1. Introduction

Magnetic skyrmions are topological quasiparticles in the magnetization process. Their compact size and topological protection characteristics make skyrmions have broad prospects for future applications in information processing, data storage and precision measurement [1–4]. Recently, there has been an increasing interest in finding optical counterparts for magnetic skyrmions. Efforts have been made to construct topological non-trivial vector vortices within the realm of optics, including the electromagnetic field vectors [5–7] and photon spin vectors [8–15].

It is well known that a light beam carries both spin angular momentum (SAM) and orbital angular momentum (OAM). The SAM is related to the polarization while the OAM is connected with the helical phase [16,17]. The strong spin-orbit interactions for the evanescent wave field at optical interfaces lead to the appearance of transverse spin, which lays the foundation for the construction of a tri-component optical spin texture [18–21]. Recently, considering the transverse and longitudinal spin of surface electromagnetic wave, Du et al. proposed a paradigm of photonic skyrmions in the system of plasmonic optical vortex (OV) [8]. Subsequently, a great number of theoretical and experimental studies have been made [9–15].

Generally, a photonic skyrmion is described through the spin texture of the light field in the cylindrical coordinates $(S_{r}, S_{\phi }, S_{z})$. In a plasmonic OV system, the photonic skyrmion is Neel-type $(S_{r}, 0, S_{z})$ [8,9]. Previous discussions about photonic skyrmions focus on their generation in different dielectric-metal hybrid systems [8,9] or dielectric systems with different materials [10,11], as well as the production of more complicated structures like skyrmion lattices [11,12] or skyrmion pairs [13]. As for the control of photonic skyrmions, former works mainly concentrate on controlling their general profiles [14] and positions [15]. Less attention has been paid on the detailed modulation of different components of the spin distribution. Moreover, most of the photonic skyrmions have a much greater $S_{z}$ value than $S_{r}$ (the ratios between their maximum values are often larger than 4.5). However, for the practical applications, it is necessary to adjust $S_{r}$ and $S_{z}$ components according to our intention. In this paper, we pay special attention to the topic of manipulating photonic skyrmion. Our study shows that, by changing the metal film thickness and the beam frequency, a photonic skyrmion with preferred spin distribution can be successfully modified. Moreover, as the metal layer is thin enough, the surface waves on both surfaces will couple with each other and form two different modes, corresponding to the symmetric coupling and anti-symmetric coupling respectively [20,22]. Therefore, we further discuss the distinctive variation characteristics of spin textures corresponding to two different coupling modes.

The paper is arranged as follows. In Section 2, we present the basic thin metal film structure and calculate the photonic skyrmion texture corresponding to the plasmonic OV excited in our system. Then, in Section 3, we discuss the detailed skyrmion patterns within the metal film, with a specific focus on the differences of symmetric and anti-symmetric coupling modes. In Section 4, we further extend our discussion to the situations when the dielectric materials of the upper and lower layers are different. Finally, we come to a general conclusion.

2. Photonic spin distribution

Our discussion starts with the thin-metal structure as Fig. 1. A metal film with the thickness $h$ and complex dielectric constant $\epsilon _{2}$ is sandwiched by two isotropic dielectric layers $\epsilon _{1}$ and $\epsilon _{3}$. Under the resonance condition, the electromagnetic wave satisfies:

 

Fig. 1. (a) Schematic diagram of the thin metal film structure, where symmetric coupling and anti-symmetric coupling modes are sketched respectively; (b) the dispersive relationship of SPP excited in (a) when $\epsilon _{1}=\epsilon _{3}=4$, $\epsilon _{2}$ is silver and $h=50 nm$, in which the upper branch corresponds to the symmetric coupling mode and the lower branch corresponds to the anti-symmetric coupling mode.

