The reflection of a TM-polarized light beam from a Kretschmann configuration with a saturable gain medium is investigated theoretically. Here, the dielectric constant of the gain medium is described by a classical Lorentzian oscillator model. When surface plasmon polaritons are effectively excited in this structure, it is demonstrated that the curves of enhanced total reflection (ETR) show different shaped hysteresis loops associated with optical bistability owing to gain saturation effect. The effects of the angle of incidence, the thickness of metal film, and the value of small-signal gain on bistable ETR are discussed in detail in a homogeneously broadened (HB) gain medium at line center. Analogous results can also be obtained in an inhomogeneously broadened (inHB) gain medium, while the two switch thresholds and the width of optical bistability hysteresis in an inHB gain medium are significantly different from those in a HB gain medium.
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Surface plasmon polaritons (SPPs) are localized electromagnetic waves, which propagate along the interface between a metal and a dielectric and are evanescent in the direction perpendicular to the interface [1–5]. Optical excitation of SPPs exists in various configurations, such as attenuated total reflection (ATR) excitation in the Kretschmann geometry , ATR excitation in the Otto geometry, diffraction excitation on a grating, diffraction excitation on a rough surface, and excitation with a scanning near-field optical microscopy probe . SPPs have gained much attention in recent years due to their unique electromagnetic properties. Typically, the features that strong enhancement and spatial confinement of the electromagnetic field of SPPs have been exploited in various fields, including plasmonics [7–9], nonlinear plasmoncis , nano-optics [3,6], and nano-photonics [8,11,12]. In fact, many applications of SPPs are limited by absorption loss associated with the imaginary part of the dielectric constant in metal. Therefore, many attempts have been made to promote the performance of optical SPP-based systems by reducing the loss of SPPs . Among them, the adjacent dielectric with optical gain has been proposed as a possible solution to this problem [4,14–22].
The Kretschmann geometry containing a gain medium is considered as a good candidate to explore the new features and the potential applications of SPPs. For example, this structure has been widely employed to study the strong coupling between SPPs and gain media [4,5,23–25], amplified spontaneous emission of SPPs [19,26], and stimulated emission of SPPs [15,27]. Based on this structure, an all-optical spatial light modulator has been successfully demonstrated , and an optical differentiator and integrator has been proposed . In addition, enhanced total reflection (ETR) with the simultaneous excitation of SPPs in this structure has been theoretically predicted by G. A. Plotz et al. . And similar results have been further illustrated by M. A. Noginov et al. . However, gain saturation effect  and spectral broadening of the gain medium are rarely considered in above mentioned works.
On the other hand, optical bistability (OB) is a nonlinear optical phenomenon and is of ongoing interest almost four decades due to its possible applications in all-optical information processing. In early studies, one of the ways to realize OB is introducing Kerr-type nonlinear materials into a feedback system . As for most nonlinear optical processes, the Kerr effect is weak, such that large input power is required to obtain a significant response. Then, the use of SPPs, which cause strong confinement and enhancement of the electric-field, is proposed to enhance the nonlinear effects  so as to reduce the thresholds of OB [33–35]. As a result, OB in plasmonic systems is one of the hot topics and has been widely studied, both theoretically [35–41] and experimentally [33,34,42,43]. By means of plasmonic platforms, bistable ATR [33,36–38,42], bistable transmission [35,39,43], bistable lateral shift  and so on, many nonlinear phenomena of bistability have been investigated. However, to the best of our knowledge, the phenomenon of bistable ETR has not been reported before.
In this paper, we theoretically investigate the reflection of a TM-polarized light beam from a Kretschmann configuration containing a saturable gain medium with the help of Fresnel formulas. The permittivity of the gain medium is described by a classical Lorentzian oscillator model. Meanwhile, gain saturation effect and spectral broadening of the gain medium are considered here. It is found that bistable ETR with the simultaneous excitation of SPPs exists in this structure due to gain saturation effect. The effects of the angle of incidence, the thickness of metal film, the value of small-signal gain and spectral broadening of the gain medium on bistable ETR are discussed in detail. These results may be helpful for the designs of all-optical switch and active plasmonic device.
