1. Introduction

When a bounded light impinges on the boundary of two media with different refractive indices, the reflected or refracted light beam undergoes displacements both in and perpendicular to the incident plane. The lateral shift was first observed by Goos and Hänchen in the experiment of totally internal reflection (TIR) [1] and then theoretically explained by Artmann [2]. The different plane wave components of a finite-sized light beam acquire different phase changes and the coherent superposition of these components then forms a reflected beam with a lateral displacement. The transverse Imbert-Fedorov shift [3,4], also known as the spin Hall effect of light (SHEL), results from the conservation of angular momentum and the spin-orbit interaction [5–7], and thus the photons with different spin (the left/right-hand circularly polarized) components accumulate on the two sides of the incident plane. Both Goos-Hänchen (GH) shift and Imbert-Fedorov (IF) shift are sensitive to the optical properties of materials and the structures of devices. The precise measurement of beam displacements has potential application in devising new types of optical devices. For example, the GH shift can be utilized for optical switching [8,9] and optical sensors [10–12]; the IF shift is applied to identify graphene layers [13], edge detection [14] and other optical sensors [15–17]. In general, the beam shifts in reflection or refraction from single interface are too small to be conveniently used in practice. Therefore, the enhancement and manipulation of beam shifts have attracted much attention in recent years. Different materials and various structures were employed to achieve large tunable GH and IF shifts, such as dielectric slab [18–20], graphene [21–23], surface plasmon resonance (SPR) systems [24–30], multi-layer nanostructures [31–34], coherent media [35–40].

Optical bistability (OB) in reflection from the layered structure containing nonlinear medium is of great interest because there is no need of an optical cavity [41–46]. Kaplan first explored the bistable reflection of light incidents on the interface of a nonlinear medium at grazing-angle [41]. The hysteretic response can result from the combined effect of the intensity-dependent nonlinear refractivity and the positive feedback from the change in the coupling efficiency of SPR [42,43] or guiding mode resonance [44,45]. Recently, multi-layer structures with graphene were proposed to realize OB [46–48]. In addition to the bistable reflection, a hysteretic behavior between the GH/IF shifts and the incident intensity of light beam can also be produced owing to the excitation of surface plasmons (SPs) [25,49]. There are two types of OB, absorptive type and dispersive type. In most of the above investigations, Kerr media with intensity-dependent refractive index were utilized and then the resulting OB is dispersive-type. Up till now, few studies focused on the absorptive OB with SPs. To the very recent, the interplay between nonlinear gain media and SPs was investigated to achieve bistable enhanced reflectance [50,51]. Nevertheless, to the best of our knowledge, the influence of saturated absorption effect on the nonlinear response of SPR system has not been reported before.

In this paper, we study the bistable reflection and beam shifts in a Kretschmann configuration containing a saturable absorbing medium. When the light incidents upon the prism-metal interface with a specific angle, SPs can be resonantly excited in the medium below the metal, resulting a dip in the reflectance spectrum, in which the phase of reflected light changes rapidly. The reflectivity and phase variation in the SPR dip depend strongly on the absorption coefficient of medium. Owing to feedback from the intensity-dependent saturable absorption effect, the reflectivity as well as the GH/IF shifts can be switched from one stable state to the other stable state, thus exhibiting hysteretic feature. We also discuss and analyze the system parameters, such as the thickness of metal film, the linear absorbing coefficient, on the hysteresis curves. The results may be helpful for devising new type of bistable devices and SPR-based devices.

2. Model and equations

We consider a three-layer Kretschmann configuration supporting SPR as shown in Fig. 1. A thin metal film with thickness ${d_{1}}$ is sandwiched between a high-index prism and a low-index dielectric medium which has the characteristic of saturable absorption. BK7 glass with permittivity ${\varepsilon _{1}} = 2.28$ and silver with permittivity ${\varepsilon _{2}} ={-} 29.384 + 0.365i$ at incident wavelength $\lambda = 780{\textrm{nm}}$ are taken as the prism and the metal layer. The saturated absorbing medium is a homogeneously broadened atomic vapor of which the dielectric constant can be described as

 

Fig. 1. Schematic diagram of Goos-Hänchen and Imbert-Fedorov effects in a three-layer Kretschmann configuration composed of a prism, a thin metal film backed by a saturable absorbing medium.