Download Full Size | PDF

The excitation of surface plasmon polaritions (SPPs) requires transverse magnetic (TM) mode, which, under the cylindrical coordinates, can be expressed as $\vec {E_{j}}=(E_{rj} \;\; E_{\phi j} \;\; E_{zj})^{T}$, $\vec {H_{j}}=(H_{rj} \;\; H_{\phi j} \;\; 0)^{T}$ ($j=1, 2, 3$ correspond to the $\epsilon _{j}$ regions respectively). Therefore, Eq. (1) can be further written as

We set

Based on the result of Eqs. (3) and (4), we may further express all the electric and magnetic field components in Eq. (2) with $E_{zj}^{(0)}$ and $f_{j}$ :

Now, we shall obtain all the components of the electromagnetic field by specifying the expression of $E_{zj}$ in Eq. (5). As we use vortex beam to excite SPP in our configuration, the independent variable $\phi$ is solely related to the vortex phase factor $e^{il\phi }$. Therefore, the continuity of the normal electric displacement vector $D$ and tangential electric field $E$ at $z=0$ and $z=h$ lead to the solution of $E_{zj}$ component in all the three regions as:

By substituting the expression of $E_{z2}$ in Eqs. (6) to (5), all the electric and magnetic components within the metal layer can be expressed by the basic variables $r$ and $z$:

According to the optical angular momentum theory [13], the SAM can be calculated through the equation $\mathbf {S_{j}} = \frac {1}{4\omega } \mathrm {Im} \left \{ \tilde {\epsilon }_{j} \left ( {\vec {E_{j}}}^{\ast } \times \vec {E_{j}} \right ) + \tilde {\mu }_{j} \left ( {\vec {H_{j}}}^{\ast } \times \vec {H_{j}} \right ) \right \}$, where in our case $\tilde {\epsilon }_{2}=\epsilon _{2}+\frac {\partial \epsilon _{2}}{\partial \omega }$ and $\tilde {\mu }_{2}=\mu _{2}+\frac {\partial \mu _{2}}{\partial \omega } \approx \mu _{2}$(the approximation is due to the high frequency of visible light region). Therefore, we may calculate the SAM of the SPP excited by the vortex beam, or the plasmonic OV within the metal region $\mathbf {S_{2}}$ by substituting Eqs. (8a) and (8b) to the SAM equation. (Note all the parameters in the equations are real numbers.)

We discuss the symmetrical configuration where the materials of the upper and lower dielectric layers are the same. We choose Printed Circuit Boards (PCBs) for the dielectric layers ($\epsilon _{1}=\epsilon _{3}=4$), silver for the metal film, and the thickness of the metal film $h=50 nm$.

Here, we adopt the lossless Drude model for silver, where $\epsilon _{2}$ can be expressed as:

We choose an angular frequency of $\omega _{0} =4.5 \times 10^{15} rad/s$. Bring $\omega _{0}$ into the dispersion relation Eq. (7), we obtain the evanescent wave vectors in the metal layer as $\kappa _{2s}= 5.85184 \times 10^{7} rad/m$ (symmetric coupling mode) and $\kappa _{2a}= 6.21992 \times 10^{7} rad/m$ (anti-symmetric coupling mode). Therefore, according to Eq. (9), we may calculate the spin distribution of the skyrmions within the metal layer. Due to the coupling effect of the upper and lower interfaces, the longitudinal distribution of the skyrmions is formed, where each z-transection of the metal layer along the $z$ direction has different skyrmion textures $(S_{r2}, 0, S_{z2})$. When $l=1$, on any z-transection except $z=-h/2$, the spin textures are Neel-type skyrmions. On $z=-h/2$ transection, however, the spin texture has the form $(0, 0, S_{z2})$.

In Fig. 2(a), we show the schematic diagram of the spin distributions of the skyrmions on the transections $z=-h/16$ and $z=-15h/16$. Figure 2(b) shows the changing law of the corresponding spin components $S_z$ and $S_r$ with the radial variable $r$. We find the pattern of the skyrmion texture possesses generality, which are valid for both the symmetric and anti-symmetric coupling modes.

 

Fig. 2. Spin distribution and corresponding radial diagram of the skyrmions at $z=-h/16$ (upper two plots) and $z=-15h/16$ (lower two plots) transections. The background color hue in (a) depicts $S_z$ component, while the direction and length of the arrows depicts $S_r$ component. The white circle shows the edge of the photonic skyrmion. (b) The radial variation of (a), in which the black line represents $S_z$ component and the red line represents $S_r$ component. The skyrmion region $|r| \leq r_{2}$ is marked with gray shade. The components are normalized to the maximal absolute value of total SAM.