2. Excitation of SPPs in the Kretschmann configuration
The prism-metal-dielectric structure, as known as the Kretschmann geometry is shown in the inset of Fig. 1. A thin metal film with thickness dm and permittivity ∊m is sandwiched between a prism and a dielectric layer. A high permittivity ∊0 prism is used here for ATR excitation of SPPs. The thickness of the adjacent dielectric layer with permittivity ∊d is of the order of 10λ, here λ is the wavelength of the incident TM-polarized light beam in vacuum.
The electromagnetic field of SPPs at the metal-dielectric interface can be obtained from the solution of Maxwell’s equations in each medium, and the associated boundary conditions. The dispersion of SPPs can be written as [1,4,6],Eq. (1) is still valid even if ∊m and ∊d are both complex numbers.
SPPs will be resonantly excited if the wavevector component of the incident beam along the interface , here θ is the angle of incidence, matches the wavevector of SPPs, i.e.,Fig. 1.Eq. (5) must be selected to enforce the physical energy propagation .
The normal ATR excitation of SPPs in the Kretschmann configuration is shown in Fig. 1. The inset of Fig. 1 is the Kretschmann geometry. The parameters are taken as those used by M. A. Noginov et al. : the prism with permittivity ∊0 = 3.18, silver film with permittivity ∊m = −15.584 + 0.424i at λ = 594nm, the dielectric with permittivity ∊d = 2.25. The dielectric without optical gain (i.e., the imaginary part of the dielectric ∊i = 0) is considered here, and the signs of the square root in Eq. (5) are all positive in three different media. In Fig. 1, is the critical angle for total internal reflection, and θSPP = 65.38°, which is determined by Eqs. (1) and (2), is the surface plasmon resonance angle. The reflectivity R as a function of the incident angle θ for different thicknesses of metal film is illustrated in Fig. 1. Figure 1 clearly demonstrates that a dip associated with the excitation of SPPs exists in the angular dependence of the reflectivity R(θ), and that the width and depth of the reflectivity dip strongly depend on the thickness of metal film. As dm increases, the full width at half maximum (FWHM) of the reflection dip decreases, while the value of reflectivity minimum decreases firstly and then increases. Figure 1 also clearly shows that an optimal thickness  of silver film dopt = dm2 = 53.9nm for SPPs excitation exists in this structure. The energy of the incident light is almost 100% transferred to SPPs at θ = θSPP and dm = dopt. Actually, SPPs can be effectively excited in this structure when the incident angle θ is close to θSPP and the thickness of metal film dm is in the vicinity of dopt.
3. The permittivity of the gain medium
Assume that the permittivity of the gain medium can be described by a classical Lorentzian oscillator model . Within the usual dipole approximation, the electric polarization P can be expressed as
The electric polarization depends on the electric field in a manner that can often be described by the relationship4,13,15] in linear optics.
The relative permittivity ∊d of the gain medium is related to the susceptibility through,46]. The small value of absorption (or gain) of the background susceptibility can be neglected, then ∊d is a complex number and can be written as,
4. ETR in the case of small-signal gain
In an optical amplifying medium, small-signal gain coefficient g0 can be expressed as ,47,48].
For the light intensity in the gain medium Id ≪ Is (here Is is the saturation intensity of the gain medium), small-signal gain coefficient g0 is almost unchanged with the variations of Id. The permittivity ∊d is independent of the light intensity Id.
In the Kretschmann geometry containing a gain medium, the signs of the square root in Eq. (5) are both positive in the prism and metal film, while in the gain medium the sign is negative , which coincide with the feature of SPPs that the electromagnetic fields are evanescent in the direction perpendicular to the interface.
Figure 2 illustrates the behavior of R as a function of small-signal gain and the incident angle at three different thicknesses of metal film, (a) dm = 39nm, (b) dm = 60nm, and (c) dm = 70nm. Other parameters are aforementioned. The color bar on the right of Fig. 2 represents log10(R). Small-signal gain coefficients are 705cm−1, 1410cm−1 and 2115cm−1 corresponding to ∊i = −0.010, ∊i = −0.020 and ∊i = −0.030, respectively. Such high optical gain may exist in semiconductor quantum wells . Figure 2 clearly demonstrates that ETR with the simultaneous excitation of SPPs exist in this structure when small-signal gain coefficient g0 above a certain value , which is not only a function of the incident angle θ but also depends on the thickness of metal film dm. Subplots (b) and (c) in Fig. 2 clearly show that a singular point and a reflectivity minimum point appear in the curves of the reflectivity R versus ∊i near the surface plasmon resonance angle at a fixed dm, and that the singular point and the reflectivity minimum point are close to each other as dm increases. The singular point of R is determined by the denominator of Eq. (3), i.e., 1 + r0mrmd exp(2ikmzdm) = 0, which is also named as the characteristic equation . Comparing subplots (a) and (c) whit subplot (b), it is found that the more dm close to dopt = 53.9nm is, the larger the value of the singular point of R is, and the smaller the value of the reflectivity minimum point is.