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Here ${k_{iz}} = {(k_0^2{\varepsilon _i} - k_x^2)^{1/2}}$ represents the normal wave vector in medium i, ${k_x} = \sqrt {{\varepsilon _1}} {k_0}\sin {\theta _i}$ denotes the wave vector along the x direction and ${k_0} = {{2\pi } / \lambda }$ is the wave vector in vacuum.

Because SPR can only be excited by light with p-polarization while the s-polarized light undergoes nearly total reflection, we focus on the reflectivity and lateral displacement of p-polarized light. According to the stationary phase theory [2], the GH shift, related to the reflected phase [${\varphi _p} = \arg ({r_p})$], is expressed as

The transverse shift is spin-dependent. The transverse displacements of different spin components are the same in magnitude but opposite in direction. The IF shift of the left-hand circularly polarized (spin) component of the reflected light beam can be described as [52]:

3. Results and discussions

Before embarking on a detailed analysis of the nonlinear response of the SPR system, we first consider the linear case that the incident light is weak ($I \approx 0$) and then the nonlinearity of the absorbing medium can be ignorable ($\alpha \approx {\alpha _0}$). The excitation of SPR is essential to the formation of OB of reflectance and beam shifts. When the tangential component of incident wave vector in the prism ${k_x} = \sqrt {{\varepsilon _1}} {k_0}\sin {\theta _i}$ matches the wave vector of SPs, SPR can be excited by a p-polarized light of which the energy penetrates into the lower medium by evanescent wave, thereby resulting in attenuated total reflection (ATR) at the resonance angle ${\theta _R} = {42.38^ \circ }$ around which a large change occurs in the reflected phase spectrum as shown by the solid lines in Figs. 2(a) and 2(b). The resonance angle can be expressed as

 

Fig. 2. (a) Reflectivity and (b) reflected phase shift as a function of the incident angle. The thickness of metal film is ${d_1} = 60\textrm{ nm}$.

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In the case of small absorption, i.e. ${\alpha _0} < < 1$, ${\theta _R}$ is approximately independent of ${\alpha _0}$. Nevertheless, the absorption coefficient has a significant influence on the minimum reflectivity ${R_{\textrm{min}}}$ at ${\theta _R}$. As ${\alpha _0}$ increases, ${R_{\textrm{min}}}$ is increased and the ATR dip is broadened, thereby reducing the slope of phase variation as shown in Fig. 2(b). Consequently, the GH and IF shifts are closed related to the absorption of the absorbing medium according to Eqs. (9) and (10).

For typical Kretschmann configuration, the reflection properties can be described approximately by the wave vector of the structure, which consists two parts: the eigen wave vector $\beta$ of the SPs and the offset $\Delta \beta$ of eigen wave vector caused by the metal layer with finite thickness ${d_1}$, which are described as [53]

When ${\Gamma _i} = {\Gamma _r}$, the minimum reflectivity goes to 0. In this case, we can obtain the optimal thickness of metal as ${d_{\textrm{opt}}}\textrm{ = }60.03\textrm{ nm}$. According to Eq. (15), the reflection spectrum is approximate to a Lorentzian profile of which the width is determined by the total loss ${\Gamma _i} + {\Gamma _r}$. In the case that $\alpha \ll 1$, the intrinsic loss can be derived from Eq. (13) by Taylor expansion of $\alpha$ (neglecting the high order items), given by

With the increase of absorption coefficient, ${\Gamma _i}$ increases gradually, thus resulting in the broadening of the ATR dip and the increase of reflectance, which are coincident with the results shown in Fig. 2.

