1. Introduction

In natural materials positive and negative volume magnetic susceptibility χ_m is measured with Evans type balances by the force change applied to a sample located in a magnetic field [1]. These balances are useful for characterization of materials with dispersionless permeability. Recently, artificial materials with THz and optical frequency magnetic resonances stronger than in natural ones were fabricated [2–5]. Metamaterials are composed of elementary cells of linear size L ≪ λ in the form of Swiss rolls with a negative real part of permeability in the microwave range [2], split ring resonators (SRR) and fishnet with a negative real part of the refractive index in the range from microwave to optical [3–5]. At frequencies higher than 100 THz, the magnetic response of such structures weakens because of internal resistance (reactance) which is inversely proportional to frequency [6, 7]. Metamaterials with electric and magnetic dispersion at THz frequencies can be made of compact dielectric particles of high permittivity [8–12], noble metals [13, 14], and aluminum [15]. Consideration of charge displacements and displacement currents in elementary cells of metamaterials leads to interpretation of the role of dipole [12] and multipole [16] resonances in periodic all-dielectric arrays of rods and arrays of metal SRRs, respectively. The recent studies on all-dielectric metamaterials were concluded by Merlin, who proved theoretically that in metamaterials composed of structures with large values of dielectric permittivity para- and diamagnetic responses are possible with susceptibilities significantly larger than existing in natural materials [17].

Development of metamaterials with a tunable magnetic response entails new characterization techniques. Recently, a single SRR located at the tip of an aperture tapered-fiber metal-coated probe was used for near-field detection of the magnetic field of light [18]. Aperture probes with [18] and without [19, 20] an SRR were used to perform passive measurements of magnetic and electric components of static evanescent fields excited in metamaterials and photonic crystals. Another experimental method of independently testing the electric and magnetic resonances of a properly oriented SRR by means of a y-polarized TEM₁₀ mode (y-polarized HG₀₁ mode [21]) with an on-axis longitudinal magnetic field H_z and off-axis E_y components was proposed by Banzer et al. [22].

2. Active magnetic near-field probe

To perform optical range measurements of magnetic responses of all-dielectric and plasmonic metamaterial elementary cells with linear dimensions up to hundreds of nanometers we propose a concentrator of the magnetic component of an electromagnetic field into subwavelength spots. The concentrator is a dielectric tapered-fiber probe able to excite local magnetic resonances in the near-field in a manner similar to conventional scanning near-field optical microscopy (SNOM), see Fig. 1a. Azimuthally polarized light interacts with the structure of the concentrator generating plasmons. This allows for the confinement of the longitudinal magnetic field component H_z to energy densities exceeding those of other components by more than an order of magnitude. The field penetrates the sample below the probe and directly (using the magnetic field H_z) induces magnetic resonances in SRRs, fishnet elementary cells, and in 3D compact all-dielectric particles [8–11], all with dimensions comparable to the size of the spot, which may be subwavelength. Then, the scattered signal is measured in the far-field using a classical intensity detector, giving information on the properties of the sample. Thus, our structured probe is an active device generating a magnetic field illuminating a sample, while in [18–20] the probes are used as pasive recorders of magnetic fields existing in samples.

 

Fig. 1 (a) A dielectric probe with radial metal stripes concentrates the longitudinal magnetic component of light, shown in yellow, into a subwavelength spot to measure the magnetic moment of individual metamaterial elementary cells. (b) The tapered at angle α part of a silver-coated fiber probe with eight equidistant slits and eight h thick metal lands of constant angular width equal π/8. (c) Azimuthal currents J_ϕ generating the longitudinal magnetic field H_z indicated by black arrowheads and an out-of-plane vector, respectively.