5. Bistable ETR in the case of gain saturation
As the light intensity in the gain medium Id further increases, gain coefficient g becomes smaller, and this phenomenon is known as gain saturation . According to Eq. (11), ∊d is not independent of Id anymore under gain saturation. ∊r and ∊is are changed simultaneously with the variations of Id, here ∊is, to distinguish ∊i in the case of small signal gain, represents the imaginary part of ∊d in the case of gain saturation. Combining Eqs. (7) and (8) with Eq. (10) provides that15].
For simplicity, in the following numerical calculations, assume that the frequency of the incident light is close to the frequency of the the harmonic oscillator, i.e., ω ≃ ω0. Near resonance, ∊r also can be viewed as independent of Id, while ∊is is decreased as Id increases.
5.1. The gain medium with a homogeneously broadened lineEqs. (9)–(11) with Eq. (15), the relative permittivity ∊d of a HB gain medium at line center can be expressed as,
Note that, the distribution of the light intensity Id in the gain medium is inhomogenous in the Kretschmann geometry owing to the evanescent character of SPP. ∊is changes gradually with increasing distance from the interface. An effective dielectric constant is proposed here, by utilizing the mean light intensity instead of the gradient light intensity Id in the gain medium.
Assume that the magnetic field at the metal-dielectric interface is Hmd, then the light intensity at the interface can be depicted by , here |Hmd| represents the amplitude of Hmd. And then the mean light intensity in the gain medium can be approximated as,Eqs. (3) and (4) with Eqs. (16)–(18), the relationship between the input intensity I and the reflectivity R can be constructed with Hmd.
To obtain the angular regime for SPPs excitation, a dielectric without optical gain is considered firstly. A reflection dip corresponding to the excitation of SPPs is shown in Fig. 3 at dm = 60nm and ∊i = 0, other parameters are aforementioned. The FWHM of the reflection dip range from 65° to 65.86°. To avoid damaging the structure, a dielectric with high optical gain and low saturation intensity is proposed here. We assume that small-signal gain coefficient ∊i = −0.020 and Is = 1MW/cm2 .
The inset (a) of Fig. 3 depicts that the reflectivity R as a function of input intensity I for different angles of incidence under gain saturation. It should be noted that θSPP = 65.38°. The blue-dashed, red-dotted and black-solid curves in the inset (a) correspond to θ1 = 65.34°, θ2 = 65.30° and θ3 = 65.26°, respectively. The inset (a) clearly demonstrates that the curves of ETR show hysteresis loops associated with OB when the incident angle θ is in close proximity to θSPP, and that the phenomenon of bistable ETR will disappear if the deviation between θ and θSPP exceeds a certain value, as shown by the black-solid curve. The inset (a) also clearly illustrates that, the width of OB hysteresis will be wider and the peak value of bistable ETR will be larger as θ tends to θSPP. In addition, as θ approaches θSPP, SPPs in this structure will be excited effectively, and the electromagnetic fields associated with SPPs will be significantly enhanced, hence both the lower switch threshold and the higher switch threshold of OB hysteresis move toward the lower input intensity. Similar results also can be obtained in the case of θ > θSPP, as shown in the inset (b) of Fig. 3.
The thickness of metal film dm plays an important role for the excitation of SPPs in the Kretschmann geometry, as shown in Fig. 1. Here, the effect of dm on bistable ETR is investigated when θ = 65.46° and ∊i = −0.020. Other parameters are mentioned above. It is worth mentioning that, the optimal thickness of silver film for SPPs excitation in this structure is dopt = 53.9nm. The red-dashed, blue-dashdotted, black-dotted and magenta-solid curves in Fig. 4 correspond to dm1 = 58nm, dm2 = 60nm, dm3 = 62nm and dm4 = 64nm, respectively. The red-dashed curve shows that, the phenomenon of bistable ETR will disappear if dm is below a certain value, which mainly relies on the incident angle θ. In Fig. 4, the blue-dashdotted and black-dotted curves of ETR show S-shaped hysteresis loops, and the magenta-solid curve of ETR shows a X-shaped hysteresis loop. Figure 4 clearly illustrates that, as dm increases, both switch thresholds and the width of OB hysteresis will increase, while the peak value of bistable ETR will decrease.