In the following, we consider the nonlinear response of the reflection to the incident light intensity. As shown in Fig. 3, the reflectivity and the GH/IF shifts exhibit hysteretic feature. When the input power increases from zero, the absorption of the substrate as well as the intrinsic loss of SPs is large, thus leading to a high reflectance. The wide ATR dip then results in a small lateral displacement. The transverse displacement is also small due to the low ratio of $|{{{{r_s}} / {{r_p}}}} |$. Near the point B, the saturated absorption effect emerges and is becoming serious. As the input intensity increases to the high threshold ${I_h}$, an abrupt switch from B down to point C takes place. Due to the saturation, a narrow ATR dip with low reflectivity occurs at point C. Accompanying with the switching process of reflectance, the GH and IF shifts jump to the state with large values as result of the steep variation of phase and the enhanced ratio of $|{{{{r_s}} / {{r_p}}}} |$. If the input power is gradually decreased, the absorbing medium remains saturated. The states of low reflectance and large beam shifts can be maintained until a low threshold ${I_l}$ is reached. As the input decreases further, the incident intensity is not large enough to make the absorbing medium saturated and the intrinsic loss increases suddenly. As a result, the reflectance is then switched from point D up to point A, the state with high reflectivity. Simultaneously, the GH and IF shifts jump to the state with small values. Over the range of incident intensity ${I_l} < {I_0} < {I_h}$, the reflectivity as well as the beam shifts has two stable outputs which are switchable by the input power.

 

Fig. 3. Hysteretic responses of (a) Reflectivity, (b) GH shift and (c) IF shift as a function of the input light intensity. The incident angle is ${\theta _i} = 42.38^\circ$ which corresponds to ${\theta _R}$, the thickness of metal film is ${d_1} = 60\textrm{ nm}$ and the linear absorption coefficient is ${\alpha _0} = 0.015$.

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In the scheme of OB with the excitation of SPs in Kerr media [25,42,43,46], the system is switched between ATR and TIR due to the movement of the resonance angle. However, in our scheme, the resonance angle remains unchanged [see Fig. 2(a)] and the two stable outputs is owing to the intensity-dependent absorption coefficient. In Fig. 4, we plot the internal electric field profile at the points marked with A, B, C and D in Fig. 3(a). In the process A-B, the internal electric field is weak due to the large absorptive loss of the absorbing medium. As the input power approaches to ${I_h}$, the saturation effect becomes remarkable. Meanwhile, the nonlinear absorption coefficient as well as the intrinsic loss of the medium is reduced, thus enhancing the electric field, which further strengthen the saturation effect. As a result, a positive feedback is established that causes an abrupt enhancement of the electric field from point B to point C. The hysteresis curve jumps from one stable state to the other stable state at the same time. In the process C-D, although the input power is decreasing, the enhanced electric field maintains the absorbing medium be deeply saturated. While the incident intensity is decreased to ${I_l}$, the saturation effect becomes weak and the intrinsic loss increases, then the electric field is reduced gradually from point C to point D. Due to the feedback mechanism, the state of low reflectance can no longer be sustained, thereby leading to the transition from point D to point A where the electric field is small and the absorption is large. The cooperative effect of the intensity-dependent absorption and absorption-dependent electric field gives rise to the inherent feedback and therefore a hysteretic response.

 

Fig. 4. Distribution of the internal electric field profile in the Kretschmann configuration. ${E_0}$ is the amplitude of the electric field of incident light. The four curves respectively correspond to the points A-D labeled in Fig. 3(a).

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The thickness of the metal film plays a crucial role in the excitation of SPs. In Fig. 5, we investigate the influence of the thickness ${d_1}$ on the bistable curves of reflectivity, GH shift and IF shift. With the increase of ${d_1}$, both ${I_l}$ and ${I_h}$ of the hysteretic response are increased, but ${I_h}$ increases faster than ${I_l}$. As a result, the region exhibiting two stable states becomes large, i.e. the width of hysteresis loop is broadened. In the Kretschmann configuration, the radiation loss ${\Gamma _r}$, due to the energy leakage from the metal film with finite thickness, decreases exponentially with the increase of ${d_1}$ as indicated by Eq. (14). According to Eq. (16), the minimum reflectivity ${R_{\min }}$ is increased and then the lower branch of hysteresis goes up as illustrated in Fig. 5(a). Meanwhile, the slope of reflected phase shift and the reflectivity ratio $|{{{{r_s}} / {{r_p}}}} |$ are reduced. Therefore, the upper branches of the hysteresis loops for GH/IF shifts are decreased correspondingly [see Figs. 5(b) and 5(c)]. Furthermore, if the thickness is large enough, the phenomenon of OB will disappear and we cannot obtain two stable outputs for any input intensity.

