1. Introduction

In type II superconductors, an external magnetic field can partially penetrate as filament like flux lines named Abrikosov vortices carrying a single quantum of flux [1]. With a nanometric core surrounded by circulating supercurrents over a distance given by the London penetration depth ${\lambda _L}$ (typically ${\lambda _L}$ ∼100 nm in Niobium), these vortices represent the most compact magnetic objects in superconductors. It has been shown that they have the potential of widespread use as quantized information bits [2] to design high-density digital cryoelectronics (tens of Josephson junctions per µm² [3,4]). The local supercurrents produced by a single vortex placed close to a Josephson junction can also induce Josephson phase shifts and thus affect the Josephson transport properties [5–8]. For the development of various schemes in superconducting electronics [6], it is crucial to implement simple and precise vortex imaging and manipulation methods.

Over the past decades, various vortex imaging techniques have been introduced, including pioneering Bitter decoration techniques [9], Lorentz microscopy [10], scanning Hall probe microscopy [11], magnetic force microscopy [12], scanning electron microscopy [13], scanning superconducting quantum interference device (SQUID) microscopy [14]. While scanning tunneling microscopy is capable of resolving a vortex core (∼10 nm) [15] and SQUID microscopy provides sensitivity down to the single electron spin level [16], these local probe techniques are heavy to implement in a cryogenic environment. On the other hand, magneto-optical (MO) imaging [17–19] provides far-field optical images with a wide field of view and can be combined with other optical tools recently developed for single vortex manipulation [20] and generation [21]. MO imaging of individual vortices is based on the Faraday rotation of light polarization in a MO indicator placed onto the superconductor, using a beam path configuration with nearly crossed polarizers [22]. The most commonly used MO indicators are ferrite garnets such as Bismuth substituted lutetium iron garnets (Bi:LuIG), that are grown with a thickness of a few micrometers on a thick (∼500 µm thickness) gadolinium-gallium-garnet (Gd₃Ga₅O₁₂, GGG) transparent substrate. However, this MO imaging method suffers from poor spatial resolution of the vortices (apparent vortex diameters $\mathop \gt \limits_\sim $ 1.5 µm [17]). Indeed, the thickness of the GGG substrate introduces severe spherical aberrations that broaden the point-spread function (PSF), i.e. the spatial distribution of a point-source image, well beyond the optical diffraction limit. Another source of degradation of the vortex images is the residual gap between the superconductor layer and the MO indicator pressed on top of it. This gap places the indicator away from the region where the magnetic field stemming from vortices is the most concentrated [18]. To eliminate both problems, it has been proposed to grow a nanometric layer of MO indicator directly on the superconductor. Using a 40 nm-thick EuSe indicator film, apparent vortex diameters as small as 800 nm have been reported [23]. Yet, this technique has limited versatility since EuSe layers are difficult to fabricate and cause strong light absorption. Moreover, they undergo a magnetic ordering phase transition at 4.6 K that restrict their range of use to liquid helium temperatures and thus superconductors with low critical temperatures [24].

The implementation of optical microscopy with a high numerical aperture (NA) remains a crucial step to achieve high-resolution vortex images associated with strong MO signals. It is challenging since the common high-NA microscope objectives have many lenses subjected to mechanical strains and phase changes at cryogenic temperatures. Moreover, such objectives would not compensate for the spherical aberrations introduced by the thick garnet substrate. In this context, implementing a solid immersion lens (SIL) combined with a single, low-NA aspheric lens is a powerful approach. SIL-microscopy allows immersion with a high refractive index adapted to the sample and was originally developed to achieve higher optical resolutions [25]. It has been successfully used to improve the light extraction efficiency of solid-state single photon sources such as individual quantum dots [26–28] or molecules [29,30]. In MO imaging, a SIL-based objective should also eliminate the strong spherical aberrations introduced by the thick garnet substrate. Here, we address the fundamental optical limitations of vortex MO imaging by coupling the indicator on its substrate to a SIL with a high refractive index n ∼2 matching that of the GGG substrate. Using the Weierstrass (i.e. super-hemispherical) SIL geometry [31] to focus and collect light from a subwavelength region in close contact with a superconducting Niobium layer, we achieve MO imaging with single-vortex apparent diameters less than 600 nm. We also demonstrate a significant increase in the vortex MO signals and image acquisition rates. The features of the vortex optical images are well reproduced with vectorial electromagnetic-field simulations, which help optimizing the performance of vortex imaging.

