1. Introduction

Spatial dispersion caused by the repulsion between free electrons inside metals is described by the first-order correction to the Drude’ s model [1], which is the zeroth-order term of the description of the free-electron gas [2,3]. This nonlocal phenomenon implies that there is a limitation to the Drude’s model. Although the impact of this nonlocal effect on optical response of metals was proposed as early as 1960s, it was not received attention continuously due to a lack of experimental confirmation. However, recently there report some interesting and exotic physical effects originated from the nonlocal characteristics, such as the existence of multipole surface-plasmon modes at simple-metal surfaces [4], the significant blueshift in quantum plasmon resonances [5,6], the field enhancement of film-coupled nanospheres [7], and the size-dependent damping in individual metallic nanoparticles [8]. These experimental works have greatly reignited one’s interest in such nonlocal effects. One approach to tackle the problem of this effect involves the use of hydrodynamic model [2], and it can provide predictions in numerical simulations of plasmonic nanostructures [7]. In the framework of this approach, several structures that can be impacted by nonlocal effect have been demonstrated, such as hourglass plasmonic waveguides [9], the interface between a metal and a dielectric with high permittivity [10], slab system [11], composite system [12], and classical prism coupler system [13]. Moreover, numerical tools based on the hydrodynamic model have also been developed simultaneously [14–17]. These researches are beneficial in the field of nanophotonics.

The Goos-Hänchen (GH) shift, defined as the lateral displacement from the position predicted by geometrical optics when the light beam is totally reflected by the dielectric interface, was first observed experimentally in 1947 [18], and then it was theoretically explained by Artmann using the stationary-phase method [19]. In view of the potential applications in optical switches [20], de/multiplexer [21], sensor [22], and polarization beam splitting [23], it has gained considerable attention in decades. However, since its tiny magnitude puts a limit to the validity of GH shift in applications, various optical structures that can result in significantly large GH shift have been proposed. The attenuated total reflection (ATR) in classical Kretschmann-Raether geometry is one of the ways applied to obtain bidirectional beam displacements when the surface plasmon resonance is properly excited [24]. Actually, a plenty of studies have demonstrated the effect of materials-dependence characteristics and structural parameters on the GH shift and Imbert-Fedorov shift in the Kretschmann-Raether model [25–27]. While, there still lacks of investigation on the influence of the nonlocal effect on GH shift in such a structure.

Here, we investigate the effect of the spatial dispersion of metals on the optical response in the KR configuration. The comparison of dispersion relations predicted by the Drude’s and hydrodynamic models reveals that the impact of the nonlocal effect is enhanced at large wavevector. In contrast to the existing results in the literature, we focus on the behavior of the optical properties of the reflected light, and the corresponding behaviors are discussed in the context of a phenomenological model and Kretchmann’s theory. These findings may be helpful for the understand the light-matter interactions in complex metallodielectric structures.

The structure of this article is as follows. In Section 2, we introduce the theoretical formalism in the framework of hydrodynamic approach. In Section 3, we compare the nonlocal response against the local response to study how the nonlocality changes the optical properties of the reflected light in the system. Finally, a conclusion is presented in Section 4.

2. Materials and methods

The Kretschmann-Raether configuration is shown in Fig. 1. A metal layer of thickness $d$ with permittivity $\varepsilon _{m}$ is bounded by the upper and down semi-infinite dielectric media characterized by $\varepsilon _{1}$ and $\varepsilon _{2}$, here $\varepsilon _{1}>\varepsilon _{2}$. Because the nonlocality has an impact on $p$-polarization only [14], here we consider the $p$-polarized light beam to be incident upon the system with an angle of incidence $\theta$ greater than the critical angle of total internal reflection. The light will be reflected from the dielectric-metal interface. Assuming the time dependence of the fields is $e^{-i\omega t}$, where $\omega$ is the angular frequency of light, the electromagnetic fields in medium I can be written as

 

Fig. 1. Schematic diagram of lateral shift $S_{r}$ in the Kretschmann-Raether configuration. $\varepsilon _{1}$, $\varepsilon _{2}$, and $\varepsilon _{m}$ represent the permittivity of dielectric medium I, II and the metal with the thickness $d$, respectively, here $\varepsilon _{1}>\varepsilon _{2}$. $E_{i}$ and $E_{r}$ are the electric fields of incident and reflected light, respectively, and $\theta$ is the angle of the incidence.

