Introduction

In the last few years, nano-structures have been studied and developed for magnetic responses. In particular, enhancing magnetic responses of dielectric layered structures at visible wavelength range has drawn significant interest due to their promising applications in sensors and exploitation of the non-linear properties that they offer [1,2,3]. Naturally, the magnetic response of most dielectric materials is weak, especially in the optical frequency range [4]. Additionally, planar multi-layer dielectric structures also show a relatively non-magnetic behavior with a very small magnetic permeability (μeff) [5,6,7]. However, multi-layer dielectric structures are promising in enhancing μeff due to their immense capability of being engineered in geometry and layer thicknesses. Recently, significant magnetic responses have been achieved in metal-insulator-metal structures by exciting Fano resonances and in planar dielectric-metal multi-layer structures, where dielectric layers are isolated by air and silver layers [8]. Although the significant magnetism obtained from these structures makes these structures promising for sensing applications, they are not favorable for sensing due to creating multiple Fano resonances through anti-phase dipole oscillations, and hence, forming destructive interference, and also due to the complexity in the structures. A U-shaped split ring resonator designed by metal-dielectric multi-layer structure has also been proposed for enhanced magnetization, which has potential for sensing applications as well [9]. However, applications of these structures could be limited due to the complexity in the fabrication of the practical devices.

The property of evanescent electromagnetic fields of surface plasmon polaritons (SPPs) in a planar metal-dielectric structure is used in many sensing techniques. However, the applications of surface plasmon resonance (SPR)–based sensors become limited while detecting small molecular weight (< 8 kDa) and low concentration (< 1 pM) analytes, which is often the case for several critical biological samples [10]. Recently, several SPR sensors have been proposed that show hyper sensitivity (S) so that molecules with less than 8 kDa weight and 1 pM concentration can be detected [11,12,13]. Although S is increased in the hyper sensitive structures by making the structures complex, e.g., by adding additional layers, the angle-resolved reflectivity (R) profile is significantly broadened. The increase of full-width at half-maximum (FWHM) of the R profile causes the figure-of-merit (FoM) of the structure to decrease, which is the most critical performance parameter for such SPR sensors [14]. Additionally, several hybrid SPR sensors [15, 16] and SPR sensors with additional complexity by including black phosphorus, metamaterials, silicon, and MoS2 nano-structures have shown to enhance S and FoM [17,18,19,20]. Recently, the addition of graphene and BaTiO3 layers in SPR sensor has shown that the resonance angle can be shifted significantly with a small change of refractive index of the target sample [21]. Caballero et al. proposed magneto-optical SPR sensor, where the detection is based on the calculations of the transverse magneto-optical Kerr effect [22]. However, these hybrid SPR sensors often include complex arrangements of nano-structures, and need complicated and expensive manufacturing techniques.

SPR sensors based on Kretschmann configuration are the best choice for many sensing applications because of their simple structural arrangements. In SPR sensors, the presence of a biomolecule or any other target molecule in the sensing layer changes the SPR electromagnetic fields. The plasmonic characteristics and applications of SPR sensors have been explored well. However, SPR sensors usually show weak magnetic property and their magnetic properties have never been used for sensing. Optical magnetism in the optical frequency range using insulator-metal-insulator (IMI) Kretschmann plasmonic structure has not been explored till now. Similar to SPR resonances, strong magnetic resonances, also known as magnetic plasmons, can be excited in a multi-layer structure [23]. The concept is based on designing IMI planar multi-layer structures that support localized surface plasmons (LSPs), which act as magnetic dipoles and thus create strong optical magnetism. Thus, the enhanced localization of light by LSPs increases the magnetic response of the IMI planar multi-layer structure [24].

In this work, we show that an SPR sensor based on Kretschmann configuration can create strong magnetic response in optical frequency range in the presence of a target sample, and thus, magnetic property of these structures can be used to increase the performance in sensing the target sample. Using finite difference time domain (FDTD) simulations and scattering parameters (S-parameters) retrieval method, we calculate angle-resolved μeff and R profiles of a SPR sensor that has a structure based on Kretschmann configuration. We show that the FoM of the sensor based on the magnetic resonance increases significantly compared to that of a conventional purely SPR-based sensor. Exploitation of the magnetic property of the simple Kretschmann configuration has the potential of obviating the need of a more complex sensor structure while achieving high sensor performance.