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The active part of the probe are the radial metalic stripes made of silver arrayed in an azimuthal grating, shown in detail in Fig. 1b. The metal lands and the slits have constant angular width equal π/8 what means that their lattice periodicity decreases toward the apex assuring momentum matching for excitation of plasmons by light for a wide range of wavelengths. The probe is internally illuminated by an azimuthally polarized, cylindrically symmetric Laguerre-Gauss TE₁₀ mode [21]. Plasmons, which have a strong azimuthal component confined between the metal lands, propagate towards the apex where they form an azimuthal current in the metal and azimuthal virtual displacement current in slits circulating the apex. This results in generation of a strong longitudinal magnetic field H_z, as indicated in Fig. 1c. This magnetic concentrator is analogous to SNOM metal-coated tapered-fiber probes with a corrugated core surface [23–25], where grooves assure coupling of incident light to surface plasmons.

We establish the properties of the concentrator using the finite-difference time-domain method in 3-dimensional cylindrical coordinates where we analyze steady state and transient responses. The transient calculations are done for a temporal Gaussian pulse centered at 600 nm modulated by a sine finction and the spectra are calculated by Fourier-transformation. The concentrator has a silica fiber core (n = 1.45) of 3.2 μm diameter tapered at a half-angle α = 40° and a silver coating of thickness h. Silver is modeled using Drude dispersion $\varepsilon (\omega )={\varepsilon }_{\infty }-{\omega }_p^2/\left[\omega (\omega +i\Gamma )\right]$ fitted to experiental data [26] for wavelengths λ 190 ÷ 1900 nm with parameters equal ε_∞ = 3.70, ω_p = 13673 THz, and Γ = 27.35 THz [23].

3. Magnetic field needle

Figure 2 shows how a radially corrugated probe transforms an incident azimuthally polarized beam of λ = 500 nm (chosen as an example) into a strong magnetic focus of the longitudinal H_z component on the axis. The incident field couples to plasmons at the metal–dielectric interface of the silver stripes. Plasmons propagate toward the apex where the electric field has a dominant, circulating azimuthal component and, in accordance to Maxwell’s equations, generates a strong longitudinal magnetic field with a spot size of about 185 nm. In addition to H_z, other components of the EM field also exist, most importantly E_ϕ, E_z, and H_r. However, their energy density is the highest away from the axis for E_ϕ and H_r, while for E_z is more than one order of magnitude weaker than H_z. In the case of E_ϕ and H_r, the strong side lobes at r = 800 nm result from part of the light passing directly through the gaps between the metal stripes, because at that radius the gaps are wider than 300 nm. A probe with a bigger core radius and a wider beam will have side lobes farther away, so optical fibers with larger cores than considered here will have an even smaller overlap of the components than obtained here.

 

Fig. 2 Electromagnetic field in the tapered part of the probe with 50 nm silver layer: energy density of (a) azimuthal electric field $\varepsilon E_{\varphi }^2$, (b) longitudinal field $\varepsilon E_z^2$, (c) radial magnetic field $\mu H_r^2$, (d) longitudinal $\mu H_z^2$. The pseudocolor scale is logarithmic and in arbitrary units, wavelength λ = 500 nm is used as an example.

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Light emission from the magnetic probe is not restricted to only an aperture as in Al-coated aperture SNOM probes, thus the spot size, defined as the full-width at half-maximum (FWHM), varies depending on various parameters. We analyze how the silver thickness h affects the spot size and the magnetic focus intensity. Shown in Fig. 3, the size of the focused magnetic field H_z radiated by radially corrugated probes is smaller than for an all-dielectric probe (h = 0 nm) for virtually all analyzed silver thicknesses in the considered wavelength range. The FWHM decreases slowly with decreasing wavelength and the lowest FWHM is observed for the largest h. Formation of a narrow H_z needle is governed by azimuthal currents J_ϕ flowing through the set of coupled stripes, virtual displacement currents in grooves, and by the azimuthal component E_ϕ of part of the electric field refracted leaving the probe. As h increases, the contribution from the refracted field diminishes leaving only J_ϕ and associated plasmonic edge modes. Far from the apex plasmons form a coupled mode at both inner and outer edges, however, as they propagate forward and the grooves become narrower and approach the z-axis, the mode decouples and the inside one experiences a cut-off, not reaching the apex, and only the outer one remains. If h is large enough, then the mode is confined at the outer edges, similarly as discussed in [27] on V-grooves, until it reaches the probe apex and radiates. However, for small h plasmons will radiate into an azimuthally polarized mode before reaching the apex. The farther from the apex the decoupling takes place, the larger the radius of the doughnut beam is which diffracts over a longer distance before reaching a sample. Also, the longer the wavelength, the farther from the apex this decoupling takes place. Hence, for smaller silver thicknesses h and for longer wavelengths the FWHM is larger.