 

Fig. 5. (a) Reflectivity, (b) GH shift and (c) IF shift versus incident intensity with different thickness of metal film ${d_1}$. The incident angle and the linear absorption coefficient are the same as those in Fig. 3.

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As increasing the metal film thickness from ${d_1} = 60\textrm{ nm}$, the Kretschmann structure transit from the critical-coupled state to the under-coupled state, so a larger incident intensity is required to excite OB in the structure with a thick metal film. In Fig. 6, we plot the distributions of the internal electric field in the three-layer structure at the points C₁, C₂ and C₃ shown in Fig. 5(a). The inputs at these points correspond to the high thresholds of the hysteresis loops. When ${d_1}$ is close to the optimal thickness ${d_{\textrm{opt}}}\textrm{ = }60.03\textrm{ nm}$, the electric field can be greatly enhanced in the saturable absorbing medium, and therefore the thresholds for the bistable response are small. As ${d_1}$ increases, the electric field strength decreases gradually, and therefore, a higher input power is necessary to make the absorbing medium saturated, resulting in the increase of the threshold ${I_h}$. Similarly, the minimum input power required to maintain deep saturation is smaller for a thin metal film, thus the threshold ${I_l}$ decreases with ${d_1}$.

 

Fig. 6. Distribution of the internal electric field in the Kretschmann configuration at the three points C₁, C₂ and C₃ shown in Fig. 5(a).

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In the case of ${d_1} < {d_{\textrm{opt}}}$, the bistable behavior can also be observed as shown in Fig. 7. As ${d_1}$ decreases, both the high and low thresholds are decreased. The width of the hysteresis loop is getting smaller and the OB response will disappear if ${d_1}$ is further decreased. At the high threshold ${I_h}$, the absorption transits from a large value to a small value where ${\Gamma _i}$ is still larger than ${\Gamma _r}$, then the GH shift at ${I_h}$ is negative [see Fig. 7(b)]. As the input intensity increases gradually from ${I_h}$, the absorption coefficient is decreased owing to the saturation effect. At the same time, ${\Gamma _i}$ becomes smaller and gets closed to ${\Gamma _r}$. Consequently, the GH shift is increasing with the input power. However, for a large enough input, ${\Gamma _i}$ can be smaller than ${\Gamma _r}$ and a positive lateral shift is then obtained [see the tails of the dashed and dotted lines in Fig. 7(b)]. The bistable response of IF shift exhibits similar tendency and the transverse displacement turns from positive to negative for a large input as shown in Fig. 7(c). Therefore, the phenomena of bistable beam shifts can be found in both the cases ${d_1} > {d_{\textrm{opt}}}$ and ${d_1} < {d_{\textrm{opt}}}$. Nevertheless, the hysteretic behaviors are more complex for the case ${d_1} < {d_{\textrm{opt}}}$.

 

Fig. 7. (a) Reflectivity, (b) GH shift and (c) IF shift versus incident intensity with different thickness of metal film ${d_1}$. The incident angle and the linear absorption coefficient are the same as those in Fig. 3.

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The effect of linear absorption coefficient ${\alpha _0}$ on the bistable reflectance, GH shift and IF shift is shown in Fig. 8. According to the previous analysis, the minimum reflectivity increases with ${\alpha _0}$ as a result of the increasing intrinsic loss ${\Gamma _r}$. Therefore, the initial reflectivity in the case of small signal ($I \approx 0$) is lower for small ${\alpha _0}$ [see Fig. 8(a)]. As ${\alpha _0}$ increases, the thresholds and the width of hysteretic loop increase accordingly. At the points C₁, C₂ and C₃, the reflectance is smaller for large ${\alpha _0}$, which corresponds to a narrower ATR spectrum. With an input intensity a little beyond the switch-up threshold ${I_h}$, both the lateral and the transverse displacements are increased with ${\alpha _0}$ owing to the increasing angular variation of phase ${{\textrm{d}{\varphi _p}} / {\textrm{d}{\theta _i}}}$ and reflection ratio $|{{{{r_s}} / {{r_p}}}} |$. The change of the bistable curves is related to the electric field in the saturable absorbing medium. The distributions of the internal electric field at the points C₁, C₂ and C₃ in Fig. 8(a) are plotted in Fig. 9. Although the electric field in the absorbing medium increases with ${\alpha _0}$, a large input intensity is still essential to achieve deep saturation, thus the hysteretic loop moves right, the direction of increasing incident intensity.