The rest of the paper is organized as follows: In Section “Theoretical Modeling”, we describe the theoretical models that we used to calculate μeff and R using S-parameters retrieval method. In Section “Sensor Structure and Simulation Setup”, we discuss the Kretschmann configuration–based SPR sensor structure investigated in this work and the FDTD simulation approach. In Section 1, we discuss a feasible experimental setup for the proposed sensing technique. In Section “Results and Discussion”, we present and discuss the calculated magnetic response of the sensor, and also the S and the FoM of the sensor calculated from μeff and R profiles. In Section “Conclusion”, we draw conclusions on the findings.

Theoretical Modeling

The electromagnetic response of a structure with a complex μeff can be determined through systematic Drude-Lorentz representation [25] or S-parameters retrieval technique [26,27,28,29]. Often, the Drude-Lorentz analytical model is not precise, especially, if the structure is a planar multi-layer. Alternatively, the S-parameters retrieval technique depends on the reflection (S11) and transmission (S21) coefficients assuming a real index of the medium, and gives more precise results for μeff [26]. The coefficients S11 and S21 are computed from amplitude and phase of the peak fields as recorded on the detection planes. The S-parameters retrieval technique assumes that the detection planes are far away from the metal-sample layer interface so that the fields can be assumed to be propagating like a plane wave. Practically, the distance of the detection plane for the reflected waves from the metal-sample layer interface and the distance of the detection plane for the transmitted waves from the metal-sample layer interface must be much greater than the wavelength of the incident light. It is also essential to recompense for the phase that gathers as the fields spread through the background medium from the source to the multi-layer structure, and from the multi-layer structure to the detection planes. Thus, using the S-parameters retrieval technique, μeff can be calculated by the following equations [26]

$$ S_{11}=\frac{i}{2}\left( \frac{1}{z}-z\right) \sin(n_{\text{eff}}k_{0}z), $$
(1a)
$$ S_{21}=\frac{1}{\cos(n_{\text{eff}}k_{0}d)-\frac{i}{2}(z+\frac{1}{z})\sin(n_{\text{eff}}k_{0}d)}, $$
(1b)
$$ n_{\text{eff}}=\frac{1}{k_{0}d}\cos^{-1} \left[\frac{1}{2S_{21}}(1-S_{11}^{2}+S_{21}^{2})\right], $$
(1c)
$$ z=\pm\sqrt{\frac{(1+S_{11})^{2}-S_{21}^{2}}{(1-S_{11})^{2}-S_{21}^{2}}}, $$
(1d)
$$ \varepsilon_{\text{eff}}=\frac{n_{\text{eff}}}{z}, $$
(1e)
$$ \mu_{\text{eff}}=n_{\text{eff}}z, $$
(1f)

where neff is the effective refractive index, εeff is the effective permittivity, z is the normalized impedance, d is the total thickness of the multi-layer structure, i.e., the sum of the thicknesses of the different layers, and k0 = 2π/λ, where λ is the wavelength of the incident light. To determine the effective constitutive parameters precisely, we follow the condition k0d < 1 as d < λ [26]. The reflectivity R can be determined as

$$ R=\mid r_{\text{p}} \mid^{2}, $$
(2)

where rp is the reflection coefficient for p-polarized incident light calculated using S-parameters retrieval method.

In this work, we will show that μeff of the multi-layer structure at resonance can be used to detect a target sample instead of using R at resonance. Practically, μeff can be precisely measured by several off-the-shelf devices, such as by a Ferromaster, which is a handy instrument capable of measuring μeff precisely [30]. The magnetic resonance angle is identified as the angle at which μeff reaches maximum while varying the incidence angle of light. The S of the multi-layer structure can be calculated as a change in the incidence angle for per unit change in the refractive index (ns) of the sample, where the change in the angle can be due to the resonance for μeff as proposed in this work or for R as in conventional SPR excitation. The FoM depends inversely on the broadening of the FWHM of the response profiles. Thus, we can write the S and FoM considering μeff or R at resonance as

$$ \begin{array}{@{}rcl@{}} S_{\mu,R}&=&\frac{{\Delta} \theta_{\mu,R}}{{\Delta} n_{\text{s}}}, \end{array} $$
(3)
$$ \begin{array}{@{}rcl@{}} \text {FoM}_{\mu,R}&=&\frac{S_{\mu,R}}{\text {FWHM}}, \end{array} $$
(4)

where Δθ and Δns are the changes in the incidence angles and the refractive index of the target sample, respectively. The subscripts μ and R represent whether the change in the angle is due to magnetic resonance or SPR, respectively.

