmetric and anti-symmetric Lorentzian line shapes to extract the resonance fields (solid lines), which are plotted in Fig. 1(b) as a function of frequency $f$⁠. These resonance fields are then fitted with the dispersion relation for these spin waves (using $MS=0.83MA.m−1$ and $g=2.11$⁠),^21–24 which results in $Meff=0.76±0.07MA.m−1$ and thickness $t=16±5nm$⁠, in line with what we expect for Py.²² 

With the verification of spin-wave excitation complete, we move on to the optical measurement of the spin waves. A typical measurement for a device tuned to $km=9μm−1$ is displayed in Fig. 2(c), which contains the spin-wave amplitude at two different phases (0° and 90°) with respect to the microwave excitation source.²⁵ In this measurement, there is a resonance at $∼55mT$⁠, the resonance field of the $km=9μm−1$ spin wave. Furthermore, there is a small resonance at $∼100mT$⁠, which is the result of the large wavelength (⁠$ks$⁠) spin waves that are also excited. A second feature in the spectrum is the amplitude difference between the resonances at positive and negative fields, which is a well-known effect of spin-wave excitation using electrical antennas. It is the result of a difference in spin-wave excitation efficiency because of the magnetic field chirality that either matches or opposes the chirality of the propagating spin wave at positive or negative magnetic fields.²⁶ This is additional confirmation that the grating allows us to detect propagating spin waves with wavelengths of $∼700nm$ (⁠$km=9μm−1$⁠) using a $10μm$ laser spot.

We can also demonstrate the $k$-selectivity of the grating, which should act as a Fourier filter and be sensitive to only those spin-wave wavelengths that match the grating. This property is illustrated in Fig. 2(d), which contains measurements with a $km=9μm−1$ antenna and different gratings designed to detect $k=9,5μm−1$ spin waves and a device without a grating. As a function of the grating $k$-value, the resonance field of the measured spin-wave resonance increases, because the gratings are resonant to different spin waves. The predicted resonance fields for the corresponding gratings are indicated in the figure with the solid vertical lines and agree perfectly with the measured resonances. Although the antenna selectively excites spin waves with $km=9μm−1$⁠, there is still a finite excitation efficiency for spin waves with different $k$-vectors. These are measured with gratings that are tuned to different wavelength spin waves and demonstrate the extreme $k$-selectivity of the grating. Additionally, for the measurement without a grating, there is no resonance at the $km$ peak. This proves that we do indeed need the grating to detect spin waves with wavelengths smaller than the diffraction limit. The large peak visible in the measurement without a grating will be discussed in more detail later in this article.

As mentioned in the introduction, an additional benefit of this technique is the possibility to do spatially dependent measurements of the spin waves. This is demonstrated in Fig. 2(e), where several measurements are plotted with the laser spot focused on different positions along the device. There are several interesting aspects here. First, as we move away from the antenna, the propagating spin waves attenuate and resonance amplitudes decrease on a typical length scale of $5−10μm$⁠, as expected from the spin-wave attenuation length.^27,28 Second, the amplitude asymmetry between the spin waves at positive and negative magnetic fields reverses as we move to the other side of the antenna. This is completely in line with the chirality of the excitation mechanism coupled with the type of spin waves that are excited.²⁶ Last, if the laser spot is focused on or near the antenna, the spectrum becomes more complex as extra resonances are present. It is the result of a modified spatial filtering of the spin waves because the laser spot is (partially) filtered by the spin-wave antenna rather than the grating. In supplementary material note II, we show additional measurements for a $km=5μm−1$ device, which shows similar behavior. To summarize the first part of this article, we have shown experimentally that the grating technique allows us to measure propagating spin waves with a wavelength of $700nm$ using a spot size of $10μm$⁠. This method enables both wavelength-selective and spatially dependent measurements.

After the experimental demonstration, we now continue by establishing a more fundamental understanding of the spectra. A simple interpretation of the MO signal is given in Fig. 1(a), which suggests the MO signal is determined by a spatial averaging of the filtered laser spot intensity multiplied by the complete spectrum of excited spin waves. In supplementary material note IV, we describe how we can calculate the excited spin waves, the filtering of the laser spot, and the eventual MO signal using a simple model based on the interpretation given in Fig. 1(a). This model is fitted to the measurement of Fig. 2(c), as visible in Fig. 3(a).²⁹ The model is fitted for positive fields only, but is also plotted at negative magnetic fields.

We find an overall agreement between the measurements and the calculations in Fig. 3(a). At positive fields, we find a sharp intense peak around the resonance field (solid line) expected from the $km=9μm−1$ antenna. However, the agreement with the small resonance around the $ks$ peak (solid line at $∼95mT$⁠) is not as good. Although the model does produce a small resonance around this field, the exact positioning, phase and amplitude do not line up with the experimentally measured spectrum. At negative fields, the agreement between the measurement and the model is less accurate: there is a significant mismatch in signal amplitude. We do expect a large field asymmetry due to the chirality in the excitation field that either matches or opposes the spin-wave chirality for positive or negative fields. However, the measured amplitude difference between positive and negative fields (⁠$∼4$⁠) is very large compared to other electrical and optical measurements (⁠$∼2$⁠), which agree with our theoretical calculations.^28,30

The model can also reproduce the $k$-selectivity of the grating qualitatively as illustrated in Fig. 3(b), where the calculated theoretical curves for several different gratings parameters are plotted. As the grating $k$-value goes down, the main resonance moves up to higher magnetic fields. Moreover, the intensity decreases as only the top MO curve has a grating matched to the main excited spin wave. Additionally, we find that the position of these resonances for the $k=5$ and $k=9μm−1$ grating match very closely to what is expected based on the dispersion relationship (solid vertical lines) and what is observed experimentally [Fig. 2(d)]. We understand these spectra by realizing that the antenna excites a wide range of different-wave vector spin waves, which we visualize as the excitation efficiency with the gray shaded area in Fig. 3(b). Even though the grating is not matched to the spin-wave antenna, there is still a small amount of spin waves that are excited which have wavelengths matched to the grating.