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In general, the Maxwell’s equations that describe the electromagnetic fields inside metals are

In Case I, the divergence of the electric field is zero ($\triangledown \cdot E=0$), this illustrates that the nonlocal effect is overlooked for transverse wave. In this situation, Eq. (8) reduces to the usual expression

In Case II, the curl of the electric field is zero ($\triangledown \times E=0$), the underlying condition behind this case is that the magnetic field in the $y$ direction can not be existed for longitudinal wave. Hence, the expression of Eq. (8) is simplified to

According to the usual boundary conditions, the tangential components of fields are continuous at the interface of $z=0$

According to the stationary-phase theory, the GH shift is proportional to the partial derivative of the reflection phase to the angle of incidence, so if the incident beam is wide enough, the lateral shift of the reflected light can be expressed as [19]

In our calculation, we choose $\text {TiO}_{2}$ as medium I (described by a Cauchy formula [16,29]), glass of high refraction index 1.81 as medium II, the material parameters for Ag are $\omega _{p} =8.16$eV, and $\gamma =0.049$eV for damping rate [16].

3. Numerical results and discussion

To illustrate the influence of nonlocality on the optical response of the system, the dispersion relations of the system under different $d$, which correspond to the poles of the reflection coefficient, and light-cone lines for two different dielectric medium are plotted in Fig. 2. The red dashed and blue solid lines denote the dispersion relations predicted by using the Drude’s model and by taking the nonlocality into account, the pink and green-solid lines represent the light-cone lines in medium I and II, respectively. The pink and green-dotted lines denote pure bound modes at the individual interfaces of metal-medium I and metal-medium II, respectively. As a consequence of the asymmetric structure (i.e. $\varepsilon _{1} \ne \varepsilon _{2}$), there are two branches of dispersion curves in Fig. 2. The modes of the upper and lower branches denote the bound supermodes, also termed as coupled modes at the interfaces of metal-medium II and metal-medium I, respectively [30]. The crossing points of upper lines and the light-cone lines in medium I suggest that the coupled modes at the metal-medium II interface can be excited by the light from medium I coupled under special conditions.

 

Fig. 2. The normalized dispersion curves for the configuration with lossless metal shown in Fig. 1 predicted by nonlocal (blue solid lines) and local theories (red dashed lines) under $k_{p}d=0.2$ ($d\approx 5$nm) (a), $k_{p}d=0.5$ ($d\approx 12$nm) (b) and $k_{p}d=2.0$ ($d\approx 48$nm) (c). The light-cone lines in medium I and II are indicated as the pink and green solid lines, respectively. The pink and green dotted lines denote bound modes at interface of metal-medium I and metal-medium II, respectively.

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For the upper lines, as $k_{x} \to 0$, the nonlocal dispersion lines coincide with the cases of local effect, and they asymptotic with $k_{x} =k_{0} \sqrt {\varepsilon _{2} }$, the light-cone line in medium II. It shows that the impact of nonlocal effect can be generally ignored in the small $k_{x}$ region. As result of the coupling between the two pure bound modes supported by the individual metal-dielectric medium interface, at the case of $k_{p} d \ll 1$, $\omega$ increases, then decreases with the increasing of $k_{x}$, whose characteristic is different with the case of large $d$. As $d$ increases, the upper mode evolves into the pure bound mode supported by the individual metal-dielectric medium II interface (shown by the green dotted lines in Fig. 2) due to the decoupling [31]. We note that increasing $k_{x}$ leads to a higher discrepancy between local and nonlocal cases. As $k_{x} \to \infty$, $\omega$ increases linearly and slowly for the nonlocal case which is distinct from the local case where $\omega$ tends to constant $\omega _{p} /\sqrt {1+\varepsilon _{2}}$. But anyways, both of them are insensitive to the thickness of metal. This proves that the role of nonlocal effect becomes significant for large value of $k_{x}$. In order to obtain a meaningful difference, we choose $k_{x}$ as high as possible and this is reason why we choose $\text {TiO}_{2}$ as medium I, whose permittivity is the highest in the visible range [32].

The frame of a phenomenological model can be employed to illustrate the impact of the nonlocality on the GH shift near the reflection dip in such a system. The reflection coefficient can be expressed in the form of [33,34]

 

Fig. 3. (a) The trajectories of the zeros (solid lines) and poles (dashed lines) as a function of $d$ in the complex plane of propagation constant. The blue and red lines denote the cases of nonlocal and local theories, respectively, and the direction of arrow indicates the increasing change of thickness $d$. The parallel dashed lines show where the critical thicknesses are expected for zeros (black lines) and poles (blue line for nonlocal case and red line for local case). Dependence of $|r^{loc} |$ (b) and $|r^{nloc} |$ (c) on the incident angle and thickness $d$, here $\lambda =475$nm.