2. Experimental setup

The experimental setup used for single vortex MO imaging is depicted in Fig. 1. A 100 nm-thick Niobium film deposited on a sapphire substrate is cooled below its critical temperature ${T_c}$ = 8.6 K under a weak external magnetic field that penetrates the superconducting film in the form of quantized flux tubes. The MO indicator pressed on top of the Niobium film is a 2.5 µm-thick garnet of composition Bi_xLu_3-xFe_5-zGa_zO₁₂ with x ∼ 0.9 and z ∼ 1.0 [32]. It is grown on a GGG substrate with a thickness h ∼500 µm. This sample is illuminated with a white light LED source filtered at the optical wavelength $\lambda $ = 550 nm with a bandwidth of 15 nm. This wavelength corresponds to a local maximum in the MO indicator's rotational power (Verdet constant $V\; \sim $ 0.16° mT^-1µm^-1) together with a garnet transmission factor of 0.43. The SIL used here is a half-ball lens made of S-LAH79 glass with refractive index $n$ = 2.011 at $\lambda $ = 550 nm, which nearly matches the GGG index ${n_{\textrm{GGG}}}$ = 1.97 at this wavelength. We operate this SIL combined with the GGG garnet in the Weierstrass geometry, which exploits rigorous stigmatism in the conjugation of two specific points through a spherical refractive surface [33]. The radius r of the SIL is indeed chosen so that the nearest Weierstrass point coincides at best with the MO indicator. The requirement $r = n\; \times \; h$ for such geometry is nearly fulfilled with the choice of $r$ = 1 mm. The residual mismatch between n and ${n_{\textrm{GGG}}}$ produces negligible spherical aberrations: The Strehl ratio, which corresponds to the ratio of maximal intensities in the PSF and Airy spots, is indeed greater than 0.989. The overall NA of a SIL-based objective operating in the Weierstrass configuration can be as high as n, provided that the aspheric lens has a NA greater than $1/n$. We use here an aspheric lens with focal length 10 mm and NA = 0.49. However, a holder used to press the SIL onto the GGG garnet (See Fig. 1(b)) reduces the overall NA of this objective to 0.68.

 

Fig. 1. Optical design for single vortex magneto-optical imaging. (a) The general setup operates on Faraday rotation of light polarization in a Bi:LuIG MO indicator placed on the superconductor in a nearly-crossed-polarizer beam path configuration. (b) The superconductor placed in the cryostat is a Niobium film deposited on a sapphire substrate. The magnetic field stemming from vortices penetrates the indicator and is imaged with a SIL-based objective comprising the SIL and an aspheric lens. The SIL placed on top of the GGG substrate is a half-ball lens operated in the Weierstrass configuration. The white lamp has a fibered output with diameter 3 mm. A source iris controls the aperture of the illumination beam, while a NA iris is placed on the detection path in the Fourier plane of a telescope. A 15 nm bandpass filter is used to collect enough light without introducing significant chromatic aberrations. (c) Simulation of the vertical component of magnetic field ${B_z}({r,z} )= ({{\phi_0}/2\pi } )({z - {z_0}} ){[{{r^2} + {{({z - {z_0}} )}^2}} ]^{ - 3/2}}$ stemming from a vortex in the half space above the Niobium surface ($z$ = 0), r and z being cylindrical coordinates, ${z_0} ={-} 1.27{\lambda _L}$ and ${\lambda _L}$=100 nm [34].

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3. Results and discussion

3.1 Vortex images with a solid immersion lens

Figure 2(a) is a MO image of a spontaneous vortex distribution in a Niobium film at 4 K after field cooling under an external magnetic field of 0.3 Oe, using the aspheric lens alone as in conventional MO imaging setups. A background subtraction scheme is used: An image is acquired above ${T_c}$ (at 10 K), then subtracted from the image taken at 4 K. The vortices are located at pinning sites that are randomly distributed in the Niobium film. While the real vortex size given by ${\lambda _L}$ is ∼100 nm in Niobium, the single vortex spots have a full width half maximum (FWHM) diameter of 1.7 µm (Fig. 2(a)). The addition of the SIL drastically reduces the size of the vortex images by a factor of three (Fig. 2(b)). As demonstrated in Fig. 2(c), a FWHM diameter of 583 nm is achieved. The threefold increase in resolution with the SIL design is evidenced in the case of a higher concentration of vortices: In Fig. 2(d)-(e), the magnetic field applied during field cooling of the Niobium film is raised to 7.1 Oe, which sets the average inter-vortex distance in the micrometer range. While individual vortices can not be distinguished with a single aspheric lens, they are distinctly resolved with the SIL-based objective.