Sensor Structure and Simulation Setup

The SPR Kretschmann configuration sensor structure consists of three different layers, as schematically shown in Fig. 1. The first layer is a semi-infinite prism of glass material, e.g., BK7. The exciting light is incident on the metal layer through the semi-infinite prism, and also the reflected light is collected from the semi-infinite prism. The second layer is metal, e.g., silver (Ag), which has a thickness of 45 nm. The third layer is the sample layer, which has a thickness of 50 nm. We assume that the incident light has a wavelength of 633 nm, which is often used in SPR-based systems for excitation [31, 32]. The refractive indices of different sensor layers are frequency dependent. The refractive index of BK7 has been calculated using the approach described in Ref. [33] and found to be 1.515 at 633 nm. The refractive index of Ag has been calculated using the Drude-Lorentz model [34].

Fig. 1
figure 1

Schematic illustration of a feasible experimental setup for the proposed technique

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We take the refractive index of the sample layer as a parameter and vary from 1.3 to 1.9. There are several organic samples with a refractive index in this range. While different proteins and biomarkers have refractive indices in the range of 1.3–1.45 such as human fibrinogen (Fb) protein molecules and thyroglobulin (Tg) have refractive indices 1.39 and 1.45, respectively [35, 36], several other important organic samples such as benzene (C6H6), potassium hydrogen phthalate (C8H5KO4), air-dried herring DNA (H-DNA), cyclotetramethylenete-tranitramine (HMX), and tetrahydrofuran (THF) clathrate have refractive indices in the range of 1.5–1.9 [37, 38].

To find out the dynamics of the incident light with the sensor in the presence of the target sample, we carry out two-dimensional full-field FDTD simulations. The simulation domain is 7 μm in the direction of the layer interfaces, i.e., in the y-direction, and 8.5 μm in the direction perpendicular to the interfaces, i.e., in the x-direction. We use a non-uniform meshing scheme for the computational domain to optimize the computational efficiency and accuracy of FDTD solutions. We use the perfectly matched layer boundary condition at the edges of the simulation domain in the direction perpendicular to the layer interfaces. We use Bloch boundary condition in the direction of the interfaces. The incident light has a transverse magnetic polarization and the incidence angle varies from 54° to 64°. The detection planes for the reflected and transmitted light are at \(\sim 7\lambda \) from the metal-sample interface.

Feasible Experimental Setup

In this section, we discuss a feasible experimental setup for practically implementing the proposed technique. The experimental setup for the proposed technique will not be much different from that for a conventional Kretschmann configuration–based sensor. In the proposed technique, the objective will be to measure the change in the optical magnetism rather than the change in the reflection profile as is measured in the conventional technique. We present the feasible experimental setup for the proposed technique in Fig. 1. In experiments, a He–Ne laser can be used as the 633-nm-wavelength light source. A neutral-density filter will be used to change and control the intensity of incident light. A polarizing prism cube (PC) will be used to isolate the p-polarized light from the incident unpolarized light. The p-polarized light will be incident on a convex lens so that it converges and focuses on the metal layer of the sensor. The sensor will be placed on a rotation stage so that the incidence angle of light can be changed. A photometer can record the reflected light intensity and a Ferromaster can record the optical magnetism of the sensor structure. Finally, the reflected light intensity or the optical magnetism of the sensor will be analyzed by a digital analyzer or oscilloscope.

Results and Discussion

Figure 2 illustrates μeff,εeff, neff, and z of the sensor based on Kretschmann configuration that we study in this work as described in the previous section. We vary the refractive index ns of the sample layer from 1.3 to 1.9. In Fig. 2a, we find that μeff < 1 and εeff < 0 when \(n_{\text {s}}\lesssim 1.69\). Therefore, the sensor structure is non-magnetic when \( n_{\text {s}}\lesssim 1.69\). We note that the response of the structure is non-magnetic, i.e., \(\mu _{\text {eff}}\leqslant 1\), when z ⋅ neff < 1, as shown in Fig. 2b [39]. However, the sensor structure becomes magnetic, i.e., μeff > 1, when ns > 1.69. The sensor structure shows a magnetic resonance and μeff reaches maximum at ns ≈ 1.75. At magnetic resonance, neff is minimum and z ⋅ neff > 1, as shown in Fig. 2b, and therefore, the magnetic field becomes strong to create optical magnetism. Also, we note that the magnetic response of the structure is Lorentzian, which verifies the optical magnetism as has also been observed before for planar multi-layer structures [40, 41].