Download Full Size | PDF

The boundary of the skyrmions is defined at the second zero point ($r=r_{2}$) of $S_{z}$. Within this range, $S_z$ component undergoes a spin reversal and $S_{r}$ component also reaches its extremum, then returning to 0. It can be found from Eq. (9) that $r_{2}$ is only correlated with $\beta$. In the expression of $S_{z}$, the component pertaining to the radius of skyrmion (the second zero point) is $\frac {1}{r}{J_{l}}'\left ( \beta r \right ) J_{l}\left ( \beta r \right )$. Our numerical calculation shows that the second zero point of this component is at $R = \beta r_{2} = 3.8317$. Thus, the boundary of the skyrmion is $r_{2}= 3.8317/\beta$. Under the same angular frequency $\omega$, it is obvious from the dispersive relation in Fig. 2 that $\kappa _{s} < \kappa _{a}$. So we have $\beta _{s} < \beta _{a}$, and $r_{2s} > r_{2a}$. In addition, if we change the materials of the upper and lower dielectric layers simultaneously, only the size and the total SAM intensity of the skyrmion texture will decrease with the increase of the $\epsilon$ of the dielectric layers.

3. Modulation of spin texture

3.1 Effect of thickness of the metal film on spin texture

By changing the thickness of the metal layer, we find that the general patterns of the skyrmion texture shown in Fig. 2 remain unchanged. In order to infer the changing pattern of the overall spin distribution of the skyrmions, we select two characterization quantities in Fig. 2(b) and discuss their change with the film thickness. For $S_{z}$ component, we choose the first extreme value (peak/valley) $S_{zm} = S_{z} (r=0)$. While for $S_{r}$ component, we choose its first extremum $S_{rm}$.

From Eq. (9), it can be found that $S_{zm}$ is determined by $\kappa _{2}$, $\omega$ and $z$, satisfying the equation:

Setting $\omega = 4.5 \times 10^{-9}$, the variation of $S_{zm}$ with $z$ is shown in the left column of Fig. 3, (a), (b) and (c) plot the situation when $h = 50 nm$, $30 nm$ and $10 nm$ respectively. The blue curve refers to the symmetric coupling mode, while the red curve refers to the anti-symmetric coupling mode. We can see that both of the two $S_{zm} \sim r$ curves are always symmetric with respect to $z=-h/2$ and reach their maximal values at $z=-h/2$. As the thickness $h$ gradually decreases, the absolute value of $S_{z}$ increases rapidly. Moreover, the gap between symmetric and anti-symmetric coupling branches is becoming increasingly large.

 

Fig. 3. Variation of $S_{zm}$ and $S_{rm}$ with $z$ as the film thickness $h$ changes to (a) $50 nm$ (b) $30 nm$, and (c) $10 nm$. The left column shows $S_{zm}\sim z$ relationship, and the right column shows $S_{rm}\sim z$ relationship. The blue lines correspond to symmetric coupling mode, while the red lines correspond to anti-symmetric coupling mode.

Download Full Size | PDF

For $S_{rm}$, we found that the curves of both modes are anti-symmetric with respect to $z=-h/2$, as shown in the right column of Fig. 3. This result is easy to understand as we consider the symmetry of our structure: the $S_r$ components of the upper and lower interfaces are opposite in sign. Therefore, for either symmetric or anti-symmetric coupling modes, there must be a sign-changing zero point at the central transection $z=-h/2$, on which the spin texture is no longer a skyrmion.

As the thickness $h$ decreases, the absolute value of $S_{rm}$ decreases simultaneously, which is exactly opposite to the changing pattern of $S_{zm}$. Meanwhile, the gap between symmetric and anti-symmetric coupling curves also becomes larger, which is, in turn, same as the changing pattern of $S_{zm}$. Hence, by adjusting the metal film thickness $h$, we can effectively control the proportion of $S_z$ and $S_r$ components to the total spin texture. For example, when $h=50 nm$, the normalized absolute value for different component extrema at the interfaces $z=0$ and $z=-h$ are $S_{zm}=0.41$ and $S_{rm}=0.91$; As the thickness decreases to $h=10 nm$, the $S_{zm}=0.94$ and $S_{rm}=0.17$. The ratio between the components $S_{zm} / S_{rm}$ changes from 0.45 to 5.53. For other transections within the metal film, the variation range of the ratio may reach infinity.