 

Fig. 2. Optical resolution of vortex images. All scale bars are 2 µm. Single vortices in a Nb film with thickness 100 nm are imaged at 4 K with the aspheric lens alone (a) and with the SIL-based objective (b), under magnetic fields of 0.3 Oe and 2.2 Oe, respectively. The vortices are imaged by a CCD (charge-coupled device) camera (MicroMAX EEVx512 FT CCD57, Princeton Instruments) having 512 ${\times} $ 512 pixels with an area of 13 µm ${\times} $ 13 µm. One pixel maps an area of 500 nm ${\times} $ 500 nm in (a). When adding the SIL (b), the magnification of the imaging system is increased by a factor ${n^2}$, so that one pixel maps an area $4 \times 4$ times less. (c) Profiles of the smallest vortex spots imaged without (blue circles) and with (red circles) the SIL, and fitted with Lorentzian curves (blue and red solid curves, respectively). The FWHM diameters are respectively 1.7 µm and 583 nm. Vortices in greater concentration are imaged after field cooling at 7.1 Oe (d) without and (e) with the SIL. (f) Vectorial field simulations of the PSF with the aspheric lens alone (left inset) and with the SIL-based objective (right inset). The blue and red solid curves are the intensity profiles of the respective PSFs calculated from the Debye integral. The black circles represent the cross-section of the left-inset image, taking pixilation into account (pixel size 500 × 500 nm). The dashed blue curve is a Lorentzian profile fitting the circles. Its FWHM of 1.7 µm is in accordance with the experimental vortex spot diameters measured with the aspheric lens used alone.

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In order to model the vortex images without and with the SIL, we have computed the vectorial field propagation all along the optical path (z-axis) from the source to the sample and back to the camera with a homemade ray-tracing program [35], taking into account the optical aberrations and changes in polarization at each interface with their Fresnel coefficients. A vortex is considered as a punctual object diffracting light isotropically and producing Faraday rotation of light polarization. Initializing the incident polarization along the x-axis, the PSFs associated to the intensity components ${I_x}$ and ${I_y}$ along the x- and y-axes are calculated at the output of the aspheric lens, then projected onto the analyzer axis oriented as in the experiment. The corresponding simulated PSFs are shown in Fig. 2(f) for the configurations without and with the SIL, keeping the same aspheric lens in both imaging systems. In the case of the aspheric lens used alone (blue curve), the pronounced lobes are due to severe spherical aberrations introduced by the propagation through the GGG substrate. We indeed estimate the peak-to-valley wave-front distortion of ∼0.6 $\lambda $ in this configuration. For a better match with the experimental images, we have added a slight contribution of wavefront distortion 0.03λ caused by deformation of the GGG substrate under non-uniform pressure exerted on its edges to minimize the gap between the Nb film and the MO indicator. This deformation is not taken into account in the SIL configuration, since the flat surface of the SIL exerts a uniform pressure over the central area of the GGG substrate. With the SIL-based objective (red curve), the PSF is nearly diffraction-limited due to the inherent elimination of spherical aberrations.

The evolution of the FWHM vortex spot size, averaged over ∼30 vortices, is shown in Fig. 3(a) as a function of the NA of the SIL-based objective. The NA is varied by controlling the aperture of a diaphragm placed on the detection path in the Fourier plane (See Fig. 1). As expected, the increase in NA leads to sharpening of the vortex spots, whose average size follows the NA-dependence of the diffraction-limited spot diameter $0.61\lambda /\textrm{NA}$ arising from a point source (solid curve in Fig. 3(a)). The deviations between the experimental spot sizes and these lower bounds are attributed to residual Nb-BiLuIG gap and to the spreading of the vortex field lines penetrating the MO indicator (see Fig. 1(c)). At the highest NAs, stronger deviations are observed. They may be due to enhanced optical aberrations, e.g. induced by a slight misalignment of the aspheric lens and SIL optical axes.

 

Fig. 3. (a) Average FWHM of vortex images as a function of the effective NA controlled with a diaphragm. The diffraction-limited FWHM is given by the solid curve. (b) The SNR (red squares) of the vortex images is derived as the average ratio of the peak vortex signals to the noise in the background. It is plotted as a function of the analyzer offset angle for the case without (blue) and with the SIL (red) under similar illumination powers. The error bars are indicative of the signal variations amongst vortex spots in the recorded image. The solid curves are plots of Eq. (1), taking $\varepsilon = 4.4\; {10^{ - 2}}$ (red curve) and $\varepsilon = 5\; {10^{ - 3}}$ (blue curve), using the SNR proportionality factor as the only fitting parameter.