Fig. 2
figure 2

Effective constitutive parameters a μeff and εeff, and b neff and z for the Kretschmann-based SPR sensor at 633 nm incident wavelength with ns varying from 1.3 to 1.9

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While we investigate the effects of sample layer refractive index on the sensor constitutive parameters in Fig. 2, we present the dynamics of the incident light with the sensor structure in Fig. 3. The results presented in Fig. 3 help to understand the physics of the proposed technique. Figure 3a and b show magnetic field profiles in the vertical cross-section, i.e, in the direction perpendicular to the layer interfaces, of the sensor when ns = 1.3 and ns = 1.546, respectively. We find that dual mode SPPs are excited at the metal-dielectric interface due to the interaction of the incident light with the sensor structure when the structure is non-magnetic. When ns ≈ nprism, strong SPPs are excited. However, we find that the response of the conventional multi-layer planar structure based on Kretschmann configuration is non-magnetic when \(n_{\text {s}}\lesssim 1.69\), due to the excitation of dual mode SPPs. By contrast, single mode SPPs are excited when ns > 1.69, and as a result, the structure becomes magnetic. In Fig. 3c, we note the excitation of single mode SPPs when ns = 1.75. The excited single mode SPPs at ns = 1.75 are similar to LSPs, and hence, confine the electromagnetic fields more strongly at the metal-sample layer interface. When ns = 1.75,neff is minimum due to the maximum power confinement in the metal film. When ns varies from 1.75 to 1.82, strong LSPs are excited as shown in Fig. 3d. For sensing purpose, both SPPs and LSPs are similar from the detection point-of-view. However, the excitation of LSPs support stronger light confinement, and as a result, can enhance the sensor performance, especially, the FoM [42].

Fig. 3
figure 3

Magnetic field profiles polarized in the z-direction for 633 nm incident wavelength when ans = 1.30, b ns = 1.546, c ns = 1.75 and d ns = 1.82

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The excited LSPs at the metal-prism and metal-sample layer interfaces of the sensor structure interact with each other through the fields that penetrate into the metal. Magnetic dipoles are created due to anti-phase electric dipole oscillations at the top and bottom of the metal layer. Although the anti-symmetric resonance is also observed when ns > 1.80, the magnetic resonance weakens and μeff decreases due to interactions among magnetic dipoles, and therefore, due to the phase retardation of the scattered field with respect to the the incident field.

Now, in Fig. 4, we present the changes in μeff and R when the incidence angles vary. The change in μeff and R with the incidence angle is crucial in the determination of sensing for such a sensor. In Fig. 4a, we show μeff as a function of the incidence angle (θμ) when ns = 1.70 and 1.71. We note that μeff has a resonance at θμ = 56.56° and reaches a maximum value of ∼6.63 when ns = 1.70, whereas μeff has a resonance at θμ = 56.86° and reaches a maximum value of ∼ 7.29 when ns = 1.71. Figure 4b shows R profiles against incidence angles (θR) of the excitation light when ns = 1.70 and 1.71. The R profiles are commonly used in conventional sensors for the calculation of S and FoM. We note that the resonance angles are slightly different for μeff and R in Fig. 4a and b. This is because when the resonance occurs, the phase between the incident and reflected light alters. S-parameters retrieval method computes μeff at resonance that depends not only on the multi-layer thicknesses but also on the distance between the source and the metal-sample layer interface, and also on the distance between the metal-sample layer interface and the reflection and transmission profile detection planes. By contrast, the calculation of R profile depends only on the multi-layer thickness.