 

Fig. 8. (a) Reflectivity, (b) GH shift and (c) IF shift versus incident intensity with different linear absorption coefficient of absorbing medium ${\alpha _0}$. The incident angle and the thickness of metal film are the same as those in Fig. 3.

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Fig. 9. Distribution of the internal electric field in the Kretschmann configuration at the three points C₁, C₂ and C₃ shown in Fig. 8(a).

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The thresholds of the hysteresis loop are related to the saturable intensity of the medium. The saturation effect can be explained by a two-level system, of which the saturable intensity is ${I_s} = {{{\hbar ^2}} / {({{\mu^2}{T_1}{T_2}} )}}$, where $\mu$ is the electric dipole moment of the transition, ${T_1} = {1 / {{\gamma _\parallel }}}$ represents the decay time of the population from the upper level to the lower level, and ${T_2} = {1 / {{\gamma _ \bot }}}$ denotes the decoherence time of the transition. ${\gamma _\parallel }$ and ${\gamma _ \bot }$ are the decay rate of population and the dephasing rate of the transition, respectively. ${\gamma _ \bot }$ is also the linewidth of the absorption spectrum. Therefore, the medium with a narrow absorption linewidth corresponds to a low saturable intensity and low thresholds of OB. For example, the saturable intensity is about $2.5\textrm{ mW/c}{\textrm{m}^2}$ for the D₂ line of cold rubidium atoms of which the natural linewidth is ${\gamma _ \bot } = 2\pi \times 6\textrm{ MHz}$. However, for a homogeneously broadened media, the linewidth, mainly determined by the collision process between gas atoms or the lattice vibration of solid crystals, is of the order of GHz or even THz, then the saturable intensity is of the order of W/cm² or kW/cm². Moreover, the response time of OB with the excitation of SPs, a critical parameter for the operation of optical switching, is mainly determined by the nonlinear material and the structure [54]. For the saturable absorbing media, the response time is several times of ${T_2}$, which is of the order of $\mathrm{\mu s}$ for cold gas media, and $\textrm{ps - ns}$ for hot gas media and solid materials. If the response time of nonlinear absorption is short enough (sub-ps or fs), the switching time also depends on the structure of which the switching time can be sub-ps for OB with the excitation of SPs [55].

4. Conclusions

In summary, the nonlinear reflection of a light beam is explored in a Kretschmann configuration containing saturable absorbing medium. With the excitation of SPs, the reflectivity and the angular variation of reflected phase at resonance depend strongly on the intrinsic loss of the system, then the lateral and transverse displacements of the reflected beam are closely related to the absorption coefficient of medium. Owing to the intensity-dependent saturated absorption and the inherent feedback effect, the reflectance and the GH/IF shifts exhibit nonlinear hysteretic behavior. We also study the factors that affect the bistability of reflection. The thickness of metal film ${d_1}$ has influence on the radiation loss of the structure, which then leads to the change of switching thresholds as well as the width of hysteresis loop. The phenomenon of OB is easy to occur when ${d_1}$ is around the optimal thickness ${d_{\textrm{opt}}}$. The linear absorption coefficient ${\alpha _0}$ also affect the thresholds and the shape of bistable curve. Large ${\alpha _0}$ leads to the increase of thresholds, while the phenomenon of OB will disappear for a small ${\alpha _0}$.

It is worthy to make a comparison between our scheme and the previous schemes of OB with the excitation of SPs in Kerr nonlinear medium [25,42,43,46]. For the Kretschmann structure with Kerr medium, the system is switched between two modes, attenuated total reflection and totally internal reflection, by tuning the resonance angle via power-dependent refractive index of Kerr medium. Hence OB is dispersive-type in these investigations. However, the resonance angle in our scheme remains unchanged, and an absorptive OB is obtained. The two stable states of system results from the intensity-dependent intrinsic loss which is controlled by the saturation effect of absorbing medium. The results may provide an alternative avenue for controlling light by light.