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From Fig. 3(a), we note that the pole and zero begin from the same point corresponding to the single interface between the metal and medium II, namely $d\to \infty$, as $d$ decreases, they move away from each other. The poles move in the upper half of the complex plane only, and they can not cross the real axis for a physical reason. As interpreted in Ref. [33], if it did, there is infinite reflection coeffience for some real angle of incident light. While both the local zero and nonlocal zero shift from the lower half to the upper half of the complex plane with the increasing of $d$. At critical $d_{c}$, they cross the real axis, which corresponds to the minimum reflection caused by the total absorption shown in Fig. 3(b) and 3(c). From Fig. 3(a), the critical thickness moves to a large value due to the nonlocality of metal. Furthermore, the new critical thickness can be expressed as

In previous works, the critical thickness $d_{c}^{loc}$ has been theoretically and experimentally verified for local case interpreted via the balance between the internal damping and radiation damping [24,35,36]. As discussed in Refs [35,36], the reflectance of light can be transform to

Due to the loss of metal, the dispersion relation is complex-valued transcendental equations of the implicit form $C_{1}^{loc}C_{2}^{loc}- B_{1}^{loc}B_{2}^{loc}\varphi _{m}^{2}=0$ for the local case. In this situation, the value $k_{x}$ makes the denominator of Eq. (28) zero, and $k_{x}^{sp}$ corresponds to the movements of pole $\tilde {k}_{xp}$ in Fig. 3(a). As $d$ increases, the value of ${k_{xp}^{loc}}''$ decreases, so that the radiation damping ${{k_{x}^{R}}}''={{k_{x}^{sp}}}''-{{k_{x}^{0}}}''$ also decreases. The coupling between the incident light and the surface plasmon wave changes between the overcoupled regime and the undercoupled regime. When the nonlocality of metal is taken into account, the change of two dampings leads to the movement of the critical thickness.

Figure 4 displays the dependence of $|r^{loc}|$ and $|r^{nloc}|$ on the angle of incidence for different values of thickness. For the local case (see the red lines in Fig. 4), when $d<d_{c}^{loc}$ (for $d=44$nm to $47.65$nm in Fig. 4(a)), the $|r_{min}^{loc}|$ is decreasing near the resonance as $d$ increases. Conversely, when $d>d_{c}^{loc}$ (for $d=47.90$nm to $50$nm in Fig. 4(c) and 4(e)), the value of $|r_{min}^{loc}|$ increases. Taking into account the effect of the nonlocality (see the blue lines in Fig. 4), as $d$ increases, the value of $|r_{min}^{nloc}|$ also change from decreasing (for $d=44$nm to $48$nm in Fig. 4(b) and (d)) to increasing (for $d=48.2$nm to $50$nm in Fig. 4(f)) with the change from overcoupled to undercoupled. In the range of $d_{c}^{loc}<d<d_{c}^{nloc}$ from Fig. 4(c) and 4(d), $|r^{loc}|$ corresponds to the undercoupled regime, while, $|r^{nloc}|$ corresponds to the overcoupled regime, hence there is another critical $d$ satisfied the condition of $|r_{min}^{loc}|=|r_{min}^{nloc}|$. In the meanwhile, the corresponding phase shifts show the opposite behaviors.

 

Fig. 4. Dependence of amplitude of the reflected light on the incident angle under different $d$ calculated by local (red lines) and nonlocal (blue lines) theories. The other parameters are the same as in Fig. 3.

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The dependence of phase $\phi _{r}^{loc}$ and $\phi _{r}^{nloc}$ on the angle of incidence for different values of thickness is plotted in Fig. 5. According to Eq. (35), the phase can exhibit opposite behaviors as the function of $\theta$ depending on the location of the imaginary parts of both the zero and the pole in the complex plane. However, the poles are located in the upper half of the complex plane only. Consequently, the behaviour of the phase shift for the reflected light is determined by the movement of zero in the complex plane. For the case of $\phi _{r}^{loc}$ (see the red lines in Fig. 5), when $d<d_{c}^{loc}$ (for $d=44$nm to $47.65$nm in Fig. 5(a)), the phase is monotonically decreasing near the surface plasmon resonance. Conversely, when $d>d_{c}^{loc}$ (for $d=47.90$nm to $50$nm in Fig. 5(c) and 5(e)), it is increasing function of the angle of incidence, which is in accordance with the movement of zeros changing from lower half to upper half of the complex plane. With effect of the nonlocality (see the blue lines in Fig. 5), as $d$ increases, the phases also change from monotonically decreasing (for $d=44$nm to $48$nm in Fig. 5(b) and 5(d)) to increasing (for $d=48.2$nm to $50$nm in Fig. 5(f)). We conclude that the behaviours of phase for both cases show similar changes in the range of $d<d_{c}^{loc}$ (Fig. 5(a) and 5(b)) and $d>d_{c}^{nloc}$ (Fig. 5(e) and 5(f)), but they are different in the range of $d_{c}^{loc}<d<d_{c}^{nloc}$ from Fig. 5(c) and 5(d) due to the zeros are located in the different half of the complex plane, which lead to the different behaviours for the lateral shift of the reflected light.