Figure 5 illustrates the effect of the value of small-signal gain g0 associated with ∊i by Eq. (11) on bistable ETR when dm = 60nm and θ = 65.46°. Other parameters are forementioned. The red-dashed, blue-dashdotted, black-dotted and magenta-solid curves in Fig. 5 correspond to ∊i1 = −0.018 (), ∊i2 = −0.020 (), ∊i3 = −0.022 () and ∊i4 = −0.024 (), respectively. The red-dashed curve shows that, if g0 is below a certain value, the phenomenon of bistable ETR will disappear. In Fig. 5, the blue-dashdotted and black-dotted curves of ETR show S-shaped hysteresis loops, and the magenta-solid curve of ETR shows a X-shaped hysteresis loop. Figure 5 clearly demonstrates that, both switch thresholds and the width of OB hysteresis will synchronously increase as g0 increases, but the peak value of bistable ETR is independent of g0.
5.2. The gain medium with an inhomogeneously broadened line
The gain profile has a Gaussian lineshape in an inhomogeneously broadened (inHB) gain medium [31,49]. When the spectral line is inHB, the saturation phenomenon is more complicated. But at line center, the gain coefficient can be simplified as ,
Figure 6 illustrates the effect of spectral broadening of the gain medium on bistable ETR when dm = 56nm, ∊i = −0.020, and θ = 65.40°. Here, assume that the saturation intensity in an inHB gain medium is the same as that in a HB gain medium, Is = 1MW/cm2. Other parameters are mentioned above. The blue-dashed curve and the red-solid curve in Fig. (6) respectively correspond to bistable ETR in a HB and an inHB gain media. Figure 6 shows that the phenomenon of bistable ETR also exist in an inHB gain medium. As shown in Fig. 5, the peak value of bistable ETR is determined by dm and θ, therefore the blue-dashed and the red-solid curves in Fig. (6) have the same peak value. According to Eqs. (16) and (20), the decrement of ∊is in an inHB gain medium is littler than that in a HB gain medium under the same light intensity . At a fixed dm and θ, the lower (or higher) switch threshold of OB hysteresis corresponds to a certain value of ∊is. Comparing with a HB gain medium, an inHB gain medium need a higher light intensity to obtain the same value of ∊is. Therefore, both switch thresholds and the width of OB hysteresis in the red-solid curve are respectively larger than those in the blue-dashed curve, as shown in Fig. 6.
In the Kretschmann geometry for SPPs excitation, the dielectric with a variable real part of ∊d, such as a nonlinear Kerr medium  and a refractive index thermosensitive dielectric , contributes to the formation of bistabe ATR; the dielectric with a negative and fixed imaginary part of ∊d, e.g., a typical gain medium in the case of small-signal gain, gives rise to the appearance of ETR [17, 30]. In this work, we theoretically demonstrate that the dielectric with a negative and variable imaginary part of ∊d, as an example, an optical gain medium in the case of gain saturation at line center, leads to the emergence of bistabe ETR.
In conclusion, the reflection of a TM-polarized light beam from a Kretschmann configuration containing a saturable gain medium is theoretically investigated with the help of Fresnel formulas. It is found that bistable ETR with the simultaneous excitation of SPPs exists in this structure due to gain saturation effect. The regime that is convenient for the emergence of bistable ETR in this structure can be summarized as follows: 1) the incident angle is close to θSPP; 2) the thickness of metal film is a little bigger than dopt; 3) small-signal gain is above a certain value; 4) significant gain saturation exists in the gain medium. In addition, to avoid damaging the structure, an optical amplifying medium with high optical gain and low saturation intensity is proposed here. The exact bistability condition  for bistable ETR in this structure will be further studied. Bistable ETR should be more expected than bistable ATR for the design of all-optical switch.
National Natural Science Foundation of China (NSFC) (11304274, 11404186, 51661029, 61665014, 21661036); Foundation of Yunnan Educational Committee (2013Y430); Doctoral Scientific Research Foundation of YNNU.
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