Resonant Faraday effect using high-order harmonics for the investigation of ultrafast demagnetization

During the past few years high-order harmonic generation (HHG) has opened up the field of ultrafast spectroscopy to an ever larger community by providing a table-top and affordable femtosecond extreme ultraviolet (EUV) and soft-x-ray source. In particular, the field of femtomagnetism has largely benefited from the development of these sources. However, the use of x-ray magnetic circular dichroism (XMCD) as a probe of magnetization, the most versatile and reliable one, has been constrained by the lack of polarization control at HHG sources, so studies have relied on more specific magneto-optical effects. Even the recent developments on the generation of elliptically polarized harmonics have only resulted in a few time-resolved experiments relying on this powerful technique since they add complexity to already-difficult measurements. In this article we show how to easily probe magnetization dynamics with linearly polarized EUV or soft-x-ray light with a versatility similar to XMCD by exploiting the Faraday effect. Static and time-resolved measurements of the Faraday effect are presented around the Co $M$ edges. Using simple theoretical considerations, we show how to retrieve the samples magnetization dynamics from the Faraday rotation and ellipticity transients. Ultrafast demagnetization dynamics of a few nanometers in Co-based samples are measured with this method in out-of-plane as well as in-plane magnetization configurations, showing its great potential for the study of femtomagnetism.

Figure 1

The Faraday effect leads to a transmitted beam elliptically polarized with a rotation of the polarization ${\theta }_F$ and an ellipticity ${\varepsilon }_F$. The ellipse major and minor axes, $a$ and $b$, define the ellipticity as $tan{\varepsilon }_F=b/a$. The $p^{\prime }$ and $s^{\prime }$ axes are rotated axes parallel to $a$ and $b$. The electric field of the beam is easily described in this coordinate system.

Figure 2

Polarization of the harmonic beam after a magnetized sample for two opposite magnetization directions, $M^{+}$ (blue) and $M^{-}$ (orange), starting from three different configurations of the incident polarization: (a) linearly $p$ polarized, (b) linearly polarized with an angle $\theta$ compared with the $p$ axis, and (c) linearly polarized with an angle $-\theta$ compared with the $p$ axis. Because of the Faraday rotation ${\theta }_F$ and ellipticity ${\varepsilon }_F$, the $s$ component of the electric field strongly depends on the sample's magnetization. For a $p$-polarized incident polarization, the $s$ component of the transmitted electric field is opposite for opposite magnetization, leading to the same measured intensity. A small rotation of the incident polarization $\theta$ allows us to differentiate between these two magnetization directions.

Figure 3

Schematic diagram of the experimental setup. The magnetic sample is pumped by an IR laser pulse and probed by a delayed linearly polarized harmonic pulse, generated by the same laser in a gas cell filled with neon. Both pump and probe are focused at the same position onto the sample and are almost in collinear geometry perpendicular to its surface. The analyzer is set at the Brewster angle for 60 eV radiation, i.e., reflecting mainly the $s$ component of the harmonic beam. The harmonics are then separated and detected by a spectrometer composed of a concave grating and a CCD camera. A set of three ${\mathrm{SiO}}_2$ mirrors and aluminum filters shield the detector from the IR light. (inset) The Co based samples are surrounded by a rotating assembly of four permanent magnets which can magnetize the sample at saturation out-of-plane in both directions.

Figure 4

(a) Harmonics spectrum: vertical distribution of the beam as function of the photon energy (30 s acquisition time). (b) Energy projection of the harmonics spectra showing the intensity of each harmonic for two configurations, with and without the sample in the harmonics path (30 and 2 s acquisition time, respectively).

Figure 5

Static Faraday effect of the CoDy sample as a function of photon energy in out-of-plane geometry. (a) Average of the energy projection of the recorded harmonics spectra showing the intensity of each harmonic for two opposite directions of the sample's magnetization ($M^{-}$ in orange and $M^{+}$ in blue lines). (b) Asymmetry of the Faraday effect [see Eq. (15)] measured for three different polarization axis angles of the incident light ($\theta , -\theta$, and 0).

Figure 6

Harmonic intensity as a function of time delay obtained from the CoDy sample magnetized out-of-plane ($M^{-}$) for a pump fluence of $5\mathrm{mJ}{\mathrm{cm}}^{-2}$ and at a photon energy of 57 eV. The points have been obtained by an average over three acquisitions of 20 s. The inset shows data from another scan focused on the short delays. The orange solid lines in both curves represent the best fits obtained.

Figure 7

Asymmetry as a function of photon energy for two opposite incident polarization axes ($\theta$ and $-\theta$) recorded for the CoPt-1 sample showing a strong Faraday rotation below and above 60 eV and a vanishing Faraday rotation close to 60.5 eV. The calculated asymmetry of a 12-nm-thick Co layer is shown (dashed blue line) for comparison and is in good agreement with the measurements.

Figure 8

Harmonic intensity as a function of time delay for three different photon energies: (a) 57.4 eV, (b) 60.5 eV, and (c) 63.6 eV recorded for a Co/Pt multilayer sample (Co-Pt2). For each energy, the Faraday effect was measured for two opposite magnetization directions of the sample ($M^{+}$ in blue and $M^{-}$ in orange). Each point represents an acquisition of 20 s. At 57.4 and 63.6 eV the two curves start at different levels and get closer upon excitation by a femtosecond IR pulse. This can be understood in term of a reduced magnetization and hence Faraday rotation after the pump [panels (g) and (i)] compared with the unpumped state [panels (d) and (f)]. As expected from the static Faraday measurements (Fig. 7), the transient variations of the signal are reversed for 57.4 and 63.6 eV. As shown earlier, the Faraday rotation vanishes at 60.5 eV (Fig. 7) and the two transients overlap. These two curves reveal a strong variation of the Faraday ellipticity, [panels (e) and (h)]. Since our setup is not sensitive to the helicity of the light, the response for opposite magnetizations is the same.

Figure 9

Normalized magnetization as a function of time delay recorded with three different probe photon energies (57.4, 60.5, and 63.6 eV) for the CoPt-2 sample magnetized out-of-plane. The curves are obtained from Fig. 8 and Eqs. (12) and (14). The value of $I(M=0)$ is the average obtained over a full delay scan. No features are visible around $t=0$ on this scan showing that $I(M=0)$ is constant. The pump fluence used was about $5\mathrm{mJ}{\mathrm{cm}}^{-2}$. While the curves recorded at 57.4 and 63.6 eV present similar evolutions, the curve recorded at 60.5 eV presents a slightly faster demagnetization time and a reduced demagnetization rate, as best revealed by the fits obtained for 57.4 and 63.6 eV (solid line) and for 60.5 eV (dashed line).

Figure 10

Normalized magnetization as a function of time delay recorded with at 57.4 eV for the Co sample magnetized in-plane. The solid line corresponds to the best fit. The inset represents the experimental magnetic geometry, in which the incidence angle on the sample is 30°. It has to be noted that, in this geometry, the apparent magnetization of the sample (the magnetization seen by the probe beam) is about half of the saturation magnetization, resulting in a lower Faraday signal. Nevertheless, the demagnetization is clearly visible, revealing a slower dynamic than the Co/Pt multilayer at same probe energy. The pump fluence used was also $5\mathrm{mJ}{\mathrm{cm}}^{-2}$.