 

Fig. 5. Dependence of phase shift of the reflected light on the incident angle under different $d$ calculated by local (red lines) and nonlocal (blue lines) theories. The other parameters are the same as in Fig. 3.

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Figure 6 demonstrates the lateral shifts for different thicknesses. The red and blue lines represent the results from local and nonlocal theories, respectively. From Fig. 6, it can be seen that the lateral shift changes from positive to negative (local case shown by red lines in Fig. 6(a) and 6(c), nonlocal cases shown by blue lines in Fig. 6(d) and 6(f)) result from the corresponding phase changes from decrease to increase around the reflection dip, as $d$ increases. When $d$ approaches to $d_{c}$, the sharp slope of the phase shift leads to the giant magnitude of lateral shift, for example, from Fig. 6(a) and (6(c)) at the case of $d=47.65$nm (or $47.9$nm), $S_{r}^{loc}=1000 \lambda _{1}$ (or $S_{r}^{loc}=-2500 \lambda _{1}$) for local lateral shift. While, from Fig. 6(d) (6(f)) at the case of $d=48$nm (or $48.2$nm), $S_{r}^{nloc}=3000 \lambda _{1}$ (or $S_{r}^{nloc}=-1500 \lambda _{1}$) for nonlocal lateral shifts. It can be found that the nonlocality of metal has a sensitive impact on the lateral shifts. When $d<d_{c}^{loc}$, shown in Fig. 6(a), compared with the large positive $S_{r}^{loc}$ (red lines), the value of $S_{r}^{nloc}$ is small (blue lines). In the range between $d_{c}^{loc}$ and $d_{c}^{nloc}$ in Fig. 6(b) and 6(c), the negative $S_{r}^{loc}$ becomes positive $S_{r}^{nloc}$ with the effect of nonlocality. When $d>d_{c}^{nloc}$ in Fig. 6(d), the nonlocal effect makes the small negative $S_{r}^{loc}$ change to large negative $S_{r}^{nloc}$.

 

Fig. 6. Dependence of lateral shifts on the incident angle under different $d$. The red and blue lines correspond to the local and nonlocal case, respectively. The other parameters are the same as in Fig. 3.

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The results can be applied to other wavelength of the incident light. Figure 7 shows the critical thickness as a function of wavelength. From Fig. 7, we can see that both $d_{c}^{loc}$ and $d_{c}^{nloc}$ increase with the increasing of $\lambda$. Nonetheless, $d_{c}^{nloc}>d_{c}^{loc}$ is always satisfied.

 

Fig. 7. Dependence of critical thickness on the wavelength of incident light. The blue and red lines correspond to the nonlocal and local theory, respectively.

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4. Conclusion

We have investigated the optical response near the surface plasmon excitation in the Kretschmann-Raether structure by considering metals in the framework of the hydrodynamic model. In contrast to the dispersion relations predicted by the Drude model, we have found that the impact of the nonlocal effect on the dispersion is significantly more evident at large wavevector. We have used the traditional Kretchmann’s theory and phenomenological model to investigate the behaviors of optical properties, such as amplitude and phase of the reflected light, and its GH shift, and the critical thickness can be existed where the critical coupling and real zero in the complex plane of propagation constant are expected.

The nonlocal effect makes the critical thickness shift to a large value, and there are three regimes of thickness according to different influence of nonlocality on the behaviors of optical properties. In the region between the two different critical points predicted by the local and nonlocal theories, respectively, the lateral shift of the reflected light show opposite behaviours. When the thickness of the metal is below the critical thickness predicted by the local model, the lateral shift for the nonlocal model is smaller than that for the local model, and it reverses completely when the metal’s thickness is beyond the critical thickness predicted by the nonlocal model.

We have also investigated the influence of nonlocality on the optical properties for a simple insulator-metal-insulator structure in our work. Strictly speaking, the impact of nonlocality cannot be considered dominant for the single or two interface problems, but it can be more pronounced in multilayered systems. The calculations and results provide a solid foundation for the understanding of nonlocal effect in the complex plasmonic waveguide structures and the design of the optical devices.