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3.5 Signal-to-noise ratio

The quality of magneto-optical imaging is also characterized by the signal-to-noise ratio (SNR) of the vortex images. In Fig. 3(b), the SNR averaged over ∼20 vortices is plotted for both cases, without and with the SIL, as a function of the analyzer offset angle $\phi $ with respect to the crossed polarizers configuration. Assuming that the noise essentially arises from the photon shot-noise, and that the Faraday rotation angle ${\theta _F}$ of light polarization after the double pass through the indicator is small (${\theta _F}\; \sim $ 5 mrad), the SNR can be modeled as

For both configurations, the experimental data are well reproduced with this expression, using the extinction ratios $\varepsilon = 5 \times {10^{ - 3}}$ and $\varepsilon = 4.4 \times {10^{ - 2}}$ measured without and with the SIL, respectively. On one hand, the higher NA of the SIL-based objective introduces undesired components of light polarization that degrade the extinction ratio. On the other hand, it offers higher photon collection efficiency due to its enhanced NA, which ultimately results in a SNR three times stronger under the same illumination beam, as shown in Fig. 3(b).

An important experimental parameter in vortex MO imaging is the aperture of the light source. While increasing the aperture enhances the illumination intensity in the sample plane, it also promotes depolarization effects that degrade $\varepsilon $ and impact the SNR. To model the aperture dependence of $\varepsilon $, the fibered source with a diameter of 3 mm is modeled by a sum of incoherent, non-polarized and uniformly distributed point sources. Launching 10,000 rays, the three field components are propagated along the optical z-axis towards the sample and reflected towards the analyzer and camera. These fields are then summed incoherently to obtain the intensity components ${I_x}$, ${I_y}$ and ${I_z}$ as a function of the diameter of the source aperture. The spatial distributions of these three components in the camera plane are displayed as insets in Fig. 4(a), in the case of a single source point. As expected, the ${I_y}$ and ${I_z}$ components are mainly created when propagating the beam through the aspheric lens and the SIL, and take their greatest values at the largest incidence angles. From these intensity profiles, the extinction ratio $\epsilon \; = {I_y}/{I_x}$ is computed and shown in Fig. 4(a) as a function of the source diameter. A degradation of $\epsilon $ by two orders of magnitude is obtained at the largest diameters. The experimental extinction ratio is displayed in Fig. 4(b) as a function of the diameter of the source iris. As the source size is reduced below 10 mm down to 2 mm, there is a gain in the quality of extinction by an order of magnitude and a threefold gain in the SNR (Fig. 4(c)). On the other hand, using large source apertures allows high illumination intensities of the sample and photon flux collection with the SIL, and thus fast vortex imaging. This is illustrated in the series of single-shot images displayed in Figs. 4(d)–(f), recorded for various acquisition times and a fixed aperture of 10 mm. Sub-second imaging with a SNR above unity and a sub-micron resolution is evidenced, which offers great prospects for fast and precise vortex imaging.

 

Fig. 4. Influence of the light source aperture. (a) The extinction ratio is computed in the Weierstrass configuration for various diameters of the source beam, using 100 point sources. The insets show the spatial distribution of the intensity components along the x, y and z axes, calculated with a single point source. Experimental evolutions of the extinction ratio (b) and the SNR (c) as a function of the source beam diameter. d-f: Normalized images taken with an integration time of 200 ms (d), 400 ms (e) and 1 s (f), for a source diameter of 10 mm. The SNRs are respectively 0.73, 1.23, 2.45. The scale bars are 2 µm.

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4. Conclusion

In conclusion, we have developed a MO imaging setup based on a SIL operating in the Weierstrass configuration and leading to sub-600 nm optical resolution of single Abrikosov vortices. This represents a three-fold improvement in the resolution with respect to single lens imaging systems. Such performance is combined with a higher efficiency in light collection, which is essential for fast acquisition of vortex images. Among the avenues for improvement of the vortex image quality, higher SNRs could be achieved using a quasi-punctual source for aberration-free illumination of the sample. Greater vortex resolutions and faster vortex imaging could also be reached by increasing the effective NA of the SIL, for instance using a thinner indicator substrate. High-NA MO imaging of vortices with a SIL is thus promising for precise optical localization and manipulation of single vortices applied to the optical operation of Josephson transport [6]. Achieving optical resolutions of the order of the London penetration depth will also enable real-time studies of the forces involved in dense vortex lattices.