Fig. 4
figure 4

a μeff and b R as functions of resonant incident angles for ns = 1.70 and 1.71 at 633 nm incident wavelength

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The detection accuracy of a SPR sensor is influenced by the FWHM of μeff or R profiles. A narrow FWHM of the angle-resolved μeff and R profiles indicates a high signal-to-noise ratio and precision in the measurement of resonance angles [43]. The FWHM of μeff profiles shown in Fig. 4a are 1.17° and 1.18° when ns = 1.70 and 1.71, respectively. On the contrary, the FWHM of R profiles shown in Fig. 4b are 2.42° and 2.52° when ns = 1.70 and 1.71, respectively. The increase of FWHM for R from that for μeff is attributed to the faster damping of SPPs than LSPs. The FWHM values of μeff and R profiles are plotted in Fig. 5 when ns varies from 1.70 to 1.82. We note that the FWHM for R is more than a factor of two greater than that for μeff. This is because the confinement of light at SPR is comparatively lower than that at the magnetic resonance. A better confinement supports lower FWHM value. As ns increases, FWHM for μeff decreases while that for R increases.

Fig. 5
figure 5

Variation of FWHM for μeff and R profiles against ns. The subscripts μ and R are used when the parameters are calculated using μeff and R profiles, respectively

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In Fig. 6a, we show the calculated S from both μeff and R profiles. The calculated S from μeff profile is smaller than that from R profile. This is due to low angle shifting property of magnetic resonance. The maximum S calculated using μeff is 30°/RIU when ns = 1.75. When ns > 1.75,S gradually decreases due to the excitation of anti-symmetric LSP modes. As a result, angle shifting property is reduced. On the other hand, the calculated S from R profile is 47°/RIU when ns = 1.75 and 54°/RIU at the peak when ns = 1.78.

Fig. 6
figure 6

Variation of a S and b FoM against ns. The subscripts μ and R are used when the parameters are calculated using μeff and R profiles, respectively

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The change in FoM due to the variation of ns is shown in Fig. 6b. Since the accuracy of detection of a sensor depends on FWHM of the response profile, the most applicable parameter to judge the performance of this kind of devices is FoM. FoM is the mostly used measure for the performance of SPR-based sensors [44,45,46]. When ns varies from 1.70 to 1.75, FoM does not vary much and remains ∼ 24.5 RIU− 1 when calculated using μeff profile. Both S and FoM are maximum when ns = 1.75. When ns > 1.75, FWHM of μeff increases, and therefore, FoM decreases. Conversely, FoM ≈ 16 RIU− 1 at ns = 1.75, when calculated using R profile. However, the maximum FoM of 17.5 RIU− 1 is observed at ns > 1.75, when calculated using R profile.

In Table 1, we present a comparison of the FoM obtained from our proposed technique with that of several recently reported sensors that are based on SPR. The FoM of our proposed technique is significantly greater than that of the techniques in Table 1. Moreover, most of the recently reported SPR sensors are based on complex hetero-structures with novel two-dimensional materials. As a result, the fabrication complexity and the cost of these sensors significantly increase, and the performance of these sensors often suffers too. For example, the MoS2 material included in Ref. [47] incurs energy loss, and hence, a reduced S. Similarly, Refs. [11, 16, 49, 51] include significant complexity without much enhancement in the sensor performance compared to the enhancement obtained by our proposed technique using a simple Kretschmann configuration.

Table 1 Comparison of FoM with previously reported sensors
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It is important to note that the prospect of achieving optical magnetism at visible wavelength by IMI planar structures has recently been a subject of significant interest. In this work, we have determined magnetic resonance of a Kretschmann configuration SPR-based sensor at a specific ns of the target sample. This magnetic resonance can be created at a lower ns than that demonstrated in this work, if required, using different 2-D materials such as graphene and MoS2 on metal surface. We can create magnetic resonance at a different ns also by nano-structuring of metal and dielectric layers, and optimization of thickness and refractive index of the metal and dielectric layers. Additionally, we can shift the magnetic resonance at a lower ns also by using a smaller wavelength for the incident light.

Conclusion

We show that strong magnetic resonance can be created in SPR sensors based on Kretschmann configuration. The magnetic response μeff of the structure can be approximately an order of magnitude greater at resonance than that when out of resonance, which shows promises of the magnetic property of these structures in sensing applications. Thus, using μeff profiles for sensing could lead to a novel approach for detection of different critical organic and inorganic samples. Our calculations based on S-parameters retrieval method show that the sensor FoM can be greatly increased using the optical magnetism compared to that obtained using the conventional technique in state-of-the-art sensors, which often employ much more complex structures. We believe that the increased sensor performance using optical magnetism resonance in the presence of a target sample from a very simple planar layered structure can lead this approach to applications of sensing of variety of organic and inorganic materials.