Yet, the spectrum for the case without a grating differs from the experimental situation [2(d)]. In the experiments, there is a resonance at approximately the uniform FMR mode. The calculations, however, show that this resonance should not be excited by the spin-wave antenna. Something similar is also observed around $−120mT$ (⁠$k0$⁠) in the measurements where a grating is present, as shown in Fig. 3(a). As electrical spin-wave excitation should not occur at these magnetic fields, we cannot explain the presence of these peaks. In supplementary material note V, we provide additional details on these peaks from which we conclude that it is not a spin-wave mode that is excited by the laser.

Although we do not know the origin of the uniform FMR-like peak in the measured spectra, we do not believe it affects the main conclusions of this article. The $k$-selectivity of the grating, the exact location of the resonance fields and the frequency dependence of the resonances (not shown) all suggest that the grating can be used to detect spin waves with a wavelength smaller than the laser spot. Finally, in supplementary material note II, we also show calculations for a position sweep and compare them with the experimental results, showing that the model is also able to qualitatively explain this behavior.

Thus, overall, the theoretical model can qualitatively explain the most relevant data. Yet, the inability of the model to fully explain the data suggests that the simple approach we advocate here for a qualitative understanding of the measured spectrum is incomplete. There are some indications that this might be related to the complex optical behavior of the grating. As we demonstrate in supplementary material note III, light polarized parallel to the grating produces a finite MO signal, while for light polarized perpendicular to the grating lines, there is no MO contrast. Moreover, in this same supplementary material note, we also demonstrate that the measured MO contrast depends on the $k$ value of the grating, with a maximum signal around $k=9μm−1$⁠. Last, we also show that if we place the grating directly on top of the strip, rather than on top of the insulating layer, the measured signal increases significantly, suggesting the extra optical path length through the insulating layer leads to the spreading out of the light, which reduces the $k$-sensitivity of the grating. To fully understand this behavior, more detailed theoretical and experimental work should be performed to investigate the true influence of the grating. Near-field effects, such as plasmonic resonances, need to be carefully considered to form a complete picture of the behavior of the grating.^31,32 In this case specifically, the effect of the grating on the polarization of the light needs to be addressed. Last, we have made no attempt to optimize the thickness of the grating, although one can imagine that this can also impact its behavior.

Nonetheless, theoretical work from contact lithography suggests that the resolution in the near field can be as high as $λ/20$⁠,³¹ limited only by the distance between the grating and the magnetic strip, and required MO contrast to produce a measurable signal. This demonstrates that our technique has enormous potential for the detection of spin waves with wavelengths as small as several tens of nm. One can then imagine a wealth of applications for this method that range from fundamental investigations into spin-wave behavior to the practical investigations into magnonic devices. For example, one can investigate $k$-dependent behavior as a function of position using an optical technique with wavelengths that are usually only accessible using x-ray based techniques.^12,16 This grating-based method could also be used in combination with phase-resolved MOKE techniques.^33–35 Furthermore, on the practical side it allows one to, for example, measure the interfacial Dzyaloshinskii–Moriya interaction (iDMI) in films thinner than reachable using all-electrical propagating spin-wave spectroscopy.³⁶ As ultrathin films are routinely measured with MOKE, we expect the layer thickness to be less of an issue for the optical technique. Additionally, the inverse of the process demonstrated here should also be possible: using a grating to $k$-selectively excite spin waves with an ultra-fast laser pulse.^37,38 This brings nano-scale all-optical propagating spin-wave spectroscopy one step closer to realization.³⁹ 

Concluding, in this article, we have demonstrated an optical technique that can measure propagating spin waves beyond the diffraction limit using a metallic grating. This technique works down to wavelengths of at least $700nm$ (⁠$km=9μm−1$⁠) using a diffraction-limited laser spot size of $10μm$⁠. Additionally, we could also detect spin waves with extreme $k$-selectivity and investigate the propagation of spin waves as a function of position. This opens up a route for the integration of photonics and magnonics⁴⁰ and brings us one step closer to the optical detection of spin waves in the exchange-wave regime.

See the supplementary material for a more detailed description of the experimental setup, additional measurements on devices with a $km=5μm−1$ antenna, additional measurements on the optical behavior of the grating, details of the model that is used to describe the observed behavior, and details of the FMR-like mode that is visible in our experiments.

This work is part of the research programme of the Foundation for Fundamental Research on Matter (FOM), which is part of the Netherlands Organisation for Scientific Research (NWO). We also thank L.G.T. van de Coevering from March Microwave Systems B.V. for lending us microwave equipment.

The authors have no conflicts to disclose.

Juriaan Lucassen: Investigation (equal); Writing – original draft (lead). Mark J. G. Peeters: Investigation (equal); Writing – original draft (equal). Casper F. Schippers: Methodology (lead); Writing – review & editing (equal). Rembert A. Duine: Supervision (equal); Writing – review & editing (equal). Henk J. M. Swagten: Supervision (equal); Writing – review & editing (equal). Bert Koopmans: Conceptualization (lead); Supervision (equal); Writing – review & editing (equal). Reinoud Lavrijsen: Supervision (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding authors upon reasonable request.