 

Fig. 3 FWHM of H_z calculated 10 nm from the apex for h ∈ [0, 100] nm.

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At wavelengths around 480 nm for h ≥ 40 nm the FWHMs become equal due to an equilibrium between the enlargement of the spot size by an earlier decoupling for small h and direct enlargement due to a large h. For h ≥ 60 nm and λ < 450 nm the FWHM order is reversed due to the decoupling cut-off being smaller than the thickness for short wavelengths. However, as h decreases (h ≤ 50 nm) below the cut-off, plasmons decouple before the apex and their FWHM at the sample position increases. The wide FWHM values observed for h = 10÷30 nm are explained by very weak guiding provided by the silver lands. Light instead of coupling to plasmon modes in waveguides formed by silver stripes leaks out of the dielectric core forming a strong background signal comparable to that of an all-dielectric probe.

The dominant contribution of the longitudinal magnetic component to the total energy within the focus is illustrated in Fig. 4a–d, where energy density is integrated over an area of diameter equal to the FWHM of H_z. The integrated energy density of all field components increases as h increases, because thick stripes are more efficient at confining and guiding energy toward the apex than thin ones. For small h, effects generated by the dielectric core become strong compared to those of the silver stripes. Hence, cross-coupling between polarizations is weak, as expected for a structure composed mostly of an all-linear-dielectric. The electric component E_z for h = 20 nm is almost non-existent, while the integrated energy densities of E_ϕ, H_r, and H_z are not similar to those of larger h values, and h = 20 nm may be regarded as intermediate between a mostly dielectric-dependent behavior of the probe and a predominantly plasmonic response.

 

Fig. 4 Integrated energy densities of (a) E_ϕ (b) E_z (c) H_r (d) H_z 10 nm from the apex in the focus where H_z is dominant. Units are arbitrary and the same for all subfigures. (e) Ratio of integrated energy densities of H_z to E_ϕ.

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Maximum energy density of H_z as a function of silver thickness does not guarantee the most efficient active sensing of magnetic properties. More important is the magnetic energy density of H_z to electric energy density of E_ϕ ratio, what decides on magnetic excitation of a sample, shown in Fig. 4e. The reference probe without silver has a linear dependence of $\mu H_z^2/(\varepsilon E_{\varphi }^2)<2.5$ in the optical range. Adding the silver stripes increases this ratio up to 5.5 for h = 60 nm at λ = 450 nm, however, this peak is narrow and for λ = 790 nm falls below 3.5. For larger thicknesses of 80 and 100 nm the ratio is lower in the respective peaks than for h = 60 nm, but over a large wavelength range stays above 4. The physics of the peak in Fig. 4b,e and its dependence in h are the same as for the FWHM minimum in Fig. 3: a trade-off between the outer mode cut-off and a large h guiding energy away from the axis. This is also modulated by diffraction which is small for short wavelengths and greater for larger if the size of the system (h) is the same. Thus, radially corrugated probes with thick metalisation are more efficient at focusing the magnetic field in terms of localizing magnetic energy density as well as generating small FWHMs.

4. Conclusions

The proposed device for concentrating the magnetic component of the electromagnetic field may be employed as an active probe of a scanning near-field magnetic microscope with a classical shear force control of the probe-sample distance. Near-field illumination generates magnetic responses of elementary cells and the energy of scattered light is recorded with intensity detectors in the far-field. The shear force controlled distance of tens of nanometers allows for illumination of individual elementary cells of an arrayed metamaterial with a field of FWHM reaching down to 160 nm for short wavelengths. It may also be feasible to use the proposed groove/stripe structure in combination with hollow cantilever probes. Once materials with optically induced magnetism are developed the proposed magnetic field concentrator may be used for magnetic write/read operations.