3.2 Effect of the frequency of the beam on spin texture

From Eq. (8), we can see that the frequency $\omega$ has direct influence on $S_{zm}$. It has also indirect influence on $S_{zm}$ through its influence on $\kappa _2$. We fix the film thickness $h$ at $50 nm$ and change the frequency $\omega$. The changing patterns of $S_{zm}$ and $S_{rm}$ at the $z=0$ and $z=-h/2$ transections are shown in the left column of Fig. 4. (The patterns at upper interface and the central transection are enough to reflect the general change of the longitudinal skyrmion distribution because of the structural symmetry.) We can see that as $\omega$ increases, the absolute value of $S_z$ increases under symmetric mode and decreases under anti-symmetric mode. The gap between the two modes becomes increasingly smaller.

 

Fig. 4. Variation of $S_{zm}$ and $S_{rm}$ with $\omega$ at (a) $z=0$ transection and (b) $z=-h/2$ transection as the frequency of the vortex beam source changes. We have omitted the $S_{rm}\sim \omega$ plot in (b) since $S_r$ component is always $0$ at $z=-h/2$. The blue curves correspond to symmetric coupling mode, while the red curves correspond to anti-symmetric coupling mode.

Download Full Size | PDF

Furthermore, it is worth noting that for the transection near $z=0$, there is a sign-changing zero point on the curve of symmetric mode, indicating that the sign of $S_{zm}$, or the central spin of the skyrmion $S_{z}(r=0)$, can be controlled by $\omega$ in the neighborhood of the zero point ($\omega =5.121 \times 10^{15} rad/s$). The result is also valid for the transection near the lower interface $z=-h$. However, it does not apply to the anti-symmetric mode, which always has a negative $S_{zm}$.

For $S_{rm}$, as mentioned in Section 2, the value of $S_r$ at $z=-h/2$ is $0$, so we only plot $S_{rm}$ curves at $z=0$. As depicted in the right plot of Fig. 4(a), the anti-symmetric curve is always above the symmetric curve, and the two curves are gradually approaching each other.

To summarize, we can see that with the increase of the frequency $\omega$, the absolute values of both $S_z$ and $S_r$ increase under symmetric mode and decrease under anti-symmetric mode. That is to say, by controlling the frequency, the absolute value of the total spin can be regulated. Meanwhile, we can regulate the relative proportion of $S_z$ and $S_r$ components by adjusting the thickness of the metal layer. Therefore, by controlling the thickness $h$ and frequency $\omega$, we can achieve the manipulation of both components $S_z$ and $S_r$, thus realizing the control of the skyrmion texture in our configuration.

4. Asymmetrically disposed structure

In this section, we discuss the asymmetrically disposed structure, where the materials of the upper and lower dielectric layers are different. In our configuration, we keep the upper layer as PCB ($\epsilon _{1}=4$) and change the lower layer to air ($\epsilon _{3}=1$). The material of the thin metal film is still silver, with a thickness of $h=50 nm$. By substituting these parameters to Eq. (6), we obtain the dispersion relation of the new configuration as shown in Fig. 5(b). Since the structural symmetry has been destroyed, symmetric and anti-symmetric coupling are supplanted by short-range (SR) and long-range (LR) coupling correspondingly [17].

 

Fig. 5. (a) Asymmetric set of the thin metal film structure, in which we change the material of lower dielectric layer to air; (b) the dispersive relationship of SPP excited in (a) when $\epsilon _{1}=4$, $\epsilon _{3}=1$, $\epsilon _{2}$ is silver and $h=50 nm$, in which the upper branch corresponds to the SR coupling mode and the lower branch corresponds to the LR coupling mode.

Download Full Size | PDF

We select $\omega =5.536 \times 10^{15} rad/s$, corresponding to the evanescent wave vectors $\kappa _{2s}= 4.654 \times 10^{7} rad/m$ (SR coupling mode) and $\kappa _{2a}= 8.932 \times 10^{7} rad/m$ (LR coupling mode). Numerical calculations show that the photonic skyrmions will still appear on each z-transection of the metal layer, but the longitudinal distribution is no longer symmetric with respect to $z=-h/2$. We still use the variation of $S_{zm}$ and $S_{rm}$ to characterize the new patterns of distribution. The results are shown in Fig. 6.

 

Fig. 6. Variation of (a) $S_{zm}$ and (b) $S_{rm}$ with $z$ under the asymmetrically disposed structure Fig. 5(a). The blue curves correspond to SR coupling mode, while the red curves correspond to LR coupling mode.

Download Full Size | PDF

As we can see, the absolute values of both $S_z$ and $S_r$ components corresponding to SR coupling mode are about two orders of magnitude larger than that of LR coupling mode. The extrema of the SR mode in Fig. 6(a) and (b) both appear at the lower (air-metal) interface, while the extrema of the LR mode both appear at the upper (PCB-metal) interface.

Moreover, it is worth noting that the $S_{rm}$ curve of the LR mode has a zero point at $z=-442h/512$, on which transection the skyrmion texture disappears. However, there is no zero point for the $S_r$ curve of SR mode, indicating that for SR coupling, $S_{r}=0$ cannot occur on any transection. Further numerical simulations show that the specific no-skyrmion plane for LR mode is related to the ratio between the dielectric constants of the upper and lower layers and independent of their absolute values. This suggests that we may adjust the dielectric constant of the upper or lower layer under LR coupling mode, so that the no-skyrmion plane can reach the z-transection we prefer. For SR mode, there is no such advantage. Adjusting the ratio between the dielectric constants of the upper and lower layers can only strengthen the total spin of the skyrmion on one side, and weaken it on the other side. However, a no-skyrmion plane cannot appear under SR mode.

5. Conclusion

In conclusion, the vortex beam can excite photonic skyrmions in the thin metal film. The coupling effect of the upper and lower layers results in a longitudinal (z-direction) photonic skyrmion distribution, where in every z-transection within $[-h, 0]$ there is a 2D photonic skyrmion. There are two coupling modes: symmetric and anti-symmetric coupling modes, and the distributions of spin are different for the two modes.

The preferred modification of photonic skyrmions can be achieved by modulating the thickness of the metal layer $h$ and the frequency of the incident beam $\omega$. On one hand, changing the thickness $h$ can effectively modulate the proportions of $S_{z}$ and $S_{r}$ components. With the decrease of $h$, the ratio $S_{z}$ to total spin $|S|$ increases. The differences of $S_z$ and $S_r$ between symmetric coupling and anti-symmetric coupling gradually increase. On the other hand, adjusting the frequency $\omega$ can manipulate $|S|$. As $\omega$ increases, the absolute value of $S_z$ component and $S_r$ component will decrease simultaneously with the same order of magnitude. The difference between symmetric and antisymmetric coupling modes becomes less pronounced. Moreover, for the symmetric coupling mode, the spin inversion of $S_z$ component near the surface of the metal layer can be effectively controlled by $\omega$ in the neighborhood of $\omega =5.121 \times 10^{15} rad/s$.

Moreover, we also discussed the influence of the relative dielectric constants $\epsilon _{1}$ and $\epsilon _{3}$, corresponding to the upper and lower dielectric layers. If we keep $\epsilon _{1}=\epsilon _{3}$ and increase the two constants simultaneously, it can be found that $S_z$ and $S_r$ components will decrease with the same proportions. If we let $\epsilon _{1}=4\epsilon _{3}$, It can be observed that the short range coupling mode has a spin intensity two orders of magnitude higher than the long range coupling mode. For the short-range coupling, the spin intensity in the upper half exceeds that in the lower half. For the long-range coupling, the situation is exactly the opposite. It is worth mentioning that, in the case of short-range coupling mode, there is no z-transection with $S_{r}=0$, which only exists for the long-range coupling mode, and the position of z-transection can be modulated by modulating the ratio of the dielectric constants of the upper and lower dielectric layers.